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feb longitudinal face aging wild recent deep learning approaches chi nhan duong concordia university montreal quebec canada email duon khoa luu cylab biometrics center dept electrical computer engineering carnegie mellon university pittsburgh usa email kluu kha gia quach concordia university montreal quebec canada email tien bui concordia university montreal quebec canada email bui aging raised considerable attentions interest computer vision community recent years numerous approaches ranging purely image processing techniques deep learning structures proposed literature paper aim give review recent developments modern deep learning based approaches deep generative models face aging task structures formulation learning algorithms well synthesized results also provided systematic discussions moreover aging databases used methods learn aging process also reviewed aging face age progression deep generative models ntroduction recent years age progression received considerable interest computer vision community starting predominant approaches require lots time professional skills support forensic artists several breakthroughs achieved numerous automatic age progression approaches anthropology theories deep learning models proposed general age progression methods technically classified four categories modeling reconstruction prototyping deep learning based methods methods first three categories usually tend simulate aging process facial features adopting prior knowledge anthropometric studies representing face geometry appearance set parameters via conventional models active appearance models aams morphable models manipulate parameters via learned aging functions although achieved inspiring synthesis results face representations still linear facing lots limitations modeling aging process meanwhile fourth category introduces modern approaches deep generative models dgm face modeling aging embedding figure given input images age progression task predict future faces subject many years later process since deep learning structures capabilities interpreting transferring highly features input signals suitable modeling human aging process result superior synthesized facial images generated inspired results paper aim provide review recent developments face age progression structures formulations several deep generative models restricted boltzmann machines rbm deep boltzmann machines dbm generative adversarial network gan well way adopted age progression problem presented moreover several common face aging databases also reviewed face aging databases database collection face aging also challenging problem several requirements collecting process subject images different ages also covered age range large therefore face aging databases still limited terms age labels number available databases characteristics age distributions several current existing face aging databases summarized table table roperties different aging databases wild images ones collected unconstrained real world conditions database morph album morph album adiencefaces cacd agedb agfw figure images subjects label type years old years old years old age groups years old years old years old age groups age groups normalized age distribution face aging databases fig databases dataset named aging face agfw also introduced work facial images individual ages sampled ranging database images divided age groups span years group contains images average database extended double scale images average images per age group image type mugshot mugshot subject type celebrities celebrities celebrities clean label public learning aging functions relationship parameter sets different age groups particular pattersons lanitis employed set active appearance models aams parameters four aging functions model general specific aging processes four variations aging functions introduced global aging function appearance specific aging function asa weighted appearance aging function waa weighted person specific aging function wsa also employing aams modeling step luu later incorporated familial facial cues process face age progression another direction modeling proposed definition aging pattern subspace ages approach authors construct representative subspace aging patterns chronological sequence face images given image proper aging pattern determined projection subspace produces smallest reconstruction error finally synthesized result target age obtained reconstructed faces corresponding age position subspace tsai enhanced ages using guidance faces corresponding subject characteristics produce stable results suo introduced graph aog smaller parts eyes nose mouth model face face aging process learned part using markov chain iii onventional pproaches section provide brief review conventional age progression approaches including modeling prototyping reconstructing based approaches properties also summarized table approach approach among earliest categories presented face age progression methods usually exploit kinds appearance models active appearance models aam morphable models represent shapes texture input face set parameters aging process simulated prototyping approach main idea methods category predefine types aging prototypes transfer difference prototypes produce synthesized face images usually aging prototypes defined average faces age groups input face image progressed target age incorporating differences prototypes two age groups notice approach requires good alignment faces order produce plausible results kemelmachershlizerman proposed construct high quality average prototypes set images table summary onventional pproaches age rogression method lanitis pattersons luu geng tsai suo suo kemelmachershlizerman shu yang approach representation architecture model based aams model based aams summary model generic specific aging processes four types aging functions learning effects morphological changes efforts adult aging stage model based aams incorporated familial facial cues model based aging patterns aging pattern subspace grammatical face model model based aging patterns aging pattern subspace guidance faces according subject feature model based markov chain parse graphs model based composition graph evolution prototype image pixel reconstructing sparse representation coupled dictionaries reconstructing sparse representation hidden factor analysis illumination normalization subspace alignment transferring difference prototypes dictionary learning coupled reconstruction loss sparse reconstruction feature shu proposed use aging coupled dictionaries cdl model personalized aging patterns preserving personalized facial features dictionaries learned using face pairs neighboring age groups via personalityaware coupled reconstruction loss yang represented factors independently using sparse representation hidden factor analysis hfa since gradually changes time age factor transformed target age group via sparse reconstruction combined identity factor achieve aged face figure examples faces obtained illuminationaware age progression approach sharper average faces obtained via collection flow method introduced align normalize images one age group illumination normalization subspace alignment technique employed handle images various lighting conditions figure illustrates results obtained approach rather constructing aging prototypes age group methods focus constructing aging basis age group model aging faces combination bases dictionary learning techniques usually employed type approach eep enerative odels face aging section firstly provide overview structures formulations common deep generative models going age progression techniques developed structures linear models deep structures compared linear models aams deep structures gained significant attention one emerging research topics representing higherlevel data features learning distribution observations example designed following concepts probabilistic graphical models pgm rbmbased models organize latent variables multiple connected layers energy function layer learn different factor represent data variations section introduces structures formulations several deep generative models including rbm deep boltzmann machines dbm generative adversarial networks gans figure structures rbm trbm dbm restricted boltzmann machines rbm undirected graphical models consisting two layers stochastic units visible hidden units simplified version boltzmann machines intra connections units layer created rbm structure bipartite graph visible hidden units pairwise conditionally independent given binary state energy rbm joint distribution visible hidden units computed exp denotes parameter set rbm including connection weights biases visible hidden units respectively conditional probabilities rbm structure computed wij logistic function original rbm visible hidden units binary make powerful able deal data extension rbm named gaussian restricted boltzmann machine introduced gaussian rbm visible units assumed values normally distributed mean variance another extension rbm temporal restricted boltzmann machines trbm designed model complex structure structure trbm shown fig major difference original rbm trbm directed connections visible hidden units previous states current states new connections short history activations act memory able contribute inference step current states visible units deep boltzmann machines dbm extension rbm one hidden layer structure dbm contains several rbms organized layers thanks structure hidden units higher layer learn complicated correlations features captured lower layer another interesting point dbm higher representations built training data unsupervised fashion unlike models deep belief network deep autoencoders connections units two consecutive layers undirected result unit receives topdown information therefore better propagate uncertainty inference process let set units two hidden layers energy state given follows weights connections notice bias terms visible hidden units ignored eqn simplifying representation exploiting advantages dbm deep appearance models dam robust deep appearance models rdam introduced proven superior classical models aams inferencing representation new face images various challenging conditions generative adversarial networks gan order avoid intractable markov chain sampling training stage rbm goodfellow borrowed idea adversarial system design generative adversarial networks gan intuition behind approach set game generator discriminator one hand discriminator learns determine whether given data generator real samples hand generator learns fool discriminator generated samples game continues learning process takes place learning process stop point discriminator distinguish real data ones produced generator also indication generator already learned distribution input data formally let input data distribution learned generator prior distribution variable gan defined two neural networks representing two differentiable functions generator discriminator denotes probability comes data distribution rather parameters cnns representing respectively training process formulated maximizing probability minimizing log min max log log original gan use fully connected neural network generator makes hard generate highresolution face images numerous extensions gan focusing different aspects structure proposed literature laplacian pyramid generative adversarial networks lapgan deep convolutional generative adversarial networks dcgan wasserstein gan deep aging models age progression thanks power deep learning models terms variations modeling many deep learning based age progression approaches recently developed achieved considerable results face age progression table iii summarizes key features deep learning based approaches model addition single face modeling trbm based age progression model introduced embed temporal relationship images face sequence taking advantages objective function avoiding reconstruction error training model able efficiently capture aging process automatically synthesize series faces various age ranges table iii roperties eep enerative odel pproaches age rogression eep earning ikelihood nverse einforcement earning irl robabilistic raphical odels pgm dversarial adv method approach architecture loss function non tractable model subject dependent wang zhang antipov irl rnn gan gan gan trbm pgm cnn pgm cnn adv adv adv age figure multiple input support architectures model steps age progression model aging details approach presented carefully designed architecture combination rbm trbm age variation modeling age transformation embedding fig illustrates aging architecture trbm proposed approach longterm aging development considered composition changes represented sequence subject faces different age groups decomposition set rbms employed model age variation age group well wrinkles presented faces older ages trbm based model constructed embed aging transformation faces consecutive age groups particularly keeping similar form energy function original trbm rbm bias terms defined bti pli atj qlj model parameters denote reference faces produced set learned rbm structure linear nonlinear interactions faces efficiently exploited finally wrinkle enhancement together geometry constraints incorporated steps consistent results therefore plausible synthesized results achieved using technique comparison term synthesis quality model conventional approaches shown fig recurrent neural model approaching age progression similar way decomposition instead using trbm wang proposed use recurrent neural network gated recurrent unit gru model aging sequence recurrent connection hidden units model efficiently exploit information previous faces memory produce smoother transition faces synthesizing process fig illustrates architecture proposed rnn age progression particular let input face young age network firstly encodes latent representation units bottom gru decodes representation older face subject using top gru relationship interpreted follows wzh wzx wrh wrx tanh wch wcx similar formulations also employed relationship difference aged face computed form loss function system trained obtain synthesis capability finally order generate wrinkles approach adopted wrinkle transferring although approach produced improvements comparing classical approaches use fixed reconstruction loss function limited figure structures model temporal preserving tnvp approach deep aging path sdap sdap shares mapping function tnvp aims embedding aging transformation whole aging sequence new objective function adapted min max dimg log log log dimg log dimg figure comparison sequence generated trbm based model iaap approach synthesis ability usually resulted blurry faces model rather synthesis previous approaches antipov zhang turned another direction age progression direct approach adopted structure gan architectures fig illustrates structure conditional adversarial autoencoder caae figure one easily see authors adopted gan structure presented section additional age label feature representation latent variables way encode relationship subject identity related features input face age label training simply changing aging label according target age generator able synthesize aged face age compared eqn denotes vector represented age label latent feature vector decoder function norm total variation functions respectively pdata denotes distribution training data one see conditional constraint age label represented last two terms loss function although model type avoid requirement longitudinal age database training easy converged due step maintaining good balance generator discriminator hard achieve moreover similar approach models also incorporate objective functions therefore synthesized results limited terms image sharpness temporal preserving transformation recently addressing limitation intractable learning process trbm based model well image quality approaches temporal preserving tnvp approach introduced embedding feature transformations faces consecutive stages keeping tractable density function exact inference evaluation unlike previous approaches incorporate pgm cnn structures proposed model enjoys advantages architectures improve image synthesis quality highly feature generation idea model start pgm relationships variables image latent domains see fig given denote bijection functions mapping latent variables respectively figure comparison deep learning approaches tnvp trbm conventional approaches including iaap exemplar based eap craniofacial growth cgap models function embedding aging transformation latent variables probability density function derived denote conditional distribution respectively specific design mapping functions two terms eqn computed exactly effectively result authors form deep cnn network optimized concepts pgm keeping tractable density estimation objective function model turns age progression architectures new direction cnn network avoid using fix reconstruction loss function obtain synthesized faces fig illustrates synthesized results achieved tnvp comparison approaches deep aging path sdap model inspiring advantages tvnp inverse reinforcement irl learning also taken account structure deep aging path sdap model hypothesis subject facial development duong proposed use additional aging controller structure tnvp rather embedding aging transformation pairwise relationship consecutive age groups sdap structure learns aging transformation whole face sequence better aging synthesis goal achieved via aging policy network guarantees provide appropriate planning aging path age controller corresponding subject features interesting point sdap one pioneers incorporating irl framework age progression task approach let xti age sequence subject xti face sequence true acceptance rate sdap tnvp false acceptance rate figure example age invariant face recognition using age progression models tvnp sdap incorporated synthesized results accuracy face recognition system improved significantly note results methods provided megaface website representing facial development subject aji denote variables control aging amount added xji become probability formulated via energy function exp partition function notice formulation eqn similar joint distribution variables rbm eqn goal learn aging policy network predict aji xji synthesized process objective function defined arg max log finally specific design irl framework proposed learn policy network experimental results sdap shown potential outperform tnvp approaches synthesis results verification accuracy shown fig sdap help significantly improve accuracy face recognition system onclusion paper reviewed main structures deep generative models age progression task compared classical approaches deep learning shown potential either learning highly nonlinear age variation aging transformation embedding result synthesized faces improve image quality also help significantly boost recognition accuracy face verification system several common aging databases support facial modeling aging embedding process also discussed eferences aging database http antipov baccouche dugelay face aging conditional generative adversarial networks arxiv preprint arjovsky chintala bottou wasserstein gan arxiv preprint bengio learning deep architectures foundations trends machine learning burt perrett perception age adult caucasian male faces computer graphic manipulation shape colour information proc soc lond biol sci chen chen hsu reference coding face recognition retrieval eccv chen duan houthooft schulman sutskever abbeel infogan interpretable representation learning information maximizing generative adversarial nets nips pages denton chintala fergus deep generative image models using laplacian pyramid adversarial networks nips pages duong luu quach bui beyond principal components deep boltzmann machines face modeling cvpr pages ieee duong luu quach bui longitudinal face modeling via temporal deep restricted boltzmann machines cvpr duong quach luu savvides temporal preserving approach facial ageprogression face recognition iccv duong quach luu savvides learning longitudinal face tractable deep modeling meets inverse reinforcement learning arxiv preprint geng zhou automatic age estimation based facial aging patterns pami goodfellow mirza wardefarley ozair courville bengio generative adversarial nets nips pages hinton training products experts minimizing contrastive divergence neural computation hinton salakhutdinov reducing dimensionality data neural networks science seitz miller brossard megaface benchmark million faces recognition scale cvpr suwajanakorn seitz age progression cvpr pages ieee krizhevsky hinton learning multiple layers features tiny images lanitis taylor cootes toward automatic simulation aging effects face images pami levi hassner age gender classification using convolutional neural networks cvprw sun global local consistent age generative adversarial networks arxiv preprint liu yuen torralba sift flow dense correspondence across scenes applications tpami luu suen bui ricanek automatic childface based heritability factors familial faces bids pages ieee moschoglou papaioannou sagonas deng kotsia zafeiriou agedb first manually collected age database hawaii patterson ricanek albert boone automatic representation adult aging facial images proc iasted intl conf visualization imaging image processing pages quach duong luu bui robust deep appearance models icpr pages radford metz chintala unsupervised representation learning deep convolutional generative adversarial networks arxiv preprint ramanathan chellappa modeling age progression young faces cvpr ricanek tesafaye morph longitudinal image database normal adult fgr pages ieee rothe timofte gool deep expectation real apparent age single image without facial landmarks ijcv rowland perrett manipulating facial appearance shape color ieee shen shih liao age progression prediction children faces ism pages ieee shu tang lai liu yan personalized age progression aging dictionary iccv december suo chen gao dai concatenational graph evolution aging model pami suo zhu chen compositional dynamic model face aging pami sutskever hinton learning multilevel distributed representations sequences aistats pages tsai liao lin human face aging guided prediction detail synthesis multimedia tools applications wang cui yan feng yan shu sebe recurrent face aging cvpr yang huang wang wang tang face aging effect simulation using hidden factor analysis joint sparse representation tip zhang song age conditional adversarial autoencoder cvpr july
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deterministic policy optimization combining pathwise score function estimators discrete action spaces daniel levy stefano ermon nov department computer science stanford university danilevy ermon abstract policy optimization methods shown great promise solving complex reinforcement imitation learning tasks methods broadly applicable often require many samples optimize complex policies modelbased methods greatly improve cost poor generalization requiring carefully handcrafted model system dynamics task recently hybrid methods successful trading applicability improved however limited continuous action spaces work present new hybrid method based approximation dynamics expectation next state current policy relaxation allows derive novel hybrid policy gradient estimator combining score function pathwise derivative estimators applicable discrete action spaces show significant gains sample complexity ranging learning parameterized policies cart pole acrobot mountain car hand mass method applicable discrete continuous action spaces competing pathwise methods limited latter introduction reinforcement imitation learning using deep neural networks achieved impressive results wide range tasks spanning manipulation levine levine abbeel locomotion silver games mnih silver autonomous driving ermon song ermon methods search optimal policies without explicitly modeling system dynamics williams schulman algorithms build estimate policy gradient sampling trajectories environment perform gradient ascent however methods suffer either high sample complexity due generally large variance estimator restricted policies parameters hand reinforcement learning methods learn explicit model dynamics system policy optimized model reward signal learned dycopyright association advancement artificial intelligence rights reserved namics approaches greatly improve sample efficiency dynamics model needs carefully task recently hybrid approaches interface attempted balance generalizability notable success robotics levine existing methods however limited continuous action spaces levine abbeel heess work introduce hybrid reinforcement learning algorithm deterministic policy optimization applicable continuous discrete action spaces starting class deterministic policies relax corresponding policy optimization problem one carefully chosen set stochastic policies approximate dynamics relaxation allows derivation novel policy gradient estimator combines pathwise derivative score function estimator enables incorporating assumptions remaining applicable discrete action spaces perform optimization larger class policies slowly annealing stochasticity converging optimal deterministic solution additionally justify bound dynamics approximation certain assumptions finally complement estimator scalable method estimate dynamics first introduced levine abbeel general extension rewards rendering applicable large class tasks contributions follows introduce novel deterministic policy optimization method leverages model dynamics applied action space whereas existing methods limited continuous action spaces also provide theoretical guarantees approximation show estimator applied broad class problems extending discrete rewards utilizing dynamics estimation method levine abbeel show method successfully optimizes complex neural network deterministic policies without additional variance reduction techniques present experiments tasks discrete action spaces hybrid methods applicable tasks show significant gains terms ranging related work sample efficiency key metric methods especially used physical systems methods improve sample efficiency cost defining learning dynamics models modelfree methods typically require significantly samples broadly applicable methods approximate dynamics using various classes models spanning gaussian processes deisenroth rasmussen neural networks levine abbeel assuming reward function known differentiable policy gradient computed exactly differentiating dynamics model thus enabling optimization heess extends idea stochastic dynamics policies expressing randomness exogenous variable making use trick requires differentiability dynamics function state action limiting applicability continuous action spaces hand algorithms broadly applicable require significantly samples well variance reduction techniques control variates mnih gupta ermon trust regions schulman converge reasonable time recently hybrid approaches attempted balance sample efficiency applicability levine abbeel locally approximate dynamics fit locally optimal policies serve supervision global policies good dynamics model also enables generating artificial trajectories limit strain simulator however works limited continuous action spaces work also considered hybrid algorithm handle improve sampleefficiency discrete action spaces well background section first present canonical formalism review score function pathwise derivative estimators respective advantages show applications first reinforce estimator williams followed standard method modelbased policy optimization consisting dynamics equation notations definitions let denote state action spaces respectively refers deterministic dynamics function deterministic policy function stochastic policy conditional distribution actions given state denoted clarity considering parameterized policies stochastic ones functions deterministic ones functions throughout work dynamics considered deterministic consider standard setting agent interacts environment obtains rewards actions agent goal maximize expected cumulative reward formally exists initial distribution collection reward functions sampled step agent presented chooses action according policy computed finite horizon setting episode ends agent thenpprovided cumulative reward goal agent find policy maximizes hpthe expected cumulative reward score function estimator pathwise derivative review two approaches estimating gradient expectation let probability distribution measurable set interested obtaining estimator following quantity score function estimator score function estimator relies relies following identity given appropriate regularity assumptions log last quantity estimated using montecarlo sampling estimator general applied even discrete random variable long log differentiable however suffers glasserman pathwise derivative pathwise derivative estimator depends exists function distribution independent sampling equivalent sampling computing given observation quantity estimated using sampling conversely lower variance glasserman requires differentiable function override resp resp aim maximizing objective function using gradient ascent reinforce using score function estimator derive reinforce rule williams applicable without assumptions dynamics action space stochastich policy iwant maximize reinforce rule log estimate quantity using sampling requires differentiability assume knowledge estimator however applicable deterministic policies method policy optimization let deterministic policy differentiable assuming knowledge directly idifferentiate first term easx ily computed given knowledge reward functions second term given knowledge dynamics computed recursively differentiating method extended stochastic dynamics policies using pathwise derivative estimator reparameterizing noise heess method applicable deterministic differentiable policy differentiable dynamics variables differentiable reward function implies must continuous method aims utilizing dynamics differentiating dynamics equation relaxing policy optimization problem deterministic policies optimal theoretic setting objective work perform hybrid policy optimization deterministic policies continuous discrete action spaces order accomplish present relaxation optimization problem relaxation allows derive subsequent section policy gradient estimator differentiates approximate dynamics leverages assumptions action spaces contrary traditional hybrid methods assume differentiability dynamics function action variable thus elegantly handling discrete action spaces previous section place setting assume terminal rewards convex state space relaxing dynamics constraint section describe relaxation starting class deterministic policies parameterized construct class stochastic policies parameterized chosen close desired adjusting extended class derive gradient estimator first explain relaxation detail construct finally provide guarantees approximation formally problem deterministic policy dynamics written maximize subject given deterministic constraint equivalently rewritten made observation proceed relaxing optimization constraint approximate particular differentiable dynamics relaxed optimization program therefore maximize subject relaxation casts optimization stochastic policies allows derive policy gradient theorem different approximated dynamics later describe project solution back element construction show construct stochastic policies deterministic ones providing parameterized knob control randomness closeness deterministic policy discrete action spaces natural parameterization softmax model however requires careful parameterization order ensure policies included use deterministic policy prior control strength formally choose class parameterized functions define thehfollowing stochastic policy exp therefore defined easily verify choose arbitrary define continuous action spaces continuous setting simple parameterization adding gaussian noise control variance formally given let surjection derived setting complicated stochastic parameterizations easily derived long density remains tractable rounding assume surjection stochastic policy made deterministic setting certain value examples mapping consists setting similar spirit simulated annealing optimization kirkpatrick optimize slowly annealing converge optimal solution theoretical guarantees presented relaxation provide theoretical justifications show first given conditions stochastic policy trajectory computed approximate dynamics stochastic policy close trajectory computed true dynamics deterministic policy additionally present connections case dynamics discretization system bounding deviation real dynamics paragraph assume action space continuous given terminal reward setting amount approximation defined divergence trajectory following deterministic policy trajectory corresponding approximate dynamics policy allow relate optimal value relaxed program optimal value initial optimization problem theorem approximation bound let deterministic stochastic policy respectively let suppose assume supx following guarantee furthermore distributions fixed variance approximation converges towards proof see appendix know solving relaxed optimization problem provide expected terminal reward policy given reward function bound shows optimal value true optimization program within optimal value relaxed one equivalence systems relaxation strong theoretical guarantees dynamics discretization system analogous notations let consider dynamical system system written thus write dynamics relaxed problem intuitively discretization step tends policy converges deterministic one limit approximation true dynamics continuous time systems proposition formalizes idea proposition let trajectory obtained relaxed system following stochastic policy let continuous time trajectory probability proof see munos relaxed policy gradient relaxation presented previous section allows differentiate dynamics action space section derive novel deterministic policy gradient estimator relaxed policy gradient rpg combines pathwise derivative score function estimators applicable action spaces show apply estimator using scalable method estimate dynamics first presented levine abbeel conclude extending estimator discrete reward functions effectively making algorithm applicable wide variety problems simplicity omit stochasticity derivations consider fixed estimator place relaxation parameterized stochastic policy objective arg aim wish ascent setting letting define find perform gradient theorem relaxed policy gradient given trajectory sampled following stochastic policy define following quantity computed recursion defined log log unbiased estimator policy gradient relaxed problem defined equation proof see appendix presented rpg estimator effectively combination pathwise derivative score function estimators intuitively pathwise derivative used discrete action spaces requires differentiability action end differentiate pathwise handle discrete portion score function estimator benefits pathwise derivative gradient estimates pathwise derivatives considered lower variance glasserman context intuitively reinforce suffers high variance requires many samples properly establish credit assignment increasing decreasing probabilities actions seen trajectory based reward conversely rpg estimates dependency state gradient directly adjust steer trajectory high reward regions possible estimator utilizes model dynamics reinforce assigns credit indifferently actions rpg adjusts entire trajectory however examining expression rpg computation requires gradient dynamics state unlike reinforce next section present scalable dynamics fitting method detailed levine estimate term scalable estimation dynamics relaxed policy gradient estimator computed exclusively sampled trajectories provided dynamics however information available agent thus needs estimated samples review method first presented levine abbeel provides estimation dynamics incorporating information multiple sampled trajectories although dynamics deterministic utilize stochastic estimation account modeling errors formally sampled trajectories form xit ait dynamical system goal accurately model dynamics system also estimate correctly prevents simply fitting parametric function samples indeed approach would provide guarantee good approximation especially important variance estimator dependent quality approximation approximation order fit dynamics model choose locally model dynamics discretized stochastic process parameterized choose approach model global dynamics instead good local model around latest sampled trajectories corresponds term want estimate model term interested estimating approach simple requires large number samples accurate greatly increase sampleefficiency assume prior sampled trajectories use gmm approach levine abbeel corresponds softly piecewise linear dynamics iteration update prior using current trajectories fit model allows utilize past trajectories well nearby time steps included prior another key advantage stochastic estimation elegantly handles discontinuous dynamics models indeed averaging dynamics locally around discontinuities effectively smooths discontinuities underlying deterministic dynamics refer levine detailed derivations prior work estimators incorporating models dynamics heess levine abbeel constrained continuously differentiable rewards present extension type estimators class rewards make assumptions form reward approximate smooth function assume reward written sum indicator functions approximate smooth function given surrogate reward apply estimator practice however discrete reward functions often defined function approximate reward function sigmoid function pointwise converges present appendix example approximate reward functions mountain car task given arbitrarily close surrogate rewards prove continuity result sequence optimal policies surrogate rewards converges optimal policy proposition optimal policies surrogate reward functions let reward function defined optimal policy reward let define value state policy exists sequence smooth reward functions optimal reward proof see appendix practical algorithm algorithm present practical algorithm perform deterministic policy optimization based previously defined rpg summary given construct extended class stochastic policies perform optimization stochastic policies slowly annealing converging policy easily extend estimator case rewards extension discrete rewards proof see appendix compact subsets assumption covers large collection tasks goal end compact subspace going approximate arbitrarily close smooth function proposition smooth approximation indicator function let compact neighborhood exists smooth experiments empirically evaluate algorithm investigate following questions much rpg improve sample complexity rpg learn optimal policy true reward using smooth surrogate reward effective approximation dynamics attempt evaluate benefits relaxation compared estimators fairly possible use additional techniques trust regions schulman method compared ones incorporate improvements leave study sophisticated estimators future work classical control tasks apply relaxed policy gradient rpg set classical control tasks discrete action spaces cart pole barto sutton anderson acrobot sutton mountain car sutton hand mass munos algorithm relaxed policy gradient inputs environment giving given deterministic class policies stochastic class policies number training episodes nep learning rate schedule annealing schedule initial parameters initialize nep sample policy update gmm prior dynamics fit dynamics log log end end end return first three tasks used openai gym brockman implementation followed munos implementation hand mass diagram tasks presented figure baselines compare methods two different baselines derivative free optimization algorithm called method cem szita algorithm mnih cem algorithm samples different set parameters step simulates using parameters keeps top using distribution parameters cem known target movable hand origin mass figure top row cart pole acrobot bottom row mountain car hand mass work well duan lacks sample efficiency algorithm exploit structure variant reinforce williams learns value function reduce variance estimator task algorithm evaluate distinct random seeds results analysis present learning curves tasks figure even training done surrogate rewards instances report actual reward show cem learning curves comparable due nature algorithm cem episode equivalent rpg episodes instead report final performance tables tasks policy parameterized neural network tanh hidden units softmax layer outputting distribution actions practice optimize stochastic policies fixed experiments converged near deterministic policy since optimization wellconditioned enough case policies trained using adam kingma evaluate differently depending task since cartpole acrobot fixed reward threshold considered solved present number training samples needed reach performance table contrast mountain car hand mass report table reward achieved fixed number samples metrics meant evaluate sample complexity shown tables algorithm clearly outperforms across tasks requiring times less samples solve tasks showing significantly better performance number samples shown table rpg performs better cem hand mass within mountain car despite using times less samples also note cem particularly well suited acrobot method explore space parameters fast find parameters quickly optimal policies fairly simple explaining impressive specific task additionally point full plots performance number samples reported tasks figure numbers tables extracted robustness variance training stability overall method robust choice hyperparameters learning rate architecture indeed policy trained tasks learning rate whereas different learning rates crossvalidated examining training curves estimator demonstrates significantly less variance stable training process tasks challenges exploratory acrobot mountain car rpg exploration process guided model dynamics completely undirected often leading total failure phenomenon observed control average reward cart pole mountain car hand mass acrobot episodes rpg figure mean rewards random seeds classical control tasks performance rpg shown method samples solve cart pole threshold cem rpg acrobot threshold cem rpg table average numbers samples task solved cart pole acrobot tasks rpg cem method samples performance hand mass episodes cem rpg mountain car episodes cem rpg table average mean rewards hand mass mountain car tasks rpg cem tasks hand mass cart pole favors bad actions seen high reward trajectories leading unstable control limitations section explore limitations method leverage dynamics problem flexibility class problems tackle better sample complexity also seen instance limitations reward function presented general way extend estimator rewards discrete domain function approximations difficult construct atari games indeed one would fit indicator function low dimensional manifold corresponding set images encoding state given game score living space order limitations type tasks show results classical control tasks estimator broadly applicable tasks dynamics estimated reasonably well shown possible number locomotion robotics tasks levine abbeel heess however work directly applicable raw pixel input tasks atari domain computational overhead compared reinforce model presents computational overhead requires evaluating well fitting dynamics matrices practice minor compared training operations sampling trajectories computing necessary gradients experiments computing amounts less overhead dynamics estimation constitutes discussion approximate reward tasks except hand mass estimator trained using approximate smoothed rewards performances reported figure table show impair training regarding baselines interesting note since cem leverage structure problem natural train real rewards experimented smoothed rewards obtained comparable numbers work presented method find optimal deterministic policy action space particular discrete actions method relies relaxation optimization problem carefully chosen larger class stochastic policies class policies derive novel policy gradient estimator relaxed policy gradient rpg differentiating approximate dynamics perform optimization slowly annealing stochasticity policy converging terministic solution showed method successfully applied collection tasks presenting significant gains training stability existing methods furthermore introduced way apply algorithm reward functions learning policy smooth surrogate reward function provided construction method also important note method easily amenable problems stochastic dynamics assuming one reparameterize noise work also opens way promising future extensions estimator example one could easily incorporate imaginary rollouts estimated dynamics model trust regions schulman finally work also extended broadly gradient estimation discrete random variables acknowledgments authors thank aditya grover jiaming song steve mussmann useful discussions well anonymous reviewers insightful comments suggestions research funded intel corporation fli tri nsf grants references alexandroff urysohn alexandroff urysohn zur theorie der topologischen mathematische annalen barto sutton anderson barto sutton anderson neuronlike adaptive elements solve difficult learning control problems ieee transactions systems man cybernetics brockman brockman cheung pettersson schneider schulman tang zaremba openai gym arxiv preprint deisenroth rasmussen deisenroth rasmussen pilco approach policy search proceedings international conference machine learning duan duan chen houthooft schulman abbeel benchmarking deep reinforcement learning continuous control international conference machine learning levine abbeel levine abbeel learning manipulation skills online dynamics adaptation neural network priors intelligent robots systems iros international conference ieee glasserman glasserman monte carlo methods financial engineering volume springer science business media lillicrap sutskever levine continuous deep modelbased acceleration arxiv preprint heess heess wayne silver lillicrap erez tassa learning continuous control policies stochastic value gradients advances neural information processing systems ermon ermon generative adversarial imitation learning advances neural information processing systems gupta ermon gupta ermon imitation learning policy optimization proceedings international conference machine learning kingma kingma adam method stochastic optimization arxiv preprint kirkpatrick kirkpatrick gelatt vecchi optimization simulated annealing science levine abbeel levine abbeel learning neural network policies guided policy search unknown dynamics advances neural information processing systems levine levine finn darrell abbeel training deep visuomotor policies journal machine learning research song ermon song ermon infogail interpretable imitation learning visual demonstrations advances neural information processing systems mnih mnih kavukcuoglu silver rusu veness bellemare graves riedmiller fidjeland ostrovski control deep reinforcement learning nature mnih mnih badia mirza graves lillicrap harley silver kavukcuoglu asynchronous methods deep reinforcement learning international conference machine learning munos munos policy gradient continuous time journal machine learning research may schulman schulman levine moritz jordan abbeel trust region policy optimization international conference machine learning icml silver silver lever heess degris wierstra riedmiller deterministic policy gradient algorithms proceedings international conference machine learning silver silver huang maddison guez sifre van den driessche schrittwieser antonoglou panneershelvam lanctot mastering game deep neural networks tree search nature sutton sutton generalization reinforcement learning successful examples using sparse coarse coding advances neural information processing systems szita szita learning tetris using noisy method neural computation williams williams simple statistical algorithms connectionist reinforcement learning machine learning appendix tasks specifications cart pole classical cart pole setting agent wishes balance pole mounted pivot point cart due instability agent perform continuous cart movement preserve equilibrium agent receives reward time step spend upwards task considered solved agent reaches reward acrobot first introduced sutton system composed two joints two links initially hanging downwards objective actuate links swing end lower link fixed height path reward determined negative number actions required reach objectif task considered solved agent reaches reward mountain car consider usual setting presented sutton objective get car top hill learning policy uses momentum gained alternating backward forward motions state consists horizontal position car reward expressed task difficulty lies limitation maximum tangential force need exploration agent receives reward use surrogate smoothed reward dctpg evaluate performance fixed number episodes hand mass consider physical system first presented munos agent holding spring linked mass agent move four directions objective bring mass target point keeping hand closest origin point state described dynamics described munos final time step agent receives reward proof theorem lemma let random variable bounded variance let varx proof simply use inequality let define sxz varxi let prove recursion obtained triangular inequality triangular inequality lemma uniform bound variances given inequality fact compute geometric sum concludes proof proofs theorem method presented background want directly differentiate require reward function differentiable computing need differentiate relaxed equation dynamics log log recursion corresponds estimate trajectories obtained running stochastic policy environment thus concluding proof veloc ity continuity distance compact proving result reward ion osit figure surrogate smooth reward mountain car task left true reward right smoothed surrogate reward proof proposition given compact open neighborhood let show exists according urysohn lemma alexandroff urysohn continuous functions exist general setting normal spaces given pair disjoint closed sets show explicitly construct simpler assumptions sufficient show construct function open set containing closed unit ball equal unit ball null outside given function easily construct scaling translating let define exp function smooth rand let define function strictly positive null outside large enough function satisfy desired property thus proving construct figure shows example smooth surrogate reward mountain car task proof proposition let consider deterministic case reward terminal let optimal policy reward exists sequence open neighborhoods diameter tends let smooth reward function constructed neighborhood let deterministic optimal policy reward start state distinguish two cases former policy reach compact thus surrogate reward make difference latter policy take final state xnt xnt however xnt diameter tends xnt cauchy sequence complete space thus converges limit since xnt
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fast distributed algorithms connectivity mst large graphs gopal peter michele jul july abstract motivated increasing need understand algorithmic foundations distributed graph computations study number fundamental graph problems messagepassing model distributed computing machines jointly perform computations graphs nodes typically input graph assumed initially randomly partitioned among machines common implementation many systems communication goal minimize number communication rounds computation main result almost optimal distributed randomized algorithm graph connectivity algorithm runs rounds notation hides polylog factor additive polylog term improves best previously known bound klauck soda optimal polylogarithmic factor view existing lower bound improved algorithm uses bunch techniques including linear graph sketching prove useful design efficient distributed graph algorithms using connectivity algorithm building block present fast randomized algorithms computing minimum spanning trees approximate many graph verification problems algorithms take rounds optimal polylogarithmic factors also show almost matching lower bound rounds many graph verification problems leveraging lower bounds communication complexity department computer science university houston houston usa gopalpandurangan michele supported part binational science foundation grant nsf grant nsf grant department computer science royal holloway university london united kingdom introduction focus paper distributed computation graphs increasingly becoming important rise massive graphs web graph social networks biological networks data consequent need fast algorithms process graphs several graph processing systems pregel giraph recently designed based distributed computing model study number fundamental graph problems model abstracts essence systems present almost tight bounds time complexity needed solve problems model introduced explained detail section input graph distributed across group machines pairwise interconnected via communication network machines jointly perform computations arbitrary input graph typically input graph assumed initially randomly partitioned among machines common implementation many real world graph processing systems communication via message passing computation advances synchronous rounds constraint amount data cross link network round goal minimize time complexity number rounds required computation model aimed investigating amount possible number available machines following sense machines used time complexity scale problems admit linear scaling possible achieve scaling klauck present lower upper bounds several fundamental graph problems model particular assuming link bandwidth one bit per round show lower bound rounds graph connectivity also present algorithm graph connectivity spanning tree verification algorithm thus exhibits scaling linear number machines question existence faster algorithm particular algorithm matching lower bound left open paper answer question affirmatively presenting algorithm graph connectivity thus achieving speedup quadratic optimal polylogarithmic factors result important two reasons first shows graph problems obtain superlinear elaborate point shall take closer look proof lower bound connectivity shown using communication complexity techniques proof shows possibly randomized algorithm graph connectivity problem exchange bits information across machines since links complete network machines link carry polylog bits per round single round network deliver bits information thus lower bound rounds follows result paper thus shows possible exploit full available bandwidth thus achieving second implies many important graph problems solved rounds well include computing spanning tree minimum spanning tree mst approximate many verification problems spanning connected subgraph cycle containment bipartiteness important note different output requirement explained next exists throughout paper denotes polylog polylog denotes polylog lower bound computing spanning tree graph also implies lower bound fundamental problems computing mst tree shortest paths tree however lower bound holds requirement vertex machine hosts vertex must know end computation status incident edges whether belong output respective status output criterion usually required distributed algorithms proof lower bound exploits criterion show algorithm requires machine receiving bits information since machine incident links results lower bound hand relax output criterion require final status edge known machine show accomplished rounds using fast connectivity algorithm paper model describe adopted model distributed computation model big data model introduced investigated model consists set machines pairwise interconnected bidirectional communication links machine executes instance distributed algorithm computation advances synchronous rounds round machines exchange messages communication links perform local computation link assumed bandwidth polylog bits per round polylog bits transmitted link round discussed theorem easy rewrite bounds scale terms actual bandwidth machines share memory means communication alternate equivalent way view communication restriction instead putting bandwidth restriction links put restriction amount information machine communicate round results obtain model also apply latter model local computation within machine considered happen instantaneously zero cost exchange messages machines costly operation however note algorithms paper every machine every round performs computation bounded polynomial assume machine access private source true random bits although model fairly general model computation mostly interested studying graph problems specifically given input graph vertices associated unique integer edges avoid trivialities assume typically initially entire graph known single machine rather partitioned among machines balanced fashion nodes edges partitioned approximately evenly among machines assume model whereby vertices along information incident edges partitioned across machines specifically type partition assume throughout random vertex partition rvp vertex input graph assigned randomly one machine typical way used many real systems pregel partition input graph among machines easy accomplish via however notice upper bounds section discuss alternate partitioning model random edge partition rep model edge assigned independently randomly one machines show results random vertex partition model related random edge partition model also hold much weaker assumption whereby required nodes edges input graph partitioned approximately evenly among machines hand lower bounds rvp clearly apply partitions well formally random vertex partition variant vertex assigned independently uniformly random one machines vertex assigned machine say home machine slight abuse notation write vertex assigned machine incident edges assigned machine well home machine know ids neighbors vertex well identity home machines neighboring vertices weights corresponding edges case weighted note immediate property rvp model number vertices machine balanced machine home machine vertices high probability convenient way implement rvp model hashing vertex hashed one machines hence machine knows vertex also knows hashed eventually machine must set designated local output variable need depend set vertices assigned output configuration must satisfy certain feasibility conditions problem hand example minimum spanning tree problem corresponds set edges edges union sets must form mst input graph paper show results distributed algorithms monte carlo recall monte carlo algorithm randomized algorithm whose output may incorrect probability formally say algorithm computes function every input outputs correct answer probability least probability random partition random bit strings used algorithm round time complexity algorithm maximum number communication rounds termination problem node graphs let time complexity solving error probability model denoted minimum exists protocol solves terminates rounds graph problem function say exists integer similarly say exists integer real upper bounds usually use imply high probability algorithms succeeding probability least case sometimes omit simply say time bound applies high contributions techniques main result paper presented section randomized monte carlo algorithm model determines connected components undirected graph correctly high probability terminates improves upon previous best bound since strictly superior wide range parameter constants improving bound since various attempts get faster connectivity algorithm fail due fact end congesting particular machine since focus scaling time complexity respect omit explicitly stating polylogarithmic factors run time bounds however hidden polylogarithmic factor much bits may need machine leading bound machine links example simple algorithm connectivity simply flooding vertex floods lowest labeled vertex seen far end vertex label lowest labeled vertex shown algorithm takes rounds graph diameter model using conversion theorem hence new techniques needed break barrier connectivity algorithm result application following three techniques randomized proxy computation technique similar known techniques used randomized routing algorithms used congestion given machine redistributing evenly across machines achieved roughly speaking executions individual nodes uniformly random among machines crucial distribute computation communication across machines avoid congestion particular machine fact allows one move away communication pattern imposed topology input graph cause congestion particular machine balanced communication distributed random ranking drr drr simple technique used build trees low height connectivity algorithm connectivity algorithm divided phases following current component first phase vertex component chooses one outgoing edge components combined merging along outgoing edges done naively may result long chain merges resulting component tree high diameter communication along tree take long time avoid resort drr suitably reduces number merges drr component chooses random rank simply random number say interval component merges component side selected outgoing edge rank larger rank otherwise merge thus becomes root drr tree tree induced components set outgoing edges used merging procedure shown height drr tree bounded log high probability linear graph sketching linear graph sketching crucially helpful efficiently finding outgoing edge component sketch vertex component short polylog bit vector efficiently encodes adjacency list vertex sampling sketch gives random outgoing edge vertex component useful property linearity sketches adding sketches set vertices gives sketch component obtained combining vertices edges vertices edges automatically cancelled leaving sketch outgoing edges linear graph sketches originally used process dynamic graphs streaming model distributed setting use reduce amount communication needed find outgoing edge particular graph sketches avoid checking whether edge edge crucially reduce communication across machines note earlier distributed algorithms classical ghs algorithm mst problem would incur much communication since involve checking status edge graph observe seem straightforward effectively exploit techniques algorithm implemented variant giraph model example linear sketches easily applied distributed streaming model sending coordinator machine sketches partial stream added obtain sketch entire stream mimicking trivial strategy model model would cause much congestion one node leading time bound using techniques fast connectivity algorithm section give algorithms many important graph problems particular present algorithm computing mst hence also present algorithms approximate many graph verification problems including spanning connected subgraph cycle containment bipartiteness algorithms optimal polylogarithmic factor section show lower bound rounds many verification problems simulating model model communication complexity inputs randomly assigned players related work theoretical study graph computations distributed systems relatively new several works devoted developing mapreduce graph algorithms see references therein note flavor theory developed mapreduce quite different compared one model minimizing communication also key goal mapreduce algorithms however usually achieved making sure data made small enough quickly small number mapreduce rounds fit memory single machine see mapreduce algorithm mst comparison model models parallel distributed processing including parallel bsp model mapreduce congested clique refer particular according among models restricted communication big data model one similar mapreduce model model closely related bsp model considered simplified version bsp costs local computation synchronization happens end every round ignored unlike bsp refinements thereof several different parameters make analysis algorithms complicated model characterized one parameter number machines makes model simple enough analytically tractable thus easing job designing analyzing algorithms time still captures key features distributed computations model related classical con gest model particular congested clique model recently received considerable attention see main difference model aimed study computations size input significantly bigger number available machines thus many vertices input graph mapped machine whereas two aforementioned models aimed study distributed network algorithms thus vertex corresponds dedicated machine local knowledge available per vertex since access free information vertices machine model compared two models hand vertices assigned machine communicate links incident machine limit bandwidth unlike two models vertex dedicated processor differences manifest time complexity particular fastest known distributed algorithm congested clique model given problem may give rise fastest algorithm model example fastest algorithms mst congested clique model require messages implementing algorithms model requires rounds conversely slower ghs algorithm gives bound model recently developed techniques see used prove time lower bounds standard con gest model congested clique model directly applicable work closest spirit recent work woodruff zhang paper considers number basic statistical graph problems distributed model similar model however important differences first model asynchronous cost function communication complexity refers total number bits exchanged machines computation second distribution input assumed assume random distribution third important difference assume edge partition model problems graphs edges graph opposed vertices partitioned across machines particular connectivity problem show message complexity lower bound essentially translates round lower bound model shown using proof technique lower bound also applies random edge partition rep model edges partitioned randomly among machines well hand easy show upper bound connectivity rep model connectivity hence rep model tight bound connectivity related problems mst however contrast rvp model arguably natural partition model show tight bound results step towards better understanding complexity distributed graph computation partition model technical point view king also use idea similar linear sketching technique might also useful context model connectivity algorithm section present main result monte carlo randomized algorithm model determines connected components undirected graph correctly high probability terminates rounds high probability algorithm optimal polylog virtue lower bound rounds delving details algorithm briefly discuss simpler less efficient approaches easiest way solve problem model first collect available graph data single machine solve problem locally example one could first elect referee among machines requires rounds instruct every machine send local data referee machine since referee machine needs receive information total links bounded bandwidth requires rounds refined approach obtain distributed algorithm model use conversion theorem provides simulation congested clique algorithm idea mst algorithm rep model first filter edges assigned one machine using cut cycle properties mst leaves machine edges convert edge distribution rvp accomplished rounds via hashing vertices randomly machines routing edges appropriately apply rvp bound rounds model message complexity round complexity upper bound total number messages sent received single node single round parameters refer performance congested clique model unfortunately existing algorithms typically require scale maximum node degree thus converted time complexity bound model becomes best therefore order break barrier must develop new techniques directly exploit additional locality available model next subsection give high level overview algorithm formally present technical details subsequent subsections overview algorithm algorithm follows strategy repeatedly merges adjacent components input graph connected subgraphs form larger connected components output phases labeling nodes nodes belong current component label beginning first phase node labeled unique forms distinct component also component proxy component note phase component contains nodes might spread across different machines use term component part refer nodes component held machine hence phase every component partitioned component parts end algorithm vertex label two vertices label belong connected component algorithm relies linear graph sketches tool enable merging multiple components intuitively speaking random linear sketch node graph neighborhood returns sample chosen uniformly random incident edges interestingly linear sketch represented matrices using polylog bits crucial property sketches linear given sketches combined sketch refers matrix addition property yields random sample edges incident graph contracted edge single node describe technical details section describe communicate graph sketches efficient manner consider component split parts nodes hosted machines find outgoing edge first instruct machine construct linear sketch graph neighborhood nodes part sum sketches yielding sketch spi neighborhood part combine sketches distinct parts select random component proxy machine current component round see section next machine sends spi machine note causes messages sent component proxy finally machine computes spi uses sample edge incident node construction guaranteed endpoint distinct component see section point component proxy sampled edge inducing edges component graph vertex corresponds component enable efficient merging components employ distributed random ranking drr technique break long paths manageable directed trees depth log end every component chooses rank independently uniformly random component virtually connects neighboring component according via conceptual directed edge latter higher rank thus process results collection disjoint rooted trees rooted node highest local rank show section trees depth log merging components tree proceeds leafs upward parallel tree first merging phase leaf merges parent relabeling component labels nodes label note proxy mcj knows labeling computed outgoing edge vertex vertex therefore machine mcj sends label machines hold part section show done parallel leafs trees rounds repeating merging procedure log times guarantees tree merged single component finally section prove log repetitions process suffice ensure components end last phase correspond connected components input graph communication via random proxy machines recall algorithm iteratively groups vertices components subsequently merges components according topology components may split multiple component parts spanning multiple machines hence ensure efficient load balancing messages machines need send behalf component parts hold algorithm performs communication via proxy machines algorithm proceeds phases phase consists iterations consider phase algorithm construct sufficiently random hash function component machine selected proxy machine component first machine generates random bits private source randomness distribute random bits machines via following simple routing mechanism proceeds sequences two rounds selects bits set private random bits remain distributed sends bit across link machine upon receiving machine broadcasts machines next round ensures bits become common knowledge within two rounds repeating process distribute bits takes rounds machines random bits generated leverage result formulation theorem tells generate random hash function independent using log true random bits instruct machine disseminate log polylog random bits according routing process machine locally constructs hash function used determine component proxies throughout iteration phase show communication via proxy machines fast model lemma suppose machine generates message size polylog bits component part residing let denote message part let component part addressed proxy machine component messages delivered within rounds high probability easy see accuracy log bits suffices break ties proof observe except first phase algorithm claim immediately follow standard argument destinations messages chosen independently uniformly random two distinct messages component destination let stipulate component part held machine component part component denote part means machine component part component suppose algorithm phase iteration construction hash function independent component parts held single machine parts different components since distinct component parts follows proxy machines selected component parts held machine distributed independently uniformly random let number distinct component parts held machine consider link connecting another machine let indicator variable takes value component proxy part let otherwise let number component parts chose proxy machine endpoint link since expected number messages sent machine specific link first consider case log independent proxies component parts chosen independently thus apply standard chernoff bound see gives log applying union bound machines conclude every machine sends messages proxy machine requires rounds consider case log holds log log thus standard chernoff bound log log analogously first case applying union bound machines yields result linear graph sketches see section algorithm proceeds merging components across randomly chosen edges subsection show provide sampling capabilities way model implementing random linear graph sketches description follows notation recall vertex associated unique integer known home machine simplicity also denote vertex define incidence vector describes incident edges follows otherwise note asymptotics results change size space polylog note vector corresponds incidence vector contracted edge intuitively speaking summing incidence vectors zeroes edges corresponding vertices hence vector represents outgoing edges component since incidence vector requires polynomial space would inefficient directly communicate vectors component proxies instead construct random linear sketch polylog property allowing sample uniformly random nonzero entry edge incident referred streaming literature see shown performed linear projections therefore beginning phase algorithm instruct machine create new common polylog sketch matrix call phase sketch machine creates sketch vertex resides hence represented polylogarithmic number bits observe linearity words crucial property sketches sum sketch allows sample edge incident contracted edge summarize properties following statement lemma consider phase let subgraph induced vertices let associated sketches vertices constructed applying phase sketch matrix respective incidence vectors combined sketch sui represented using polylog bits querying possible sample random edge incident endpoint constructing linear sketches without shared randomness construction linear sketches described far requires fully independent random bits would need shared machines shown theorem also corollary possible construct linearity properties using random bits log independent analogously section generate required true random bits machine distribute among machines rounds invoke theorem machine parallel generate required shared log independent random bits constructing sketches outgoing edge selection know construct sketch graph neighborhood set vertices describe combine sketches way model goal step current component find outgoing edge connects component recall might split parts across multiple machines therefore first step machine locally constructs combined sketch part resides lemma resulting sketches polylogarithmic size present sketch incidences respective component parts next combine sketches individual parts component sketch instructing machines send sketch part component proxy machine virtue lemma messages delivered component proxies within rounds finally component describe construction nodes access source shared randomness create sketch matrix later show remove assumption proxy machine combines received sketches yield sketch randomly samples outgoing edge see lemma thus end procedure every component randomly selected exactly one neighboring component show complexity procedure lemma every component select exactly one outgoing edge rounds high probability proof clearly since every moment node unique component label machine holds component parts parts selected one edge thus machine selected edges edges sent corresponding proxy lemma requires rounds procedure completed proxies communicate decision components parts entails many messages first part routed using exactly machines links used first part difference messages travel opposite direction lemma follows merging components proxy machine component selected one edge connecting different component neighboring components merged become new bigger component accomplished relabeling nodes graph nodes new component label notice merging thus virtual component parts compose new component moved common machine rather nodes incident edges remain home machine get possibly assigned new label think components along sampled outgoing edges component graph use distributed random ranking drr technique avoid long chains components long paths conceptually construct forest directed trees subgraph modulo edge directions component graph tree depth log component proxy component chooses rank independently uniformly random easy show log bits provide sufficient accuracy break ties proxy machine virtually connects neighboring component rank chosen latter proxy higher case say becomes parent child lemma rounds structure completed high probability proof need show every proxy machine component knows parent component every root proxy machine knows root note step proxy machines child components communicate respective parent proxy machines moreover number messages sent determining ordering guaranteed high probability since edges instantiating lemma follows delivery messages completed rounds instead using drr trees alternate simpler idea following let every component select number merging done outgoing edge obtained sketch connects component component one show merging procedure also gives time bound since links bidirectional parent proxies able send replies within number rounds message schedule communication reverse order component highest rank among neighbors call root component since every component except root components connects component higher rank resulting structure set disjoint rooted trees next step merge components tree single new component vertices part component tree receive label root consider tree proceed parallel trees start merging components leafs connected parent component lemma distributed algorithm merges trees drr forest rounds high probability largest depth tree proof proceed iterations merging current leaf components parents tree thus sufficient analyze time complexity single iteration end describe procedure changes component labels vertices leaf components drr forest label respective parent rounds beginning iteration select new proxy component querying shared hash function current iteration number ensures dependencies proxies used iteration know lemma message schedule leaf proxies communicate respective parent proxy rounds vice versa thus every leaf proxy knows component label parent already shown lemma deliver message component part respective proxy combining sketches rounds hence message schedule broadcast parent label leaf proxy component part time machine receives parent label locally changes component label vertices corresponding part following result proved theorem keep paper also provide direct simpler proof result see appendix lemma theorem depth drr tree log high probability analysis time complexity show number phases required algorithm determine connected components input graph log beginning phase distributed across machines distinct components beginning algorithm node identified component thus algorithm ends completion phase smallest integer denotes number connected components input graph pairs components merged phase would straightforward show process would terminate log phases however algorithm component connects neighboring component latter higher rank nevertheless difficult show slightly different process also terminates log phases components gets merged often enough intuition result since components ranks taken randomly component probability neighboring component higher rank exactly one half hence average half components merged neighbor components thus becomes root one component means average number new components half well lemma log phases component labels vertices correspond connected components high probability proof replace corresponding random variables consider stochastic process defined sequence let random variable counts number components actually participate merging process phase outgoing edge another component call components participating components clearly definition show every phase end fix generic phase random ordering participating components define random variables takes value participating component root participating phase otherwise number participating components phase noticed probability participating component merged neighboring component thus become root phase exactly one half therefore hence linearity expectation using linearity expectation leverage result prove claimed statement let call phase successful reduces number participating components factor markov inequality probability phase successful thus probability phase algorithm successful least consider sequence log phases algorithm shall prove within many phases algorithm reduced number participating components sufficient number times algorithm terminated log let indicator variable takes value phase successful otherwise also includes case phase log number take place algorithm already terminated let successful phases log phases algorithm since linearity expectation log log log log independent apply standard chernoff bound gives log log log hence high probability log phases enough determine components input graph theorem distributed algorithm model determines connected components graph rounds high probability proof lemma algorithm finishes log phases high probability analyze time complexity individual phase recall takes rounds sample outgoing edge see lemma building drr forest requires additional rounds according lemma merging drr tree fashion parallel takes rounds see lemma depth virtue lemma bounded log since time bounds hold high probability algorithm consists log phases high probability union bound conclude total time complexity algorithm high probability conclude section noticing easy output actual number connected components termination algorithm every machine needs send yes directly proxies components labels holds subsequently proxies send labels components received yes one predetermined machine since communication performed via components proxies follows lemma first step takes rounds second step takes log rounds applications section describe use fast connectivity algorithm building block solve several fundamental graph problems model time constructing minimum spanning tree given weighted graph edge associated weight initially known home machines minimum spanning tree mst problem asks output set edges form tree connect nodes minimum possible total weight klauck show rounds necessary constructing spanning tree assuming every spanning tree edge home machine home machine must output part show break barrier slightly less stringent requirement spanning tree edge returned least one machine necessarily home machines algorithm mimics mst construction procedure originally devised centralized streaming model end modify connectivity procedure section ensuring component proxy chooses outgoing edge minimum weight outgoing edge mwoe high probability describe phase mst construction detail analogously connectivity algorithm section proxy component determines outgoing edge guarantees sketch construction lemma chosen uniformly random possible outgoing edges repeat following process log times proxy broadcasts every component part recall lemma communication possible rounds upon receiving message machine part constructs new sketch first zeroes entries refer edges weight see section detailed description combine sketches vertices parts proxy turn samples new outgoing edge since time sample randomly chosen edge eliminate higher weight edges easy see edge mwoe thus proxy machine includes edge part mst output note additional elimination procedure incurs logarithmic time complexity overhead end phase proceed virtually merging components along mwoes similar manner connectivity algorithm see section thus requiring rounds total let set added outgoing edges since components connectivity algorithm eventually match actual components input graph graph vertices induced connects vertices moreover since components merged according trees see section follows fully classify complexity mst problem model theorem exists algorithm model outputs mst rounds output least one machine rounds output machines hold endpoint bounds tight polylogarithmic factors log show following result problem model theorem exists log algorithm problem model runs rounds high probability proof use exponentially growing sampling probabilities sampling edges check connectivity leveraging result karger procedure proposed classic con gest model implemented model well use fast connectivity algorithm place thurimella algorithm used time complexity dominated procedure thus algorithms graph verification problems well known graph connectivity important building block several graph verification problems see analyze problems formally defined section model theorem exist algorithms model solve following verification problems rounds high probability spanning connected subgraph cycle containment containment cut connectivity edge paths cut bipartiteness proof discuss problem separately cut verification remove edges given cut check whether resulting graph connected connectivity verification run connectivity algorithm verify whether connected component checking whether label edge paths verification since lies paths iff disconnected simply use connectivity verification algorithm previous point cut verification verify subgraph cut simply verify connectivity graph removing edges subgraph bipartiteness verification use connectivity algorithm reduction presented section spanning connected subgraph cycle containment containment verification also follow reductions given lower bounds verification problems section show rounds fundamental lower bound many graph verification problems model end use results classical theory communication complexity popular way derive lower bounds distributed models even though many verification problems known satisfy lower bound classic distributed con gest model reduction encodes set disjointness requiring least one node receive information across single short highway path via longer paths length moreover assume random vertex partition model whereas results assume worst case distribution lastly pair machines communicate directly model thus breaking bound con gest model complexity bounds follow communication complexity set disjointness random input partition model see standard model communication complexity players alice bob alice bob receives input vector bits random input partition model alice receives addition bit probability revealed alice bob input defined similarly respect set disjointness problem alice bob must output index following result holds lemma lemma constant every randomized communication protocol solves set disjointness random input partition model communication complexity probability least requires bits show main result section theorem exists constant algorithm round complexity vertex graph diameter model solves following problems connectivity spanning connected subgraph cycle containment containment cut edge paths proof idea proof similar simulation theorem present argument spanning connected subgraph problem defined remaining problems reduced scs problem using reductions similar spanning connected subgraph scs problem given graph subgraph want verify whether spans connected show reduction set disjointness algorithm scs model requires rounds given instance set disjointness problem random partition model construct following input graphs nodes consist special nodes nodes clarity presentation assume integers edges consist edges let set machines simulated alice let set machines simulated bob first alice bob use shared randomness choose machines receive vertices alice assigns machine chosen randomly bob assigns random machine otherwise denote machine alice bob output terminate simulation subgraph determined disjointness input vectors follows contains nodes edges recall random partition model randomly distributed alice bob alice knows bob knows hence alice bob mark corresponding edges part according respective input bits alice received bob receive assigns node random machine adds edge similarly edge added either alice bob depending receives see figure note since assigned according random input partition model resulting distribution vertices machines adheres random vertex partition model clearly scs disjoint describe simulation alice point view simulation bob similar alice locally maintains counter initialized represents current round number simulates run machines yielding set messages figure graph construction spanning connected subgraph problem given set disjointness instance thick edges edges subgraph subgraph contains edges remaining edges determined input vectors set disjointness instance polylog bits need sent bob simulate algorithm machines next round construction send messages asynchronous random partition model communication complexity alice sends message bob tuple corresponds message generated machine simulated alice destined machine simulated bob upon receiving message bob increases round counter locally simulates next round machines delivering messages appropriate machines adding source destination fields message incurs overhead log log bits hence total communication generated simulating single round upper bounded therefore takes rounds solve scs model gives polylog communication complexity protocol set disjointness random partition model communication alice bob determined communication across links required simulation carry polylog bits per round note errs probability simulation errs probability extra term comes possibility machines refer machine large enough small enough follows need simulate least many rounds since lemma set disjointness problem requires bits random partition model error smaller interestingly lower bounds hold even graphs diameter contrast analogous results classic distributed con gest model assumed remark lower bound connectivity verification already shown conclusions several natural directions future work connectivity algorithm randomized would interesting study deterministic complexity graph connectivity model specifically graph connectivity admit deterministic algorithm investigating connectivity connectivity also interesting research direction general question motivated algorithms presented paper whether one design algorithms superlinear scaling fundamental graph problems recent results directions acknowledgments authors would like thank mohsen ghaffari seth gilbert andrew mcgregor danupon nanongkai sriram pemmaraju helpful discussions references giraph http kook jin ahn sudipto guha andrew mcgregor analyzing graph structure via linear measurements proceedings annual symposium discrete algorithms soda pages kook jin ahn sudipto guha andrew mcgregor graph sketches sparsification spanners subgraphs proceedings acm symposium principles database systems pods pages noga alon babai alon itai fast simple randomized parallel algorithm maximal independent set problem algorithms noga alon ronitt rubinfeld shai vardi ning xie local computation algorithms proceedings annual symposium discrete algorithms soda pages otakar boruvka certain minimal problem mor spol brne iii keren petteri kaski janne korhonen christoph lenzen ami paz jukka suomela algebraic methods congested clique proceedings acm symposium principles distributed computing podc pages chen gopal pandurangan aggregate computation siam fan chung olivia simpson distributed algorithms finding local clusters using heat kernel pagerank proceedings workshop algorithms models waw pages graham cormode donatella firmani unifying framework algorithms distributed parallel databases atish das sarma stephan holzer liah kor amos korman danupon nanongkai gopal pandurangan david peleg roger wattenhofer distributed verification hardness distributed approximation siam andrew drucker fabian kuhn rotem oshman power congested clique model proceedings acm symposium principles distributed computing podc pages michael elkin hartmut klauck danupon nanongkai gopal pandurangan quantum communication speed distributed computation proceedings acm symposium principles distributed computing podc pages robert gallager pierre humblet philip spira distributed algorithm spanning trees acm trans program lang mohsen ghaffari fabian kuhn distributed minimum cut approximation proceedings international symposium distributed computing disc pages james hegeman gopal pandurangan sriram pemmaraju vivek sardeshmukh michele scquizzato toward optimal bounds congested clique graph connectivity mst proceedings acm symposium principles distributed computing podc pages hossein jowhari mert saglam tardos tight bounds samplers finding duplicates streams related problems proceedings acm symposium principles database systems pods pages david karger random sampling cut flow network design problems proceedings annual acm symposium theory computing stoc pages david karger philip klein robert tarjan randomized algorithm find minimum spanning trees acm howard karloff siddharth suri sergei vassilvitskii model computation mapreduce proceedings annual symposium discrete algorithms soda pages valerie king shay kutten mikkel thorup construction impromptu repair mst distributed network communication proceedings acm symposium principles distributed computing podc pages hartmut klauck danupon nanongkai gopal pandurangan peter robinson distributed computation graph problems proceedings annual symposium discrete algorithms soda pages eyal kushilevitz noam nisan communication complexity cambridge university press shay kutten gopal pandurangan david peleg peter robinson amitabh trehan sublinear bounds randomized leader election theoret comput silvio lattanzi benjamin moseley siddharth suri sergei vassilvitskii filtering method solving graph problems mapreduce proceedings acm symposium parallelism algorithms architectures spaa pages christoph lenzen optimal deterministic routing sorting congested clique proceedings acm symposium principles distributed computing podc pages christoph lenzen roger wattenhofer tight bounds parallel randomized load balancing distrib jure leskovec anand rajaraman jeffrey david ullman mining massive datasets cambridge university press zvi lotker boaz elan pavlov david peleg spanning tree construction log log communication rounds siam nancy lynch distributed algorithms morgan kaufmann publishers grzegorz malewicz matthew austern aart bik james dehnert ilan horn naty leiser grzegorz czajkowski pregel system graph processing proceedings acm international conference management data sigmod pages andrew mcgregor graph stream algorithms survey sigmod record michael mitzenmacher eli upfal probability computing randomized algorithms probabilistic analysis cambridge university press danupon nanongkai distributed approximation algorithms weighted shortest paths proceedings acm symposium theory computing stoc pages danupon nanongkai atish das sarma gopal pandurangan tight unconditional lower bound distributed randomwalk computation proceedings annual acm symposium principles distributed computing podc pages rotem oshman communication complexity lower bounds distributed proceedings international colloquium structural information communication complexity sirocco pages gopal pandurangan david peleg michele scquizzato message lower bounds via efficient network synchronization proceedings international colloquium structural information communication complexity sirocco appear gopal pandurangan peter robinson michele scquizzato tight bounds distributed graph computations corr david peleg distributed computing approach society industrial applied mathematics judy qiu shantenu jha andre luckow geoffrey fox initial big data stack http towards available isabelle stanton streaming balanced graph partitioning algorithms random graphs proceedings annual symposium discrete algorithms soda pages ramakrishna thurimella distributed algorithms sparse certificates biconnected components algorithms yuanyuan tian andrey balmin severin andreas corsten shirish tatikonda john mcpherson think like vertex think like graph pvldb leslie valiant scheme fast parallel communication siam leslie valiant bridging model parallel computation commun acm sergei vassilvitskii models parallel computation hitchhikers guide massively parallel universes http david woodruff qin zhang distributed computation communication expensive distrib appear omitted proofs proof lemma proof consider one phase algorithm suppose phase components one phase components thus setting gives valid upper bound height drr tree phase component picks random rank thus ranks distinct high probability target component rank higher source component connects otherwise source component becomes root drr tree consider arbitrary component graph consider unique path starting form node represents component root tree contains let number nodes assign indexes nodes according position path selected node root see figure figure one drr tree one path one node root tree nodes path labeled indicator variable associated indexed position node path define indicator variable takes value node root otherwise length path random choice outgoing edge made components parts outgoing edge component random distinct component means ranks first nodes path form set random values hence probability new random value higher rank node path probability new random value higher previously chosen random values probability value highest among first values path probability thus hence linearity expectation expected height path tree produced drr procedure log notice independent identically distributed random variables since probability smallest ranked node root depends random neighbor picks independent choices nodes thus applying standard chernoff bound see log log applying union bound paths concludes proof
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nov label efficient learning transferable representations across domains tasks zelun luo stanford university zelunluo yuliang zou virginia tech ylzou judy hoffman university california berkeley jhoffman stanford university feifeili abstract propose framework learns representation transferable across different domains tasks label efficient manner approach battles domain shift domain adversarial loss generalizes embedding novel task using metric approach model simultaneously optimized labeled source data unlabeled sparsely labeled data target domain method shows compelling results novel classes within new domain even labeled examples per class available outperforming prevalent approach addition demonstrate effectiveness framework transfer learning task image object recognition video action recognition introduction humans exceptional visual learners capable generalizing learned knowledge novel domains concepts capable learning examples recent years computational models based learnable convolutional networks made significant improvements visual recognition shown demonstrate generalizations enabling faster learning subsequent tasks frequently evidenced finetuning however efforts focus supervised learning scenario closed world assumption made training time domain interest tasks learned thus generalization ability models observed byproduct large push research community address generalizing adapting deep models across different domains learn tasks data efficient way shot learning generically transfer information across tasks approaches consider scenarios isolation aim directly tackle joint problem adapting novel domain new tasks annotations given large labeled source dataset annotations task set seek transfer knowledge sparsely labeled target domain possibly wholly new task set setting line intuition able learn reusable general purpose representations enable faster learning future tasks requiring less human intervention addition setting matches closely common practical approach training deep models use large labeled source dataset often imagenet train initial representation continue supervised learning new set data often new concepts conference neural information processing systems nips long beach usa approach jointly adapt source representation use distinct target domain using new multilayer unsupervised domain adversarial formulation introducing novel within domain class similarity objective new objective applied even target domain classes source domain evaluate approach challenging setting joint transfer across domains tasks demonstrate ability successfully transfer reducing need annotated data target domain tasks present results transferring subset google street view house numbers svhn containing digits subset mnist containing digits secondly present results challenging setting adapting imagenet images videos action recognition related work domain adaptation domain adaptation seeks learn related source domains well performing model target data distribution existing work often assumes domains defined task labeled data target domain sparse several methods tackled problem maximum mean discrepancy mmd loss source target domain weight sharing cnn parameters minimizing distribution discrepancy network activations also shown convincing results adversarial generative models aim generating data target data training generator discriminator simultaneously adversarial discriminative models focus aligning embedding feature representations target domain source domain inspired adversarial discriminative models propose method aligns domain features information transfer learning transfer learning aims transfer knowledge leveraging existing labeled data related task domain computer vision examples transfer learning include try overcome deficit training samples categories adapting classifiers trained categories power deep supervised learning imagenet dataset learned knowledge even transfer totally different task image classification object detection image classification semantic segmentation achieve performance paper focus setting source target domains differing label spaces label spaces share structure namely adapting classifying different category sets transferring classification localization plus classification task learning learning seeks learn new concepts annotated examples deep siamese networks trained rank similarity examples matching networks learns network maps small labeled support set unlabeled example label aside metric methods also served essential part ravi propose learn lstm learn update rule learner finn tries find good initialization point easily new examples new tasks exists domain shift results prior learning methods often degraded unsupervised learning many unsupervised learning algorithms focused modeling raw data using reconstruction objectives probabilistic models include restricted boltzmann machines deep boltzmann machines gans autoregressive models also popular alternative approach often terms learning defines pretext task predicting patch ordering frame ordering motion dynamics colorization form indirect supervision compared approaches unsupervised learning method rely exploiting spatial temporal structure data therefore generic method introduce learning algorithm transfers information large labeled source domain sparsely labeled target domain goal learn strong target source cnn supervised loss source labeled data domain transfer adversarial loss target unlabeled data semantic transfer pairwise similarity softmax entropy loss target labeled data supervised loss target cnn figure proposed learning framework joint transfer across domains semantic transfer across source target across target labeled unlabeled data introduce domain discriminator aligns source target representations across multiple layers network domain adversarial learning enable semantic transfer minimizing entropy pairwise similarity unlabeled labeled target images use temperature softmax similarity vector allow label spaces classifier without requiring large annotation overhead required standard supervised learning approaches fact setting commonly explored convolutional network convnet based recognition methods learning convnets usual learning procedure use large labeled dataset imagenet initial training network parameters termed learned weights used initialization continued learning new data new tasks called broadly applied reduce number labeled examples needed learning new tasks recognizing new object categories imagenet learning new label structures detection classficiation pretraining focus transfer case shared label structure classification different category sets assume source domain contains images associated labels similarly target domain consists unlabeled images well images associated labels assume target domain sparsely labeled number pairs much smaller number unlabeled images additionally number source labeled images assumed much larger number target labeled images unlike standard domain adaptation approaches transfer knowledge source target domains assuming marginal conditional distribution shift shared label space tackle joint image feature space adaptation well transfer across semantic spaces namely consider case source target label spaces equal even challenging case sets joint domain semantic transfer approach consists unsupervised feature alignment source target well semantic transfer unlabeled target data either labeled target labeled source data introduce new domain discriminator used domain alignment following recent domain adversarial learning approaches next introduce new semantic transfer learning objective uses cross category similarity tuned account varying size label set overlap depict overall model figure take source labeled examples target labeled examples unlabeled target images input learn initial layered source representation classification network depicted blue figure using standard supervised techniques initialize target model depicted green figure source parameters begin adaptive transfer learning model jointly optimizes target supervised loss lsup domain transfer objective ldt finally semantic transfer objective lst thus total objective written follows lsup hyperparameters determine influence domain transfer loss semantic transfer loss respectively following sections elaborate domain semantic transfer objectives domain adversarial loss define novel domain alignment objective function called domain adversarial loss recent efforts deep domain adaptation shown strong performance using feature space domain adversarial objectives methods learn target representation target distribution viewed model aligned source distribution viewed source representation alignment accomplished adversarial minimization across domain analogous prevalent generative adversarial approaches particular domain discriminator trained classify whether particular data point arises source target domain simultaneously target embedding function defined application layers network trained generate target representation distinguished source domain representation domain discriminator similar consider representation domain invariant domain discriminator distinguish examples two domains prior work considers alignment single layer embedding time learns domain discriminator takes output corresponding source target layers input separately domain alignment methods focus first second order statistics shown improved performance applying domain alignment independently multiple layers network rather learning independent discriminators layer network propose simultaneous alignment multiple layers discriminator layer domain discriminator information accumulated output previous discriminator layer well source target activations corresponding layer respective embeddings thus output discriminator layer defined current layer activation function decay factor represents concatenation summation taken either source data target data notice intermediate discriminator layers share structure corresponding encoding layers match dimensions thus following loss functions proposed optimize domain discriminator embeddings respectively according domain transfer objective log ext log log ext log dsl dtl outputs last layer source target domain discriminator note losses placed final domain discriminator layer last embedding layer produce gradients throughout relevant lower layer parameters two losses together comprise ldt iterative optimization procedure involved discriminator shown figure yellow allows deeper alignment source target representations find empirically results improved target classification performance well stable adversarial learning figure illustrate purpose temperature pairwise similarity vector consider example target unlabeled point similarity four labeled source points show original unnormalized scores leftmost well similarity scores applying softmax different temperatures notice entropy values higher variance scores normalized small temperature softmax cross category similarity semantic transfer previous section introduced method transferring embedding source target domain however enforces alignment global domain statistics class specific transfer define new semantic transfer objective lst transfers information labeled set data unlabeled set data minimizing entropy softmax temperature similarity vector unlabeled point labeled points thus loss may applied either source unlabeled target data labeled unlabeled target data unlabeled target image compute similarity labeled example prototypical example per class labeled set simplicity presentation let consider semantic transfer source target domain first target unlabeled image compute similarity vector ith element similarity target image ith labeled source image xsi semantic transfer loss defined follows lst information entropy function softmax function temperature softmax note temperature used directly control percentage source examples expect target example similar see figure entropy minimization widely used unsupervised learning encouraging low density separation clusters classes recently principle entropy minimization applied unsupervised adaptation source target domains assumed share label space unlabeled target example passed initial source classifier entropy softmax output scores minimized contrast assume shared label space source target domains assume target image maps single source label instead compute pairwise similarities target points source points per class averages source points across features spaces aligned domain adversarial transfer tune softmax temperature based expected similarity source target labeled set example source target label set overlap small temperature encourage target point similar one source class whereas larger temperature allow target points similar multiple source classes semantic transfer within target domain utilize cross entropy loss labeled target examples stabilize improve learning labeled target example addition traditional cross entropy loss also calculate cross entropy loss assume labeled examples class target domain compute embedding refer cue reader cross entropy within label space example centroid cti class embedding space thus compute similarity vector labeled example ith element similarity labeled example centroid class cti calculate metric based cross entropy loss lst sup exp log exp similar scenario also unsupervised part lst unsup cross entropy loss introduce constraint target domain data similar embedding space also find loss provide guidance unsupervised semantic transfer learn stable way lst combination lst unsupervised equation lst supervised equation lst unsupervised equation lst lst lst sup lst unsup experiment section structured follows section show method outperform approach large margin parts method necessary section show method generalized bigger datasets section show domain adversarial method outperforms domain adversarial approaches datasets perform adaptation experiments across two different paired data settings first adaptation across different digit domains use mnist google street view house numbers svhn mnist handwritten digits database training set examples test set examples digits centered images svhn image dataset machine learning object recognition algorithms minimal requirement data preprocessing formatting digits training digits testing second experimental setup consider adaptation object centric images imagenet action recognition video using dataset imagenet large benchmark object classification task use task split action recognition dataset collected youtube videos action categories provides large diversity terms actions presence large variations camera motion object appearance pose object scale viewpoint cluttered background illumination conditions etc implementation details source domain embedding function loss domain adversarial loss discriminator takes last three layer activations input number output classes source target tasks takes second last third last layer activations different similarity score chosen dot product normalized support features unnormalized target feature use temperature semantic transfer within target transfer label space shared use objective function network trained adam optimizer learning rate conduct experiments pytorch framework svhn mnist experimental setting experiment define three datasets labeled data source domain labeled data target domain iii unlabeled data target domain take training split svhn dataset dataset fairly compare traditional learning paradigm episodic training subsample examples class construct dataset perform traditional training episodic learning experiment corresponds labeled examples total training data respectively since approach involves using annotations small subset data randomly subsample different subsets training split mnist dataset use remaining data note source domain target domain classes utilize digits svhn digits mnist figure illustration task model effectively transfer learned representation svhn digits left mnist digits right baselines prior work compare six different methods target model trained scratch model pretrained iii matching networks first pretrain model use support set matching networks matching networks baseline iii except model learning examples class randomly selected support set last example class used query set adversarial addition baseline model also trained domain adversarial loss full model model proposed domain adversarial loss results analysis calculate mean standard error accuracies across sets data shown table due domain shift matching networks perform poorly without marginally better training scratch method adversarial improves overall performance sensitive subsampled data method achieves significant performance gain especially number labeled examples small reference full target dataset gives accuracy table test accuracies baseline models method row row correspond order six methods proposed section note accuracies two matching net methods calculated based nearest neighbors support set report mean standard error method across different subsampled data method target matching nets matching nets adv full model figure visualization different feature embeddings source domain embedding target domain embedding using encoder trained source domain domain target domain embedding using encoder target domain data target domain embedding using encoder trained method overlap overlap best viewed color zoom image object recognition video action recognition problem analysis many recent works study domain shift images video object detection settings compared still images videos provide several advantages motion provides information foreground background segmentation videos often show multiple views thus provide information hand video frames usually suffer motion blur compression artifacts iii objects experimental setting experiment focus three dataset splits imagenet training set labeled data source domain video clips per class randomly sampled training labeled data target domain set iii remaining videos training set unlabeled data target domain experiment corresponds video clips total training data respectively experiment run times baselines prior work compare method two baseline methods target model trained scratch model first reference report performance fully supervised method results analysis accuracy model shown table also model labeled data comparison performance img performance vid reported note calculate performance averaging softmax score frame video method achieves significant improvement performance standard value note compared method bigger gap accuracy believe due semantic transfer entropy loss encourages sharper softmax variance among softmax scores per video variance zero accuracy accuracy making confident predictions among key frames method achieves significant gain respective performance even little change prediction table accuracy action classification results spatial model taken vary depending hyperparameters report mean standard error method across different subsampled data method target img target vid img vid spatial img vid ablation unsupervised domain adaptation validate domain adversarial loss objective conduct ablation experiment unsupervised domain adaptation compare multiple recent domain adversarial unsupervised adaptation methods experiment first pretrain source embedding cnn training split svhn adapt target embedding mnist performing adversarial domain adaptation evaluate classification performance test split mnist follow training strategy model architecture embedding network models training strategy share first stage adda optimizes encoder classifier simultaneously also propose similar method optimize encoder try model classifier last layer perform domain adversarial training feature space choose decay factor model accuracy model shown table find method achieve performance gain best competing domain adversarial approach indicating multilayer objective indeed contributes overall performance addition experiments found multilayer approach improved overall optimization stability evidenced small standard error table experimental results unsupervised domain adaptation svhn mnist results gradient reversal domain confusion adda results methods experiments across different subsampled data method accuracy source gradient reversal domain confusion adda conclusion paper propose method learn representation transferable across different domains tasks data efficient manner framework trained jointly minimize domain shift transfer knowledge new task learn large amounts unlabeled data show superior performance popular approach hope keep improving method future work acknowledgement would like start thanking sponsors stanford computer science department stanford program care pac next specially thank huang kenji hata serena yeung ozan sener members stanford vision learning lab insightful discussion feedback lastly thank anonymous reviewers valuable comments references yusuf aytar andrew zisserman tabula rasa model transfer object category detection computer vision iccv ieee international conference pages ieee konstantinos bousmalis nathan silberman david dohan dumitru erhan dilip krishnan unsupervised domain adaptation generative adversarial networks arxiv preprint lluis castrejon yusuf aytar carl vondrick hamed pirsiavash antonio torralba learning aligned representations weakly aligned data proceedings ieee conference computer vision pattern recognition pages gabriela csurka domain adaptation visual applications comprehensive survey arxiv preprint virginia learning classification unlabeled data advances neural information processing systems pages deng dong socher imagenet hierarchical image database carl doersch abhinav gupta alexei efros unsupervised visual representation learning context prediction proceedings ieee international conference computer vision pages jeff donahue yangqing jia oriol vinyals judy hoffman ning zhang eric tzeng trevor darrell decaf deep convolutional activation feature generic visual recognition proceedings international conference international conference machine learning volume icml pages jeff donahue philipp trevor darrell adversarial feature learning arxiv preprint vincent dumoulin ishmael belghazi ben poole alex lamb martin arjovsky olivier mastropietro aaron courville adversarially learned inference arxiv preprint chelsea finn pieter abbeel sergey levine fast adaptation deep networks arxiv preprint yaroslav ganin victor lempitsky unsupervised domain adaptation backpropagation arxiv preprint yaroslav ganin evgeniya ustinova hana ajakan pascal germain hugo larochelle laviolette mario marchand victor lempitsky training neural networks journal machine learning research ross girshick jeff donahue trevor darrell jitendra malik rich feature hierarchies accurate object detection semantic segmentation computer vision pattern recognition ian goodfellow jean mehdi mirza bing david sherjil ozair aaron courville yoshua bengio generative adversarial nets advances neural information processing systems pages yves grandvalet yoshua bengio learning entropy minimization nips volume pages gretton smola huang marcel schmittfull borgwardt candela sugiyama schwaighofer lawrence covariate shift kernel mean matching dataset shift machine learning kaiming xiangyu zhang shaoqing ren jian sun deep residual learning image recognition proceedings ieee conference computer vision pattern recognition pages geoffrey hinton ruslan salakhutdinov reducing dimensionality data neural networks science geoffrey hinton terrence sejnowski learning releaming boltzmann machines parallel distrilmted processing judy hoffman saurabh gupta trevor darrell learning side information modality hallucination proceedings ieee conference computer vision pattern recognition pages judy hoffman saurabh gupta jian leong sergio guadarrama trevor darrell crossmodal adaptation detection robotics automation icra ieee international conference pages ieee judy hoffman dequan wang fisher trevor darrell fcns wild adversarial adaptation arxiv preprint vicky kalogeiton vittorio ferrari cordelia schmid analysing domain shift factors videos images object detection ieee transactions pattern analysis machine intelligence diederik kingma jimmy adam method stochastic optimization arxiv preprint diederik kingma max welling variational bayes arxiv preprint gregory koch siamese neural networks image recognition phd thesis university toronto alex krizhevsky ilya sutskever geoff hinton imagenet classification deep convolutional neural networks neural information processing systems nips yann lecun bottou yoshua bengio patrick haffner learning applied document recognition proceedings ieee yanghao naiyan wang jianping shi jiaying liu xiaodi hou revisiting batch normalization practical domain adaptation arxiv preprint joseph lim ruslan salakhutdinov antonio torralba transfer learning borrowing examples multiclass object detection proceedings international conference neural information processing systems pages curran associates liu thomas breuel jan kautz unsupervised translation networks arxiv preprint liu oncel tuzel coupled generative adversarial networks advances neural information processing systems pages wei liu dragomir anguelov dumitru erhan christian szegedy scott reed alexander berg ssd single shot multibox detector european conference computer vision pages springer jonathan long evan shelhamer trevor darrell fully convolutional networks semantic segmentation proceedings ieee conference computer vision pattern recognition pages mingsheng long yue cao jianmin wang michael jordan learning transferable features deep adaptation networks icml pages mingsheng long jianmin wang michael jordan deep transfer learning joint adaptation networks arxiv preprint mingsheng long han zhu jianmin wang michael jordan unsupervised domain adaptation residual transfer networks advances neural information processing systems pages zelun luo boya peng huang alexandre alahi unsupervised learning motion dynamics videos arxiv preprint ishan misra lawrence zitnick martial hebert shuffle learn unsupervised learning using temporal order verification european conference computer vision pages springer yuval netzer tao wang adam coates alessandro bissacco andrew reading digits natural images unsupervised feature learning nips workshop deep learning unsupervised feature learning volume page aaron van den oord nal kalchbrenner koray kavukcuoglu pixel recurrent neural networks arxiv preprint maxime oquab leon bottou ivan laptev josef sivic learning transferring midlevel image representations using convolutional neural networks proceedings ieee conference computer vision pattern recognition pages gintautas palubinskas xavier descombes frithjof kruggel unsupervised clustering method using entropy minimization aaai sinno jialin pan qiang yang survey transfer learning ieee transactions knowledge data engineering deepak pathak ross girshick piotr trevor darrell bharath hariharan learning features watching objects move arxiv preprint sachin ravi hugo larochelle optimization model learning international conference learning representations volume page ali sharif razavian hossein azizpour josephine sullivan stefan carlsson cnn features astounding baseline recognition ieee conference computer vision pattern recognition cvpr workshops columbus usa june pages joseph redmon santosh divvala ross girshick ali farhadi look unified object detection proceedings ieee conference computer vision pattern recognition pages shaoqing ren kaiming ross girshick jian sun faster towards object detection region proposal networks advances neural information processing systems pages artem rozantsev mathieu salzmann pascal fua beyond sharing weights deep domain adaptation arxiv preprint olga russakovsky jia deng hao jonathan krause sanjeev satheesh sean zhiheng huang andrej karpathy aditya khosla michael bernstein imagenet large scale visual recognition challenge international journal computer vision ruslan salakhutdinov geoffrey hinton deep boltzmann machines artificial intelligence statistics pages simonyan zisserman deep convolutional networks image recognition corr karen simonyan andrew zisserman convolutional networks action recognition videos advances neural information processing systems pages jake snell kevin swersky richard zemel prototypical networks learning arxiv preprint khurram soomro amir roshan zamir mubarak shah dataset human actions classes videos wild arxiv preprint baochen sun kate saenko deep coral correlation alignment deep domain adaptation computer workshops pages springer yaniv taigman adam polyak lior wolf unsupervised image generation arxiv preprint kevin tang vignesh ramanathan daphne koller shifting weights adapting object detectors image video advances neural information processing systems pages tatiana tommasi francesco orabona barbara caputo safety numbers learning categories examples multi model knowledge transfer computer vision pattern recognition cvpr ieee conference pages ieee eric tzeng judy hoffman trevor darrell kate saenko simultaneous deep transfer across domains tasks proceedings ieee international conference computer vision pages eric tzeng judy hoffman trevor darrell kate saenko simultaneous deep transfer across domains tasks international conference computer vision iccv eric tzeng judy hoffman kate saenko trevor darrell adversarial discriminative domain adaptation computer vision pattern recognition cvpr eric tzeng judy hoffman ning zhang kate saenko trevor darrell deep domain confusion maximizing domain invariance arxiv preprint aaron van den oord nal kalchbrenner lasse espeholt oriol vinyals alex graves conditional image generation pixelcnn decoders advances neural information processing systems pages laurens van der maaten accelerating using algorithms journal machine learning research laurens van der maaten geoffrey hinton visualizing similarities multiple maps machine learning pascal vincent hugo larochelle yoshua bengio manzagol extracting composing robust features denoising autoencoders proceedings international conference machine learning pages acm oriol vinyals charles blundell tim lillicrap daan wierstra matching networks one shot learning advances neural information processing systems pages karl weiss taghi khoshgoftaar dingding wang survey transfer learning journal big data richard zhang phillip isola alexei efros colorful image colorization european conference computer vision pages springer zhang felix xinnan chang shengjin wang deep transfer network unsupervised domain adaptation arxiv preprint network architecture svhn mnist name layer type kernel stride padding name layer type kernel stride padding name layer type kernel table embedding network structure max pool max pool name layer type kernel table discriminator structure max pool max pool image object recognition video action recognition embedding network structure name layer type kernel stride padding table discriminator structure relu relu conv ablation unsupervised domain adaptation name layer type kernel stride padding name layer type kernel name layer type kernel table embedding network structure max pool max pool table discriminator structure refer readers pytorch implementation https
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resource allocation communications underlaying cellular network yijin pan cunhua pan zhaohui yang ming chen abstract oct letter investigates power control channel assignment problem communications underlaying multiple access noma cellular network successive interference cancellation decoding order constraints target maximize sum rate pairs guaranteeing minimum rate requirements cellular users specifically optimal conditions power control cellular users subchannel derived first based results propose iterative algorithm solve resource allocation problem simulation results validate superiority proposed resource allocation algorithm existing orthogonal multiple access scheme index terms noma power control channel assignment ntroduction communications considered promising way alleviate upcoming traffic pressure core networks due short transmission distance pairs spectrum efficiency significantly improved spectrum reuse cellular users cus conventional networks uplink resources normally provided communications since traffic downlink significantly heavier uplink however uplink applications high rate requirements becoming popular future networks skype video call hence traffic uplink downlink becoming less asymmetric resource allocation problem communications underlaying downlink celluar network studied well apart communications multiple access noma another emerging technology handle transmission pressure near future cellular network multiple cus allowed share subchannel via different power pan yang chen national mobile communications research laboratory southeast university nanjing china email panyijin yangzhaohui chenming pan school electronic engineering computer science queen mary university london london email levels successive interference cancellation sic adopted cus decoding way cellular network greatly increase system throughput allow massive connectivities recently several approaches proposed combine communications noma technology users grouped noma way achieve better rate performance channel allocation problem groups modeled matching furthermore assisted noma scheme proposed enhance system throughput performance pairs merely assumed transmit exclusive channels without sharing channels cus even though frequency reuse pairs cus spectrum efficient however pairs reuse spectrum cus interference sic decoding become complicated may destroy original sic decoding order cus conquer issue one impose additional restriction power control channel assignment pairs studied current literature motivates reconsider resource allocation problem communications pairs share spectrum cus letter consider power control channel assignment pairs underlaying cellular networks consideration sic decoding constraints scenario different noma implemented transmissions furthermore since power control considered either cus groups approach directly applied solve problem target maximize sum rate pairs guaranteeing minimum rate requirements cus derive optimal conditions power control cus first propose iterative algorithm solve resource allocation problem specifically adopting auxiliary variables relaxing binary constraints formulated optimization problem transformed convex one optimally solved dual method finally simulation results show significant sum rate gains proposed algorithm conventional orthogonal multiple access oma scheme ystem odel roblem ormulation consider downlink cellular network base station serves cus subchannels scs adopting noma cus multiplexed splitting power say total number cus meanwhile underlaid pairs randomly distributed cell denote sets pairs scs respectively superposition symbol transmitted cus pni sni sni pni transmit signal transmit power respectively implement noma needs inform sic decoding order strong cus decode remove signal weak cus current work generally assumed sic decoding order follows increasing order channel gains let hni denote channel successfully decode remove interference however work underlaid pairs also contribute interference affects noma decoding order case received sinr decode signal pnj pnt binary variable denotes whether assigned pair term represents interference underlaid pairs qkn transmit power pair represents channel gain pair specifically desires decode signal superposition symbol interference cancellation successful received sinr larger equal received sinr therefore protect given sic decoding order following conditions satisfied set represents set cus index note constraints form simplify decoding order constraints following equivalent inequalities implied assumed cus served already scheduled grouping issues see generally task beyond scope paper similar receivers assumed perfect channel state information channel feedback form constraints achievable rate although spectrum efficiency improved allowing multiple pairs reusing design efficient resource allocation schemes paradigm requires high computation complexity moreover heavy signaling overhead exchange occurs since channel state information interference channels different pairs scs estimated given theses concerns assume one allocated one pair consequently following channel assignment constraints sinr receiver pair sinrnk qkn channel gain transmitter receiver pair interference channel gain receiver pair case achievable rate pair rkn sinrnk meanwhile guarantee rate fairness among cus minimum rate requirements cus imposed rate requirement transmit power constraints pairs cus qkn pmax pni pmax maximize sum rate pairs following optimization problem obtained max rmax rkn iii ower ontrol hannel ssignment solve sum rate maximization problem optimal conditions power control cus given first investigated propose iterative method obtain power control channel assignment pairs optimal power control cus assigned pair first determine optimal transmit power conditions cus simplicity superscript omitted following analysis subsection solve power control problem define easy know constraint hold equality optimal transmit power denoted otherwise sum rate pairs improved decreasing setting equality accordingly optimal transmit power easy obtain explicit expression assist solving issue define represents summation transmit powers substituting according recursive relations obtain define obviously based simplified addition optimal transmit power obtained using defining power control channel assignment based previous analysis transmit powers cus determined transmit power multiplexed pair assigned pair according transmit power constraint equivalent qkn pmax following inequality holds pmax according sic successful decoding order constraints rewritten qkn qkn recall since note feasible qkn hence transmit power pair satisfy qkn minn remark according find one additional transmit power constraint imposed pair protect sic decoding order cus hand condition holds sic decoding order constraints alway satisfied substituting achievable rate pair dnk qkn enk dnk enk problem simplified max rmax given results original rkn qkn qkn qnk qnk min max pmax minn easy see qkn concave respect qkn consequently rkn qkn concave qkn due logarithmic function increasing concave page however problem convex due introducing auxiliary variable xnk qkn temporarily relaxing integer constraints problem transformed max rmax xnk pmax inferred rmax xnk concave xnk within triangular region due perspective property problem convex therefore optimal solution problem obtained using standard dual method lagrangian obtained pmax xnk rkn dual variables associated constraints respectively taking derivative xnk respectively xnk xnk rkn derivative rkn applying conditions obtain following necessary conditions optimal solution xnk xnk xnk xnk qnk xnk qnk xnk obtained solving qnk denoting equivalent following quadratic equation dnk dnk enk enk dnk enk note discriminant quadratic equation enk dnk indicating two real roots according quadratic solution formula define tnk given xnk qnk conclude xnk tkn tkn tnk min max according follows hkn hkn hkn rkn tkn tkn rkn tkn hkn different according constraint arg maxhkn pair largest hkn assigned note value determined method updating procedure iteration pmax xnk max positive step size according proposition method converges optimal solution problem sufficient small step size thus transmit power pairs obtained qkn xnk overall analysis summarized following iterative resource allocation dbira algorithm solve problem imulation esults performance proposed resource allocation scheme evaluated simulations section cell square area located center maximum distance transmitter receiver rate requirements cus denoted set pmax dbm pmax dbm dbm loss model adopted standard deviation shadow fading results averaged random realizations comparison adopt orthogonal frequency division multiple access ofdma system multiple cus benchmark labeled scheme joint power control channel assignment algorithm applied system also shared cus allowed access fraction bandwidth multiplexed pair also interfered cus algorithm dual based iterative resource allocation dbira algorithm initialize xnk initialize step size set precision repeat calculate xnk according respectively end update according update rmax qkn according rmax calculate pnk according output qkn pnk rmax fig illustrates convergence behavior proposed dbira algorithm versus number iterations different expected shown sum rate pairs monotonically increases initial iterations moreover sum rate performance converges within iterations considered three cases validates effectiveness proposed dbira algorithm fig shows sum rate pairs cus minimum rate requirements different expected scheme outperforms scheme especially number multiplexed large cus need larger transmit power scheme satisfy rate requirement compared scheme leads larger interference pairs scheme scheme since interferences pairs summed multiplexed cus noma schemes moreover sum rate pairs decreases rate requirements cellular links also due larger transmit power cus required higher data rate requirements onclusion resource allocation problem communications underlaying cellular network investigated letter although additional power constraints introduced pairs sake noma decoding order shown underlaying noma cellular network still outperforms conventional scheme network high data requirements myriad users rate rate noma noma noma noma noma noma rate requirement iteration number fig convergence performance dbira algorithm fig sum rate cus rate requirements eferences dang coon chen resource allocation multicarrier systems ieee wireless commun vol april hoang resource allocation communication underlaid cellular networks using approach ieee tans wireless vol october luo zhang lim downlink uplink energy minimization user association beamforming ieee trans wireless vol jan zhu wang swindlehurst zhao downlink resource reuse communications underlaying cellular networks ieee signal process vol may yang huang pan chen downlink resource allocation power control communication underlaying cellular networks ieee commun vol july malandrino limani casetti chiasserini downlink uplink resource allocation hetnets support ieee tans wireless vol may dai wang yuan han wang multiple access solutions challenges opportunities future research trends ieee commun vol september zhao liu chai chen elkashlan communications towards proc ieee global commun conf globecom dec zhang xiao ding fan aided cooperative multiple access ieee trans veh vol yang pan pan chen optimality power allocation noma downlinks individual qos constraints ieee commun vol ding yang fan poor performance multiple access systems randomly deployed users ieee signal process vol dec boyd vandenberghe convex optimization cambridge university press wong cheng lataief murch multiuser ofdm adaptive subcarrier bit power allocation ieee sel areas vol oct bertsimas weismantel optimization integers dynamic ideas belmont vol
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jun accumulated persistence function new useful functional summary statistic topological data analysis view brain artery trees spatial point process applications christophe biscio department mathematical sciences aalborg university denmark jesper department mathematical sciences aalborg university denmark june abstract start simple introduction topological data analysis popular tool called persistent diagram briefly persistent diagram multiset points plane describing persistence topological features compact set scale parameter varies since statistical methods difficult apply directly persistence diagrams various alternative functional summary statistics suggested either contain full information persistence diagram functions suggest new functional summary statistic hence easier handle mild conditions contains full information persistence diagram usefulness illustrated statistical settings concerned point clouds brain artery trees appendix includes additional methods examples together technical details used examples available http keywords clustering confidence region global rank envelope functional boxplot persistent homology test introduction statistical methods make use algebraic topological ideas summarize visualize complex data called topological data analysis tda particular persistent homology method called persistence diagram used measure persistence topological features expect many readers may familiar concepts section discusses two examples without going technical details though times unavoidable refer terminology used persistent homology section discusses use persistence diagram related summary statistics motivates section new functional summary statistic called accumulative persistence function apf introduced remainder paper demonstrates use apf various statistical settings concerned point clouds brain artery trees examples tda mathematics underlying tda uses technical definitions results persistent homology see fasy references therein theory needed present paper instead provide understanding examples notion persistence topological features sequence compact subsets euclidean space either section section recalling set two points connected curve topological feature compact set maximal subset also called connected component meaning topological feature simply understood appeal remark end section birth time apf death time figure four first panels show simple example spherically growing circles centred radii time fifth panel shows persistence diagram connected components loops final panel shows corresponding accumulated persistence functions toy example let union three circles depicted panel figure three circles topological features connected components curve goes circle another outside complement four connected components one unbounded whilst three others topological features also called loops boundary bounded connected component closed curve crossings example closed curve circle let subset points within distance thinking time results point grows disc constant speed one algebraic topology technically speaking betti numbers changes exactly times see first four panels figure topological dimension let denote time ith change first three connected components three loops given say born time imaging nothing time second loops disappear two connected components merge one connected component say loops one connected components die time since two merging connected components born time decided uniformly random one die respectively survive one survives represent new merged connected component third time new loop born two connected component merge one represented oldest first born connected component whilst connected component one retained two merged time dies remaining connected component lives forever time discarded analysis finally time loop dies hence multiset points specifying appearance disappearance topological feature grows scatter plot points called persistence diagram see figure panel connected components points discarded diagram multiplicities respectively loops points multiplicities respectively term persistence refers distant connected components large loops present long time course corresponds three components loops persist long whilst last appearing loop short lifetime hence considered noise usually practice known finite subset points collected sample possibly noise redefine union closed discs radius centres given point cloud hence connected components points loops general difficult directly compute connected components loops graph constructed connected components correspond moreover triangles graph may filled way loops obtained triangulation correspond remark construction also created case union closed balls radius centres given finite point pattern construction simplicial complex may much larger alphacomplex technical result nerve theorem establishes possible identify topological features see edelsbrunner harer unnecessary paper understand precise definition notions small examples computationally convenient use may still think topological feature loop closed curve crossings simplicial complex used book keeping determining persistence loop example sphere loops torus two finally topological feature manifold closed surface filled paper omit precise definition since technical needed persistent homology brain artery trees left panel figure shows example one brain artery trees analysed bendich data tree specifies graph consisting dense cloud vertices points together edges line segments connecting neighbouring vertices details data given section bendich tree consider topological features using different types data sets described consider tree figure let denote union edges following bendich let set height function tree level assuming empty thus topological features connected components death time death time birth time birth time figure brain artery tree water level indicated left panel persistence diagrams connected components middle panel loops right pane illustrated left panel figure instead time may think water level water level increases connected components part surrounded water part blue may born die refer persistence section represent births deaths connected component persistence diagram shown figure middle panel persistence connected components brain artery trees studied several examples later bendich let represented point pattern points subsampled redefine union balls radii centres given considered remark end section loops determined corresponding right panel figure shows corresponding persistence diagram persistence loops trees studied examples follow background objective persistence diagram popular graphical representation persistence topological features sequence compact sets exemplified consists topological dimension multiset pdk points multiplicities pair times topological feature obtained time grows majority literature tda including analysis bendich brain artery trees long lifetimes main interest whereas short lifetimes considered topological noise short lifetimes interest study complex structures branch polymers fractals see macpherson schweinhart brain artery trees bendich noticed one case persistence distinguishing power specific application examples demonstrate short lifetimes also key interest many situations including analysing brain artery trees dataset bendich chazal chen note difficult apply statistical methodology persistent diagrams alternative functional summary statistics suggested bubenik introduces sequence functions called persistent landscape first function denoted considered main interest since provides measure dominant topological features longest lifetimes therefore call dominant function chazal introduce silhouette weighted average functions persistent landscape weights control whether focus topological features long short lifetimes moreover chen consider kernel estimate intensity function persistent diagram viewed point pattern dominant function silhouette intensity estimate functions hence easier handle persistence diagram however provide selected full information persistence diagram section introduce another functional summary statistic called accumulative persistence function discuss advantages differs existing functional summary statistics accumulated persistence function simplicity specificity topological dimension always assume persistence diagram pdk assumption satisfied examples least probability one often tda literature pdk transformed rotated rescaled persistence diagram rrpd given rrpdk meanage lifetime transformation useful defining accumulative persistence function apf apfk indicator function suppress notation apfk function rrpdk remainder section comments definition formally speaking rrpdk considered random viewed finite point process multiplicities see daley follows always clear context whether pdk rrpdk considered random observed hence whether apfk deterministic random function latter case apfk accumulative function random fluctuations typically increase increases depending application jumps shape apfk may interest demonstrated later examples large jump apfk corresponds large lifetime long persistence simple example shown figure jumps large indicate three connected components circles whilst first jump large indicates three original loops complicated examples considered following may hard recognize individual jumps particular remark end section suppose union balls radius centres given finite point pattern roughly speaking may following features lustrated later example small meanages jumps correspond balls merge together small values thus point pattern aggregated clustering expect jumps hence large small meanages whilst point pattern regular typically inhibition points expect jumps happen large modest values illustrated later middle panel figure considering curves cluster process determinantal point process large meanages jumps likely happen case aggregation accordingly shape different two cases illustrated first panel figure similar considerations lead expect different shapes different types point patterns expect large respective small case aggregation respective regularity small opposite happens large illustrated last panel figure clearly rrpdk correspondence pdk turn pairwise distinct correspondence rrpdk corresponding apfk correspondence would easily lost used place need careful possible correspondence example imagine want compare two apfs respect let pdk pdk underlying persistence diagrams however points close diagonal considered topological noise see section usually bot tleneck distance pdk pdk used see fasy briefly let set points distance diagonal persistence diagram let square center sides parallel side length pdk pdk pdk point pdk exactly one point square repeating condition times however small values pdk pdk correspond closeness two corresponding apfs respect note dominant function silhouette intensity estimate see tion general correspondence rrpdk like functions apfk function easier handle sequence functions persistent landscape bubenik intensity estimate chen confidence regions become easier plot contrary dominant function silhouette apf provides information topological features without distinguishing long short lifetimes outline paper discusses various methods based apfs different contexts illustrated simulation studies related spatial point process applications reanalysing brain artery trees dataset previously analysed bendich section specifies setting examples sections consider case single apf sample apfs two samples apfs respectively examples details appear appendix datasets simulated data simulation studies consider planar point cloud finite point pattern study end section topological features union closed discs radii centred change grows realisation point process count finite thus pdk rrpdk viewed finite planar point processes multiplicities apfk random function note may random conditional points necessarily independent identically distributed iid common situation spatial statistics focus point process purpose assess goodness fit specified point process model observed brain artery trees dataset dataset bendich comes brain artery trees included within cube side length one tree excluded function crashed correspondence sean skwerer want capture arteries bend space detect age gender effects persistence connected components tree represented union line segments considered section meanages number connected components always persistence loops union growing balls centres point cloud representing tree considered section loops finite death time die allocated time bendich stop growth balls thus shall consider meanages number loops always tree sometimes tree bendich use largest lifetimes analysis whereas principal component analysis clearly reveal age effects permutation test based mean lifetimes male females subjects shows clear difference considering accordingly demonstrating usefulness focus gender effect consider trees bendich two transsexual subjects excluded obtained female subjects male subjects contrast bendich consider observed meanages lifetimes accordance allocated time need redefine simplicity use notation although methods results paper presented definition mind apply well considering finally write apfkf apfkm distinguish apf females males respectively single accumulated persistence function exists several constructions results confidence sets persistence diagrams aim separate topological signal noise see fasy chazal references therein appendix accompanying example discuss obvious idea transforming confidence region one accumulate persistence function potential problem bottleneck metric used persistence diagrams corresponding closeness apfs section section focus instead spatial point process model assessment using apfs traditional tools suppose realization finite spatial point process observed copies simulated claimed model joint distribution exchangeable permutation claimed distributed case iid common situation model assessment spatial point process analysis distribution specified estimated see baddeley waagepetersen denote apfk respectively null hypothesis joint distribution exchangeable adapting ideas discuss construct test based global rank envelope usefulness demonstrated example functional data analysis measure extreme comparison depth function used ranking see romo suggest using depth ordering called extreme rank let parameter chosen behaviour interest define bounding curves alow min aupp max min max denote smallest largest values respectively extreme rank respect max alow aupp larger deeper central among given extreme rank ordering used define global rank envelope band delimited curves alow aupp max probability least alow aupp see therefore rank envelope specifying conservative statistical test called extreme rank envelope test accepts level satisfied equivalently plot extreme rank envelope allows graphical interpretation extreme rank envelope test may case rejection suggest alternative model exist alternatives extreme rank envelope test particular liberal extreme rank envelope test global scaled maximum absolute difference envelope see also possible combine several extreme rank envelopes instance combining see following example focus briefly remark results obtained combining example simulation study recall homogeneous poisson process model complete spatial randomness csr see waagepetersen simulation first panel figure consider apfs corresponding independent point processes defined unit square csr given intensity mean number points suppose claimed csr intensity however model given one following four point process models refer true model csr hence true model agrees claimed model cell process moment properties csr see baddeley silverman though mathematical point view cluster process simulated realisations exhibit aggregation regularity different scales see second panel figure cluster process model clustering cluster homogenous poisson process within disc centers discs observed constitute stationary poisson process see waagepetersen third panel figure repulsive determinantal point process dpp model regularity see lavancier biscio lavancier fourth panel figure let specifies completely models whereas figure simulated point patterns homogeneous poisson process first panel cell process second panel cluster process third panel repulsive dpp fourth panel remaining parameters cases defined used robins turner cases figure finally following recommendation let value simulate point process rpackage spatstat dimension compute extreme rank envelopes extreme rank envelope tests spptest repeat times table shows case percentage rejection hypothesis homogeneous poisson process known intensity case csr type one error test small except expected case power test increased increased process dpp power high even two cases cluster process power also case instead radius cluster becomes times larger hence easy distinguish clusters third panel figure combine extreme rank envelopes results better close best results obtained considering one extreme rank envelope figure illustrates one repetitions dimension deviation apfk extreme rank envelope obtained true model csr three models apfk outside extreme csr dpp cluster table percentage point patterns rank envelope test rejects hypothesis csr homogeneous poisson process unit square intensity true model either csr one three alternative point process models rank envelope particular meanage lifetime small middle panel means small lifetimes noise particular importance discussion section using obvious notation small may expect apfdpp apfcsr agreement middle panel large may expect apfdpp apfcsr apfcsr last relation detected extreme rank csr envelope left panel similarly may expect apfmc small csr whereas apfmc large cases detected right panel note cell process apfbs rather similar behaviour apfdpp like regular point process probably clustering rare phenomena similar simulation study discussed robins turner models notice fix number points use testing procedure based persistent homology rank function contrast onedimensional apf function summarizing topological features represented persistent diagram robins turner show test csr based persistent homology rank function useful compared various tests implemented spatstat concern first secondorder moment properties method particular useful true model cell process first moment properties csr comparing figure robins turner results determinantal matern cluster matern cluster extreme rank envelope bounds matern cluster determinantal extreme rank envelope bounds extreme rank envelope bounds determinantal figure rank envelope apfi left panel enlargement shown middle panel right panel together curves three models cell process cluster process dpp envelope obtained realisations csr model unit square intensity table true model cell process extreme rank envelope test seems less powerful test suggest hand robins turner observe latter test performs poorly true model strauss process model inhibition cluster process noticed cluster process obtain perfect power using extreme rang envelope test single sample accumulated persistence functions functional boxplot section discusses use functional boxplot sun genton sample apfk joint distribution exchangeable plot provides representation variation curves given around central curve used outlier detection detection curves extreme respect others sample illustrated example brain artery trees dataset appendix accompanying example concerning simulation study functional boxplot based ordering apfk obtained using depth function specificity make standard choice called modified band depth function mbd romo sun genton parameter define min max denote lebesgue measure mbd respect mbdr average proportion possible pairs thus larger value mbd curve central deeper sample call region delimited central curves central envelope often assumed curve outside central envelope inflated times range central envelope outlier abnormal curve generalisation similar criterion boxplot sample real numbers range may changed suitable application hand see discussion sun genton example example brain artery trees brain artery trees dataset section figure shows functional boxplots apfk females first third panels respective males second fourth panels first second panels third fourth panels central curve plotted black central envelope purple upper lower bounds obtained curves except outliers dark blue comparing two left panels concerned connected components shape central envelope clearly different females males particular interval upper lower bounds closer male female male female figure functional boxplots apfs females males obtained brain artery trees dataset first panel second panel third panel fourth panel dashed lines show outliers detected criterion figure brain artery tree male subject detected outliers criterion central region females particular interval two right panels concerned loops main difference observed interval central envelope larger females males dashed lines figure show apfs detected outliers criterion first panel third panel second panel fourth panel females one point pattern outliers steep dashed line panel males two point patterns outliers one case steep dashed line panel case figure reveals obvious issue large part right corresponding tree missing examples discuss extent analysis brain artery trees sensitive whether include exclude detected outliers confidence region mean function section considers asymptotic confidence region mean function sample iid apfk assume underlying iid rrpdk sample probability one exists upper bound death times exists upper bound nmax number topological features note state space rrpdk nmax nmax existence actual values nmax play role applying method example settings section suffices assume included bounded region number points bounded constant follows two versions nerve theorem presented fasy edelsbrunner harer respectively adapt empirical bootstrap procedure see van der vaart wellner chazal used confidence region mean dominant function persistent landscape case works follows mean function given estimated empirical mean function let independent form draws replacement set set given integer independently repeat procedure times obtain quantile distribution estimated inf following theorem verified appendix theorem let situation described large values functions provide bounds asymptotic conservative region confidence region bounds female confidence region bounds male confidence region bounds female confidence region bounds male figure bootstrap confidence regions mean apfkm mean apfkf left panel right panel mean apf lim lim example brain artery trees brain artery trees contained bounded region presented bounded number points obvious nmax exist establish confidence regions mean apfkm respective apfkf apply bootstrap procedure result shown figure trees considered left panel approximatively half confidence region overlap confidence region clear difference genders right panel difference pronounced particular interval similar results conclusions obtained exclude apfs detected outliers example course supply statistical test assess gender effect test established section applied example appendix provides additional example simulated dataset along discussion geometrical interpretation confidence region obtained two samples accumulated persistence functions section concerns test comparison two samples apfs appendix presents clustering method appendix including example unsupervised classification method appendix including example two samples consider two samples independent rrpdk distribution distribution suppose want test null hypothesis common distribution unknown section assume concentrated nmax integer nmax number adapt test statistic studied van der vaart wellner let let apfk corresponding denote empirical means respectively let interval used defining test statistic sup large values critical may rewritten sup lemma apr pendix independence samples converge distribution two independent gaussian processes denoted respectively assume sense convergence distribution lim sup follows distribution true see van der vaart wellner therefore letting inf asymptotic test rejects level power depends unknown distribution estimate bootstrap method let independent uniform draws replacement define empirical mean functions compute sup given integer independently repeat procedure times obtain estimate empirical distribution inf next theorem direct application theorem van der vaart wellner noticing apfk uniformly bounded tnmax form socalled donsker class see lemma proof appendix theorem let situation described lim lim whilst true lim lim therefore test rejects asymptotic level power remarked van der vaart wellner theorem possible present permutation test critical value bootstrap test asymptotic properties critical value permutation test test statistics constructed considering measurable functions may consider test statistic similar arguments redefining test rejects asymptotic level power example brain artery trees distinguish male female subjects brain artery trees dataset use test statistic three different settings let subset pdk corresponding largest lifetimes rrpdk obtained associated female male subjects respectively setting used bendich consider lifetimes let rrpdk associated female male subjects respectively samples setting except exclude rrpdk corresponding apfk detected outlier example hence bendich perform permutation test based mean lifetimes male female subjects conclude gender effect recognized considering setting setting setting table estimated given percentage test based used different intervals distinguish male female subjects settings described example comparison setting perform test different intervals case estimate smallest test significance level reject table shows results setting using smaller bendich even larger setting seven times larger pvalue bendich else similar smaller large difference settings indicates presence outliers violates result theorem care hence taken opinion better trust results without outliers contrast bendich see clear gender effect considering connected components notice also agreement discussion figure example setting dimension table smallest considering smaller interval appendix provides additional example illustrating use test simulation study acknowledgements supported danish council independent research natural sciences grant statistics point processes space beyond centre stochastic geometry advanced bioimaging funded grant villum foundation helpful discussions lisbeth fajstrup persistence homology acknowledged connection brain artery trees dataset thank james stephen marron sean skwerer helpful discussions casilab university north carolina chapel hill providing data distributed midas data server kitware grateful editors referees useful comments appendix appendix contain complements additional examples sections setting notation follows examples based simulated point pattern described section iid points fixed positive integer section setting corresponds applications typically considered tda aim obtain topological information compact set unobserved possibly noise appears specificity let iid points support noise iid independent follows restriction square bivariate normal distribution iid coordinates standard deviation noise denote distribution restriction imposed technical reasons practical importance let union closed discs radii centred study topological features changes grows use mentioned section finally denote circle center radius transforming confidence regions persistence diagrams used separating topological signal noise noted section exists several constructions results confidence sets persistence diagrams aim separate topological signal noise see fasy chazal references therein avoid presenting technical description constructions results depend different choices complexes precisely filtrations specificity appendix consider discuss transformation confidence region one accumulate persistence function use following notation aforementioned references consider persistence diagram pdk unobserved compact manifold obtained section considering persistence grows topological features set consisting points within distance note pdk considered unknown course simulation study presented example pretend pdk unknown let random persistence diagram obtained section iid points support let set points distance diagonal persistence diagram let square center sides parallel side length finally let fasy chazal suggest various ways constructing bound asymptotic conservative region pdk respect bottleneck distance given lim inf pdk pdk bottleneck distance defined section confidence region given consists persistence diagrams pdk exactly one point arbitrary number points square point falling noise set fasy consider points remaining points representing significant topological feature using asymptotic conservative region apfk corresponding pdk immediately obtained region bounded two funcbmin bmax specified due accumulating nature apfk tions span bounds increasing function meanage using chazal show span decreases increases illustrated example example simulation study let suppose point uniformly distributed figure shows example simulated point pattern points use bootstrap method implemented tda presented chazal compute region see panel figure two squares diagonal correspond two loops squares correspond topological noise thereby regions panel panel obtained confidence region decreases increases demonstrated bottom panels increased noticed section must careful using results based bottleneck metric small values bottleneck metric correspond closeness two corresponding apfs although close persistence diagram respect bottleneck distance imply two corresponding apfs close respect converse true hence possible apf confidence region plotted figure corresponding persistence diagram different truth figure set example left panel simulated point pattern independent uniformly distributed points right panel additional example related section functional boxplot functional boxplot described section used exploratory tool curves given sample apfk provides representation central curve variation around also used outliers detection illustrated following example example simulation study consider sample independent apfk joint distribution first apfk exchangeable whereas last play role outliers suppose apfk corresponds point process iid points point follows one following distributions unit circle uniform point perturbed gaussian mixture let follow probability otherwise two circles uniform point perturbed circle radius uniform point perturbed noise lifetime death time confidence region bounds confidence region bounds meanage birth time figure regions obtained bootstrap method panel corresponding panel figure panel shows corresponding region panel shows region larger point cloud points used let first point processes obtained distribution next following final figure shows simulated realization four types point processes figure shows functional boxplots considering left panel right panel curves detected outliers corresponding distributions plotted red blue green respectively panels outliers detected criterion agree true outliers left panel curve accumulation small jumps corresponding moments points associated circle connected growing discs sequence curves corresponding realisations jump corresponds moment circle gaussian mixture two circles circle figure simulated realizations four types point processes consisting iid points distribution either black dots blue crosses red triangles rotated green crosses points associated two circles used defining connected growing discs sequence points following distribution generally closer ones following distribution radius underlying circle smaller corresponds smaller jumps small meanages hence curves lower correspond realisations realisations expected large meanages curves larger correspond realisations realisations note redefine replaced curves would rescaling right panel observe clear jumps obtained jumps correspond first time loops circles covered union growing discs sequence used place definition curves would rescaling repeat everything distribution redefined replaced support closer becomes harder case detect outliers distribution omit corresponding plot thus simulations determining stronger criterion would gaussian mixture two circles circle gaussian mixture two circles circle figure functional boxplots apfs based topological features dimension left panel right panel panel apfs obtained iid points distribution respectively apfs detected outliers plotted red blue green case respectively needed additional example related section confidence region mean function appendix provides yet example illustrate bootstrap method section obtaining confidence region mean function sample iid apfk example simulation study consider iid copies point process consisting independent uniformly distributed points union three circles radius centred respectively circles also considered example section simulated realization point process shown left panel figure next two panels show simulated confidence regions respectively bootstrap procedure used middle panel confidence region bounds empirical mean confidence region bounds empirical mean figure simulation independent uniformly distributed points union three circles dashed lines radius centred left panel regions mean middle panel mean right panel based iid simulations mulation small jumps corresponding moment circle covered union growing discs sequence interpret small jumps topological noise jump corresponds moment circles centred connected growing discs jump three circles connected growing discs right panel accumulation small jumps corresponding moment three circles connected growing discs form loop figure disappearance loop figure corresponds jump figure additional example section two samples accumulated persistence functions section considered two samples independent rrpdk distribution distribution studied bootstrap test asses null hypothesis connection brain artery trees additional example showing circle circle figure simulation independent uniformly distributed points circle centred radius perturbed red dots together independent uniformly distributed points circle centred radius perturbed blue crosses performance test presented example simulation study let distribution rrpdk obtained independent uniformly distributed points perturbed noise define similar way circle radius simulated realisation point process shown figure seems difficult recognize underlying circles different let consider test statistics simulations two samples rrpdk obtain following percentage rejection much better results observed namely high percentage mainly caused largest lifetime loop methods two samples accumulated persistence functions clustering suppose apfk want label groups using method clustering unsupervised classification methods studied many places literature functional data see survey jacques preda particular chazal chen robins turner consider clustering connection rrpdk whereas rrpdk functions becomes easy use clustering apfk illustrated example simplicity consider standard technique known clustering algorithm hartigan wong complicated applications considered example may needed clustering algorithm noticed referee avoid use modify thereby construct matrix different apfs used perform hierarchical clustering however example results using hierarchical clustering omitted better algorithm assume pairwise different functions parameter example rrpdk nmax see section apfk clustering algorithm works follows chose uniformly random subset functions call functions centres label assign apfk label closer centre label centre respect group reassign centre mean curve group may apfk sample iterate steps assignment centres change algorithm known convergent however may several drawbacks discussed hartigan wong bottou bengio example simulation study consider groups consisting associated point processes consisting iid points point follows one following distributions groups respectively unit circle uniform point perturbed two circles uniform point perturbed circle radius uniform point perturbed noise start simulating realization point processes left panel figure shows one realization type point process seems difficult distinguish underlying circles groups three associated three point patterns fact assigned right groups right panel figure shows result clustering algorithm using kmeans algorithm takes seconds evaluating equidistant values expected see overlap curves assigned groups next repeat times simulation point processes clear distinction groups obtained algorithm applied connected ponents percentage wrongly assigned among average standard deviation assignment error fact mostly caused incorrect labelling associated expected underlying circles used definitions rather close whereas underlying set definition different two connected components represented jump middle panel figure even better results obtained considering loops instead connected components percentage wrongly assigned among average standard deviation mainly due sets underlying distinctive loops results clear distinct jumps seen right panel figure circle two circles circle circle two circles circle circle two circles circle figure left panel simulated example three point processes consisting iid points drawn distribution black dots red triangles blue crosses middle panel obtained simulation point processes associated colouring black red blue specifies whether algorithm assigns group associated right panel middle panel supervised classification suppose want assign apfk training set different groups sample independent apfk airi purpose supervised classification methods functional data may adapted consider particular method suppose believe least apfk group iid whereas remaining apfk group follow different distribution considered outliers see section parameter define mean respect mean function apfk largest mbdri see assuming apfk assigned argmin denotes trimmed mean used robustness allows control curves may like omit outliers median could used instead example simulation study consider following distributions point unit circle uniform point perturbed noise two circles radii uniform point perturbed circle radius uniform point perturbed two circles radii uniform point perturbed consider following simulation study consists apfk associated simulations point processes consisting iid points distribution apfk obtained way outliers specified way replacing figure top panels mean functions respect considering left right obtained based iid points following distribution solid curve dotted curve bottom panels examples point patterns associated assigned wrong group together circles radius respectively correctly specified simulate apfk associated apfk associated finally use assign apfk either top panels figure show means left right difference means clearest expect assignment error lower case fact wrong assignments happen mainly support well covered point pattern illustrated bottom panels repeating simulation study times percentage wrongly assigned among repetitions mean standard deviation whereas mean standard deviation investigate results depend radius smallest circle repeat everything radius place defining distributions proportion wrong assignments mean standard deviation mean standard deviation similar example error lowest due largest lifetime loop proof theorem proof theorem follows along similar lines chazal soon verified lemma note proof lemma covered approach chazal first need recall following definition denotes topological space bounded real valued borel functions defined topology induced uniform norm definition sequence random elements converges distribution random element bounded continuous function converges lemma let situation section converges tribution towards gaussian process covariance function cov proof need notation recall concepts empirical process theory dtk nmax denote apfk let class functions dtk nmax given see connection empirical process theory consider empirical process denote norm respect distribution bracket set functions smallest integer functions show log finite theorem van der vaart donsker class implies convergence distribution gaussian process statement lemma sequence dtk nmax let empty set exists consequently prove sequence chosen write random confused sections empty let nmax conditioned let uniformly selected nmax nmax hence nmax max nmax moreover lemma exists finite sequence thus choosing nmax max nmax hence definition therefore log log completes proof proof theorem donsker property established proof lemma theorem gine zinn converge distribution process quantile converges quantile therefore provides bounds asymptotic region stated theorem lemma let positive random variable exists finite sequence proof denote cumulative distribution function limit generalised inverse inf verify lemma finally references baddeley rubak turner spatial point patterns methodology applications chapman press baddeley silverman cautionary example use secondorder methods analyzing point patterns biometrics bendich marron miller pieloch skwerer persistent homology analysis brain artery trees annals applied statistics biscio lavancier quantifying repulsiveness determinantal point processes bernoulli bottou bengio convergence properties algorithms advances neural information processing systems volume mit press bubenik statistical topological data analysis using persistence landscapes journal machine learning research chazal guibas oudot gromovhausdorff stable signatures shapes using persistence computer graphics forum chazal fasy lecci rinaldo singh wasserman bootstrap persistence diagrams landscapes modeling analysis information systems chazal fasy lecci michel rinaldo wasserman robust topological inference distance measure kernel distance available chen wang rinaldo wasserman statistical analysis persistence intensity functions available arxiv daley introduction theory point processes volume elementary theory methods new york edition edelsbrunner harer computational topology american mathematical society providence fasy lecci rinaldo wasserman balakrishnan singh confidence sets persistence diagrams annals statistics gine zinn bootstrapping general empirical measures annals probability hartigan wong algorithm clustering algorithm journal royal statistical society series applied statistics jacques preda functional data clustering survey advances data analysis classification lavancier rubak determinantal point process models statistical inference journal royal statistical society series statistical methodology romo concept depth functional data journal american statistical association romo torrente robust tools analysis gene expression data biostatistics macpherson schweinhart measuring shape topology journal mathematical physics spatial variation lecture notes statistics berlin waagepetersen statistical inference simulation spatial point processes chapman boca raton waagepetersen recent developments statistics spatial point patterns annual review statistics application appear hahn multiple monte carlo testing applications spatial point processes statistics computing grabarnik seijo hahn global envelope tests spatial processes journal royal statistical society series statistical methodology available permutation bootstrap tests equality two distributions scandinavian journal statistics robins turner principal component analysis persistent homology rank functions case studies spatial point patterns sphere packing colloids physica nonlinear phenomena sun genton functional boxplots journal computational graphical statistics van der vaart asymptotic statistics cambridge university press cambridge van der vaart wellner weak convergence empirical processes springer series statistics new york
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deep learning framework breast cancer histology image classification feb yeeleng vang zhen chen xiaohui xie university california irvine irvine ysvang xhx abstract work present deep learning framework multiclass breast cancer image classification submission international conference image analysis recognition iciar grand challenge breast cancer histology images bach histology images large fit gpu memory first propose using inception perform patch level classification patch level predictions passed ensemble fusion framework involving majority voting gradient boosting machine gbm logistic regression obtain image level prediction improve sensitivity normal benign predicted classes designing dual path network dpn used feature extractor extracted features sent second layer ensemble prediction fusion using gbm logistic regression support vector machine svm refine predictions experimental results demonstrate framework shows improvement model introduction united states breast cancer continues leading cause cancer death among women races studies shown improvement survival rate last decade attributed early diagnosis awareness better treatment options common screening test includes clinical breast exam involves visual check skin tissue manual check unusual texture lump mammography requires taking image breast look changes breast mri uses radio waves obtain detailed image inside breast latter two diagnostic modals many diagnosis cad systems developed assist radiologists effort identify breast cancer early stages side screening toolbox biopsies minimally invasive procedures whereby tissue samples physically removed stained hematoxylin eosin visualized microscope histopathology slides allow pathologists distinguish normal malignant lesions assist diagnosis however even among trained pathologists concordance unanimous agreement mere high degree discord motivates development automatic cad systems using machine learning assist professionals diagnosis november january international conference image analysis recognition iciar held grand challenge breast cancer histology images bach solicit submissions automatic image analysis systems task classification breast cancer histology images present deep learning framework task breast cancer histology image classification approach uses inception googlenet architecture discriminate invasive carcinoma situ carcinoma benign lesion normal tissue patches fuse patch level predictions obtain image level prediction using ensemble framework system improves sensitivity benign normal classes using dual path network dpn extract features input second level ensemble framework involving gbm svm logistic regression experimental results held set demonstrate framework shows improvement model relate work several works published area applying machine learning algorithms cancer histology image detection classification specific area breast cancer histopathology classification camelyon competition led numerous new approaches utilizing techniques deep learning obtain results comparable highly trained medical doctors winning team used inception create tumor probability heatmap perform geometrical morphological feature selection heatmaps input random forest classifier achieve near area receiver operating characteristic curve auc score however competition involved binary class prediction tumor normal whole slide images breast cancer classification araujo published bespoke convolutional neural network architecture achieved accuracy results high sensitivity carcinoma detection grand challenge datasets evaluation metric section describe dataset provided organizers subchallenge breast cancer histology image classification evaluation metric used score submissions interested reader encouraged refer competition page details regarding subchallenge dataset breast cancer histology image classification subchallenge consist hematoxylin eosin stained microscopy images shown table dataset extended version one used araujo images digitized acquisition conditions resolution pixels pixel size image labeled one four classes normal tissue benign lesion iii situ carcinoma invasive carcinoma according predominant cancer type image images labeled two pathologists provided diagnostic image contents without specifying area interest total microscopy images even distribution four classes randomly perform split training validation sets used model development test set held used evaluation table histopathology dataset type training validation test total normal benign microscopy situ invasive evaluation metric challenge consists automatically classifying breast cancer histology images four classes normal benign situ carcinoma invasive carcinoma performance challenge evaluated based overall prediction accuracy ratio correct predictions total number images method section describe framework approach problem breast cancer histology image classification classification microscopy images stain normalization stain normalization critically important step stain images known cell nucleus stained large amount pure hematoxylin small amount eosin whereas cytoplasm stained large amount pure eosin small amount hematoxylin variations images attributed factors differences lab protocols concentration source manufacturer scanners even staining time variations makes difficult software trained particular stain appearance therefore necessitates careful preprocessing reduce variances many methods proposed stain normalization including based color devolution rgb pixel values decomposed basis vectors addition color information bejnordi takes advantage spatial information perform deconvolution step however approach currently works whole slide images framework utilized macenko used singular value decomposition svd vahadane normalizations used sparse nonnegative matrix factorization snmf part ensemble framework due fact initial empirical results showed images obtained high sensitivity invasive situ classes whereas vahadanenormalized images showed high sensitivity benign normal classes set normalized datasets normalized using target image example normalization schemes shown fig fig target image original image image macenko normalization image vahadane normaliztion classification framework microscopy classification framework consists patch level classification stage image level heatmap postprocessing stage possibly refinement stage depicted fig model training classifer patch input size extracted patches microscopy slide training validation sets datasets patches comes sliding normalized microscopy image strides remaining patches randomly assumption used patches given label original slide image obtained pretrained inception model modified accept image patch size trained discriminate four classes training time images data dynamically augmented fed model similar color perturbation scheme used brightness perturbed delta contrast delta saturation delta hue delta addition color perturbation images randomly flipped vertically horizontally randomly rotated degrees obtain eight valid orientations inception model gpus nvidia titan gpus nvidia gtx gpus receive batch images model trained epochs learning rates set bottom convolution layers eleven inception modules top fully connected layer learning rate decreased every epochs rmsprop optimizer momentum used best performing model validation set saved inference time single microscopy image heatmap tensor size obtained first dimension corresponds valid orientations image second dimension classes third fourth dimension corresponds spatial dimensions image using patching npu mage insitu mvlrgbm bneonrimganl hes mal edi mage googl ene edi lrgbm svm dualpa hne wor fig framework classification normalized input image patched twelve patches sets patches generated corresponding valid orientations sets patches passed inception googlenet model generate patch level heatmap probability tensor heatmap tensor fused using majority voting gradient boosting machine gbm logistic regression across version input image model predicts invasive situ carcinoma model outputs prediction otherwise normalize images pass dpn network extract features second fusing step involving gbm support vector machine svm output prediction benign normal class three data fusion strategies investigated competition first strategy involved finding average probabilities along first dimension heatmap assigning labels patches corresponding probable class call class map class map final label microscopy obtained majority voting second third strategies involved finding class map orientation separately first obtaining histogram classes across orientations histogram data used train two separate models logistic regression regularization gradient boosting machine gbm classifier num max depth learning rate ultimately classify image similar model predicts benign normal image passed refinement stage describe next section refinement model benign normal classes since inception model yielded low sensitivity normal benign classes many interclass misclassification two classes proposed training version dual path network dpn serve feature extractor use images dpn chosen due compact size beneficial characteristics architectures using features extracted dpn train three additional models gbm support vector machine svm logistic regression binary classification results entire pipeline presented table experimental results performance framework classification shown table baseline compare araujo although using smaller subset dataset tested set roughly size best accuracy performance classification problem framework achieves accuracy score improvement baseline score even without refinement model model offers improvement baseline table classification results gbm vahadane normalization gbm ensemble ensemble refinement macenko normalization accuracy validation set test set comparing sensitivity araujo see achieved sensitivities normal benign situ invasive classes respectively table showed higher sensitivity across four classes using framework noticeable improvement benign class saw almost improvement validates decision incorporate binary class refinement phase specifically benign normal classes table test set contingency table prediction invasive situ benign normal sensitivity ground truth invasive situ benign normal discussion work proposed deep learning framework problem multiclass breast cancer histology image classification leverage advances computer vision field propose using successful inception model initial classification propose new ensemble scheme fuse patch probabilities classification improve sensitivity benign normal class propose refinement stage using dual path network first extract features images using gradient boosting machine support vector machine logistic regression fuse predictions final result experimental results grand challenge dataset demonstrates improvement system references cancer statistics working group united states cancer statistics incidence mortality report atlanta department health human services centers disease control prevention national cancer institute available saadatmand sepideh influence tumour stage breast cancer detection survival modern times population based study bmj berry donald effect screening adjuvant therapy mortality breast new england journal medicine gelder rianne effects populationbased mammography screening starting age presence adjuvant systemic international journal cancer hadjiiski lubomir berkman sahiner chan advances cad diagnosis breast current opinion obstetrics gynecology elmore joann diagnostic concordance among pathologists interpreting breast biopsy jama macenko marc method normalizing histology slides quantitative biomedical imaging nano macro isbi ieee international symposium ieee vahadane abhishek color normalization histological biomedical imaging isbi ieee international symposium ieee wang dayong deep learning identifying metastatic breast arxiv preprint liu yun detecting cancer metastases gigapixel pathology arxiv preprint arajo teresa classification breast cancer histology images using convolutional neural plos one nayak nandita classification tumor histopathology via sparse feature biomedical imaging isbi ieee international symposium ieee gorelick lena prostate histopathology learning tissue component histograms cancer detection ieee transactions medical imaging yan weakly supervised histopathology cancer image segmentation medical image analysis ciompi francesco importance stain normalization colorectal tissue classification convolutional arxiv preprint hou convolutional neural network whole slide tissue image proceedings ieee conference computer vision pattern recognition bejnordi babak ehteshami diagnostic assessment deep learning algorithms detection lymph node metastases women breast jama tieleman tijmen lecture divide gradient running average recent magnitude szegedy christian wei liu yangqing jia pierre sermanet scott reed dragomir anguelov dumitru erhan vincent vanhoucke andrew rabinovich going deeper proceedings ieee conference computer vision pattern recognition otsu nobuyuki threshold selection method ieee transactions systems man cybernetics wang wei john ozolek gustavo rohde detection classification thyroid follicular lesions based nuclear structure histopathology cytometry part bejnordi babak ehteshami stain specific standardization histopathological ieee transactions medical imaging khan adnan mujahid nonlinear mapping approach stain normalization digital histopathology images using color ieee transactions biomedical engineering chen yunpeng dual path advances neural information processing systems
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semantic segmentation colon glands deep convolutional neural networks total variation segmentation philipp michael martin oct institute biophysics center physiological medicine medical university graz graz austria institute neuroinformatics university zurich eth zurich zurich switzerland institute computer graphics vision graz university technology graz austria ludwig boltzmann institute clinical forensic imaging graz austria graz austria abstract segmentation histopathology sections ubiquitous requirement digital pathology due large variability biological tissue machine learning techniques shown superior performance standard image processing methods part glas colon gland segmentation challenge present algorithm segment glands tissue benign malignant colorectal cancer images preprocessed according hematoxylineosin staining protocol two deep convolutional neural networks cnn trained pixel classifiers cnn predictions regularized using segmentation based weighted total variation produce final segmentation result two test sets approach achieves tissue classification accuracy making use inherent capability system distinguish benign malignant tissue introduction variability glandular structures biological tissue poses challenge automated analysis histopathology slides become key requirement quantitative morphology assessment supporting cancer grading considering nonpathological cases automated segmentation algorithms must already able deal significant variability shape size location texture staining glands moreover pathological cases gland objects tremendously differ benign glands exacerbates finding general solution segmentation problem previous work gland segmentation colon tissue used graphical models textural features others worked segmentation prostatic cancer tissue using integrated contextual segmentation model probabilistic markov models clustering region growing spatial association nuclei gland lumen reader referred work sirinukunwattana detailed description work related glandular structure segmentation deep learning methods especially convolutional neural networks cnns found applications biomedical image analysis different tasks semantic segmentation mitosis detection classification blood cell counting fig samples benign malignant colorectal cancer sections dataset ground truth labels image available pixel overlaid different colors individual objects work propose strategy semantically segment glands dataset presented glas contains annotated images benign malignant colorectal adenocarcinoma stained scanned magnification fig shows example images ground truth annotation image individual objects annotated label illustrated different colors challenge participants information whether image shows benign malignant tissue available training dataset three datasets released contest total number images training set test set test set respectively datasets contained individual glands contributions work twofold present novel deep learning scheme generate classifier predictions malignant benign object background pixels accompanied dedicated refinement classifier able distinguish touching objects pose challenge later segmentation use classification results input simple yet effective globally optimal segmentation approach based convex geodesic active contour formulation regularizes classifier predictions according minimal principle technological contributions described section subsequent sections show discuss results novel approach applied datasets glas challenge methods present segmentation method stained histopathological sections proceeds three steps raw rgb images preprocessed extract robust representation tissue structure subsequently two classifiers trained predict glands structures image finally outputs classifiers combined segmentation based weighted total variation used produce segmentation result preprocessing slides prior classification rgb images preprocessed shown fig standard color deconvolution performed specific staining used provided separates tissue components according staining emphasizes structure inherently performs data whitening first red channel deconvolved rgb image contains tissue structure information channels omitted order account different staining contrasts lighting conditions image acquisition contrast limited adaptive histogram equalization clahe applied http used setting implementation landini available fiji fig preprocessing rgb images color deconvolution separates stained tissue components red channel deconvolved image processed clahe taken input pixel classifiers learning pixel classifiers given large variability benign malignant tissue dataset opted cnns due recently shown convincing performance pixelwise classification histopathology images learn rich set features directly images general architecture cnns motivated classical architecture consists layers four convolutional layers convk feature learning three fully connected fck layers feature classifier see fig rectified linear unit relu nonlinearity max used activation function throughout layers networks convolutional layers consist set learnable square filters pixel stride followed relu activation subsampling layers subk accounting translation invariance used first three convolutional layers counted part convolutional layer final pixelwise classification input image obtained sliding window image classifying center pixel window training minibatch stochastic gradient descent mbsgd momentum weight decay dropout regularization used minimize negative loss function classifying gland objects goal predict probability pixel belonging gland background one could define binary classification problem malignant benign tissue express express unique features found tissue type thus complicate learning problem therefore formulate alternative classification problem distinguish background benign gland benign background malignant gland malignant order necessary transform provided ground truth labels reflect benignity malignancy well annotation images binarized new label assigned pixels belonging class see fig input cnn image patch size pixels centered image location denotes image domain given patch convolved filters first convolutional layer second layer filters third layer filters last layer filters see fig three subsequent fully connected layers classifier contain four output units respectively output fed softmax function producing center pixel probability distribution labels probability class stored corresponding map icl input convolutions output full connection convolutions convolutions convolutions architecture input convolutions output full connection convolutions convolutions convolutions architecture fig cnn classifier architectures separatornet architectures layers identical number convolutional convk subk fully connected fck layers differ convolution kernel size size number feature maps well number output units probability distribution labels center pixel marked red cross input patch predicted cnns fig ground truth transformation learning classification preprocessed images first row shows benign case second row shows malignant case preprocessed images overlaid individual ground truth object annotations provided annotations transformed four labels benign background benign gland malignant background malignant gland classifying structures initial experiments shown taking pixelwise predictions objectnet insufficient order separate close gland objects hence second cnn trained predict structures image separating objects learning problem formulated binary classification task depicted fig cnn structure similar given input image patch size pixels convolved filters first convolutional layer second layer filters third layer filters last layer filters three subsequent fully connected layers classifier contain two output units respectively output last layer fed softmax function produce probability distribution labels center pixel probability pixel belonging structure stored corresponding probability map refining cnn outputs probability maps obtained predictions icl refined predictions emphasize gland borders prevent merging close objects subsequent segmentation algorithm requires single foreground background map produce final segmentation result outputs combined follows foreground probability map constructed max icl controls influence refinements done separator predictions similarly evaluating produces background probability map pbg min icl total variation segmentation generate final segmentation following continuous energy functional eseg minimized min eseg min cbox denotes image domain smooth first term denotes total variation reformulation geodesic active contour energy edge function defined gradient input image thus attracting segmentation towards large gradients second term data term describing weighting map values chosen negative foreground positive background values set zero pure weighted energy minimized seeking minimal contour length segmentation use refined outputs previous classification step eqs introduce threshold ensure minimum class confidence map otherwise weighting map derived applying logit transformation log log log pbg log pbg pbg pbg regularization parameter defines data term weighted stated convex problem solved global optimum efficiently using algorithm implemented efficiently using nvidia cuda thus making use parallel computing power recent gpus segmentation continuous final segmentation achieved thresholding value optimize free parameters performing grid search suitable range values annotated training images used tune parameters based dice coefficient implementation details training dataset sampling sake execution speed using sliding window approach images rescaled half resolution prior classification upsampled bilinear interpolation original size afterwards size input patch chosen pixels sufficient contextual information available classify center pixel majority training images size pixels resizing reduces pixels considered valid part without border extension sampling patches training dataset would actually lose approximately labeled pixels using patch size pixels hand would introduce significant number boundary artifacts artificially extending border make use labeled pixels fortunately images tiles bigger image thus stitched seamlessly obtain total fig sample enough patches without heavily relying artificial border extension principle pursued sampling strategy required create ground truth labels manually annotated pixels belong structure close two gland borders green lines fig illustrate additional manual annotation separating structures due low number foreground samples compared number foreground samples artificially increased exploiting problem requirement adding nine additional rotated versions patch every fig manual ground truth annotations structures stitched images four tiles numbers red boxes red lines denote tile borders manual annotations pixels belonging structures shown green lines thickness lines increased better illustration cnn training cnns trained balanced training set image patches per class patches training sets sampled random available pool training one case stitching possible since tiles available tiles remaining images part bigger image treated individual images training test training test classification error best best epochs fig cnn training progress classification error epochs subset training data training error test set reaches test error epochs reaches test error epochs images training test sets reflect approximately distribution samples images size minibatches mbsgd set samples networks trained stopping criterion met improvement error rate test set epochs set initial learning rate linear decay saturating epochs layers weight decay chosen dropout rate set used adaptive momentum term starting increasing epochs progressing training updates influenced larger number samples beginning fig shows classification error rate function training duration epochs class represented samples test set objectnet respectively training error actually estimated fixed subset training data samples get intuition overfitting starts achieves best performance epochs minimum training error minimum test error training continued lowest training error test error reached epochs fig shows learned filters first convolutional layer networks cnn models implemented machine learning library built top theano results colon gland segmentation grid search resulted parameters optimizing segmentation based dice score confidence threshold foreground background determined empirically fixed separator predictions fully considered refining predictions table report performance detection precision recall evaluation scripts kindly provided contest organizers fig cnn training results filters first layer filters table segmentation performance metrics dataset used glas challenge dataset precision recall hausdorff without separator refinement training test test separator refinement training test test metrics reported mean standard deviation best results printed bold performance training set reported training images test set consists images test set images except values hausdorff distance higher values superior score segmentation dice shape hausdorff distance training set well test set mean standard deviation blobs area less pixels removed remaining blobs labeled unique identifiers computing measures compared using predictions segmentation performance improved separator refinement malignant cases harder segment due irregular shape pathological variations tissue fig illustrates qualitative example segmentation results training data set fig fig show results test set respectively able http average total runtime segmenting image minutes using nvidia geforce titan black gpu benign malignant benign malignant malignant benign fig qualitative segmentation results images training dataset even rows show outline ground truth green segmentation result blue numbers refer unique objects within image odd rows show segmentation difference false negative pixels colored cyan false positives colored yellow show examples segmentation algorithm works well show different types segmentation errors benignity malignancy classification proposed approach inherently learns discrimination benign malignant tissue since labels benign malignant available training dataset defined classification problem instead combining probability maps glands background done segmentation combine maps benignity malignancy subsequently benign malignant benign benign benign malignant fig qualitative segmentation results images test dataset even rows show segmentation blue outline ground truth green outline odd rows show differences false negative pixels cyan false positive pixels yellow show reasonable segmentation results different segmentation errors shown average probabilities benign case computed malignant case number pixels image domain maximum values finally indicates prediction arg max evaluated classification performance benign malignant tissue two test sets achieved accuracy average malignant benign malignant benign malignant malignant fig qualitative segmentation results images test dataset even rows show segmentation blue outline ground truth green outline odd rows show differences false negative pixels cyan false positive pixels yellow show reasonable segmentation results different segmentation errors shown decision confidence test set benign malignant test set respectively discussion conclusions paper presented method segment glands stained histopathological images colorectal cancer using deep convolutional neural networks total variation segmentation main contribution showed segmentation results greatly improved predictions refined learned structures adding separators regulate precision recall generally improves performance scores detection segmentation dice shape hausdorff final ranking well test set performance results algorithms participating challenge available online contest continuously updated algorithms new participating groups approach inherently allows accurately discriminate benign malignant cases trained labels cases average confidence decision towards benignity malignancy acceptable nevertheless distinguish detailed histologic grades among cases since information available addition segmentation ground truth acknowledgements authors grateful organizers glas challenge providing image dataset matlab evaluation scripts computing performance measures comparable among participating teams thanks goes julien martel fruitful discussions early phases challenge references kandemir tosun sokmensuer automatic segmentation colon glands using medical image analysis tosun graph matrices histopathological image segmentation ieee transactions medical imaging march sirinukunwattana snead rajpoot stochastic polygons model glandular structures colon histology images ieee transactions medical imaging farjam zoroofi image analysis approach automatic malignancy determination prostate pathological images cytometry part clinical cytometry february naik doyle agner madabhushi feldman tomaszewski automated gland nuclei segmentation grading prostate breast cancer histopathology ieee international symposium biomedical imaging isbi pages may monaco tomaszewski feldman hagemann moradi mousavi boag davidson abolmaesumi madabhushi detection prostate cancer histological sections using probabilistic pairwise markov models medical image analysis august peng jiang eisengart healy straus yang identification prostatic adenocarcinoma segmentation glandular structures journal pathology informatics nguyen sarkar jain structure context prostatic gland segmentation classification medical image computing intervention miccai pages rashid fazli boag siemens abolmaesumi salcudean separation benign malignant glands prostatic adenocarcinoma medical image computing intervention miccai pages http lecun kavukcuoglu farabet convolutional networks applications vision ieee international symposium circuits systems iscas pages may pang zhang chen gao peng cell nucleus segmentation color histopathological imagery using convolutional networks chinese conference pattern recognition ccpr pages october ciresan giusti gambardella schmidhuber mitosis detection breast cancer histology images deep neural networks medical image computing intervention miccai pages malon cosatto classification mitotic figures convolutional neural networks seeded blob features journal pathology informatics may habibzadeh fevens white blood cell differential counts using convolutional neural networks low resolution images artificial intelligence soft computing pages springer berlin heidelberg ruifrok johnston quantification histochemical staining color deconvolution analytical quantitative cytology histology august schindelin frise kaynig longair pietzsch preibisch rueden saalfeld schmid tinevez white hartenstein eliceiri tomancak cardona fiji platform biologicalimage analysis nature methods july zuiderveld contrast limited adaptive histogram equalization graphics gems pages academic press professional lecun bottou bengio haffner learning applied document recognition proceedings ieee november reinbacher pock bauer bischof variational segmentation elongated volumetric structures ieee conference computer vision pattern recognition cvpr pages hammernik ebner stern urschler pock vertebrae segmentation images based variational framework recent advances computational methods clinical applications spine imaging pages springer bresson esedoglu vandergheynst thiran osher fast global minimization active model journal mathematical imaging vision chambolle pock algorithm convex problems applications imaging journal mathematical imaging vision goodfellow lamblin dumoulin mirza pascanu bergstra bastien bengio machine learning research library arxiv preprint bergstra breuleux bastien lamblin pascanu desjardins turian bengio theano cpu gpu math expression compiler proceedings python scientific computing conference scipy june bastien lamblin pascanu bergstra goodfellow bergeron bouchard bengio theano new features speed improvements deep learning unsupervised feature learning nips workshop
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novel progressive learning technique classification rajasekar venkatesan meng joo school electrical electronic engineering nanyang technological university singapore email emjer paper progressive learning technique multi classification proposed newly developed learning technique independent number class constraints learn new classes still retaining knowledge previous classes whenever new class knowledge learnt thus far encountered neural network structure gets remodeled automatically facilitating new neurons interconnections parameters calculated way retains knowledge learnt thus far technique suitable realworld applications number classes often unknown online learning real data required consistency complexity progressive learning technique analyzed everal standard datasets used evaluate performance developed technique comparative study shows developed technique superior key machine learning sequential learning progressive learning introduction study feedforward neural network fnn gained prominence since advent back propagation algorithm several improved optimized variants algorithm developed analyzed past two decades single hidden layer feedforward neural networks slfns gained significant importance due widespread applications recognition classification function approximation area several learning techniques proposed since effective training slfn learning techniques grouped two basic categories batch learning sequential learning batch learning algorithms require pre collection training data collected data set used training neural network network parameters calculated updated processing training data together several batch learning algorithms literature one relatively new batch learning scheme called extreme learning machines elm proposed huang special nature elm input weights hidden node biases chosen random key feature elm maintains universal approximation capability slfn gained much attention several research works made due special nature random input weight initialization unique advantage extreme learning speed advantages elm traditional feedforward neural network analyzed literature many new variants developments made elm significant results achieved approximation classification regression areas batch learning involves processing complete data set concurrently updating weights technique limited applications batch learning techniques time consuming requires complete data set prior training hand learning algorithms network parameters updated new training data arrives overcome shortcomings batch learning techniques several sequential online learning algorithms developed many cases sequential learning algorithms preferred batch learning algorithms require retraining whenever new data sample received learning method combines elm recursive least square rls algorithm later developed called extreme learning machine several variants elm developed proposed literature issue existing classification techniques elm svm trained classify specific number classes learning new classes possible order learn new class data requires retraining classes anew existing techniques require priori information number classes present training dataset information number classes required either specified directly identified analyzing complete training data set based parameter network model designed parameters weights networks updated depending sequential input data makes existing techniques static respect number classes learn existing techniques suited applications dataset might well suited applications cognitive robotics involving data nature training data unknown data number classes learnt often unknown learning technique must meet dynamic needs overcome shortcoming novel learning paradigm proposed called progressive learning progressive learning next stage advancement online learning methods existing online sequential techniques learn classify data among fixed set classes initialized initialization phase algorithm fail dynamically adapt introduced new run progressive learning technique independent number class constraint learn several new classes retaining knowledge previous classes achieved modifying network structure upon encountering new class updating network parameters way learns new class retains knowledge learnt thus far existing online sequential learning methods require retraining new data sample received fails new class data unknown existing knowledge encountered progressive learning technique overcomes shortcoming allowing network learn multiple new classes alien existing knowledge encountered point time preliminaries section gives brief review elm techniques provide basic background information extreme learning machines condensed overview batch learning elm technique proposed huang given consider training samples represented varies denotes input data vector tjm denotes target class labels let number hidden layer neurons network output standard slfn given denotes weight vector input nodes ith hidden node denotes weight vector connecting ith hidden node output nodes hidden layer bias value standard slfn mentioned equation perform classifier output network equal corresponding target class input data given classifier hence slfn equation classifier exist therefore equation output network written denotes target class corresponding input data vector equation written compact form called hidden layer output matrix neural network column gives corresponding output hidden layers given input mathematical framework training process extensively described literature key results restated lemma given standard slfn hidden nodes activation function infinitely differentiable interval arbitrary distinct samples randomly chosen intervals respectively according continuous probability distribution probability one hidden layer output matrix slfn invertible lemma given small positive value activation function infinitely differentiable interval exists arbitrary distinct samples randomly chosen intervals respectively according continuous probability distribution probability one thus seen elm input weights hidden layer neuron bias randomly assigned training elm involves estimating output weights relation true output weight elm estimated using generalized inverse inverse hidden layer output matrix overall batch learning algorithm elm training set form hidden layer neurons summarized step random assignment input weights hidden layer bias step computation hidden layer output matrix step estimation output weights using inverse online sequential extreme learning machine based batch learning method elm sequential modification performed online proposed literature operates online data batch learning method elm output weight estimated using formula inverse hidden layer output matrix written hth stated solution gives least square solution uses rls algorithm update output weight matrix sequentially data arrives online well studied literature summary given let number samples initial block data rovided network calculate subsequent sequentially arriving data output weights updated steps elm based rls algorithm summarized initialization phase step input weights hidden layer bias assigned random step initial block samples data hidden layer output matrix calculated step value initial values estimated sequential learning phase step subsequent sequentially arriving data hidden layer output vector calculated step output weight updated based rls algorithm theory formulation behind operation elm discussed detail several papers standard variants activation function used elm special mapping functions variants discussed detail literature variants elm includes elm kernel elm imbalanced data elm noisy data incremental elm elm ensemble many variants summarized pro gressive learning techniq learning like children proposed progressive learning algorithm adapted natural learning process exhibited children peter jarvis book described detail nature human learning process opposed traditional machine learning algorithm cycle human learning continuous process learning training phase never ending whenever human brain stumbled upon new phenomenon learning resumes key feature human learning learning new phenomenon affect knowledge learnt new knowledge leant added along existing knowledge though several online sequential learning methods information number classes fixed initialization restricts possibility learning newer classes run existing machine learning algorithms fails resume learning entirely new class classes data encountered initializatio applications cognitive robotics learning etc system robust dynamic learn new classes run number classes encounters known beforehand system able redesign adapt meet learning new class arrives proposed learning method introduces novel technique progressive learning showcases continuous learning progressive learning technique enables learn new classes dynamically run whenever new class encountered neural network grows redesign interconnections weights incorporate learning new classification another key feature proposed method newer classes learnt addition existing knowledge present beginning proposed algorithm foreshadowed key objective progressive learning technique dynamically learn new classes run suppose network initially trained classify number classes consider network encounters number new classes alien previously learnt class progressive learning technique plt adapt automatically starts learn new class maintaining knowledge previously learnt classes introduction new class network results changes dimension output vector output weight matrix also newly formed matrices increased dimension evaluated way still retains knowledge learnt thus far also facilitates learning newly introduced class method increasing dimension matrix weight update matrix recalibration methods proposed algorithm significantly different class extreme learning machine proposed algorithm learn sequential introduction single new class also simultaneous multiple new classes block online data sequential introduction multiple new classes proposed algorithm also independent time introduction new class consider hidden layer neurons training data form steps plt algorith initialization phase step input weights hidden layer bias assigned random step initial block samples data hidden layer output matrix calculated step value initial values estimated sequential learning phase subsequent data arrives network trained either basis let chunk size unity value results training network one basis new data data arrived fall either two categories absence new class data presence new class classes data new classes current set data plt similar usual process calculating updating output weights performed subsequent algorithm steps case new classes current chunk data follows step hidden layer output vector calculated step output weight updated based rls algorithm new class chunk data arrived novel progressive learning technique used recalibrate network accommodate new class retaining old knowledge algorithm maintains classes learnt thus far separate set new data data arrives data analyzed class belongs target class new data block equal subset existing classes new classification encountered new data block target class set subset existing classes means system encountered new classification special recalibrate routine initiated recalibration routine number new classes encountered determined class labels identified let number new classes encountered upon identifying number new classes introduced set containing classes learnt thus far updated accordingly neural network redesigned number output neurons increased accordingly interconnections redone weights new network determined current previous weights old network weight update made knowledge learnt old network retained knowledge new classes included along consider hidden layer neurons network classes data currently learnt network chunk size sequential learning introduction new classes instant modify dimensions output weight matrix output weight matrix critical importance elm based networks since input weights hidden layer bias randomly assigned values matrix control number classes learnt accuracy class algorithm steps continued follows step values calculated based current values bxp pxp current matrix dimension new classes introduced therefore accommodat given output weight matrix increased number output layer neurons matrix transformed equation rectangular identity matrix dimension zero matrix upon extending weight matrix accommodate increased number output neurons learning learnt thus far incorporated newly upgraded weight matrix appending zero matrix trivial way increase dimensions matrix values updated network retains knowledge existing classes learn new classes available beginning training phase equation seen error difference arget class predicted class scaled learning factor added since new classes introduced time instant initial data samples target class label value corresponding new class therefore step update new classes written matrix step update new classes incorporated provide upgraded matrix recalibrated adapt learning new classes recalibrated output weight matrix calculated upon simplification expressed represents knowledge previously learnt dimension increased opposed populating increased dimension identity matrix values new entries calculated way newly introduced classes appear neural network present beginning training procedure training data samples thus far belong newly introduced class network recalibrated matrix represents learning new class beginning training phase current data sample considering none previous data samples belong newly introduced class computed equivalent step equivalent new classes beginning training phase therefore updated matrix represents network classes previously existin classes new classes step hidden layer output vector calculated step output weight matrix increased dimension facilitate learning new class updated based rls algorithm whenever new class encountered training resume learning new retaining existing knowledge algorithm also supports recalibration multiple new classes introduced simultaneously sequentially also new classes introduced instant time number times network algorithm progressive learning technique plt summarized fig experimentatio proposed progressive learning algorithm exhibit dynamic learning new class data current multiclass classification algorithms fails adapt encountered new class hence accuracy drops introduced one new classes proposed algorithm redesigns adapt new classifications still retaining knowledge learnt thus far proposed progressive learning algorithm tested several real world standard datasets standard datasets general uniformly distributed test performance progressive learning effectively presented conditions new classes introduced non uniform manner different time instants hence standard datasets used directly test progressive learning algorithm efficiently datasets way subset classes available training initially new classes introduced arbitrary time instances latter part training thus standard datasets modified used testing proposed algorithm default classification problems involve two classes presence class absence class two trivial classes available classification problem since minimum number classes classification two learning new classes absent bivariate datasets binary classification datasets since two classes new classes introduced proposed algorithm performs similar existing online sequential algorithm unique feature progressive learning clearly evident multiclass classification thus proposed algorithm tested multiclass classification datasets iris balance scale waveform wine satellite image digit character datasets specifications datasets shown table proposed technique experimented balanced unbalanced datasets balanced dataset one class equal almost equal number training data unbalanced dataset skewed dataset subset classes high number training samples classes fewer training samples number hidden layer neurons experimentation chosen overfitting problem mitigated test dataset consists data samples corresponding class labels used prog ressive learning network algorithm progressive learning technique classification parameters network initialized raw input data processed classification elm training initial phase processing initial block data elm training sequential phase case new classes introduced case new classes introduced elm testing estimation raw output values using class corresponding index maximu value predicted target class fig algorithm progressive learning technique proposed algorithm also works introduction multiple new classes number classes increased testing multiple new classes proposed method tested character recognition dataset described latter part section introduction multiple classes sequentially simultaneously multiple time instances experimented verified results discussions functionality technique consistency complexity three key features tested new technique functional testing used validate proposed algorithm functional results expected behavior functionality technique tested using iris waveform datasets operational working concept progressive learning proposed algorithm tested functionality test consistency another key feature essential new technique proposed algorithm provide consistent results multiple trials minimal variance elm based algorithm consistency proposed method across several trials dataset also consistency across cross validation tested complexity analysis essential new technique number operations performed calculations involved proposed method computed compared existing thod also performance proposed algorithm evaluated introducing new classes different time instances early training middle training towards end training evaluated sequential simultaneous introduction new classes experimented results analyzed discussed able ecifications multiclass classification datasets dataset number classes number remarks iris dataset basic benchmark dataset balance scale dataset benchmark dataset unbalanced data waveform dataset basic benchmark dataset wine dataset basic benchmark dataset satellite image dataset basic benchmark dataset digit dataset basic benchmark dataset character dataset dataset sequential introduction two new classes character dataset dataset sequential introduction three new classes character dataset dataset simultaneous introduction new classes functionality proposed technique experimented iris waveform balance scale datasets verify basic intended functionality technique iris dataset consists three classes uniformly distributed instances facilitate testing progressive learning dataset redistributed first samples consists two classes sentosa versicolor third class virginica introduced sample type redistribution closely emulates real time scenario encountering new class run ability proposed algorithm recognize adapt learn new class verified testing distribution details dataset used given table able ecifications ris dataset data range number classes new class added point introduction new class class labels sentosa versicolor sentosa versicolor virginica iris dataset testing accuracy samples fig esting accuracy iris dataset progressive learning algorithm tested specified iris dataset learning curve testing accuracy graph plotted result obtained shown fig testing accuracy continuously calculated test data set every new training data seen graph sample index testing accuracy implies system learnt two three classes thus far third new class introduced sample system recognizes introduction new class recalibrates sufficiently increasing number neurons output layer weight matrix suitably increased dimension weights updated based special recalibration technique proposed new network structure weight parameters formed current network parameters data obtained new class forthcoming iterations system trains recognition new class addition previously learnt classes reaches steady state testing accuracy process results sudden rise testing accuracy network settles final testing accuracy value sudden increase testing accuracy due fact network recognize newly encountered class data procedure repeated waveform balance scale dataset dataset specifications waveform balance scale dataset shown table result obtained progressive learning method shown fig respectively able ecifications aveform dataset data range number classes new class added point introduction new class class labels waveform waveform waveform waveform waveform waveform dataset testing accuracy samples fig esting accuracy waveform dataset able ecifications balance scale dataset data range number classes new class added point introduction new class class labels left right left right balanced balance dataset testing accuracy samples fig esting accuracy balance dataset test results expected behavior progressive learning method verified result shows algorithm able learn new classes dynamically run learning new class significantly affect accuracy classes previously learnt consistency performance proposed method evaluated using six benchmark datasets consistency consistency critical characteristic tested new technique proposed technique verified consistency results consistency key virtue technique exhibit learning technique provides inconsistent results reliable practical applications elm based technique input weights hidden layer bias values initialized random hence multiple executions dataset specification results different results therefore dataset specification executed multiple times determine consistency across multiple executions consistency results repeated multiple execution three datasets shown table able consistency across multiple trials trials testing accuracy iris dataset waveform dataset balance scale dataset wine dataset satellite image dataset digit dataset cross validation common method evaluate consistency given technique proposed algorithm tested datasets cross validation cross validation resulting testing accuracy tabulated table gives consistency proposed algorithm cross validation performance seen table proposed algorithm consistently accurate attempts deviation testing accuracy order mean value nominal thus results show proposed method gives consistent reliable testing accuracy balanced unbalanced datasets able consistency cross validation cross validation iris dataset waveform dataset balance scale dataset wine dataset satellite image dataset digit dataset computational reduction number computations required proposed progressive learning technique analyzed compared existing method though learning new classes dynamically run causes overhead computations seemingly increases complexity technique actual computational complexity proposed technique lesser method decrease complexity due two reasons overhead computations responsible increasing number output neurons creating new interconnections recalibration weights occur samples new class introduced thus recalibration routine invoked new class henceforth causing minimal increase computation complexity example one new class introduced recalibration procedure invoked progressive learning method also provides another distinct advantage since new classes learnt dynamically results lesser number weight calculations compared static online sequential training techniques like example iris dataset considered traditional algorithm needs update weight three output neurons entire instances training set proposed method two output neurons till occurrence third class third output neuron introduced recalibration stage triggered introduction new class thus number weight calculations effectively reduced effectively reduces number computations performed thereby reducing computat ional complexity reduction number weight calculations shown table number computations normalized computational complexity progressive learning method compared fig number computations normalized iris waveform balance scale wine satellite image digit progressive learning fig comparison computational reduction able reduction number calculations proposed method weight calculations oselm nhidden point introduction new class weight calculations proposed method nhidden calculations saved iris dataset waveform dataset balance scale dataset wine dataset satellite image dataset digit dataset though new classes learnt halfway datasets testing accuracy algorithm nearly maintained even improved compared algorithms static number classes reason change accuracy due fact new classes learnt run learning previous classes previously learnt classes new class fairly distinctive learning accuracy improved hand cases due feature set learnt class new class learning new class affect existing knowledge little extent thereby marginally reducing overall accuracy testing accuracy proposed algorithm compared existing variants voting based enhanced robust robust bayesian elm generalized pruning elm tabulated shown table seen table despite learning new classes dynamically later stage training testing accuracy either improved maintained nearly equal testing accuracy based methods proposed method provides two key advantages existing methods reduction computational complexity flexibility learn new classes instant time results obtained thus far evident proposed progressive learning algorithm learns new class data dynamic way able comp arison testing accuracy proposed method iris dataset waveform dataset balance scale dataset wine dataset satellite image dataset digit dataset introduction new class different time instants new class introduced different stages training period effect learning rate analyzed order analyze response three different test cases experimented performance measured new class introduced three different time instances early training middle training towards end training testing accuracy curve test case plotted results evaluated compared point introduction new class test case tabulated given table performance proposed network test cases given fig seen figure independent point introduction new class system network capable learning new class final steady state testing accuracy across test cases able oint introduction new class test cases point introduction new class total number samples early middle towards end fig introduction new class training early middle towards end multiple new classes performance proposed technique introduced multiple new classes sequentially simultaneously discussed section learning multiple new classes proposed algorithm tested using character recognition dataset several combinations tests made sequential introduction new classes classes sequential introduction new classes classes simultaneous introduction new classes along one new class sequentially classes performance proposed algorithm test case observed sequential introduction new classes character dataset classes used test sequential introduction two new classes proposed algorithm dataset redistributed meet testing requirements progressive learning specifications dataset given table able ecifications character dataset new classes data range number classes new class added class labels initially network sequentially trained two classes samples new class introduced training data sample fourth class introduced sample proposed algorithm identifies new classes recalibrates time continues learning results two sudden rise learning curve network first rise corresponding occurring sample corresponds learning class second rise occurring sample corresponds learning class learning curve graph shown fig characters testing accuracy samples fig sequential learning two new classes sequential introduction new classes character dataset classes used testing sequential introduction new classes network initially trained recognize two classes three new classes introduced one another initial training two classes specifications dataset shown table able ecifications character dataset three new classes data range number classes new class added class labels new classes introduced sequentially later time instants algorithm adapts new class time also maintains testing accuracy level testing accuracy curve shown fig verify learning new class independent previously learnt classes overall testing accuracy broken individual testing accuracy classes shown fig seen testing accuracy classes remains also whenever new class introduced new learning curve formed contributes towards overall accuracy along existing classes network initially trained two classes third class introduced samples learning curve class shown black line another new class testing accuracy shown red introduced sample fifth class introduced sample learning curve shown light blue graph seen class introduced learnt anew without affecting much existing knowledge learning accuracy classes collectively responsible overall accuracy network seen testing accuracy classes overall accuracy achieved characters testing accuracy samples fig sequential learning three new classes fig individual overall testing accuracy sequential introduction testing accuracy obtained introducing one two three new classes summarized table table observed learning multiple new classes affect testing accuracy previously learnt class hence method used learn large number multiple new classes progressive manner without affecting testing accuracy previously learnt classes able summary testing accuracy sequential introduction multiple new classes number classes introduced sequentially testing accuracy two base class one new class two base class two new classes two base class three new classes simultaneous introduction new classes verify proposed algorithm performs effectively multiple classes introduced simultaneously introduced block character dataset specifications shown table used two classes introduced together new class later stage testing accuracy shown fig able ecifications character dataset simultaneous new classes data range number classes new class added class labels first rise observed sample instant testing accuracy curve corresponds introduction two new classes characters algorithm identifies new classes recalibrates facilitate multiple class addition second rise curve corresponds introduction third class character order show previous knowledge retained new knowledge added along existing testing accuracy split five alphabets shown fig seen two new learning curves corresponding new class introduced sample newly introduced classes learnt simultaneously along existing classes learning curve sample index corresponds introduction class also graph clear learning additional classes significantly affect testing accuracy classes previously learnt thus enabling proposed algorithm learn multiple new classes sequentially simultaneously progressive manner characters testing accuracy samples fig esting accuracy simultaneous new classes fig individual overall testing accuracy simultaneous new classes proposed algorithm introduces new neurons output layer recalibrates network facilitat learning new classes since output layer neurons increased number hidden layer eurons learning new classes progressively learnt limited number classes learnt given number hidden layer neurons proposed algorithm extended output neurons hidden layer neurons increased number new classes learnt progressively nclusions paper novel learning technique progressive learning classification developed progressive learning enables network learn multiple new classes dynamically run new classes learnt sequential simultaneous manner hence technique much suited applications number classes learned unknown progressive learning enables network recalibrate adapt encountered new class data proposed progressive learning technique perform effectively applications cognitive robotics system trained real time experienced based data ackno wledgment first author would like thank nanyang technological university singapore ntu research student scholarship references rumelhart hinton williams learning representations errors nature vol hagan menhaj training feedforward networks marquardt algorithm ieee trans neural networks vol wilamowski neural network learning without backpropagation ieee trans neural networks vol chen cowan grant orthogonal least squares learning algorithm radial basis function networks ieee trans neural networks vol peng irwin fast nonlinear model identification method ieee trans automatic control vol branke evolutionary algorithms neural network design raining proc first nordic workshop genetic algorithms applications yao review evolutionary artificial neural networks international journal intelligent systems vol wang song improved sequential learning algorithm extreme learning machine advances neural networks isnn vol huang zhu siew extreme learning machine theory applications neurocomputing vol rumelhart hinton williams learning internal representations error propagation california university san diego jolla inst cognitive science ferrari stengel smooth function approximation using neural networks ieee transactions neural networks vol huang chen babri classification ability single hidden layer feedforward neural networks ieee transactions neural networks vol huang wang lan extreme learning machines survey international journal machine learning cybernetics vol wang han generalized layer feedforward networks regression problems ieee transactions neural networks learning systems vol liang huang saratchandran sundararajan fast accurate online sequential learning algorithm feedforward networks ieee transactions neural networks vol huang zhu siew extreme learning machine new learning scheme feedforward neural networks proceedings international joint conference neural networks vol huang chen siew universal approximation using incremental constructive feedforward networks random hidden nodes ieee transactions neural networks vol huang chen convex incremental extreme learning machine neurocomputing vol huang chen enhanced random search based incremental extreme learning machine neurocomputing vol wang cao yuan study effectiveness extreme learning machine neurocomputing vol huang zhou ding zhang extreme learning machine regression multiclass classification ieee trans systems man cybernetics part cybernetics vol zhu qin suganthan huang evolutionary extreme learning machine pattern recognition vol huang saratchandran sundararajan fully complex extreme learning machine neurocomputing wang han parsimonious extreme learning machine using recursive orthogonal least squares ieee transactions neural networks learning systems vol huang saratchandran sundararajan efficient sequential learning algorithm growing pruning rbf networks ieee transactions systems man cybernetics part vol huang saratchandran sundararajan generalized growing pruning rbf ggap neural network function approximation ieee transactions neural networks vol wang sun liu hybrid recursive least squares algorithm online sequential identification using data chunks neurocomputing vol wang han dong constructive extreme learning machine application large tanker motion dynamics identification neurocomputing vol rong huang sundararajan saratchandran online sequential fuzzy extreme learning machine function approximation classification problems ieee transactions systems man cybernetics part vol huang liang rong saratchandran sundararajan sequential extreme learning machine computational intelligence vol huang huang song rends extreme learning machines review neural networks vol daliri hybrid automatic system diagnosis lung cancer based genetic algorithm fuzzy extreme learning machines journal medical systems vol zhang fuzzy extreme learning machine classification electronics letters vol avci coteli new automatic target recognition system based wavelet extreme learning machine expert systems applications vol malathi marimuthu baskar ramar application extreme learning machine series compensated ransmission line protection engineering applications artificial intelligence vol freney verleysen kernel extreme learning support vector regression neurocomputing vol zong huang chen weighted extreme learning machine imbalance learning neurocomputing vol man lee wang cao miao new robust training algorithm class single hidden layer feedforward neural networks neurocomputing vol cao lin huang liu voting based extreme learning machine information sciences vol lei zhu xia wang prediction interactions amino acid sequences ensemble extreme learning machines principal component analysis bmc bioinformatics zhai wang dynamic ensemble extreme learning machine based sample ntropy soft computing vol jarvis peter towards comprehensive theory human learning vol psychology press zhao chen chen wang wang class incremental extreme learning machine activity recognition cognitive computation vol
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measure combined effects morphological parameters inclusions within composite materials via stochastic homogenization determine effective mechanical properties vladimir salnikov sophie lemaitre daniel philippe nicolas oresme mathematics laboratory apr university caen lower normandy juin caen cedex france abstract previous papers described efficient reliable methods generation representative volume elements rve perfectly suitable analysis composite materials via stochastic homogenization paper profit methods analyze influence morphology effective mechanical properties samples precisely study dependence main mechanical characteristics composite medium various parameters mixture inclusions composed spheres cylinders top introduce various imperfections inclusions observe evolution effective properties related main computational approach used throughout work homogenization technique validated however comparison direct finite elements method give details features method validation campaign well keywords composite materials cylindrical spherical reinforcements mechanical properties stochastic homogenization introduction motivation paper study influence morphological parameters composite materials effective mechanical properties usage composite materials industrial applications motivated huge amount publications subject recent years concern experimental modelling results reason address question twofold well one hand explore existing modelling techniques hand mind concrete applications related project industry aeronautics need modelling analysis composite materials applied domains comes fact experimental work usually expensive difficult carry thus important develop modelling approaches efficient reliable sufficiently flexible outcome validated experiment strategy adopt related notions stochastic homogenization key idea consider sample composite material sufficiently large capture behavior compute macroscopic parameters mechanical properties example young modulus poisson ratio eventually whole stiffness tensor take account possible imperfections random factors one average result series tests representing macro characteristics usual technology generate series samples representative volume elements randomly controlling though parameters perform computation average result generally accepted main characteristic affecting effective properties composite material morphology combination geometric characteristics inclusions distribution supporting matrix analyze phenomenon one needs thus tool generate rves capturing various morphological parameters developed implemented tool described algorithms produce rves containing spheres represent globular inclusions cylinders responsible reinforcements able reach volume fraction inclusions relatively high values addition control geometric configuration sample whole namely manage intersections inclusions eventually distribution moreover extended method introduce irregularities shape inclusions paper describe results computations carried generated samples via homogenization technique presentation chosen results useful applications trends intuitively obvious particular explore influence redistribution volume fraction globular reinforcements well effects imperfections paper organized follows next section briefly recall rve generation methods proposed extended works second part give details main computational method used throughout work homogenization technique coupled stochastic methods rve generation describe convenience limitations well present results validation campaign compare direct finite elements method section iii description results analysis via mentioned methods effective properties composites depending morphological parameters inclusions conclude describing eventual industrial applications work progress expected results sample generation computational techniques outlined introduction section devoted brief description methods used perform computations reasons choose concrete methods rve generation generation samples computation important step process modelling behavior composite materials since morphology composites may quite complex challenging task one hand important able approximate rather involved geometries hand method fast reliable ideal case stage generation much shorter computation number works inclusions represented simple geometric objects like spheres ellipsoids see example one considers complicated geometry problem managing intersection inclusions arises immediately among established approaches dealing one mention two important families random sequential adsorption rsa type algorithms molecular dynamics based methods rsa based sequential addition inclusions verifying intersection main idea make inclusions move reach desired configuration let mention first method needs efficient algorithm verification intersection geometric shapes second one algorithm predicting time intersection moving objects exists limited class shapes often amounts difficult minimization problem described classical rsa version applied mixture inclusions spherical cylindrical shapes key ingredient approaches explicit formulation algebraic conditions intersection cylinder sphere two cylinders specific recapitulate ideas algorithms algorithms observed rsa approach extremely efficient relatively small volume fractions inclusions permits generate sample fractions second method powerful higher volume fractions order generates configuration second rsa get stuck example sample mixture spherical cylindrical inclusions presented figure fig view rve spherical cylindrical inclusions periodic boundary conditions algorithm rsa generation procedure input volume fractions number inclusions ncyl nsp aspect ratio compute parameters cylinders spheres numofgenspheres numofgencylinders umof gencylinders ncyl generate new cylinder using algorithm alg check overlaps cylinder generated yes redo increase numofgencylinders umof genspheres nsp generate new sphere check overlaps sphere generated yes redo using algorithm alg check overlaps cylinder generated yes redo increase numofgenspheres algorithm generation procedure input volume fractions number inclusions ncyl nsp aspect ratio compute parameters cylinders spheres fix criterion stop simulation generate nsp spheres ncyl cylinders disregarding overlapping overlappingenergy check overlapping alg couple overlapping inclusions compute interaction force table perform integration step evolution equations update value overlappingenergy outcome algorithms list inclusions vector form list coordinates centers radii eventually axes symmetry inclusions perfectly suitable various computational techniques homogenization procedures applied pixelized samples well finite element computations mesh constructed pixelization addition able introduce various imperfections inclusions without spoiling efficiency generation algorithm figure shows two examples surface inclusion waved part inclusion taken produce irregular shapes waved surface inclusion part inclusion taken fig various imperfections inclusions homogenization scheme efficient scheme generation samples proceed computation effective mechanical properties mentioned introduction evaluating properties adopted philosophy stochastic homogenization schematically one view process follows fix macroscopic parameters material volume fraction type inclusions generate series samples rves composite material parameters stochastic part perform accurate computation effective properties rves deterministic homogenization part average computed macroscopic characteristics samples let describe mechanical model behind computation well main computational method homogenization procedure consider representative volume element denote displacement field defined point system mechanical equilibrium described law strain tensor model small deformations stored mechanical energy stress tensor subject condition div linear case law simplifies stiffness tensor notice composite material stiffness tensor depend point dependence governed microscopic geometry sample namely phase matrix inclusion point belongs suppose averaged strain prescribed decompose two parts equivalent representing periodic boundary thus problem actually solving reads div periodic antiperiodic solution tensor field interested average order obtain homogenized stiffness tensor chom equation chom recover components chom space one needs perform computation six independent deformations morally correspond usual stretch shear tests couple natural approaches solving problem one construct mesh rve employ finite elements method discretize basically pixelize rve use homogenization scheme major difficulty arising applying former method rather involved geometry one needs construct fine mesh task moreover proceed finite elements one requires considerable memory resources idea latter fourier space equations acquire rather nice form case homogeneous isotropic material one construct green operator basically produce exact solution composite material containing possibly several phases one introduces artificial reference medium green operator defined computation iterative procedure approximate corrections microscopic behavior material comparison reference medium sake completeness let present method details following essentially works introduce reference medium stiffness tensor correction equations equivalently rewritten div periodic antiperiodic tensor called polarization periodicity assumptions permit rewrite first line fourier space using linearity fourier transform property respect derivation one obtains denotes fourier image coordinates fourier space key observation fourier space relation polarization deformation tensors namely green operator isotropic reference medium coefficients computed explicitly going back original variables initial problem reduces periodic integral equation equation solved iteratively using following algorithm algorithm numerical scheme initialize fix convergence criterion acc converged convergence test acc compute acc converged stop algorithm converges compute inserted equation mentioned computation repeated complete set independent global deformation fields chosen approach computational efficiency time memory consumption also convenience applied problems discuss afterwards however details worth commented first several versions schemes possibility use initial variables formulate dual problem solution produce compliance tensor one produce sort mixture two using polarization primary variable efficiency methods depends contrast characteristics different phases chosen implement direct accelerated scheme presented since one theoretical result optimal values parameters reference medium according convergence theorem formulated composite coefficients constituent phases respectively one choose compute optimality accelerated scheme also coherent recent results second carried validation campaign comparing results computations using process several types finite elements approaches analytical results possible constructed test samples simple geometries square bar different orientations plane cutting sample two parts etc addition scheme samples computed stiffness tensor using finite elements adapted mesh mesh constructed pixelized sample cases used hexahedron elements parallelepipeds adapted mesh constructed cubes corresponding voxels mesh built pixelized sample computation adapted meshes gives results good agreement theoretical estimations mesh obtained pixelization tendency overestimating parameters inclusions rigid matrix although values contrast matrix inclusions interest difference reach great importance observing trends interesting scheme produces results parameters often underestimated situation certainly reversed inclusions less rigid matrix computing mechanical properties composite materials simple way increase resolution pixelization tests describe paper pixelization around already produces reasonable results given sample like one presented figure sufficient make several tests different resolutions vector data observe result stabilizes let however note one interested thermal properties typically two phases rather fine geometry problem much complicated suggest advanced techniques adapting sample computations elsewhere third mentioned stochastic part computations due averaging results several samples macroscopic characteristics usual approach inspired monte carlo method since pioneer work metropolis ulam great number applications method various problems pure mathematics physics engineering science general concrete context homogenization estimation effective properties composite materials natural problem arises computed average given sampling need decide far real average usual way use student distribution compute confidence intervals sampling mean value standard deviation see example precise algorithm tests sufficient make runs obtain acceptably small values deviation needed often mostly high values volume fraction inclusions significant contrast two materials depict confidence intervals figures follow overload plots however verify trends exhibit due statistical errors iii mechanical properties composites section present tendencies behaviour composite materials various morphological parameters obtained making series tests using algorithms described start simple tests aim validate methods sense produce expected results simple tendencies continue involved analysis influence combinations morphological parameters algorithm generation rves able control number parameters already mentioned working spheres cylinders supposed represent respectively globular inclusions microfiber reinforcements types inclusions able assign volume fraction generated sample fsp fcyl respectively choose number inclusions type nsp ncyl cylinders extra parameter aspect ratio ratio length cylinder diameter already gives lot parameters top introduce imperfections inclusions comment end section result certainly depends mechanical parameters matrix inclusions describe concisely fixing value contrast two media output thus normalized respect matrix value homogenized parameters carried several series computations varying parameters paper present selection results qualitatively obvious first sight quantitatively important applications basic tests starting real computations need fix one detail namely typical size representative volume element according size acceptable increasing modify result computations case fixing relative size rve equivalent determine minimal acceptable number inclusions small number inclusions clearly corresponds small piece material studied large number means sample includes sufficient microscopic variety close homogeneous context effective properties analyzed dependence effective properties number inclusions parameters fixed typical trend shown figure clearly shows total number properties stabilize effect pronounced higher volume fractions computations follow thus consider number inclusions order normalized bulk modulus fig dependence mechanical parameters composite number inclusions volume fraction fixed contrast let turn analysis influence morphological properties follows depict trends effective bulk shear moduli one since usually rather similar qualitatively principal computing whole homogenized stiffness tensor turns close isotropic thus possible extract combination mechanical characteristics young modulus poisson ratio coefficients choice bulk shear moduli motivated comparison experimental measurements datasheets start let consider spherical inclusions figure cylindrical ones figure spheres contrast greater effective parameters increase volume fraction decrease contrast less contrast effect pronounced similar effect observed cylinders addition reinforcement longer inclusions higher aspect ratio efficient first tests considered preparatory results validation check methods since outcome rather predictable follows discuss interesting tests contrast contrast contrast contrast contrast contrast volume fraction volume fraction bulk modulus fig shear modulus dependence mechanical parameters composite volume fraction spherical inclusions nsp similar picture advanced morphology analysis let turn subtle questions related analysis influence morphology effective properties composite materials namely let consider composites reinforced mixture globular inclusions understand factors repartition inclusions two types influence result fraction aspect contrast normalized bulk modulus fraction contrast normalized bulk modulus aspect contrast normalized bulk modulus fig fraction aspect fraction aspect contrast normalized shear modulus dependence mechanical parameters composite volume fraction aspect ratio cylindrical inclusions ncyl first series tests fix overall number inclusions type study dependence effective properties aspect ratio cylindrical inclusions various volume fractions results figure clearly show contrast greater composite better reinforced longer cylinders effect certainly better visible higher contrast although expected presence spherical inclusions mixture permit reach values parameters homogenized medium case volume fraction formed cylinders figure opposite effect present values contrast less quantitatively less pronounced let fix volume fraction type inclusions vary number figure shows slightly efficient aspect ratio contrast normalized bulk modulus aspect ratio aspect ratio contrast normalized bulk modulus aspect ratio contrast normalized bulk modulus contrast normalized bulk modulus fig dependence mechanical parameters composite material aspect ratio cylinders mixture inclusions comparison nsp ncyl large number cylinders although looking numerical values one sees effect rather subtle neglected global analysis notice saturation phenomenon total number inclusions small perfect agreement discussion size rve figure interesting series tests group study influence repartition volume inclusions spheres cylinders diagonals plot figure represent volume fractions two types inclusions fixed sum one notice effect better observed cylinders spheres fsp fcyl contrast fsp fcyl contrast normalized bulk modulus normalized bulk modulus fsp fcyl contrast fsp fcyl contrast normalized normalized bulk modulus bulk modulus fig fsp fcyl contrast fsp fcyl contrast normalized normalized bulk modulus bulk modulus dependence mechanical parameters composite material number various inclusions fixed volume fraction note scale plots different others difference minimal maximal values almost absorbed confidence intervals except corner small number inclusions also figure contrast normalized bulk modulus contrast normalized bulk modulus contrast normalized bulk modulus contrast normalized bulk modulus contrast normalized bulk modulus contrast normalized bulk modulus fig dependence mechanical parameters composite material repartition inclusions volume spheres cylinders nsp ncyl plot left level sets map right imperfections following two series tests represent true life situations inclusions ideal shape algorithms able generate various types imperfections including perturbing surfaces inclusions waves taking parts inclusions preserving overall volume fraction figure let mention generating imperfections still suppose interface two materials perfect voids discontinuities created proves reasonable assumption mechanical properties let note though one studies example thermal electrical conductivity result subtle first case main parameter relative wave amplitude ratio amplitude perturbations surface characteristic size inclusions radii spheres cylinders performed computations typical dependence effective properties parameter several aspect ratios cylinders mixture presented figure trends observed leave doubt perturbations fact contribute efficient reinforcement material computations show effect pronounced composite already well reinforced higher volume fraction larger aspect ratio cylinders significant contrast two phases second series tests concerns simulation possible defects introducing inclusions matrix future composite material model process generating zones obtained composite spoilt namely inclusion intersects partially zone piece moved placed apart matrix main parameter perturbation thus volume fraction zones figure shows effect amount defects properties material curves less smooth dependencies presented globally one sees defects least reasonable volume fraction contribute reinforcement well studied material reinforced without imperfections clearer effect visible relative wave amplitude aspect normalized bulk modulus relative wave amplitude aspect normalized bulk modulus relative wave amplitude relative wave amplitude aspect normalized bulk modulus aspect normalized shear modulus fig dependence mechanical parameters composite waving surface inclusions nsp ncyl contrast conclusion outlook conclude let recapitulate main messages paper started presenting methods find suitable analysis mechanical properties composite materials approach find optimal called stochastic homogenization iterative scheme used computing effective properties given sample stochastic part enters level random generation samples one important advantages approach low time memory consumption comparison example finite elements methods turning concrete computational results conclude important morphological properties influencing effective properties composite materials volume fractions various types inclusions rather basic geometry fiber reinforcements moreover introducing various fraction defects aspect normalized bulk modulus fraction defects fraction defects aspect normalized bulk modulus fig fraction defects aspect normalized bulk modulus aspect normalized shear modulus dependence mechanical parameters composite volume defects material nsp ncyl contrast irregularities inclusions thus making shape geometrically complex positive effect parameters obtained material also mentioned behind paper precise motivation related industrial applications work within framework industrial project related amelioration effective properties composite materials certainly dream applied mathematician context would propose optimal scheme production contacts companies actually produce composite materials show reality fabrication process flexible one wants example paper deliberately omitted analysis influence collective orientation inclusions possible unequal distribution sample though algorithms permit control parameters contrary always checked resulting homogenized medium sufficiently isotropic often modelling results aimed validation properties rather production planning choice limited number possible strategies particular case input simulation consists parameters different phases composite geometry sample form images obtained tomography microscopy expected outcome estimation effective parameters best scenario images segmented regions belonging matrix inclusions labeled format sample perfectly suitable performing computation described however mean main applied aim paper test methods fact computations described accumulated huge database samples together already computed effective properties used various parameter fitting inverse problems acknowledgements computations described paper carried cluster center informatics resources higher normandy crihan centre ressources informatiques work supported accea project selected fonds unique fui program salnikov choi karamian efficient reliable stochastic generation rves analysis composites within framework homogenization accepted computational mechanics doi ver par dynamique variations autour pixellisation calcul des effectives des composites accepted publication proceedings csma michel moulinec suquet computational scheme linear composites arbitrary phase contrast int numer meth engng doi monchiet bonnet fft iterative scheme computing effective properties elastic composites arbitrary contrast int numer meth engng segurado llorca numerical approximation elastic properties composites journal mechanics physics solids ghossein fully automated numerical tool comprehensive validation homogenization models application spherical particles reinforced composites international journal solids structures zhao zou dense random packings spherocylinders soft matter man donev stillinger sullivan russel heeger inati torquato chaikin experiments random packings ellipsoids physical review letters widom random sequential addition hard spheres volume chem phys lubachevsky stillinger geometric properties random disk packings journal statistical physics williams philipse random packings spheres spherocylinders simulated mechanical contraction physical review ghossein random generation periodic hard ellipsoids based molecular dynamics algorithm journal computational physics michel moulinec suquet effective properties composite materials periodic microstructure computational approach computational methods applied mechanics engineering eyre milton fast numerical scheme computing response composites using grid refinement european physical journal applied physics moulinec silva comparison three accelerated schemes computing mechanical response composite materials international journal numerical methods engineering metropolis ulam monte carlo method amer statistical assoc salnikov numerical approaches analysis topology phase space dynamical integrability chaos solitons fractals vol yoshinaga kats halperin adsorption polymers macromolecules kaminski schrefler probabilistic effective characteristics cables superconducting coils comput meth appl mech engrg leclerc karamian vivet campbell numerical evaluation effective elastic properties overlapping random fibre composites technische mechanik leclerc effects fibre dispersion effective elastic properties overlapping random fibre composites comput mat leclerc vivet influence morphological parameters random short fibre composite effective elastic properties mechanics industry kanit forest galliet determination size representative volume element random composites statistical numerical approach int solids struct yvonnet toulemonde numerical modelling effective conductivities composites arbitrarily shaped inclusions highly conducting interface composites science technology list case exhaustive since range application monte carlo approach without exaggeration enormous rather time consuming numerical experiment took cpu time antares cluster crihan computational center
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divergence undistortion continuous cocycle superrigidity full shifts oct chung yongle jiang abstract article prove full topological version popa measurable cocycle superrigidity theorem full shifts precisely prove every continuous cocycle full shifts every finitely generated group one end undistorted elements divergence function cohomologous group homomorphism via continuous transfer map target group complete admits compatible metric using ideas behrstock mosher mozes sapir show class acting groups large including wide groups undistorted elements groups strong thick finite orders consequence irreducible uniform lattices higher rank connected semisimple lie groups mapping class groups surfaces richard thompson groups aut certain dimensional groups artin groups class partially extends main result introduction studying measurable orbit equivalence popa established celebrated cocycle superrigidity theorem bernoulli shifts measurable setting certain groups full shift action cocycle superrigid every finite set class polish groups arise closed subgroups unitary groups finite von neumann algebras class contains countable groups proved topological version popa theorem showing every finitely generated group one end every full shift continuous rigid every countable group shortly cohen removed assumption one expects continuous cocycle superrigidity theorem applications continuous orbit equivalence theory systematically studied popa theorem holds class groups containing countable groups one may wonder whether extend main result target groups belonging precisely study continuous cocycles shifts target groups polish groups admit compatible metrics following denote class groups ginv contains class date october chung yongle jiang direction seminal paper katok spatzier established pioneer rigidity properties hyperbolic actions higher rank abelian groups showed standard actions including known irreducible examples every continuous cocycle cohomologous constant cocycle without assumptions periodic data theory many results rigidity cocycles higher rank abelian group actions proved see recent monograph details references nature would like investigate whether results extended hyperbolic actions groups beyond paper first steps understand question consider simplest anosov action full shifts general countable groups target groups cocycles complete admits compatible metrics following investigate following class groups every continuous cocycle cohomologous group homomophism ginv finite set since continuous cocycles discrete target groups automatically continuous defined follows every continuous cocycle continuous cohomologous group homomophism countable discrete group finite set proved finitely generated belongs iff one end expect conclusion still holds replace along direction prove following theorem counterpart theorem setting theorem let finite set finitely generated infinite group let subshift assume following conditions satisfied one end divergence function grows see section slow growth distortion property abbreviated sdt property see definition topological mixing homoclinic equivalence relation strong see section every element sdt property every continuous cocycle group cohomologous homomorphism via continuous transfer map assume undistorted elements see section transfer map chosen continuous theorem deduce following divergence undistortion cont cocycle superrigidity corollary let finitely generated infinite group one end undistorted elements divergence function grows organize paper follows section review definitions elementary properties divergence functions distortion property undistorted elements translation numbers section review continuous cocycle homoclinic equivalence relation shifts also introduce modified version specification property shifts deal rigidity continuous cocycles section also establish key lemma construction invariant holonomies undistorted elements section prove main theorem combining techniques growth rate divergence functions get universal solution continuous transfer map show indeed continuous whenever undistorted elements finally using ideas illustrate class contains wide groups undistorted elements groups strong thick finite orders implies irreducible uniform lattices higher rank connected semisimple lie groups mapping class groups surfaces thompson groups aut certain coxeter groups artin groups acknowledgement chung supported nrf grant funded korea government msip jiang grateful advisor hanfeng support buffalo part paper written authors supported science research center program nrf funded ministry science ict future planning thank jason behrstock informing examples finitely generated groups whose divergences would also like thank mark sapir helpful comments specially providing information thompson groups preliminaries section recall definitions set notations throughout paper finitely generated group unless otherwise specified divergence functions let geodesic metric space typically would take cayley graph endowed word metric respect symmetric generating set main reference divergence functions use section section note several definitions divergence functions use definition taking since two constants would guarantee definitions equivalent corollary cayley graph finitely generated group chung yongle jiang define divergence pair points relative point length shortest path avoiding ball around radius min path exists define divergence infinity divergence pair points div supremum divergences relative divergence given divx max div say divx grows divx one check divx iff log divx divx polynomial grows recall given two functions write exists constant set easy check class divergence functions invariant cayley word metric also clear property growing divergence function preserved distortion sdt property first recall definition distortion functions needed definition sdt property definition let subgroup group generated finite sets respectively distortion function defined max also define compression function follows min later use would frequently work rather roughly speakg ing one think inverse function see proposition note could define inverse function directly since may strictly monotone however still use denote number sup makes sense proposition given two functions write exists let record properties functions later use leave proof exercises proposition constant open intervals depend choice finite generating sets hence also particular hgi infinite order taking infinite least linear divergence undistortion cont cocycle superrigidity hgi infinite order subadditive superadditive definition let finitely generated infinite group say slow growth distortion property abbreviated sdt property exists hgi element also say sdt property sdt element record following facts later use proposition function continuous except atrmost countably many points every let functions exist generally grows log proof proof left exercises prove observe hence prove proposition hence apply part proof similar proof paper mainly interested case hgi element later use danger confusion would simply write respectively hgi respectively undistorted elements translation numbers give explicit examples groups sdt property let first look special case exists element grows linearly general recall following definition definition let finitely generated infinite group finitely generated subgroup say undistorted exists constant another way characterize undistorted elements use translation numbers recall fix finite generating set let element infinite order every limit exists chung yongle jiang equals inf theorem translation number respect defined section elementary properties translation numbers established lemma element called undistorted element every finite generating set definition say finitely generated group satisfies property exists element finite generating set every every clear group property iff undistortion element every finite generating set following proposition proposition let finitely generated infinite group finite symmetric generating set element infinite order following statements equivalent hgi undistorted undistorted element exists positive constant exist positive constants exists positive constant exists positive constants proof proposition deduce take lim sides hence clear lemma let finitely generated infinite group sdt property subgroup generated elements sdt property subgroup generated elements undistorted infinite normal subgroup proof definition sdt element infinite order hence infinite element hence finite generating set hence sdt property respectively undistorted property normal subgroups continuous cocycles homoclinic equivalence relation specification property shifts continuous cocycles homoclinic equivalence relation let subshift finite set finite symmetric generating set write word length induced following function called continuous exist every divergence undistortion cont cocycle superrigidity every cocycle continuous continuous every say two continuous cocycles cohomologous holds continuous map let comment definition general two metric spaces function continuous positive constants holds definition depends specific metric rather topology satisfactory dealing subshift thus need meaningful definition continuous map independent specific metric definition given choosing sup one check definition reduced one note definition continuous cocycles different associated may different would use denote homoclinic equivalence relation subshift let continuous map every define maps lemma assume sdt element complete compatible metric ginv maps defined lim lim every well defined satisfy cocycle condition every furthermore exists every every chung yongle jiang every constant appeared defining continuity proof every every one hence thus every one thus cauchy sequence since sdt property complete well defined choose also get last inequality similarly also specification property shifts proof theorem need following version specification property subshift one key novelty paper since handle ginv continuous cocycles need generalize version specification used schmidt approach definition involves cone structure defined using euclidean structure extend definition general groups actions need suitable counterpart euclidean structure note need degenerate cone line structure defining specification property used definition let finitely generated group finite generating set let element write every define xak divergence undistortion cont cocycle superrigidity say equivalence class point strong respectively respectively dense exist constants following property satisfy find element say homoclinic equivalence relation strong respectively exists point strong respectively remark definition one take different ratio long less would clear proof lemma later use would take sdt undistorted element record following lemmas later use lemma let element infinite order proof suppose rji hence therefore lemma let full shift sdt element strong element every coordinate constant proof proof similar proof lemma use lemma instead lemma proof theorem recall main steps proof theorem apply lemma corollary lemma lemma lemma lemma successively section prepare corresponding lemmas setting proofs natural modification ones completeness record main changes lemma let group finite symmetric generating set assume one end divergence function sdt elements cfg cfh continuous cocycle subshift ginv proof first since passing subsequence may assume large enough chung yongle jiang may find possible since note following facts may assume increasing sequence passing subsequence since also hence min since sdt elements lemma definition need show lim lim since write let since one end find path avoids ball radius min connect write divx following equality deduce note min hence following estimation kcr divx since max max constants appeared defining continuity remark one also use gersten definition divergence functions proof still assuming growing condition functions relation function one used paper see lemma divergence undistortion cont cocycle superrigidity similar corollary following lemma lemma let group finite symmetric generating set assume one end sdt element cfg cfg continuous cocycle subshift ginv lemma let subshift continuous ginv assume point element sdt property cocycles lemma equal lim denotes diagonal proof fix fix suppose sufficiently small definition implies exists since satisfy deduce following hence constants appeared defining continuous used since sdt property next since deduce following hence thus deduce chung yongle jiang lemma assume exists point dense sdt property lim continuous ginv continuous map map constant undistorted strong cocycles lemma equal taken continuous proof observe proof lemma also works ginv left check continuous map defined continuous assumptions second part proof lemma know since undistorted proposition know hence let estimate proof lemma know constants appeared defining continuity hence therefore let change deduce constants since dense continuous simple density argument implies hence continuous lemma assume one end element sdt property exists point dense lim cfgii positive integer maybe infinity continuous map map constant depending continuous cocycle ginv assume undistorted strong taken continuous divergence undistortion cont cocycle superrigidity proof proof proof lemma replacing lemma lemma lemma lemma respectively lemma statement lemma except replace continuous continuous everywhere proof proof still works since change transfer map using lemmas finish proof theorem corollary proof theorem proof proof theorem still works note need use lemma instead lemma proof corollary lemma apply theorem full shifts examples groups remarks besides full shifts give another class subshifts satisfying assumptions theorem example generalized golden mean subshifts let finite subsets let subset consisting every exists gfj xgj subshift using argument proof lemma also obtain strong sense definition every element infinite order hence similar proof corollary know generalized golden mean subshifts examples theorem remark main theorem drop condition specification result hold even special case see example example investigate class groups undistorted elements growing divergence functions giving examples groups growing divergence functions let review definitions wide groups unconstricted groups thick groups let connected locally connected topological space point global least two connected components definition call finitely generated group unconstricted one asymptotic cones say finitely generated group wide none asymptotic cones clearly wide groups unconstricted stallings ends theorem know unconstricted groups groups recall finitely generated subgroup finitely generated group undistorted word metric equivalent word metric restricted chung yongle jiang definition algebraic network subgroups let finitely generated group finite collection subgroups let say network respect subgroups finitely generated undistorted finite index subgroup finite generating set contained two subgroups thickly connected sense exists finite sequence subgroups infinite definition algebraic thickness let finitely generated group called algebraically thick order zero unconstricted say thick order respect finite collection subgroups network respect subgroups algebraically thick order said algebraically thick order respect smallest value statement holds algebraically thick order algebraically thick order respect algebraic thickness property depend word metric let metric space embedding connected subset every subset denote dist subset called connected two points connected path say two points connected definition tight algebraic network subgroups say finitely generated group algebraic network respect collection subgroups finite index subgroup finite generating set contained two subgroups exists finite sequence subgroups intersection infinite connected definition strong algebraic thickness let finitely generated group strongly algebraically thick order zero wide given say algebraically thick order respect finite collection subgroups algebraic network respect subgroups strongly algebraically thick order divergence undistortion cont cocycle superrigidity say strongly algebraically thick order respect smallest value statement holds strongly algebraically thick order strongly algebraically thick order respect strongly algebraic thickness property depend word metric list examples growing divergence functions linear divergence proposition every group wide linear divergence examples wide groups cyclic groups satisfying law wide corollary law word letters group satisfies law whenever replaced every set elements solvable groups uniformly amenable finitely generated groups groups satisfying law corollary cyclic groups whose center contains wide theorem products arbitrary infinite groups wide example page note although example page mentioned groups unconstricted indeed get actually wide let irreducible lattice lie group higher rank assume either form sln finite set valuations number field including infinite valuations associated ring wide proposition theorem page last paragraph let connected semisimple lie group finite center nontrivial compact factors let uniform lattice hence asymptotic cone equivalent asymptotic cone conditions quotient space maximal compact subgroup symmetric space noncompact type euclidean factor thus applying theorem get every asymptotic cone euclidean building rank every two points asymptotic cone belong flat therefore every asymptotic cone cut points hence thus wide thompson groups corollary polynomial growth divergence finitely generated group strongly algebraic thick order divergence polynomial degree corollary examples strongly algebraic thick groups note although authors proved chung yongle jiang groups algebraic thick proofs also imply actually strongly algebraic thick groups mapping class groups mcg orientable surface genus punches satisfying theorem aut theorem growth corollary authors constructed groups whose divergences however groups torsion hence undistorted elements word hyperbolic groups examples groups least exponential divergence see theorem many groups satisfy property list finitely generated abelian groups lemma heisenberg group biautomatic groups proposition geometrically finite hyperbolic groups biautomatic theorem mapping class group mcg oriented surface genus punches theorem outer automorphism group finitely generated free group theorem semihyperbolic groups iii lemma note biautomatic groups semihyperbolic page let irreducible uniform lattice semisimple linear lie group bicombable semihyperbolic theorem page indeed following remark theorem prove every irreducible uniform lattice connected semisimple lie group finite center also semihyperbolic sln examples let closed symplectically hyperbolic manifold denote ham group hamiltonian diffeomorphisms every finitely generated subgroup ham undistorted elements theorem remark thompson groups proposition corollary theorem note thompson group arguments know class groups undistorted elements growing divergence functions contain class class class wide groups undistorted elements class groups strongly algebraic thick order following groups class let connected semisimple higher rank lie groups finite center nontrivial compact factors let irreducible uniform lattice wide undistorted elements divergence undistortion cont cocycle superrigidity groups heisenberg group infinite sln thompson groups see observe finite index subgroup one end linear divergence following holds lemma let subgroup inclusion finitely generated groups inclusion map property also property class includes following groups aut mapping class groups mcg closed orientable connected surface genus applying section know aut mcg property every action fixed point consequence property hence applying theorem page stallings end theorem know groups corollary ask following question question finitely generated groups whose divergences undistorted elements class groups also contains artin groups let finite simplicial graph vertex set artin group raag associated group presentation connected edge connected always polynomial divergence corollary furthermore least vertices one end connected theorem corollary projects onto number vertices adding commuting relations generators generator length greater equal length thus undistorted therefore least vertices connected raag group belongs similarly certain coxeter groups racgs also belong class given finite simplicial graph vertices associated racg group presentation connected edge graph define associated graph follows embedded loops length four vertices two vertices connected edge corresponding share pair chung yongle jiang adjacent edges given subgraph define support collection vertices appearing corresponding vertices graph said cfs component full support exists component whose support entire vertex set point vertex separating complement connected connected separating vertices edges theorem furthermore join complete bipartite graph cfs polynomial divergence theorem besides finite exist two vertices say connected edge element undistorted applying theorems corollary since unaware criterion classify groups subexponential divergence function property want end paper following question question let finitely generated amenable group virtually cyclic divergence function finitely generated infinite group property hyperbolic references translation lengths geom dedicata alonso brady cooper ferlini lustig mihalik shapiro short notes word hyperbolic groups group theory geometrical viewpoint trieste world sci river edge edited short behrstock charney divergence quasimorphisms artin groups math ann behrstock divergence thick groups short conjugators illinois math behrstock mosher thick metric spaces relative hyperbolicity quasiisometric rigidity math ann bleak bowman gordon lynch graham hughes matucci sapir centralizers thompson group groups geom dyn brady meier connectivity infinity right angled artin groups trans amer math soc bridson haefliger metric spaces curvature grundlehren der mathematischen wissenschaften fundamental principles mathematical sciences vol berlin burillo embedded subgroups thompson group algebra burillo cleary stein taback combinatorial metric properties thompson group trans amer math soc calegari freedman distortion transformation groups geom topol appendix yves cornulier chung jiang continuous cocycle superrigidity shifts groups one end math ann divergence undistortion cont cocycle superrigidity cohen continuous cocycle superrigidity full shift finitely generated torsion group arxiv culler vogtmann criterion property proc amer math soc dani thomas divergence coxeter groups trans amer math soc davis geometry topology coxeter groups london mathematical society monographs series vol princeton university press princeton mozes sapir divergence lattices semisimple lie groups graphs groups trans amer math soc sapir spaces asymptotic cones groups topology appendix denis osin mark sapir epstein cannon holt levy paterson thurston word processing groups jones bartlett publishers boston farb combing lattices semisimple lie groups pusan gruyter berlin farb lubotzky minsky phenomena mapping class groups duke math franks handel distortion elements group actions surfaces duke math furman popa cocycle superrigidity theorem int math res imrn art gersten quadratic divergence geodesics cat spaces geom funct anal gersten short rational subgroups biautomatic groups ann math golan sapir divergence thompson groups preprint gromov hyperbolic groups essays group theory math sci res inst vol springer new york asymptotic invariants infinite groups geometric group theory vol sussex london math soc lecture note vol cambridge univ press cambridge katok rigidity higher rank abelian group actions volume cambridge tracts mathematics vol cambridge university press cambridge katok schmidt cohomology expansive automorphisms compact abelian groups pacific math katok spatzier first cohomology anosov actions higher rank abelian groups applications rigidity inst hautes sci publ math kleiner leeb rigidity symmetric spaces euclidean buildings inst hautes sci publ math continuous orbit equivalence rigidity appear ergod dyn sys meier geometric invariants artin groups proc london math soc osin sapir lacunary hyperbolic groups geom topol appendix michael kapovich bruce kleiner popa cocycle orbit equivalence superrigidity malleable actions groups invent math superrigidity malleable actions spectral gap amer math soc chung yongle jiang polterovich growth maps distortion groups symplectic geometry invent math schmidt cohomology shifts finite type pacific math serre trees york translated french john stillwell walters introduction ergodic theory graduate texts mathematics vol new chung department mathematics sungkyunkwan university suwon korea tel address phuchung yongle jiang department mathematics sungkyunkwan university suwon korea address yongleji
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designing distributed consensus protocols linear systems directed graphs may zhao yongfang liu guanrong chen abstract technical note addresses distributed consensus protocol design problem systems general linear dynamics directed communication graphs using motion planning approaches class distributed consensus algorithms developed rely sampling information sampling instants linear systems proposed algorithms solve consensus problem directed graph containing directed spanning tree particular settling time according task requirements compared existing results systems best knowledge solve fixedtime consensus problems general linear systems directed graphs directed spanning tree extensions formation flying studied multiple satellites described hill equations index terms consensus linear system directed graph settling time directed spanning tree ntroduction past years coordinative control problems systems great interest various scientific engineering communities due broad applications fields spacecraft formation flying distributed sensor network automated highway systems forth compared traditional monolithic systems coordination control reduces systems cost breaches size constraints prolongs life span systems one interesting important issue arising coordination control systems design distributed protocols based local relative information guarantee states agents reach agreement known consensus problem according convergence rate existing consensus algorithms roughly categorized two classes namely asymptotic consensus consensus consensus control problem key task design appropriate distributed controllers usually called consensus protocols due practical engineering requirements networked agents designing consensus protocols hot research topic area consensus problem zhao liu school automation northwestern polytechnical university shaanxi china liuyongfangpku chen department electronic engineering city university hong kong kowloon hong kong gchen may draft previous works study consensus problem firstly shown systems signed gradient flows differential function discontinuous algorithms used since variety consensus algorithms proposed solve consensus problem different scenarios see references therein directed graphs spanning tree consensus protocol designing problems studied class consensus protocols systems given consensus problems multiple nonlinear systems settling time estimation settling time functions depend initial states agents prohibits practical applications knowledge initial conditions unavailable advance recently authors present novel class nonlinear consensus protocols networks called consensus assumes uniform boundedness settling time regardless initial conditions also multiple linear systems formation problems studied however works derived multiagent systems undirected topologies significant challenging solve consensus problem linear systems directed topologies motivated observations using motion planning approaches technical note investigates consensus problem general linear systems directed graphs main contributions technical note summed following aspects firstly using motion planning approaches novel framework introduced solve fixedtime consensus problems framework general linear systems considered technical note novel class distributed protocols designed solve consensus problems particular settling time according task requirements secondly communication topologies among networked agents technical note directed contains directed spanning tree best authors knowledge first time solve consensus problems general linear systems directed graphs finally protocols designed technical note based sampling measurements relative state information among neighbors greatly reduces cost network communication notations let sets real numbers real matrices respectively represents identity matrix dimension denote column vector entries equal one matrix inequality means positive definite denote kronecker product matrices vector let kxk denote set represents number elements otation reliminaries let set real matrices superscript means transpose real matrices represents identity matrix dimension denotes identity matrix appropriate dimension let denote vector entries equal one denote real part diag represents matrix matrices diagonal kronecker product matrices may draft defined amn satisfies following properties systems agents directed graph developed model interaction among agents vertex set edge set edge ordered pair vertices means agent receive information agent directed edge defined parent node defined child node neighbors node denoted neighbors number agent directed tree directed graph every node except root exactly one parent spanning tree directed graph directed tree formed graph edges connect nodes graph say graph contains spanning tree subset edges forms spanning tree adjacency matrix associated defined aij node adjacent node aij otherwise laplacian matrix graph associated adjacency matrix given lij lii aij lij iii istributed fixed time consensus control linear multi agent systems directed graphs section distributed consensus problem linear systems directed graphs studied consider systems agents general linear dynamics may regarded linearized model nonlinear systems dynamics agents networks described axi bui xin state agent control input constant matrices compatible dimensions controllable assumption suppose graph communication topology directed spanning tree lemma assume directed graph spanning tree zero simple eigenvalue eigenvector nonzero eigenvalues open right half plane lemma let mij stochastic matrix represents set real matrices eigenvalue algebraic multiplicity equal may draft one eigenvalues satisfy sia satisfies furthermore element nonnegative definition consensus problem systems said solved finite settling time states systems satisfy lim kxi initial conditions technical note assumed relative measurements information used develop distributed consensus protocols moreover ith agent obtain consensus error neighborhood objective technical note design distributed control law based relative information states agents networks reach consensus directed graphs order achieve control objective technical note following protocol proposed sampling time sequence settling time according task requirements consensus protocol designed considering following hamiltonian function pti axi bui represents costate hamiltonian function cost function proposed follows may draft seen initial terminal times respectively according pontryagin principle necessary condition optimality written axi bui besides according extremal condition one therefore determination optimal control boiled computing substituting one gets integrating equation form follows designing terminal condition follows one invertible followed therefore time sequence one distributed protocol summarize far control protocol states linear systems derived say distributed controller states agent systems derived average state neighbors intuitional point view enough times motion planning steps states agents systems achieve consensus may draft note proposed protocol exists invertible thus moving following lemma given lemma invertible controllable proof follows abb first proof sufficiency given follows let abb abb tbb reductio assume singular thus least one nonzero vector makes taking derivatives equation order respect time setting get abb abb simplifying equations abb let may abb draft follows matrix linearly dependent note controllable controllable contradicts condition controllable thus proved nonsingular invertible similarly proof necessity given reductio assumed uncontrollable thus least one nonzero vector makes follows abb obtained contradicts condition invertible thus proved controllable proof completed remark lemma one proposed protocol exists controllable fundamental requirement control linear systems remark existing works usually design consensus protocols using smallest real part nonzero eigenvalues laplacian matrix associated communication graph researchers used adaptive control approaches overcome requirements nonzero eigenvalues laplacian matrix technical note proposed protocol require knowledge laplacian matrix associated communication graph following theorem provides main result technical note theorem suppose assumption holds settling time distributed sampling protocol solve consensus problem linear system controllable proof first prove sampling time series states systems achieve consensus substituting one gets axi integrating one gets may draft let xtn follows diag assumption directed graph spanning tree thus stochastic matrix according lemma one gets eigenvalue algebraic multiplicity equal one eigenvalues satisfy thus followed lemma matrix exist column vector lim besides since one bounded ensures item matrix bounded follows lim let one thus lim lim note thus one discrete states achieve consensus exponential rate kxi secondly proof discrete states achieve consensus according one thus lim kxi lim kxi therefore settling time distract states achieve consensus exponential rate may draft finally proof continuous states achieve consensus integrating obtained let thus kxi besides note since length time interval upper bounded bounded furthermore since one controllable follows lemma invertible thus assuming coefficients exists least finite constant coefficient thus rewritten small presents infinitesimal since kxi polynomial rate one according therefore limtk obtained similarly one gets therefore kxi sum protocol linear systems achieve consensus fixed settling time proof completed may draft remark undirected graphs consensus problems investigated interesting papers directed graphs consensus problem systems solved however algorithms difficult develop solving consensus problem systems directed graphs technical note using motion planning approaches designed protocol successfully solves consensus problems general linear systems directed graphs remark compared existing works consensus problems article settling time according task requirements realizes consensus state space also controls settling time time axis remark worth mentioning protocols designed technical note based sampling measurements relative state information among neighbors greatly reduces cost network communication remark connected undirected graphs consensus algorithms studied special case result theorem paper directed graphs containing directed spanning tree consensus problems systems dynamics harmonic oscillators solved technical note pplication spacecraft formation flying section application preceding control laws spacecraft formation flying low earth orbit addressed spacecraft formation flying needs precise coordination among multiple spacecraft whose dynamics coupled common control law early pertinent works spacecraft formation flying variable dynamics second integrators linear systems precise nonlinear models launched ren control laws developed spacecraft asymptotically convergent desired formation order simplify analysis section assumed reference orbit circular orbit radius eccentricity reference orbit relative motion spacecraft respect reference orbit described local vertical local horizontal lvlh frame let position vector spacecraft circular reference orbit relative position dynamics spacecraft respect reference orbit written given assume relative orbit radius ith spacecraft reference orbit small compered radius reference orbit linearized equations relative dynamics ith spacecraft respect reference orbit given hill equations uxi uyi uzi may draft natural frequency reference orbit uxi uyi uzi control inputs let uxi uyi uzi thus hill equations one following form spacecraft said achieve formation flying fixed settling time velocity vectors converge value positions maintain prescribed separation terminal time denotes desired constant separation spacecraft given advance based algorithm distributed formation control law spacecraft proposed note controllable thus according theorem following theorem given theorem assume directed topology graph among satellites satisfies assumption settling time tion protocol sampling time sequence solves formation flying problem multiple satellites systems described kri example consider formation flying six spacecraft respect circular reference orbit orbital radius note gravitation constant earth gme universal constant gravity mass earth therefore natural frequency reference orbit spacecraft mass directed communication topology spacecrafts given fig desired formation six satellites maintain regular hexagon separation thus given simplify things let initial time may draft fig communication topology spacecrafts directed graph spanning tree time time fig time positions spacecrafts formation time select initial states spacecraft following depict positions velocities spacecrafts respectively control forces shown fig seen desired formation derived fixed settling time motion trajectories six spacecraft space illustrated fig shows final formation configurations six agents may draft time time time time time time fig velocities spacecrafts fig control signals spacecrafts fig trajectories spacecrafts space may draft fig final formation configurations spacecrafts space form regular hexagon side onclusions technical note studied distributed consensus protocol design problem systems general linear dynamics directed graphs using motion planning approaches class distributed consensus algorithms developed rely sampling information sampling instants linear systems proposed algorithms solve consensus problem directed graph containing directed spanning tree particular fixed settling time offline according task requirements extensions formation flying studied multiple satellites described hill equations future works focus solving distributed consensus problem mobile agents modeled nonlinear dynamics directed switching graphs eferences murray consensus problems networks agents switching topology ieee transactions automatic control ren beard consensus seeking multiagent systems dynamically changing interaction topologies ieee transactions automatic control cao morse anderson agreeing asynchronously ieee transactions automatic control hong chen bushnell distributed observers design control networks automatica duan chen huang consensus multiagent systems synchronization complex networks unified viewpoint ieee transactions circuits systems regular papers ren liu xie distributed consensus linear systems adaptive dynamic protocols automatica liu zhao chen formation tracking control multiple vehicles motion planning approach international journal robust nonlinear control doi chen ren lan chen distributed average tracking reference signals bounded accelerations ieee transactions automatic control zhao liu duan wen distributed average computation multiple signals output measurements international journal robust nonlinear control doi song cao consensus nonlinear systems via pinning control systems control letters chen wang lin adaptive consensus networked mobile agents nonlinear dynamics automatica may draft robust consensus tracking class dynamic systems systems control letters meng lin ren robust cooperative tracking multiple nonlinear systems automatica wang control integrator systems using binary information proceedings chinese control decision conference ccdc changsha china convergent gradient flows applications network consensus automatica hui haddad bhat semistability stability differential inclusions discontinuous dynamical systems continuum equilibria ieee transactions automatic control wang hong consensus networks agent dynamics proceedings world congress ifac pages lin consensus algorithm systems dynamics automatica chen lewis xie distributed consensus via binary control protocols automatica sayyaadi doostmohammadian consensus directed switching network topologies timedelayed communications scientia iranica mauro alessandro alessandro elio consensus disturbance rejection discontinuous local interactions directed graphs ieee transactions automatic control cao ren consensus networks unknown inherent nonlinear dynamics automatica zhang yang zhao consensus tracking harmonic oscillators using state feedback control output feedback control international journal robust nonlinear control zhao duan wen chen distributed tracking system leader bounded unknown acceleration systems control letters zuo tie new class nonlinear consensus protocols systems international journal control zuo nonsingular consensus tracking networks automatica wang consensus tracking systems asian journal control zhao duan wen chen distributed tracking multiple nonlinear systems settling time estimation automatica liu geng formation control linear systems motion planning approach systems control letters horn johnson matrix analysis new york cambridge univesity press godsil royle algebraic graph theory new york springer bryson applied optimal control hemisphere publishing corporation london zheng chen ren cao consensus dynamical systems sampled position data automatica huang duan chen necessary sufficient conditions consensus systems sampled position data automatica wen duan chen consensus systems nonlinear dynamics information approach international journal robust nonlinear control scharf hadaegh ploen survey spacecraft formation flying guidance control part control proceeding american control conference boston massachusetts ren formation keeping attitude alignment multiple spacecraft local interactions journal guidance control dynamics may draft
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jun learning multivariate distributions ilias university southern california diakonik daniel university california san diego dakane alistair stewart university southern california alistais june abstract study problem estimating multivariate probability density functions prove first sample complexity upper bound learning densities prior work upper bound sample complexity learning problem known case detail give estimator draws samples unknown target density outputs hypothesis high probability target total variation distance upper bound sample complexity comes close known lower bound problem introduction background motivation estimation probability density function based observed data classical paradigmatic problem statistics rich history see inference task known density estimation distribution learning informally described follows given set samples unknown distribution believed belong given family want output hypothesis distribution good approximation target distribution first arguably fundamental goal density estimation characterize sample complexity problem minimax sense number samples supported nsf award career sloan research fellowship supported nsf award career sloan research fellowship inherently required obtain desired accuracy expectation high probability words given distribution family desired accuracy interested obtaining estimator sample complexity upper bound lower bound showing estimator achieve accuracy fewer samples sample complexity unsupervised learning problem depends structure underlying family perhaps surprisingly density estimation studied several decades sample complexity learning yet various natural fundamental distribution families emphasize known simple complexity measure distribution family characterizes sample complexity learning unknown distribution total variation distance contrast dimension concept class plays role pac model learning boolean functions see noted classical quantity metric entropy variants bracketing entropy provide upper bounds sample complexity distribution learning tight general specifically upper bounds suboptimal quantitatively qualitatively various distributions families see natural example two main strands research distribution learning first one concerns learnability parametric distribution families mixtures gaussians sample complexity learning parametric families typically polynomial dimension goal design computationally efficient algorithms second strand focus paper studies learnability nonparametric distribution families various assumptions shape underlying density long line work strand within statistics since recently theoretical computer science see section overview related work majority literature studied univariate setting fairly wide range distributions hand multivariate setting specifically regime fixed dimension significantly challenging poorly understood many natural distribution families results comparison prior work work study problem density estimation family distributions distribution logarithm probability density function concave function see definition distributions constitute rich attractive family particularly appealing modeling inference encompass range interesting distributions including uniform normal exponential logistic extreme value laplace weibull gamma chi beta distributions see distributions studied range different contexts including economics statistics roughly speaking metric entropy distribution family logarithm size smallest subset metric space said respect metric every exists paper focus total variation distance distributions probability theory see recent survey theoretical computer science algebra combinatorics geometry problem density estimation distributions central importance area shape constrained inference problem received significant attention statistics literature see references therein recently theoretical computer science section provide detailed summary related work subsection confine describing prior work relevant results paper study following fundamental question many samples required learn arbitrary density total variation distance despite significant amount work density estimation understanding question even constant dimension remains surprisingly poor prior work addresses case finite sample regime specifically study estimation problem respect squared hellinger distance obtain following results sample complexity lower bound sample complexity upper bound tight logarithmic factors specifically prior work finite sample complexity upper bound known even paper obtain sample complexity upper bound total variation distance using known relation total variation squared hellinger distances sample complexity upper bound immediately implies upper bound squared hellinger distance moreover aforementioned lower bound also directly applies total variation distance hence upper bound tight multiplicative factor formally state results need terminology notation definitions let lebesgue measurable function use denote lebesgue measurable function probability density function pdf total variation distance two measures defined dtv sups supremum lebesgue measurable subsets domain probability density functions dtv definition probability density function called exists upper concave function denote set upper logconcave densities respect lebesgue measure use following definition learning total variation distance remark learning model incorporates adversarial model misspecification proposed estimators robust sense definition agnostic distribution learning let family probability density functions randomized algorithm agnostic distribution learning algorithm probability density function input sample access probability algorithm outputs hypothesis density def dtv opt opt inf dtv agnostic learning definition subsumes huber model prescribes target distribution form arbitrary distribution main result paper following theorem theorem main result exists agnostic learning algorithm family densities following performance guarantee probability density function algorithm draws samples probability least outputs hypothesis density def dtv opt opt inf dtv best knowledge estimator provides first finite sample complexity guarantees exception prior work problem provides finite sample guarantees confined case previously mentioned study case general dimension focusing squared hellinger def distance recall squared hellinger distance defined two densities holds dtv therefore sample lower bound also holds total variation distance sample upper bound immediately applies squared hellinger distance implies upper bound tight multiplicative factor proposed estimator establishing theorem robust model misspecification respect total variation distance noted estimator rely maximum likelihood opposed statistics literature problem contrast estimator relies inequality classical result empirical process theory see theorem inequality recently used obtain sharp learning upper bounds wide range distribution families including univariate densities far know first use inequality obtain learning upper bounds structured distributions multiple dimensions remark despite many desirable properties maximum likelihood estimator mle known huber contamination address downside recent work theoretical computer science statistics proposed alternative robust estimators moreover densities conjectured see mle suboptimal sample complexity even without noise facts together provide strong motivation design analysis surrogate estimators desirable properties work densities mle known robust limit weaker metrics related work area nonparametric density estimation shape constraints classical topic statistics starting pioneering work grenander monotone distributions see early recent book topic various structural restrictions studied literature starting monotonicity unimodality concavity majority literature focused univariate setting number works studied nonparametric distribution families fixed dimension see recent years body work computer science nonparametric density estimation focus sample computational efficiency past decade density estimation densities extensively investigated line work statistics obtained complete understanding global consistency properties maximum likelihood estimator mle dimension terms finite sample bounds sample complexity density estimation characterized variation distance moreover known mle univariate setting dimension show mle optimal squared hellinger distance also prove bracketing entropy lower bounds suggesting mle may recent line work theoretical computer science studies case obtains sample computationally efficient estimators total variation distance specifically gave robust estimators distributions among others based inequality technical overview subsection provide overview techniques establishing theorem approach inspired framework introduced given family structured distributions want learn proceed follows find appropriately structured distribution family approximates sense every density total variation distance density choosing family appropriately obtain tight sample upper bounds sample upper bounds estimator achieve goal see lemma leverages inequality aforementioned approach used obtain sampleoptimal computationally efficient estimators various structured distribution families particular family univariate densities chooses family densities interval pieces similarly take family densities piecewise linear interval pieces structural approximation result multivariate case viewed appropriate generalization results specifically show density total variation distance function essentially defined hyperplanes approximation established roughly speaking exploit fact families sets defined small number hyperplanes small dimension allows use inequality learn approximation thus approximation appropriate number samples upper bound dimension resulting set system number samples needed learning task prove structural approximation result densities proceed follows first make use concentration results densities implying negligible fraction probability mass comes points much smaller maximum value allow approximate function takes distinct values furthermore superlevel sets given corresponding superlevel sets convex use results convex geometry approximate convex sets respect volume inscribed polytopes facets applying approximation superlevel set gives function note number constructions possible differ exactly layers constructed overlap many constructions either incorrect difficult analyze work provide simple construction succinct proof yields sample complexity upper bound believe careful structural approximation result may lead tight sample upper bound leave interesting question future work organization section record basic probabilistic analytic ingredients require section prove main result finally conclude open problems section preliminaries def inequality denote let lebesgue measurable function given family measurable subsets define def say set shattered every exists satisfies dimension family sets defined maximum cardinality subset shattered shattered subset size say dimension let probability density function empirical distribution fbn corresponding independent probability measure samples drawn defined inequality states following theorem inequality let probability density function fbn empirical distribution obtained drawing samples let family subsets dimension fbn universal constant approximation convex sets polytopes large literature approximating convex sets polytopes see surveys make essential use following theorem provides volume approximation inscribed polytope bounded number facets theorem let convex body sufficiently large exists convex polytope facets vol vol universal constant proof theorem prove theorem make essential use following general lemma establishing existence estimator using inequality lemma let family probability density functions suppose exists family subsets following holds pair densities dtv exists agnostic learning algorithm error guarantee opt succeeds probability using samples proof lemma implicit include proof completeness estimator extremely simple correctness relies inequality draw samples output density objective function fbn show estimator agnostic learning algorithm let argmin dtv opt dtv note pair densities collection subsets dtv theorem markov inequality follows probability least samples drawn fbn straightforward suffices solve optimization problem additive error conditioning event dtv dtv dtv opt hka opt fbn fbn opt fbn since since fbn fbn opt fbn opt dtv fbn opt opt completes proof lemma view lemma prove theorem establish following proposition exists family sets whose dimension satisfies condition lemma pair densities holds dtv lemma proposition together imply exists agnostic learner sample complexity gives theorem main part section devoted proof proposition proof overview proof two main steps first step define appropriately structured family functions arbitrary density function specifically function takes log distinct values sets union intersections many halfspaces produce family sets dtv gkad dimension yields desired result proceed details proof proposition start formally defining family functions definition given let set functions following form def set log def let intersection halfspaces given define function max furthermore assume asymptotic constants used defining sufficiently large ready state prove first important lemma lemma exists proof function denote def superlevel sets note since convex set define desired approximation natural way constructing appropriate polyhedral approximations superlevel sets finite set geometric series ratio concretely given define function def follows set denote maximum value consider collection convex sets apply theorem approximate set polytope appropriate number facets theorem applied prescribes exists polytope intersection many halfspaces vol vol defines function remains prove first point definition must case condition rabove therefore prove lemma suffices show start noting vol vol similarly denote vol following claim establishes contribution small claim holds vol points proof assume without loss generality attains maximum value let notice vol vol vol vol used fact since hence vol moreover claim indeed therefore equivalently hence vol recall definition choose sufficiently large asymptotic constants holds following sequence inequalities vol vol change variable assumption since using definition completes proof claim establish following crucial claim claim vol vol def proof recall since equivalently write follows min claim indeed write second third equalities follow thus vol vol vol vol first inequality implied second inequality follows fact whenever consider index definition recalling obtain therefore vol vol vol vol vol second inequality implied third inequality uses fact fact whenever completes proof claim ready complete proof following vol vol vol vol claim vol claim proof lemma complete proceed define family subsets bound dimension particular define family sets exactly express differences two elements definition define family sets collection sets form notice dtv show following lemma lemma dimension furthermore sufficiently small constant dtv kad proof note determined completely log values log halfspaces used define convex polytopes show defined values another set halfspaces possible determine whether based solely relative ordering halfspaces belongs consider arbitrary set points wish bound number possible distinct sets obtained intersection set intersection determined relative ordering elements given intersections halfspaces defining note number orderings question formally write halfspaces similarly appear definition respectively following claim exist different set functions set given furthermore functions distributive intersection proof note given terms halfspaces equivalently written follows note viewed function halfspaces expression depends relative ordering thus express one functions halfspaces since functions defined using unions intersections differences distribute intersections halfspace number possible intersections set size claim halfspaces different intersections halfspaces different therefore number possible intersections element exp log log hand dimension must least therefore log log therefore claim comparing variation distance kad note lemma chosen sufficiently small exist dtv dtv dtv dtv dtv dtv kad dtv dtv kad completes proof lemma proof proposition theorem complete conclusions paper gave first sample complexity upper bound learning densities upper bound agrees previously known lower bound multiplicative factor sample complexity upper bound previously known problem result step towards understanding learnability densities multiple dimensions number interesting open problems remain outline two immediate ones optimal sample complexity density estimation plausible conjecture correct answer total variation distance believe sophisticated version structural approximation results could give upper bound hand seems likely adaptation construction could yield matching lower bound polynomial time algorithm function sample complexity learn densities estimator underlying work lemma previously exploited obtain computationally efficient learning algorithms fact running sample time obtaining computationally efficient algorithm case general dimension challenging important open question references acharya diakonikolas hegde schmidt fast nearoptimal algorithms approximating distributions histograms proceedings acm symposium principles database systems pods pages acharya daskalakis kamath optimal testing properties distributions nips acharya diakonikolas schmidt density estimation time proceedings annual symposium discrete algorithms soda pages available https probability distributions theory statistical testing technical report economics working paper archive wustl washington university louis bagnoli bergstrom probability applications economic theory baraud birge shape restricted density estimation stochastic processes applications barlow bartholomew bremner brunk statistical inference order restrictions wiley new york biau devroye risk estimates block decreasing densities journal multivariate analysis balabdaoui doss inference mixture symmetric distributions appear journal bernoulli available http blumer ehrenfeucht haussler warmuth learnability dimension journal acm estimating density order restrictions nonasymptotic minimax risk annals statistics risk histograms estimating decreasing densities annals statistics bronstein approximation convex sets polytopes journal mathematical sciences brunk estimation parameters restricted inequalities annals mathematical statistics canonne diakonikolas gouleakis rubinfeld testing shape restrictions discrete distributions stacs pages chan diakonikolas servedio sun learning mixtures structured distributions discrete domains soda pages chan diakonikolas servedio sun efficient density estimation via piecewise polynomial approximation stoc pages chan diakonikolas servedio sun density estimation time using histograms nips pages chen samworth smoothed maximum likelihood estimation applications statist sinica cule samworth stewart maximum likelihood estimation density journal royal statistical society series daskalakis kamath tzamos clt poisson multinomials applications proceedings annual acm symposium theory computing stoc daskalakis diakonikolas donnell servedio tan learning sums independent integer random variables focs pages daskalakis diakonikolas servedio learning distributions via testing soda pages daskalakis diakonikolas servedio learning poisson binomial distributions stoc pages devroye nonparametric density estimation view john wiley sons diakonikolas hardt schmidt differentially private learning structured discrete distributions nips pages diakonikolas kane stewart efficient robust proper learning distributions arxiv report diakonikolas kane stewart fourier transform poisson multinomial distributions algorithmic applications proceedings stoc diakonikolas kane stewart optimal learning via fourier transform sums independent integer random variables proceedings conference learning theory colt pages full version available https diakonikolas kane stewart properly learning poisson binomial distributions almost polynomial time proceedings conference learning theory colt pages full version available https devroye lugosi springer combinatorial methods density estimation dumbgen rufibach maximum likelihood estimation density distribution function basic properties uniform consistency bernoulli dumbgen samworth schuhmacher approximation distributions applications regression ann doss wellner global rates convergence mles logconcave densities ann estimation unimodales canadian journal statistics groeneboom jongbloed nonparametric estimation shape constraints estimators algorithms asymptotics cambridge university press gordon meyer reisner volume approximation convex sets polytopes constructive method stud gordon meyer reisner constructing polytope approximate convex body geometriae dedicata grenander theory mortality measurement skand groeneboom estimating monotone density proc berkeley conference honor jerzy neyman jack kiefer pages gruber aspects approximation convex bodies handbook convex geometry hanson pledger consistency concave regression annals statistics huber robust estimation location parameter ann math han wellner approximation estimation densities via renyi divergences ann jankowski wellner estimation discrete monotone density electronic journal statistics koenker mizera density estimation ann kim samworth global rates convergence density estimation ann available http kearns vazirani introduction computational learning theory mit press cambridge vempala geometry logconcave functions sampling algorithms random structures algorithms pearson contributions mathematical theory evolution skew variation homogeneous material philosophical trans royal society london prakasa rao estimation unimodal density sankhya ser scott multivariate density estimation theory practice visualization wiley new york silverman density estimation chapman hall london stanley unimodal sequences algebra combinatorics geometry annals new york academy sciences seregin wellner nonparametric estimation multivariate convextransformed densities ann saumard wellner strong review statist tsybakov introduction nonparametric estimation springer publishing company incorporated vapnik chervonenkis uniform convergence relative frequencies events probabilities theory probab van der vaart wellner weak convergence empirical processes springer series statistics new york applications statistics valiant valiant instance optimal learning discrete distributions proceedings annual acm symposium theory computing stoc pages walther inference modeling distributions stat science wegman maximum likelihood estimation unimodal density ann math wellner nonparametric estimation densities alternative maximum likelihood talk given european meeting statisticians amsterdam available https
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dynamic algorithms graph coloring sayan deeparnab monika nov danupon abstract design fast dynamic algorithms proper vertex edge colorings graph undergoing edge insertions deletions static setting simple linear time algorithms vertex coloring coloring graph maximum degree natural ask efficiently maintain colorings dynamic setting well get following three results present randomized algorithm maintains coloring log expected amortized update time present deterministic algorithm maintains coloring polylog amortized update time present simple deterministic algorithm maintains coloring log update time improves recent coloring algorithm update time corresponding author university warwick email dartmouth college usa email university vienna austria email kth sweden email danupon contents introduction techniques dynamic vertex coloring overview randomized algorithm overview deterministic algorithm randomized dynamic algorithm vertex coloring preliminaries maintaining hierarchical partition recoloring subroutine complete algorithm analysis deterministic dynamic algorithm vertex coloring notations preliminaries algorithm bounding amortized update time deterministic dynamic algorithm edge coloring extensions case changes time randomized vertex coloring deterministic vertex coloring deterministic edge coloring open problems acknowledgements references problems introduction graph coloring fundamental problem many applications computer science proper coloring graph assigns color every node way endpoints every edge get different colors chromatic number graph smallest proper coloring exists unfortunately computational perspective approximating chromatic number rather futile constant polynomial time algorithm approximates chromatic number within factor graph assuming see stronger bound positive side know chromatic number maximum degree graph simple linear time algorithm find pick uncolored vertex scan colors used neighbors assign color assigned neighbors since number neighbors pigeon hole principle color must exist paper consider graph coloring problem dynamic setting edges graph inserted deleted time want maintain proper coloring every update objective use colors possible keeping update small specifically main goal investigate whether coloring maintained small update time note greedy algorithm described previous paragraph easily modified give update time edge inserted two nodes color scan neighbors find free color natural question whether get algorithm significantly lower update time answer question affirmative design analyse randomized algorithm maintains coloring log expected amortized update difficult see colors would simple randomized algorithm amortized update time see section details challenging result maintain coloring small update time contrast randomization allowed even maintaining time seems second result deterministic vertex coloring algorithms although achieve coloring come close design analyse deterministic algorithm maintains coloring polylog amortized update time note dynamic graph maximum degree change time results hold changing well however ease explaining main ideas restrict paper setting static upper bound known algorithm section point changes needed make algorithms work case final result maintaining edge coloring dynamic graph maximum degree proper edge coloring coloring edges two adjacent edges color design analyze simple deterministic coloring algorithm log update time upon recent coloring algorithm barenboim maimon needs update time two notions update time amortized update time algorithm amortized update time insertions deletions total update time worst case update time algorithm worst case update time every update time typical amortized update time guarantees assume input graph empty initially typically done randomized dynamic algorithms assume adversary fixes sequence edge insertions deletions oblivious randomness algorithm perspective important aspect coloring following property consider graph problem need assign state color node say constraint local node defined states neighbors say problem iff following three properties local constraint every node solution feasible iff satisfies local constraint every node iii local constraint node unsatisfied change state satisfy without creating new unsatisfied constraints nodes example coloring define constraint local satisfied color different neighbors always find recoloring satisfy local constraint introduce constraint violations hand following problems seem globally optimum coloring best approximation algorithm coloring always exists brook theorem unless graph clique odd cycle coloring always exists vizing theorem observe start feasible solution problem inserting deleting edge need change states obtain new feasible solution instance case coloring need recolor nodes incident inserted edge thus number changes guaranteed small main challenge search changes efficient manner without scan whole neighborhood contrast problems main challenge seems analyzing many nodes edges need recolor even inefficient algorithm keep coloring proper question spirit recently studied shown coloring problem also see appendix given current results coloring coloring inviting ask whether deeper connections exist designing dynamic algorithms problems particular reductions possible among problems find complete problem also interesting understand power randomization problems indeed distributed computing literature deep extensive work beyond problems fact shown problem slocal complexity class studied distributed computing see appendix coincidentally like findings paper still big gap deterministic randomized distributed algorithms coloring details refer excellent monograph barenboim elkin see references therein recent results finally also note dynamic problems focused search problems solutions always exist hard part find maintain posts new challenge comes proving conditional lower bounds dynamic algorithms problems large body work devoted decision problems seems adapt existing techniques search problems related work dynamic graph coloring natural problem many works papers however proposed heuristics described experimental results two theoretical papers aware already mentioned organisation rest paper section give high level ideas behind vertex coloring result particular section contains main ideas randomized algorithm whereas section contains full details similarly section contains main ideas deterministic algorithm whereas section contains full details section contains result emphasize sections completely self contained read independently mentioned earlier sections assume parameter known maximum degree never exceeds solely better exposition main ideas algorithms easily modify give results current maximum degree see section details techniques dynamic vertex coloring overview randomized algorithm present high level overview randomized dynamic algorithm coloring log expected amortized update time full details found section start couple sketching main idea warmup maintaining expected amortized update time first observe maintaining coloring easy using randomization oblivious adversary need expected amortized time algorithm let palette colors vertex stores last time recolored edge gets deleted edge gets inserted two vertices different colors nothing next consider scenario edge gets inserted time two vertices color without loss generality suppose vertex recolored last event scan neighbors store colors used set select random color since well clearly leads proper coloring since new color design current colors neighbors time taken compute set high since neighbors let analyze probability insertion edge time leads conflict suppose time recolored color insertion time creates conflict chose color time however probability event since least choices choose color time therefore expected time spent addition edge analysis described crucially used fact insertion edge time oblivious random choice made recoloring vertex time also clear constant sacrosanct coloring obtained amortized time however fails give even coloring time constant warmup simple algorithm coloring difficult analyze previous algorithm recoloring vertex made sure never assumed color neighbors say color blank vertex iff neighbor gets color since colors every vertex least one blank color however one blank color choose adversarial sequence updates may force algorithm spend time every edge insertion idea would randomly recolor vertex without considering colors neighbors problem recoloring may lead one neighbors vertex unhappy color cascading effect hard control take middle ground define color unique vertex assigned exactly one neighbor thus recolored using unique color cascading effect unhappy vertices explode specifically recoloring need consider recoloring unique neighbor forth idea useful although number blank colors available vertex colors none neighbors using small number colors always least holds since color neither blank unique accounts least two neighbors whereas neighbors observation suggests following natural algorithm need recolor vertex first scan neighbors identify set unique blank colors pick new color uniformly random set definition set one neighbor color neighbor exists recolor recursively using algorithm state three important properties scheme recoloring vertex make one recursive call takes time recolor vertex ignoring potential recursive call neighbor recolor vertex pick new color uniformly random set size note properties served main tools establishing bound expected amortized update time discussed previous algorithm property manage upper bound length chain recursive calls might result insertion edge input graph two vertices color get upper bound overall update time algorithm however trivial fact reader observe necessary colors order ensure three properties hold even colors indeed case algorithm described might never terminate conclude another idea required achieve log update time turns concept hierarchical partition set vertices graph describe present overview final algorithm overview final algorithm fix large constant suppose partition input graph levels following property property consider vertex level vertex neighbors levels least neighbors levels clear first glance even exists partition given static graph seems obvious way assign level vertex satisfying property one main technical contributions present algorithm maintains hierarchical partition satisfying property dynamic graph initially input graph empty place every vertex level trivially satisfies property subsequently every insertion deletion edge algorithm updates hierarchical partition way ensures property continues remain satisfied algorithm deterministic using intricate charging argument show amortized update time log also gives constructive proof existence hierarchical partition satisfies property given graph explain hierarchical partition conjunction ideas warmup leads efficient randomized vertex coloring algorithm algorithm require vertex keeps neighbors levels informed color requirement allows vertex maintain set colors assigned neighbors set remaining colors say color blank iff neighbor color hand say color unique iff exactly one neighbor color note crucial change definition unique color warmup color unique enough exactly one neighbor color addition neighbor lie level strictly level using property hierarchical partition neighbors levels argument similar one used warmup show large number colors either blank unique claim every vertex least colors either blank unique implement template warmup vertex needs recolored picks new color uniformly random set blank unique colors cause vertex unhappy vertex lies level strictly lower log levels bounds depth recursive call level use blank color whenever recolor time needs inform neighbors bounded property hierarchical partition since recursive call done vertex strictly lower level total time spent recursive calls also bounded due geometric sum finally claim time picks random color palette size order edge insertions deletions oblivious randomness probability edge insertion going problematic gives expected amortized time bound overview deterministic algorithm present high level overview deterministic dynamic algorithm coloring amortized update time polylog full details section section start warmup sketching main idea warmup maintaining coloring amortized update time let palette colors partition set equally sized subsets colors colors said type let denote type color assigned node furthermore let denote number neighbors assigned type color refer neighbors type neighbors every node let denote set neighbors every node maintains set doubly linked list note node gets color proper coloring following extra property node type type neighbors property node assigned color initially input graph empty every vertex colored arbitrarily property holds note deletion edge lead violation property make existing coloring invalid discuss edge gets inserted considering three possible cases case nothing done since different types colors case insertion edge colors assigned vertices type event first set nothing color since property continues hold color pick arbitrary endpoint find type color assigned neighbors set possible colors type change color operations take time case insertion edge addition edge vertex violates property run following subroutine recolor since neighbors types must exist type type found linear scan neighbors takes time since neighbors color assigned neighbors color must set choose exist since next update set follows delete every neighbor insert every neighbor similarly update set every neighbor takes time implement step accordingly total time spent call recolor subroutine however property may violated one neighbors case recursively call recolor keep vertices satisfy property end proper coloring vertices satisfying property priori may clear procedure even argue amortized time spent calls recolor subroutine particular chain recursive calls subroutine terminates introduce potential sums vertices number neighbors type note edge inserted deleted potential increase however call recolor potential drops least moves color type told color type tnew dtold dtnew leads drop get amount drop considering edge insertions deletions starting empty graph calls recolor subroutine since call takes time get claimed amortized update time getting polylog amortized update time one way interpret previous algorithm follows think color ordered pair first coordinate analogous notion type defined previous algorithm vertex let denote consisting first coordinates color assigned ease exposition define furthermore every vertex every let denote set neighbors notations property rewritten improve amortized update time polylog think every color tuple whose coordinate take possible values total number colors given values chosen way ensures maintain invariant carefully chosen function implement generalization previous algorithm colors represented tuples using carefully chosen parameters show deterministically maintain vertex coloring dynamic graph polylog amortized update time see section details randomized dynamic algorithm vertex coloring discussed section randomized dynamic algorithm vertex coloring two main components first one hierarchical partition vertices input graph log levels section show maintain hierarchical partition dynamically second component use randomization recoloring conflicted vertex ensure one new conflict caused due recoloring new conflicted vertex lies level strictly lower describe second component section complete algorithm combines two components appears section theorem captures main result theorem randomized fully dynamic algorithm maintain vertex coloring graph whose maximum degree expected amortized update time log preliminaries start definition hierarchical partition let denote input graph changing dynamically let upper bound maximum degree vertex assume value change time section explain relax assumption fix constant simplicity exposition assume say integer vertex set partitioned subsets level vertex index subset belongs vertex two indices let set neighbors whose levels notational convenience define whenever hierarchical partition satisfies following two note since invariant trivially satisfied every vertex highest level invariant hand trivially satisfied vertices level invariant every vertex level invariant every vertex let set possible colors coloring proper graph iff every edge given hierarchical partition coloring vertex level define key subsets let colors used neighbors lying levels let denote remaining set colors say color blank vertex assigned color say color unique exactly one vertex assigned color let respectively denote blank respectively unique colors let denote remaining colors thus every color least two vertices assigned color end section crucial observation claim vertex level proof since get following two observations turn follow definitions prove claim data structures describe data structures used dynamic algorithm first set used maintain hierarchical partition second set used maintain sets colors every vertex every level maintain neighbors level doubly linked list also maintain set neighbors doubly linked list use phrase neighborhood list refer one lists every neighborhood list maintain counter stores number vertices every edge keeps two pointers one position neighborhood list vice versa therefore edge inserted deleted linked lists updated time finally keep two queues dirty vertices store vertices satisfying either two invariants maintain coloring array contains current color every vertex maintains colors doubly linked lists color vertex keep pointer color position either depending list belongs allows add delete colors lists time also maintain counter associated color vertex value equals number neighbors color otherwise set vertex keep time counter stores last time edge recolored changed maintaining hierarchical partition initially graph empty vertices level satisfies invariants vacuously subsequently ensure hierarchical partition satisfies invariants using simple note long number edge insertions deletions polynomial requires log bits store number becomes superpolynomial every rounds recompute full coloring current graph greedy heuristic procedure describe procedure define vertex dirty violates one invariants clean otherwise goal ensure every vertex hierarchical partition remains clean inductive hypothesis assume every vertex clean edge due edge vertices form might become dirty fix dirty vertices per procedure described figure procedure always fix dirty vertices violating invariant fixing dirty vertex violates invariant crucial bounding amortized update time furthermore note change level vertex one iteration hile loop figure might lead change side neighbors hence one iteration hile loop might create multiple new dirty vertices dealt subsequent iterations hile loop hard see iteration loop acting vertex ends making clean encapsulate following lemma subsequent corollary hile invariant invariant violated vertex violates invariant hen find minimum level move vertex level update relevant data structures described proof lemma lse find vertex violates invariant level hen move vertex maximum level update relevant data structures described proof lemma lse move vertex level update relevant data structures figure subroutine called edge inserted deleted lemma consider iteration hile loop figure changes level vertex vertex becomes clean satisfies invariants end iteration furthermore end iteration proof three cases consider depending vertex moves level level case vertex moves level case vertex moves minimum level implies thus vertex satisfies invariants moves level conditions hold case vertex moves level level case steps figure imply maximum level hence vertex satisfies invariants moving level conditions hold case vertex moves level level steps figure imply every level particular setting get thus vertex satisfies invariants moves level conditions hold lemma states given iteration hile loop figure pick dirty vertex make clean process neighbors become dirty handled subsequent terms vertex refer values respectively iterations hile loop hile loop terminates every vertex clean definition remains analyze time spent implementing hile loop edge input graph lemma crucial analysis intuition follows lemma guarantees whenever vertex moves level least contrast invariant figure ensure whenever vertex moves level belowdegree less thus vertex loses least moves level slack help bound amortized update time next bound time spent single iteration hile loop figure lemma consider iteration hile loop figure vertex moves level level steps takes time implement iteration proof first claim value level vertex move identified time explicitly store sizes lists next update lists counters neighbors follows every level every vertex hen time spent operations bounded number vertices since vertex moving level level update position neighborhood lists vertices also need merge lists single list process vertices becomes dirty need put correct dirty queue takes time lemma since constant conclude takes time implement iteration hile loop figure lemma consider iteration hile loop figure vertex moves level level steps takes time implement iteration proof first bound time spent identifying level vertex move since vertex violates invariant know therefore algorithm scan list find required level time next update lists counters neighbors follows every vertex hen hen time spent operations bounded number vertices since vertex moving level level update position neighborhood lists vertices also need split list lists process vertices become dirty need put correct dirty queue takes time figure ensures satisfies invariant level moves lower level thus spend time iteration hile loop corollary takes time vertex move level different level proof corollary follows immediately lemma rest proof suppose case per proof lemma time spent least size list lemma implies hence total time spent also since constant note ignored scenarios since event constant anyway theorem bound amortized update time maintaining hierarchical partition theorem maintain hierarchical partition vertex set satisfies invariants log amortized update time devote rest section proof theorem using token based scheme basic framework follows every edge input graph create tokens use one token perform units computation implies amortized update time log since constant specifically associate many tokens every vertex many tokens every edge input graph values tokens determined following equalities max max otherwise initially input graph empty every vertex level due insertion edge total number tokens increases max levels endpoints edge insertion hand due deletion edge input graph value increases tokens associated edge disappears overall total number tokens increases due deletion edge show work done one iteration hile loop figure proportional times net decrease total number tokens iteration accordingly focus single iteration hile loop figure vertex say moves level level consider two cases depending whether moves higher lower level case vertex moves level level immediately vertex moves level hence follows lemma since always nonnegative value increase moves level focus bounding change total number tokens associated neighbors note event moving level level affects tokens associated vertices specifically infer every vertex value increases hand every vertex value remains unchanged thus total number tokens associated neighbors increases inequality follows lemma summarize total number tokens associated vertices increases focus bounding change total number tokens associated edges incident infer every edge value drops least one vertex moves level level every edge value remains unchanged overall means total number tokens associated edges drops least inequality follows lemma summarize total number tokens associated edges decreases least discussion preceding two paragraphs reach following conclusion vertex moves level level total number tokens associated vertices edges decreases least contrast lemma states takes time taken implement iteration hile loop figure hence derive time spent updating relevant data structures times net decrease total number tokens concludes proof theorem case case vertex moves level level case begin observing immediately vertex moves level hence follows lemma vertex violates invariant moving level level see step figure particular vertex moves level level last inequality holds since sufficiently large constant number tokens associated drops least moves level level also infer value increase moves lower level hence conclude total number tokens associated vertices drops least focus bounding change number tokens associated edges incident infer number tokens associated edge increases max moves level level contrast number tokens associated edge change vertex moves level lower level let increase total number tokens associated edges thus max last equality follows rearrangement next recall vertex moves level level concerned iteration hile loop figure accordingly steps figure implies levels equivalent following statement levels next step figure implies vertex violates invariant level thus get note levels hence get levels combining observation get min levels plugging get derivation last inequality holds since sufficiently large constant reach following conclusion vertex moves level level total number tokens associated vertices edges decreases least contrast lemma takes time implement iteration hile loop figure hence derive time spent updating relevant data structures times net decrease total number tokens concludes proof theorem case recoloring subroutine whenever want change color vertex call subroutine recolor described figure ensure hierarchical partition change call subroutine specifically throughout duration call recolor subroutine value remains every vertex also ensure hierarchical partition satisfies invariants making call recolor subroutine call subroutine recolor randomly choose color vertex subset case random color lies find unique neighbor assigned color recursively recolor since level strictly less maximum depth recursion bound time spent call recolor choose uniformly random notations defined section set update relevant data structures described proof lemma find unique vertex recolor figure subroutine recolor lemma takes time implement one call recolor includes total time spent chain subsequent recursive calls originate call recolor proof let assume call recolor implement step figure vertex scans neighborhood list computes subset colors appear twice among vertices vertex keeps colors separate list deletes every color list completion list consists colors algorithm samples random color next algorithm adds colors back list thereby restoring list actual state algorithm also another scan check whether total time taken invariants changing color vertex step algorithm needs update data structures see section follows note might clear sample time modification required sufficient sample first elements clarity exposition ignore issue every vertex hen operations also take time per invariants finally subsequent recursive calls suppose recolor vertices note therefore total time taken bounded since geometric series sum completes proof complete algorithm analysis initially graph empty every vertex belongs level picks random color point coloring proper since edges invariants vacuously satisfied inductive hypothesis suppose insertion deletion edge guarantee proper coloring invariants satisfied handle insertion deletion edge according procedure figure specifically insertion deletion edge first update hierarchical partition calling subroutine see figure end call subroutine know sure invariants satisfied point check existing coloring proper coloring become invalid edge getting inserted event find endpoint recolored last one larger without loss generality let endpoint change color calling subroutine recolor end call subroutine know sure coloring proper thus apply inductive hypothesis next insertion deletion edge see figure case inserted suppose notation defined section recolor figure dynamic algorithm maintain vertex coloring first bound amortized time spent calls recolor subroutine lemma theorem lemma get main result section stated theorem lemma consider sequence edge starting empty graph let thp respectively denote total time spent calls recolor subroutines edge thp proof since edge deletions lead recoloring need bother edge insertions consider scenario edge inserted graph time without loss generality assume recall last times recolored suppose vertex level immediately updated hierarchical partition following insertion edge time thus point time subroutine recolor called change color endpoint hand point time vertex recolored furthermore suppose vertex level call recolor time analysis done via three cases case case point time interval subroutine raised level vertex corollary implies takes time hand even subroutine recolor called time lemma takes time implement call total time spent calls recolor subroutine thp case case use fact vertex picks random color time particular lemma expected time spent recoloring vertex time event insertion time wish bound probability evaluated past random choices algorithm adversary fixing order edge insertions oblivious using principle deferred decision let respectively denote colors assigned vertices calls subroutines recolor recolor times note event occurs iff condition random choices made algorithm till time since fixes color time vertex picks color uniformly random subset colors let denote size subset time clearly event occurs probability remains lower bound since claim invariant imply vertex gets recolored time summarize expected time spent possible call recolor time hence total time spent calls recolor subroutine case even recolor called time lemma time spent call total time spent calls recolor subroutine proof theorem theorem holds since thp log log first second inequalities respectively follow lemma theorem deterministic dynamic algorithm vertex coloring let denote input graph changing dynamically let upper bound maximum degree vertex assume value change time section explain relax assumption main result stated theorem theorem maintain vertex coloring dynamic graph deterministically amortized update time notations preliminaries throughout section define three parameters follows use colors lemma follows lemma establish couple useful bounds parameters case used trivial upper bound lemma proof infer also infer proves part lemma next note proves part lemma let denote palette colors note indeed view colors available vectors coordinate takes one values particular color assigned vertex denotes given coloring every index define denoting first coordinates notational convenience define let denote whose first coordinates whose ith coordinate define subsets words set consists neighbors vertex whose colors first coordinates color denotes status set event vertex decides change hand set going ith coordinate color particular possible choices get following also note full neighborhood define observations encapsulated following corollary partitioned corollary every vertex every index set subsets particular maintain following invariant invariant every give brief intuitive explanation invariant associate rooted tree depth every vertex shall refer vertices tree distinguish vertices input graph total number leaves tree total number available colors thus ensure root leaf path tree corresponds color natural way internal depth corresponds ith coordinate color quantity interpreted follows consider say depth unique root leaf path corresponding color let denote number neighbors metavertex also belongs root leaf paths corresponding colors note root tree depth zero follows equals degree vertex input graph thus let denote children root tree simple counting argument implies exists index following property fraction neighbors colors whose corresponding root leaf paths contain metavertex thus case root leaf path corresponding color also passes would invariant hand gives slack requires interpret invariant fashion every subsequent index iteratively applying principle reader might find helpful keep interpretation mind going formal description algorithm analysis lemma invariant holds proper vertex coloring proof claim implies invariant reduces since nonnegative integer get since number neighbors assigned color two adjacent vertices get color invariant thus ensures proper vertex coloring claim order prove claim derive step follows part lemma steps hold long step follows step follows part lemma step follows data structures every vertex dynamic algorithm maintains following data structures trivially maintain coloring update time set doubly linked list counter responsibility neighbors update list change colors using appropriate pointers ensure given node inserted deleted given list time note vertex figures satisfies invariant color assigned vertex algorithm initially since input graph empty assign vertex arbitrary color concreteness set since edges invariant vacuously holds point show ensure invariant continues remain satisfied even edge insertion deletion bound amortized update time deletion edge suppose edge gets deleted vertex changes color due deletion need update relevant data structures without loss generality suppose largest index every vertex every delete vertex set decrement value counter one takes time summarize deletion edge handled worst case update time invariant satisfied edge deletion invariant continues remain satisfied edge deletion since lhs invariant decrease insertion edge suppose edge gets inserted graph first update relevant data structures follows let largest index note colors every vertex every insert vertex set increment value counter one takes time worst case next focus ensuring invariant continues hold towards end execute subroutine described figure lemma implies correctness algorithm lemma upper bound time spent given iteration hile loop figure hile vertex violates invariant let smallest index find minimizes set update relevant data structures see lemma figure ensuring invariant edge insertion lemma consider iteration hile loop figure vertex end iteration vertex satisfies invariant proof consider iteration loop steps induction hypothesis suppose iteration due step induction hypothesis holds first iteration loop since corollary minimizes satisfies accordingly executing step get thus induction hypothesis remains valid next iteration loop end loop get hence vertex satisfies invariant time one iteration hile loop figure lemma takes defined per step figure proof consider iteration hile loop changes color vertex since store value every takes time find index defined step figure next every index every vertex set since vertex going change kth coordinate color previous step necessary ensure vertex mistakenly continues include set takes time since time taken actually last equality follows step figure point also set shall rebuild sets loop steps applying similar argument conclude also takes time remains bound time spent loop figure first iteration loop set counter takes time consider iteration loop inductive hypothesis suppose every index every vertex list consistent changes made color earlier iterations loop first goal compute index scan begins every scanning vertices scan set considering vertex zcj end scan return index minimizes value subsequently scan set vertices zcj every vertex second scan ensures every thus takes time find index next every vertex set perform two scans list takes time implement step set time stage updated relevant data structures index changed jth coordinate color total time spent concludes concerned iteration loop since per step figure iteration since loop runs iterations total time spent loop combining discussion first paragraph proof lemma infer total time spent one iteration hile loop also bounding amortized update time section prove theorem bounding amortized update time algorithm described section recall handling deletion edge takes time worst case furthermore ignoring time spent hile loop figure handling insertion edge also takes time worst case lemma thus remains bound time spent hile loop figure focus task remainder section main idea following lemma hile loop processing vertex takes long time small turn implies large however loop choose colors minimize precisely values therefore quantities large often consider iteration hile loop figure changes color vertex let resp set ordered pairs value increases resp decreases due iteration following lemma precisely bounds increases decreases values lemma single iteration hile loop figure proof throughout proof let respectively denote concerned iteration hile loop let smallest index time every index every vertex let respectively denote set vertices time time consider vertex time concerned iteration hile loop changes kth coordinate color vertex change color iteration thus time therefore infer hence every vertex value drops one due concerned iteration hile loop follows get last inequality follows step figure gives desired lower bound size set remains upper bound size set consider ordered pair since concerned iteration hile loop increases value change color vertex infer concerned iteration hile loop change ith coordinate color thus change time time furthermore vertex change color follows also therefore get next note since time accordingly also summarize ordered pair belongs set must thus get gives following guarantee note step figure picks minimizes using upper bound sum geometric series get last inequality holds due step figure gives desired upper bound use potential function based argumentpto prove theorem potential associated graph point time given note potential always nonnegative bound amortized update follows three observations stated observation due insertion deletion edge potential changes proof consider insertion deletion edge input graph value remains unchanged furthermore value changes one hence potential changes observation due one iteration hile loop figure potential decreases least defined per step figure proof note concerned iteration hile loop change color vertex thus every vertex every index value changes one lemma let resp denote set ordered pairs value increases resp decreases due iteration therefore infer net decrease equal first step derivation follows lemma second step follows invariant observation time taken implement one iteration hile loop figure net decrease potential due iteration hile loop proof follows observation lemma proof theorem beginning edge set empty edge insertions deletions total time taken algorithm total time taken loops observation know decrease potential due tth loop hand observation get putting together get total time taken algorithm proves theorem since log log log per part lemma deterministic dynamic algorithm edge coloring let input graph changing dynamically let upper bound maximum degree vertex assume value change time section explain relax assumption present simple deterministic dynamic algorithm maintaining edge coloring algorithm data structures every vertex maintain following data structures array length entry array corresponds color color entry either null points unique edge incident colored bit vector length iff null otherwise balanced binary search tree leaves refer vertices tree metanodes distinguish vertices input graph maintain counter val every tree value counter cth leaf given furthermore value counter internal given val leaf subtree rooted val use notation denote initialization initially graph empty set nulls set counters val set coloring subroutine edge inserted need assign color using following like procedure described figure set hile invariant hen set else hen set assign color edge update following data structures figure color coloring edge inserted claim subroutine color returns proper coloring edge log time proof end hile loop since assume invariant stated loop holds end case implying turn means edge incident either color thus coloring proper show invariant always hold holds beginning since point degree since added implies total sum implies invariant loop makes sure invariant holds subsequently time analysis note log iterations values stored counters val internal trees values obtained time data structures also maintained log time since tree values path root increased full algorithm initially graph empty edge added run color edge deleted color point null set update log time thus following theorem theorem dynamic graph maximum degree deterministically maintain edge coloring log worst case update time extensions case changes time ease exposition whole paper maintained parameter known algorithm front promise maximum degree graph remains times fact algorithms work work changing well maximum degree graph edge insertions deletions fact randomized algorithm maintaining vertex coloring deterministic algorithm maintaining vertex coloring deterministic algorithm maintaining edge coloring running times take slight hit first two algorithms amortized running time polylog maximum degree seen far till time edge coloring worst case update time becomes log following three subsections give brief sketch differences algorithm analysis needs modified cases achieve making requirement algorithm stronger randomized vertex coloring let degree vertex current input graph extend dynamic algorithm section scenario changes time simply ensure following property holds property every vertex color see easy ensure property need make following two observations algorithm maintains hierarchical partition section oblivious value every vertex change definition subset colors follows subset consists colors assigned neighbor maintain modified sets incurring factor cost update time finally note even modified definition claim continues hold hence recolor subroutine figure particular randomized dynamic coloring algorithm general continue remain valid deterministic vertex coloring whenever degt less large constant whenever need change color greedy step taking degt time find free color henceforth assume degt instead fixed every vertex separate parameters depend degree time note maintained insertion deletion time per update degt degt degt time vertex assigned color color assumed tuple entry takes positive integer values note dimension tuple range dimension tuple change time need careful definitions remains except range till invariant maintain every vertex similar invariant changed appropriately add condition time coordinate invariant degt let take care situations part invariant violated part similar done section positive integer define largest value degt evaluates definition parameters since degt get take care part invariant algorithm given figure needs changed step follows degt therefore search degt otherwise search whenever part invariant violated vertex time perform following changes take degt time find color satisfying invariant show time charged edge deletions incident happened past furthermore edge deletions charged firstly note violated part invariant delete edge degt therefore gone note must case third term value edge deletion suppose note degt must since time degt look last time set time modification made algorithm must therefore must least degt edge deletions incident charge degt time taken recolor due violation part invariant deletions note since choose last time charge edge deletions maintain part invariant algorithm similar previous section two changes firstly loop checks new invariant technical change described observe even delete edge run risk invariant getting violated since rhs invariant also goes analysis one line argument everything generalizes analysis fact vertex precisely version lemma replaced degt similar changes claims lemmas brevity mention subscripts instance time analysis lemma loop taking care vertex generalizes degt replacing similarly lemma exactly changes reflects observation decrease potential single loop charged running time loop rest analysis section extra change lemma technical change made get since could searching range increases update time extra factor deterministic edge coloring assert invariant every edge gets color palette max degt degt subroutine color upper bound set degt degt rest algorithm remains trees need dynamically balanced done log time using say trees place algorithm needs change edge deleted degree may lead four edges two incident two incident violating invariant need using color procedure takes log time open problems one obvious open question left work whether maintain coloring polylogarithmic time using deterministic algorithm believe important question since may help understanding develop deterministic dynamic algorithms general challenging interesting design deterministic dynamic algorithms performances similar randomized ones many dynamic graph problems maximal matching connectivity shortest paths another obvious question whether deterministic update time coloring improved try optimize polylog factors hidden inside theorem believe improved however getting log deterministic update time seems challenging also interesting get poly log update time dynamic vertex coloring finally one direction study classes problems slocal every problem admit polylogarithmic update time problems slocal maximal independent set minimal dominating set acknowledgements thank anonymous reviewer proofs lemma claim project received funding european research council erc european unions horizon research innovation programme grant agreement danupon nanongkai also partially supported swedish research council reg research leading results received funding european research council european union seventh framework programme erc grant agreement references amir abboud virginia vassilevska williams popular conjectures imply strong lower bounds dynamic problems focs ieee computer society cit aaron bernstein shiri chechik deterministic decremental single source shortest paths beyond bound stoc acm cit aaron bernstein shiri chechik deterministic partially dynamic single source shortest paths sparse graphs soda siam cit sayan bhattacharya deeparnab chakrabarty monika henzinger deterministic fully dynamic approximate vertex cover fractional matching amortized update time ipco vol lecture notes computer science springer cit luis barba jean cardinal matias korman stefan langerman van renssen marcel roeloffzen sander verdonschot dynamic graph coloring wads cit leonid barenboim michael elkin distributed graph coloring fundamentals recent developments synthesis lectures distributed computing theory morgan claypool publishers isbn cit surender baswana manoj gupta sandeep fully dynamic maximal matching log update time ieee annual symposium foundations computer science focs palm springs usa october rafail ostrovsky ieee computer society cit bhattacharya henzinger italiano design dynamic algorithms via method icalp cit bhattacharya henzinger italiano deterministic fully dynamic data structures vertex cover matching soda cit bhattacharya henzinger nanongkai new deterministic approximation algorithms fully dynamic matching stoc cit sayan bhattacharya monika henzinger danupon nanongkai fully dynamic approximate maximum matching minimum vertex cover worst case update time proceedings annual symposium discrete algorithms soda barcelona spain hotel porta fira january philip klein siam cit leonid barenboim tzalik maimon graph algorithms sublinear time inspired distributed computing international conference computational science iccs june zurich switzerland petros koumoutsakos michael lees valeria krzhizhanovskaya jack dongarra peter sloot vol procedia computer science elsevier cit aaron bernstein deterministic partially dynamic single source shortest paths weighted graphs icalp vol lipics schloss dagstuhl fuer informatik cit antoine dutot frederic guinand damien olivier yoann pign decentralized dynamic cossom complex systems selforganization modelling satellite workshop within european simulation modelling conference esm pages cit dahlgaard hardness partially dynamic graph problems connections diameter icalp vol lipics schloss dagstuhl fuer informatik cit manuela fischer mohsen ghaffari fabian kuhn deterministic distributed edgecoloring via hypergraph maximal matchings focs ieee cit uriel feige joe kilian zero knowledge chromatic number comput syst sci announced ccc cit mohsen ghaffari fabian kuhn yannic maus complexity local distributed graph problems stoc acm cit monika henzinger sebastian krinninger danupon nanongkai decremental singlesource shortest paths undirected graphs total update time focs ieee computer society cit monika henzinger sebastian krinninger danupon nanongkai dynamic approximate shortest paths breaking barrier derandomization siam comput cit monika henzinger sebastian krinninger danupon nanongkai thatchaphol saranurak unifying strengthening hardness dynamic problems via online matrixvector multiplication conjecture stoc acm cit bradley hardy rhyd lewis jonathan thompson tackling edge dynamic graph colouring problem without future adjacency information journal heuristics cit still better performance guarantee approximate graph coloring inf process lett cit bruce kapron valerie king ben mountjoy dynamic graph connectivity polylogarithmic worst case time soda siam cit subhash khot ashok kumar ponnuswami better inapproximability results maxclique chromatic number automata languages programming international colloquium icalp venice italy july proceedings part michele bugliesi bart preneel vladimiro sassone ingo wegener vol lecture notes computer science springer cit tsvi kopelowitz seth pettie ely porat higher lower bounds conjecture soda siam cit danupon nanongkai thatchaphol saranurak dynamic spanning forest worstcase update time adaptive las vegas stoc acm cit danupon nanongkai thatchaphol saranurak christian dynamic minimum spanning forest subpolynomial update time focs ieee computer society cit linda ouerfelli hend bouziri greedy algorithms dynamic graph coloring communications computing control applications ccca international conference ieee cit mihai patrascu towards polynomial lower bounds dynamic problems stoc acm cit scott sallinen keita iwabuchi suraj poudel maya gokhale matei ripeanu roger pearce graph colouring challenge problem dynamic graph processing distributed systems proceedings international conference high performance computing networking storage analysis salt lake city usa november john west cherri pancake ieee computer society cit christian minimum spanning forest improved worstcase update time stoc acm cit david zuckerman linear degree extractors inapproximability max clique chromatic number theory computing announced stoc cit problems consider graph problems way similar set states associated node node pick state problem every node picks state function determines whether given node valid invalid crucially function satisfies two properties stated goal problem assign state node way nodes become valid property output function depends states neighbors words constraint local node defined states neighbors property consider assignment states neighbors invalid without modifying states neighbors way change state makes valid make erstwhile valid neighbor invalid present two examples problems note graph problems viewed assigning states nodes definitions standard main task define function node properties satisfied coloring problem set states associated node set colors feasible coloring every node different color neighbors show problem consider function determines valid iff none neighbors state satisfies properties since node neighbors possible states coloring problem follows let denote palette colors let denote number nodes identify nodes state node set consists possible element supposed color edge edge exist thus feasible solution one every nodes iff edge node assign colors incident edges every edge agree color every edges adjacent edges different colors show problem consider function determines valid iff following three conditions hold every neighbor every neighbor every two neighbors satisfies properties lemma dynamic setting update valid solution problem making two changes insertion deletion edge furthermore changes occur endpoints edge inserted deleted proof consider locally fixable problem dynamic graph suppose every node valid insertion deletion edge property due edge endpoints potentially become invalid property consider one time order make valid without making new node invalid problems constitute subclass slocal problems roughly speaking complexity class slocal follows description closely follows slocal model nodes processed arbitrary order node processed see current state neighbors compute output arbitrary function addition locally store arbitrary amount information read later nodes part state lemma problem slocal proof initially assign arbitrary state every node scan nodes arbitrary order considering node scan using property change state way ensures node becomes valid previously scanned neighbor becomes invalid thus end scan guaranteed every node valid maximal independent set mis problem locally fixable recall standard definition mis state node either feasible solution two adjacent nodes node whose neighbors state claim mis standard definition define function every node properties satisfied see consider following graph instance defined defined follows every two edges consider following mis solution graph edge gets inserted change either without loss generality suppose set also need set every words insertion deletion edge lead many changes valid solution contrapositive lemma mis
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program transformation identify parallel skeletons venkatesh kannan hamilton school computing dublin city university ireland vkannan hamilton algorithmic skeletons used ease task parallel programming abstracting details parallel implementation developer existing libraries provide implementations skeletons defined flat data types lists arrays however parallel programming still challenging requires intricate analysis underlying algorithm often uses inefficient intermediate data structures algorithmic structure given program may match skeletons paper present method automatically transform given program one defined list likely contain instances skeletons facilitates parallel execution transformed program using existing implementations parallel skeletons using existing transformation called distillation conjunction method produce transformed programs contain fewer inefficient intermediate data structures introduction today computing systems parallel hardware architectures use cpus gpus graphics processor units ubiquitous hardware essential programs developed executed parallel order effectively utilise computing power available enable parallelism inherent given program needs identified exploited however parallel programming tedious done hand difficult compiler automatically desired level ease task parallel programming collection algorithmic skeletons often used program development abstract away complexity implementing parallelism particular map reduce zipwith primitive parallel skeletons often used parallel programming libraries eden sketo data parallel haskell dph accelerate provide parallel implementations skeletons defined flat data types lists arrays however two main challenges programming using multiple skeletons program often introduces inefficient intermediate data structures may mismatch data structures algorithms used skeletons program example consider matrix multiplication program shown example mmul computes product two matrices xss yss function map used compute dot row xss transpose yss computed function transpose note definition uses multiple intermediate data structures inefficient hamilton lisitsa nemytykh eds vpt eptcs kannan hamilton kannan hamilton example matrix multiplication original program mmul mmul xss yss mmul yss mmul xss yss dot transpose xss yss xss yss rotate yss rotate rotate yss map transpose yss dot mmul xss yss oldr zipwith xss yss xss rotate yss yss rotate yss rotate yss version program defined using map reduce zipwith skeletons shown example example matrix multiplication mmul xss yss mmul yss mmul xss yss dot transpose xss yss xss yss rotate yss map dot transpose yss mmul xss yss reduce zipwith xss yss xss rotate yss zipwith yss observe though defined using parallel skeletons implementation still employs multiple intermediate data structures instance matrix constructed transpose function subsequently decomposed map challenging obtain program uses skeletons parallel evaluation contains intermediate data structures therefore desirable method automatically identify potential parallel computations given program transform operate flat data types facilitate execution using parallel skeletons provided existing libraries reduce number inefficient intermediate data structures used paper present transformation method following aspects reduces inefficient intermediate data structures given program using existing transformation technique called distillation section automatically transforms distilled program encoding inputs single referred encoded list section allows parallel execution encoded program using efficient implementations map skeletons operate lists section section discuss results evaluating proposed transformation method using two example programs section present concluding remarks possible improvements transformation method discuss related work program transformation identify parallel skeletons language focus automated parallelisation functional programs pure functional programs free makes easier analyse reason manipulate using program transformation techniques facilitates parallel evaluation independent program functional language used work shown definition definition language grammar data type declaration type component variable constructor application function definition pkm xnk function call application let pattern program contain data type declarations form shown definition name data type polymorphic type parameters data constructor may zero components may type parameter type application expression type denoted program language also contain expression variable constructor application function definition function call application variables introduced function definition bound variables free constructor fixed arity expression must equal arity constructor patterns function definition header grouped two pkm inputs xnk inputs series patterns pkm function definition must exhaustive use short notations nil cons constructors list concatenation set free variables expression denoted definition context context expression holes place expression obtained filling holes context expressions operational semantics language defined using reduction relation shown definition definition reduction relation let kannan hamilton distillation objective given program may contain number inefficient intermediate data structures order reduce use existing transformation technique called distillation distillation technique transforms program remove intermediate data structures yields distilled program transformation makes use transformation steps unfold generalise fold potentially provide speedups programs syntax distilled program shown definition set variables introduced decomposed consequently expression fewer intermediate data structures definition distilled form grammar den den xnk dek pkm pkn xnk let den variable application constructor application function definition function application pattern example shows distilled form example matrix multiplication program example lifted definitions functions top level using lambda lifting ease presentation example matrix multiplication distilled program mmul xss yss mmul xss yss zss yss xss yss xss zss yss yss yss yss yss xss yss yss let yss xss zss yss let yss yss yss distilled program function computes product matrices xss yss functions compute row xss transpose yss version matrix multiplication free intermediate data structures particular distillation removes data structures constructed subsequently decomposed part algorithm implemented given program program transformation identify parallel skeletons consequence using distillation transformation obtain semantically equivalent version original program fewer intermediate data structures encoding transformation objective data types algorithm distilled program want parallelise may match skeletons defined lists would inhibit potential identification parallel computations could encapsulated using map skeletons resolve define transformation encodes inputs distilled program single resulting encoded program defined form facilitates identification parallel skeleton instances perform encoding transformation first lift definitions functions distilled program using lambda lifting following recursive function defined distilled program encode inputs definition inputs never definition encoded perform encoding recursive functions distilled program potential instances parallel skeletons also defined recursively three steps encode inputs function referred encoded list illustrated figure described encode inputs type new type created contain variables put transformation using encoded list type encode put figure steps encode inputs function consider definition recursive function inputs form shown definition distilled program body corresponding function header xnk definition use one recursive calls function may pkm appear recursive calls part context definition general form recursive function distilled program pkm three steps encode inputs follows kannan hamilton declare new encoded data type first declare new data type elements encoded list new data type corresponds data types inputs function encoded rules declare type shown definition definition rules declare encoded data type list data tlk type variables data types inputs fresh constructor inputs corresponding xnk otherwise pkm new constructor type created set pkm inputs function encoded stated objective encode inputs recursive function list element contains variables consumed iteration achieve variables bound constructor correspond variables pkm occur context contains recursive call expression otherwise consequently type components constructor data types variables define function encode recursive function form shown definition use rules definition define function encode build encoded list element type definition rules define function encode encode encode encode pkm encode otherwise pkm pattern pkm inputs encode function creates list element element composed fresh constructor type binds zkl variables pkm occur context contains recursive call expression otherwise encoded input recursive call appended element build complete encoded list computed encode function transform distilled program creating data type encoded list encode function recursive function transform distilled program using rules definition defining recursive function operates encoded list corresponding function program transformation identify parallel skeletons definition rules define encoded function encoded list zkl xnk otherwise pkm function definition header replace inputs pattern decompose encoded list first element encoded list matched corresponding pattern encoded type instance function header transformed pattern match first element encoded list pattern type call function replace inputs encoding instance call transformed encoding inputs encoded data types encode functions encoded program obtained distilled matrix multiplication program example shown example example matrix multiplication encoded program data data data zss xss xss zss xss zss yss yss yss yss mmul xss yss mmul xss yss xss yss yss yss yss yss let yss yss yss yss let yss yss kannan hamilton correctness correctness encoding transformation established proving result computed recursive function distilled program result computed corresponding recursive function encoded program encode proof proof structural induction encoded list type base case encoded list zkl computed encode pkm definition evaluates definition evaluates inductive case encoded list zkl computed encode pkm definition evaluates definition evaluates xnk inductive hypothesis consequence result encoding transformation inputs recursive function encoded following recursive structure function parallelisation encoded program produced transformation identifying potential instances map skeletons discussed section parallel execution encoded programs objective encoded program defined encoded list likely contain recursive functions resemble structure map skeletons encode function constructs encoded list way reflects recursive structure map skeletons defined therefore look instances skeletons encoded program work identify instances map skeletons encoded program shown property function instance reduce skeleton encoded program operates encoded list efficiently evaluated parallel reduction operator associative program transformation identify parallel skeletons property reduction operator encoded list given encoded program defined encoded list reduction operator instance reduce skeleton associative proof definition given encoded function encoded list data type data types inputs encoded output data type instance reduce skeleton type binary reduction operator given given newly created data type follows binary operator associative two input data types equal identification skeletons identify skeleton instances given program use framework labelled transition systems ltss presented definition represent analyse encoded programs skeletons lts representations enable matching recursive structure encoded program skeletons rather finding instances matching expressions definition labelled transition system lts lts given program given act set states lts state unique label start state denoted start act one following actions free variable variable constructor application expression application ith argument application set patterns function definition header let body act relates pairs states actions act act lts corresponding given program constructed jek using rules shown definition start state set previously encountered function calls mapped corresponding states set function definitions lts built using rules always finite function call corresponding state reused kannan hamilton definition lts representation program jxk jen pkm xnk start dom jfk otherwise xnk jek pkm xnk jlet let jen jek definition lts substitution substitution denoted lts result simultaneously replacing ltss corresponding lts lts ensuring bound variables renamed appropriately avoid name capture potential instances skeleton ltss identified replaced suitable calls corresponding skeletons lts encoded program jlk using rules presented definition definition extraction program lts skeletons states start jlk start jlk otherwise jlk otherwise jln xnk pkm fresh jlk let let jln jlk fresh parameter contains set new functions created associates corresponding states lts parameter contains sequence arguments application expression set initialised pairs application expression corresponding lts representation parallel skeleton identified given lts example map pair map application expression map lts representation program transformation identify parallel skeletons definitions map skeletons whose instances identify encoded program follows map map map map mapreduce mapreduce mapreduce mapreduce property encoded list given rules definition encode inputs list encode proof definition pkm matches inputs therefore list computed consequently encode zkl encode encode least singleton property evident encoded programs produced transformation always defined encoded list inputs consequently identify instances map mapreduce skeletons encoded program represent patterns corresponding inputs ltss built skeletons example ltss built map skeleton function encoded program matrix multiplication example illustrated figures respectively observe lts instance lts map skeleton similarly lts instance lts mapreduce skeleton map yss yss figure lts map skeleton let yss yss figure lts function kannan hamilton parallel implementation skeletons order evaluate parallel programs obtained method presented chapter require efficient parallel implementations map skeletons work presented paper use eden library provides parallel implementations map skeletons following forms farmb trans trans int parmapredr trans trans parmapredl trans trans farmb skeleton implemented eden divides given list creates parallel processes applies map computation parallel skeletons parmapredr parmapredr implemented using parmap skeleton applies map operation parallel element given list result parmap reduced sequentially using conventional foldr foldl functions respectively currently skeletons eden library defined using foldr foldl functions require unit value operator provided input however evident property skeletons potentially identified always applied lists therefore augment skeletons provided eden adding following parallel skeletons defined using functions defined lists thereby avoiding need obtain unit value reduction operator trans trans trans trans execute encoded program produced transformation parallel replace identified skeleton instances suitable calls corresponding skeletons eden library example replacing functions instances map mapreduce skeletons respectively suitable calls parmap obtain transformed matrix multiplication program mmul shown example example matrix multiplication encoded parallel program mmul xss yss mmul xss yss xss yss yss yss farmb nope let yss yss yss let yss yss program transformation identify parallel skeletons gxy consequence automatically identifying instances map skeletons produce program defined using parallelisable skeletons using parallel implementations skeletons available existing libraries eden possible execute transformed program parallel hardware evaluation paper present evaluation two benchmark programs matrix multiplication dotproduct binary trees illustrate interesting aspects transformation programs evaluated mac pro computer intel xeon processor clocked ghz main memory clocked mhz ghc version used sequential versions benchmark programs latest eden compiler based ghc version parallel versions parallel versions benchmark program skeletons present toplevel expression executed using parallel implementations nesting parallel skeletons avoided nested skeletons present inside skeletons executed using sequential versions objective approach avoid uncontrolled creation many threads observe result inefficient parallel execution cost thread creation management greater cost parallel execution example matrix multiplication original sequential version distilled version encoded version encoded parallel version matrix multiplication program presented examples respectively version original matrix multiplication program example presented example identify function mmul instance map skeleton therefore define using suitable call farmb skeleton available eden library example matrix multiplication program mmul xss yss mmul xss yss armb nope xss map dot transpose yss dot oldr zipwith transpose xss xss yss yss xss yss xss rotate yss rotate yss yss rotate rotate yss rotate yss rotate yss figure presents speedups achieved encoded parallel version matrix multiplication program comparison original distilled versions since avoid nested kannan hamilton parallel skeletons explained earlier encoded parallel program used evaluation contains function defined using farmb skeleton uses sequential definition input size indicated nxm denotes multiplication matrices sizes nxm mxn compared original program observe encoded parallel version achieves positive speedup input sizes except case input size speedup achieved speedups achieved input sizes due intermediate data structure transpose yss order elements input size order elements inputs absent encoded parallel program verified comparison distilled version also free intermediate data structures hence encoded parallel program linear speedup compared distilled version examples observe encoded parallel figure evaluation speedup matrix multiplication versions parallelise equivalent computations multiply rows first matrix columns second matrix using farmb skeleton however encoded parallel version marginally faster version input sizes faster input sizes except due use intermediate data structures version order input sizes except speedup achieved program transformation identify parallel skeletons version also version scales better higher number cores encoded parallel version input size encoded parallel version achieves better speedup even fewer cores due elimination intermediate data structures hence scale impressively version example binary trees example presents sequential program compute binary trees dotp computes product values corresponding branch nodes trees adds dotproducts left right distilled version program remains intermediate data structures version defined using parallel skeletons example binary trees program data btree btree btree dotp btree btree btree dotp dotp dotp dotp dotp dotp applying encoding transformation obtain encoded version program shown example example binary trees encoded program data tdotp btree btree encodedotp encodedotp encodedotp encodedotp dotp encodedotp dotp dotp dotp dotp encodedotp dotp applying skeleton identification rule encoded program identify encoded version function dotp instance mapredr skeleton example shows encoded parallel version defined using suitable call skeleton eden library explained define call using parallel skeleton use sequential nested call dotp avoid nested parallel skeletons evaluation example dot product binary trees encoded parallel program data tdotp btree btree encodedotp encodedotp encodedotp encodedotp kannan hamilton encodedotp dotp dotp dotp dotp dotp dotp figure presents speedups encoded parallel version compared original version input size indicated denotes two identical balanced binary trees nodes observe encoded parallel version achieves positive speedup upon using cores resulting maximum speedup input size input size input sizes speedup factor improve using cores also observe speedup achieved scales negatively input size increases figure evaluation speedup binary trees reason performance encoded parallel version follows example observe element encoded list contains values branch nodes arguments first recursive call dotp example consequently sizes elements encoded list progressively decrease first last element input trees balanced encoded list split fashion skeleton thread parallel execution applies sequential computation dotp result workloads parallel threads results significant idle times threads process smaller also note input binary trees would result poorer performance input binary trees result better performance encoded parallel versions conclusion summary presented transformation method efficiently parallelise given program automatically identifying parallel skeletons reducing number intermediate data structures used encoding inputs program single input facilitate identification parallel skeletons program transformation identify parallel skeletons particularly computations additionally automatically check skeleton operators desired properties thereby allowing complete automation parallelisation process importantly transformation place restriction programs transformed using method evaluate transformation method presented two interesting benchmark programs whose inputs encoded results observe two possible extreme performances one case linear speedups achieved due distillation transformation reduces use intermediate data structures well parallelisation transformation another case despite parallelising program defined using existing skeleton implementations libraries positive speedups achieved limited desired despite discussed employing additional skeletons accumulate able automatically parallelise interesting programs maximum prefix sum primary challenge lies efficient execution parallel programs produced defined using skeletons important efficient implementations parallel skeletons incorporate intelligent methods across parallel threads created execute skeletons believe better across threads facilitated polytypic parallel skeletons list array data structures support nested parallelism dynamic related work previously following seminal works cole darlington program development majority work followed catered manual parallel programming address difficulties choosing appropriate skeletons given algorithm proposed diffusion transformation capable decomposing recursive functions certain form several functions described skeleton even though diffusion transform wider range functions required form method applicable functions one recursive input proposed accumulate skeleton encapsulates computational forms map reduce skeletons use accumulating parameter build result however associative property reduce scan operators used accumulate skeleton verified unit values derived manually calculational approaches program parallelisation based propose systematic ways derive parallel programs however methods restricted programs defined lists require manual derivation operators verification certain algebraic properties enable parallel evaluation programs obtained morihata extended approach trees decomposing binary tree list called zipper defining upward downward computations zipper structure however calculational methods often limited range programs data types transform also common aspect calculational approaches need manually derive operators satisfy certain properties associativity guarantee parallel evaluation address chin proposed method systematically derives parallel programs sequential definitions automatically creates auxiliary functions used define associative operators needed parallel evaluation however method restricted language applicable functions defined single recursive linear data type lists associative decomposition operator alternative calculational approaches ahn proposed analytical method kannan hamilton form general recursive functions composition polytypic data parallel skeletons even though method applicable wider range problems need associative operators transformed programs defined composing skeletons employ multiple intermediate data structures previously authors proposed method transform input given program based recursive structure input since method use recursive structure program build transformed programs lend defined using parallel skeletons observation led creating new encoded data type matches algorithmic structure program hence enables identification polytypic parallel map reduce skeletons new encoded data type created recursively consuming inputs recursive components created new encoded input recursive call occurs function body using input arguments recursive call consequently data structure new encoded input reflects recursive structure program even though method leads better identification polytypic skeletons easy evaluate performance transformed programs defined using skeletons existing libraries offer implementations skeletons defined generic data type consequently proposed method encoding inputs list respects recursive structures programs allows evaluation transformed programs using existing implementations parallel skeletons acknowledgment work supported part science foundation ireland grant lero irish software research centre references joonseon ahn taisook han analytical method parallelization recursive functions parallel processing letters manuel chakravarty gabriele keller sean lee trevor mcdonell vinod grover accelerating haskell array codes multicore gpus proceedings sixth acm workshop declarative aspects multicore programming manuel chakravarty roman leshchinskiy simon peyton jones gabriele keller simon marlow data parallel haskell status report proceedings workshop declarative aspects multicore programming damp chin takano zhenjiang parallelization via context preservation international conference computer languages murray cole algorithmic skeletons structured management parallel computation mit press cambridge usa john darlington field peter harrison paul kelly sharp qiang lyndon parallel programming using skeleton functions lecture notes computer science international parle conference parallel architectures languages europe jeremy gibbons third homomorphism theorem journal functional programming vol horacio mario leyton survey algorithmic skeleton frameworks structured parallel programming enablers software practice experience program transformation identify parallel skeletons sergei gorlatch constructing list homomorphisms parallelism mathematik und informatik mip sergei gorlatch extracting implementing list homomorphisms parallel program development science computer programming hamilton neil jones distillation labelled transition systems proceedings acm sigplan workshop partial evaluation program manipulation zhenjiang masato takeichi chin parallelization calculational forms proceedings acm symposium principles programming languages popl zhenjiang masato takeichi hideya iwasaki diffusion calculating efficient parallel programs acm sigplan workshop partial evaluation program manipulation pepm zhenjiang tetsuo yokoyama masato takeichi program optimizations transformations calculation form gttse hideya iwasaki zhenjiang new parallel skeleton general accumulative computations international journal parallel programming kluwer academic publishers venkatesh kannan hamilton extracting data parallel computations distilled programs fourth international valentin turchin workshop metacomputation meta venkatesh kannan hamilton program transformation identify parallel skeletons euromicro international conference parallel distributed processing pdp rita loogen eden parallel functional programming haskell lecture notes computer science central european functional programming school springer berlin heidelberg matsuzaki takeichi parallel skeletons manipulating general trees parallel computing matsuzaki iwasaki emoto library constructive skeletons sequential style parallel programming proceedings acm international conference scalable information systems infoscale matsuzaki kakehi iwasaki akashi skeleton library parallel processing lecture notes computer science springer berlin heidelberg kiminori matsuzaki hideya iwasaki kento emoto zhenjiang library constructive skeletons sequential style parallel programming proceedings international conference scalable information systems trevor mcdonell manuel chakravarty gabriele keller ben lippmeier optimising purely functional gpu programs acm sigplan notices akimasa morihata kiminori matsuzaki zhenjiang masato takeichi third homomorphism theorem trees downward upward lead popl alberto pettorossi maurizio proietti rules strategies transforming functional logic programs acm computing surveys skillicorn formalism parallel model software parallel computation nato asi series skillicorn domenico talia models languages parallel computation acm computing surveys
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programming example hila peleg sharon shoham eran yahav technion hilap tel aviv university technion yahave oct abstract recent years tremendous progress automated synthesis techniques able automatically generate code based intent expressed programmer major challenge adoption synthesis remains programmer communicate intent expressed intent coarsegrained example restriction expected type expression synthesizer often produces long list results programmer choose shifting user alternative approach successfully used synthesis programming example pbe user leverages examples interactively iteratively refine intent however using examples expressive enough programmers observe generated program refine intent directly relating parts generated program present novel approach interacting synthesizer using granular interaction model approach employs rich interaction model synthesizer decorates candidate program debug information assists understanding program identifying good bad parts user allowed provide feedback expected output program also underlying program user identifies program partially correct incorrect also explicitly indicate good bad parts allow synthesizer accept discard parts program instead discarding program whole show value approach controlled user study study shows participants strong preference using granular feedback instead examples able provide granular feedback much faster introduction development ecosystem programmers frequently asked take tasks involving unfamiliar apis complex data transformations program synthesis tool shorten development times aid small tasks api programming synthesis tools available wide variety purposes creating formulae microsoft excel formulating sql queries tools expert users encode full specifications also matured enough practical expressing intent despite significant progress synthesis expressing user intent remains major challenge expert user write full specifications express intent fully programmers trying solve small tasks often use partial specifications partial specifications available different forms depending synthesizer source target types pairs tests logical specifications models synthesis present user possible results satisfy criteria leads challenging task user must compare large number similar programs select solution expressing intent examples alternative proven extremely useful use examples way express intent programming example pbe form program synthesis desired behavior generalized specific instances behavior often example pairs allows iterative process synthesized program acceptable additional examples provided target program reached technique often used either synthesizers way refine results synthesis insufficiency examples programmers pbe geared towards also useful advanced users behavior difficult describe effect however interaction model user one two things accept program inspection reject differentiating example rule next iteration synthesis wasted knowledge forcing programmer work within interaction model synthesized programs bad may part program overfitted examples another right track allowing full accept full reject ignores ability programmer read understand program express directed granular feedback deeming parts desirable undesirable rather program whole fact hypothesize cases easier programmer explicitly indicate good bad candidate program instead implicitly trying express information examples moreover prove sometimes impossible express information examples programming example motivated insufficiency examples present new granular interaction model allows programmer interact synthesizer example also provide feedback parts synthesized program interaction model granular directions programmer synthesizer back synthesizer programmer candidate program presented together debug information showing execution values different program points helps programmer understand whether candidate program behaves expected intermediate states instead relying final output programmer synthesizer programmer provide inputoutput examples pbe granular feedback candidate program explicitly parts code tested granular interaction model controlled userstudy developers academia industry conduct study developed synthesizer interacts user three different ways holistic pbe granular september synthesizer also measures interaction times records later analyze implementation synthesizes functional programs scala scala popular functional programming language used many processing frameworks spark akka functional compositions considered scala way approach coding tasks aim synthesize advantages granular interaction user study strongly supports hypothesis beneficial let programmers communicate understanding program explicitly synthesizer marking parts desirable undesirable rather implicitly examples several participants user study faced inability rule undesired operation program using examples expressed extreme frustration show common one would imagine due introduction redundant superfluous operations synthesizer result undesirable operation may part several candidate programs along process holistic pbe model allow ruling study shows granular interaction model gim easier use supported strong preference participants granular feedback examples significantly shorter iteration time using granular feedback important note granular feedback completely replace examples participants restricted granular feedback sometimes forced use larger number iterations prone error accepting program therefore conclude future synthesizers integrate interaction models main contributions contributions paper synthesis framework programmers granular interaction model gim allows user approve disapprove specific parts code candidate program rather respond whole allows synthesizer present candidate programs debug information theoretical result shows examples sometimes insufficient reaching desired program show insufficiency occurs practice throughout real pbe sessions controlled user study showing programmers strong preference granular feedback instead examples able provide granular feedback much faster outline section shows examples inconvenient insufficient communicate intent programmer allow expressive power introduce three additional granular operations section full interaction model consists granular operations well examples addition also introduce debug information every example provided user section section detail experiments number iterations necessary solve set benchmarks different interaction models also detail controlled user study programmers academia industry study shows advantage granular predicates examples iteration time preference discuss need granular predicates examples order help user reach correct target program hila peleg sharon shoham eran yahav task find frequent bigram string initial example abdfibfcfdebdfdebdihgfkjfdebd question input problem takeright take right given string idea frequent bigram needs placed middle answer cababc question input problem program crops given input constant position idea vary position frequent bigram examples answer bcaaab question input problem examples output lexicographical minimum bigrams string idea frequent bigram large lexicographic order answer xyzzzy table difficulty finding differentiating example overview section provide overview granular interaction model gim synthesis simple example start showing interaction model programming example pbe shortcomings describe gim overcomes shortcomings using richer interaction model motivating example consider task writing program finds frequent string assume program constructed combining operations predefined set operations refer vocabulary assume vocabulary contains standard operations strings characters lists addition assume provide initial partial specification form example abdfibfcfdebdfdebdihgfkjfdebd example bigram frequent appears times thus expected output synthesized program interaction classical pbe synthesizer table shows interaction programmer pbe synthesizer complete task synthesizer poses question programmer candidate program consistent examples programmer provides answer form accept additional examples refine result based initial example synthesizer offers candidate program consists single method vocabulary takeright returns rightmost characters applied input programmer responds providing example inconsistent candidate program therefore differentiates target program point synthesizer offers new candidate program consistent interaction proceeds similar manner additional example may reduce number candidate programs programming example september task find frequent bigram string initial abdfibfcfdebdfdebdihgfkjfdebd specifications question abdfibfcfdebdfdebdihgfkjfdebd problem takeright take right given string idea takeright never useful since always want consider every element remove takeright result program answer remove takeright question programs debug information assists programmer understanding programs identifying good bad parts hand user restricted providing semantic examples also mark parts program code parts must must appear future candidate program allows user provide explicit syntactic feedback program code expressive cases allows synthesizer aggressively prune search space gim interaction model task finding frequent bigram demonstrated table question synthesizer produces candidate program contrast classical pbe granular interaction model provides additional debug information user showing intermediate values program examples shown comments next lines synthesized program input output values initial example next steps information would far valuable given programmer responds providing granular feedback using gim possible narrow space programs using syntactic operations presented user exclude sequence operations vocabulary instance takeright ruling program takeright appears also significantly reduces space candidate programs considered synthesizer synthesizer responds note cases debug information assists programmer understanding program determining whether correct case identifying incorrect rule user rules sequence drop debug information shows effect take second third character string user deems undesirable point computation truncate string characters considered synthesizer responds candidate program contains something programmer would like preserve using debug information see prefix creates bigrams string user mark prefix affix make sure candidate programs displayed begin prefix removes programs start function effectively slicing size search space another option available user multiple operations stemming program allowed encouraged exclude take since resulting truncation list undesirable eventually synthesizer produces following program input abdfibfcfdebdfdebdihgfkjfdebd bdfibfcfdebdfdebdihgfkjfdebd problem program crops given input constant position idea want crop anything functions place result program answer remove drop question input abdfibfcfdebdfdebdihgfkjfdebd problem beginning program actually good dividing program bigrams mapping string take truncates bigram list idea preserve good program remove take part sequence affix zip remove take answer retain map table providing granular syntactic feedback required satisfy examples careful choice examples user process terminates total examples finding differentiating examples may hard consider candidate program make progress user provide example differentiates behavior desired program find differentiating example user must understand program wrong provide examples overrule preferably also similar programs examining code easy see min problem calculating minimum part finding frequent bigram even programmer understands problem still need find differentiating example rules use min minimum list bigrams input programmer comes example desired bigram largest one lexicographically interaction programmer express explicit knowledge use min implicitly examples coming examples avoid min requires deep understanding program leveraged implicitly examples even guarantee min recur show section impossible completely remove model case since programmer already knows want avoid programs using min beneficial let communicate information explicitly synthesizer interaction granular interaction model gim improves pbe employing richer granular interaction model one hand synthesizer supplements candidate abdfibfcfdebdfdebdihgfkjfdebd list list discard bigram counts number occurrences retrieves maximum program accepted summarize key aspects gim demonstrated example key aspects september interaction model granular interaction directions synthesizer provides debug information intermediate states program programmer provide feedback parts program addition inputoutput examples assisting user approach assists user two ways first synthesizer supplements candidate programs debug information helps programmer understand good bad parts candidate program second ability give explicit feedback code provides alternative complementary way interact system without crafting potentially complicated differentiating examples insufficiency examples examples inconvenient insufficient communicate programmer intent operations needed allow programmer filter programs according semantic equivalence also according additional criteria readability best practices performance background work address synthesis functional programs provide necessary background notation functions interchangeably use mathematical notation functional composition called object scala notation scala function application arguments require parentheses functional program denote nmo function program computes formally nmo maps every element domain either element program outputs error compilation runtime vocabulary candidate program space candidate program space consists programs form scala notation input mathematical notation method predefined vocabulary object methods accept arguments handled partially applying predefined arguments constants lambda functions variables context leaving self reference argument generally candidate program space includes every program notice programs compilation errors applicable objects programming example pbe programming example program synthesis communication synthesizer done using examples classic pbe problem defined pair initial examples target language example pair input expected output result pbe problem program valid program satisfies every example nmo every since might one program language matches specifications iterative pbe problem introduced iterative model candidate program presented user may accept terminate run answer synthesizer additional examples direct continuing search hila peleg sharon shoham eran yahav insufficiency examples section show importance extending user answer model beyond examples examine formal details scenario described section user seen undesirable program component would like exclude specifically show always possible examples insufficient communicate user intent seen section user wishes rule function min simply providing example rule current program might enough remove min candidates ensure never recurs formally prove impossible completely remove methods like min search space using examples recall definition equivalence programs programs equivalent using prove following claim laim let letter exists program equivalent contains examples alone rule letter candidate programs proof follows since examples distinguish programs compute different functions next show claim applicable methods prevalent programming languages extremely useful contexts therefore likely find way vocabularies used synthesis consider two classes methods invertible methods nullipotent methods invertible methods methods exists inverse method applying two pair order leads back initial input instance reverse list iterator invertible inverse identical reverse inverse method another example includes zipwithindex map cancel invertible method always added target program along inverse resulting equivalent program hence never ruled examples nullipotent methods methods applied lead result applied often calling tolist list mkstring string calls always nullipotent takewhile true methods nullipotent certain context may synthesizer vocabulary end program space contexts nullipotent easy construct program contains nullipotent methods equivalent target program hence similarly invertible methods methods eliminated examples returning example target program section let construct equivalent program appending invertible pair functions sequence sliding function sliding applied string length return list min applied list size return member list means program equivalent every input contains min given number examples applied min letter ruled entirely programming example september construction possible many target programs showing often impossible discard undesirable member alphabet undesired sequence using examples alone furthermore since many existing pbe synthesizers prune aggressively based observational equivalence equivalence based given examples programs include undesired component may available anymore removed space properties leave need define expressive granular model practical implications claim discussed section examines existence method sequences deemed undesirable users candidate programs data well opinions collected users show inability remove undesirable letter alphabet consequences affect user frustration synthesizer see table however since retain dependent location procedure program add additional predicate setting procedure forcing location beginning program affix predicate essentially narrow search space come desired prefix additional predicates three operations highly expressive easy understand centered experiments paper around means possible predicates many granular operations exist instance user reason intermediate states program demanding excluding certain intermediate states given input user also require error error certain kind given input section expand reasons select certain expansions interaction model others gim assumes interaction users comfortable reading code means expected assisted ways generally offered regular synthesizer way interaction user synthesizer granulated interaction synthesizer user pbe tools like flashfill show user output running program input tools show code show program guaranteeing satisfies examples functional concatenation possible show user result subprogram means even familiar user still gauge effect determine whether effect desired example granular interaction model section describe granular interaction model gim mechanism extends pbe model additional predicates namely predicates gim include examples also additional predicates key idea add broader form feedback user synthesizer available pbe begin describing operations type feedback one allows user provide synthesizer discuss observed uses granular predicates setting functional compositions choose present gim three syntactic predicates refer predicates granular since impose constraints parts program rather full behavior captured function computes input output types also discuss possible predicates given candidate program input introduce following predicates tested programs input remove hold programs retain hold programs affix hold programs remove operation rules sequence one method calls undesirable example section rule min user would simply add predicate remove min however user rule sequence longer single method would apply sequence whole predicate remove reverse reverse exclude reverse method two consecutive invocations cancel retain operation defines sequence must appear target program similarly defined sequences applied single method forces method applied sequence forces sequence seen essentially creating procedure deeming procedure desirable adding debugging view code xample let consider case input list strings user presented candidate program familiar mkstring method formats list string mapping list never encountered sliding user could look method read behavior however oftentimes behavior simple enough understand operation within program therefore user provided intermediate states program like input list list aabbcc bbccdd ccddee understand sliding returns list sublists length beginning position list sliding window size enabling user introduced formal framework predicates wish leverage create user interaction model suggest following iterative process implemented user study section candidate program displayed user alongside debug information top image fig shows user able study program accept reject user dissatisfied program would like reject goal allow directly express source dissatisfaction predicates easily possible towards end let user point portion program september hila peleg sharon shoham eran yahav syntax testing new operation set proposed section user communicate synthesizer via syntactic predicates program gim testing full model user communicate via sets predicates figure program debug information sequence selected removal mark desirable undesirable seen bottom image fig process easily providing feedback program turns predicates convenient tool feedback synthesizer enabling synthesizer seen choice predicates crucial user perspective however also important synthesizer able use maintaining updating representation search space complete section show predicates described section naturally utilized synthesizer domain linear functional concatenations enumerating synthesizer state art program synthesis hinges enumerating program space fashion domain considered paper enumeration consists concatenating method calls prefixes already enumerated starting program length input enumeration restricted types compilation search space synthesizer represented tree root program input edge labeled method name path tree represents program concatenation every label along path tree initially pruned compilation errors exist return type pruned children node representing represents candidate program space unconstrained synthesizer state see every program deemed undesirable operations affix remove extended desirable program therefore extensions discarded tree representing candidate space pruned nodes programs example predicates well suited representation state synthesizer aid user also help guide search space since combination enumeration predicates monotone program pruned search space never need looked future constrained iteration means synthesizer need restarted across iterations however even predicates allow construct much smaller search space begin evaluation evaluate approach compared three interaction models pbe replicating state art synthesis user communicate synthesizer via pairs limit test granular interaction model three operations relevant functional compositions easy understand therefore select operations detailed section basic set granular operations conducted two studies study ideal sessions different operations families predicates set benchmarks controlled user study tests usability gim synthesizer programmers benefits measured control group using pbe synthesizer implemented simple enumerating synthesizer described section scala using nsc interpreter used implement scala repl vocabulary provided algorithm programs compiled evaluated inputs order support study section synthesizer accepts input additional examples rejection current program affix remove retain predicates order support user study section also precomputes space valid programs problem set performed studies using set functional programming exercises three different domains strings lists streams exercises collected scala tutorial sites examples using mapreduce tasks described tab paired vocabulary initial set examples discussion seen tab set valid programs cantly smaller many cases space still contains thousands tens thousands programs also fair amount inherent ambiguity initial example set einit seen reject column representing set programs length match einit means even limiting search space known length target program would start hundreds thousands matching programs need filtered user ideal synthesis sessions experimental questions task problem set answer following question ideal conditions expert user knowledge target program many questions candidate programs posed user predicate family test setup order test questions task problem set run four settings reject operations rejecting current example enumeration programs match initial example set pbe syntax gim described addition reject operation examples predicates selected expert user author paper making effort create run fewer iterations aggressive pruning space iteration streams lists strings programming example benchmark dropnthletter freqbigram frequword linesinfile nonemptylines anagrams histogram median posinlist sudokusquare sumsquares bitstream numhashtags slidingavg drop every letter string frequent bigram string frequent word string number lines file number lines file group words anagrams create histogram number list find median list numbers get positive numbers list validate square sudoku sum squares list numbers next integer stream bits count hashtags stream tweets average next three values every index september eini candidate space size number candidates reject pbe syntax gim table test setup synthesis experiments showing ambiguity inherent eini number iterations target program ideal synthesis session available set operations parentheses indicate examples used results tab shows results programming tasks seen table ideal thoroughly optimized expert user runs number questions produced synthesizer pbe run lowest unexpected carefully selected examples fast way differentiate programs subject examples selected less ideal conditions left following section also see run allowing predicates number questions asked lowered substantially compared syntax without involving one example user study test interaction programmer synthesizer conducted user study compared interaction programmers synthesizer using three families operations pbe control syntax gim research questions examine following questions answers consisting syntactic predicates easier faster generate example predicates question examined two different ways first task average median iteration times synthesizer compared control group pbe syntax group second users allowed gim time spent iterations provided examples measured average time total time solution improved adding exchanging available predicates users able reach correct program using predicate sets users prefer examples question examines choices made participants gim group choice possible predicates test often examples chosen whether task solved effect preference users pbe sessions distracted undesirable sequences removed check recurrence pbe group sequences deemed undesirable users syntax gim groups try determine whether repeat enough distract user also check acceptance equivalent programs superfluous elements mentioned claim addition bring anecdotal opinions volunteered participants questions examined participants show data small set users experienced scala new scala difference interest test setup developers participated study consist undergraduates final year degree graduate students history developers outside academia industry developers employed four different companies prior experience scala programming language participants study evenly distributed three test groups pbe syntax gim participant randomly assigned one test groups participants performed tasks scheduling constraints cited part order tasks randomized user reject operation allowed group forcing users provide process new information would state art synthesizer rather iterate program space participant asked use synthesizer solve three programming questions three nonemptylines selected tasks tested section high level ambiguity based initial example requiring additional libraries definitions outside scala standard library solve programs could run scala console imports definitions participants given short introduction scala already familiar aided scala repl online sources documentation implementation participants performed tasks using shown fig space programs precomputed enumerating synthesizer detailed section initial inputs program length iteration program uphold every predicate given user selected set programs selection used criterion prevent lexicographical ordering favoring short programs order also show user complex programs end every iteration user answers added synthesizer state programs filtered accordingly precomputed set exhausted user given option starting abandoning current task user study results address question individually september number correct iterations target equiv avg med finished answer answer question average median times per iteration shown table medians also shown fig examined distributions data using test threshold statistical significance selected significant difference found time per iteration control pbe group group tests histogram nonemptylines frequword significant difference found control group gim group two three tests histogram nonemptylines frequword additionally significant difference found group gim group one test histogram nonemptylines frequword results imply exception frequword test gim group using either syntax example predicates speedup iteration time solving problem pbe alone additionally exception histogram task slowdown iteration time gim seems coincidental addition looked session users gim group within session examined time create example average iteration time slowdown iteration time example see difference statistically significant therefore answer question affirmative counts syntactic predicates faster generate examples examining test groups pbe group examining users access question noticed change median total time control pbe groups syntax gim indicating possible slowdown however none individual tests well unification tests change statistically significant therefore answer question affirmative total time improved either test groups also say evidence slowdown may coincidental question correctness results table visualized fig aside histogram task completed users tasks users stopping without accepting program success percentage reaching response highest pbe lowest syntax users familiar scala users unfamiliar scala sessions used task sessions examples histogram nonemptylines frequword histogram nonemptylines frequword histogram nonemptylines frequword percent examples per user avg med min max table proportional part examples predicates provided gim group users used examples none used examples num lines table summary three tasks performed user study users users frequent word iteration time sec avg med histogram task group sessions pbe histogram syntax gim pbe lines text syntax gim pbe frequent syntax word gim hila peleg sharon shoham eran yahav users saw removed sequence removed tail takewhile filternot filter takeright drop drop takeright dropright take last drop takeright tomap map zipwithindex map pbe times seen distracting session occurrences users min max average distracted table method sequences syntax gim groups occurrence pbe group rebounds gim levels close control even overtaking nonemptylines task question summary often users chose examples appears table fig see distinction users familiar scala users users familiar scala used examples every task users unfamiliar scala every task included least one user avoiding sometimes many users proportional part examples total predicates used task fairly low entire test group ranging median looking data familiarity scala see preference examples inverse two groups users familiar scala preferred examples overwhelmingly examples users histogram task preferred predicates frequword frequent word task conversely users unfamiliar scala preferred examples overwhelmingly half participants frequword task favored predicates half participants examples solving histogram task seems suggest relationship difficulty task histogram less trivial problem frequword despite even cases examples favored tool used programming example question test whether users distracted undesirable sequences removed first located undesirable sequences many users ability access remove predicate removed counted appearance sessions users pbe group table shows results important note sequences commonly removed appeared pbe group indicator syntax operations vastly change traversal search space undesirable sequences appeared user sessions times single session sequences distracted kept reappearing users performing task average users furthermore distracting sequence appeared average around times session shows inability remove letter sequence discussed claim theoretical problem problem end process seen table real distraction ability synthesize expressive vocabulary examining presence distracting elements final program accepted participants see fig two tasks histogram nonemptylines pbe users ended accepting program superfluous elements example many histogram sessions accepted program call tomap map many nonemptylines sessions accepted program called filternot list strings nullipotent elements tomap creates map map filternot compares strings characters always filters nothing addition cases pbe users stopped equivalent program rather target program tested number iterations spent equivalence class presented candidate program accepting program users accepted equivalent program immediately one user performing histogram task tried additional iteration one user tried two additional iterations nonemptylines two users tried additional iteration one user tried two additional iteration altogether sessions users tried unsuccessfully improve upon program already either trying get rid nullipotent element realizing influence finally accepting chose tackle questions user preference measures distraction questionnaire asking users approximate preference sticking empirical results despite wish bring several anecdotes course experiment may help shed light behavior observed users pbe test group expressed specific frustration several occasions insists using take matter solving frequent word task get rid nonsense functions wanted shake solving lines task discussion conclusions section discuss results study speed ease use see speedup iteration time examples predicates change largest pbe syntax groups smaller speedup examining gim group may attribute difference two tests fact users gim group resort september examples convenient readily apparent therefore take shorter amount time create combined low preference examples conclude syntax predicates easier user use general addition combine improvement iteration time change number iterations statistically significant lack significance change total time conclude changing type predicates simply leads time spent synthesis task using shorter easier iterations distracting elements user frustration much frustration users pbe group expressed recurring program elements thought useless recurring undesirable sequences showed users recurred average times session definitely reason frustration addition pbe users wasted time effort trying remove elements removed therefore conclude avoiding distraction giving users tools would least make content users helpfulness debug information attribute success rate relatively short use times set developers never seen scala guidance offered debug information target specifically experiment separate users volunteered experiment anywhere helpful lifesaving understanding unfamiliar methods keeping track examples approval included people familiar scala developers develop scala correctness syntax operations dip correctness functionally correct program reached pbe group syntax group improvement gim attribute helpfulness debug information seems easier make correct decision program presented breakdown several examples rather single initial example available syntax group preferred operations strong preference users syntactic predicates examples however may subordinate trend within preference separating users familiar scala preference reverses users familiar scala preferred examples rest preferred examples predicates harder task histogram predicates examples easier task frequword may ability better understand candidate program savvy programmers could easily read programs preferred break observed behavior examples whereas programmers harder time reading code focus individual program elements remains conjecture set users familiar scala within gim group threats validity cross validation study user performing tasks several groups predicate families include felt would create bias toward operations based order cross validation used creates bias decided tried negate differences individual programmers tested drawing september figure median iteration time per task test group significant change pbe indicated hila peleg sharon shoham eran yahav figure examples used med min max gim users operations none used examples participants similar backgrounds year university developers department dividing evenly sampling population external validity issue mentioned section relatively small percentage participants familiar scala participants study due random assignment groups groups already pointed allow make general claims differences programmers new language however still generalize claims regards programmers working language encountered majority participants additionally sample undergraduates random rather students felt familiar enough functional programming agree participate may skew ability generalize hopefully significant compiled results undergraduates large percentage participants less related work synthesis domain program synthesis target program derived target programming language according syntax rules fall within scope implementation gim presented paper target language functional subset scala specified synthesis algorithms often use interaction model gim extends programming example pbe interaction user synthesizer demonstrating desired behavior restricted examples initial specifications refinement flashfill pbe tool automating transformations excel data set included microsoft excel show users program application data set resulting program never inspected might still suffer overfitting examples reusable escher pbe tool synthesizing recursive functions like flashfill escher decomposes synthesis task based examples searching programs could used condition blocks escher parameterized set operations used synthesis like flashfill allows refinement process synthesis category synthesis algorithms perform synthesis mainly driven types variables methods construction program performed rules methods tend figure users reached target program equivalent differences result many require initial specifications user manually chooses multiple candidate programs match specification philosophy behind gim user consider many programs could dozens time additional data rather programs considered one time additional information help user consider program depth direct search adding examples synthesis recent work connects pbe synthesis tools accept initial specifications examples inherent type information use type derivations produce candidate programs verify examples synthesizes mapreduce processes via sketching type derivations lambda calculus vocabulary examples also used verify determinism synthesis algorithm uses petrinets represent type relationships finds possible programs reachability candidates tested using tests provided user requires full test cases rather examples descriptive still require user learn lot library order program test case effort may equal learning methods required solve programming task hand sketching user restrict search space via sketches structural elements conditions loops includes holes synthesized sketching way leverage programmer knowledge expected syntactic elements used conjunction restrictions syntax allow intricate synthesis however since general sketch program single hole usually unconstrained synthesizer user must come armed least knowledge expected structure rather iteratively build gim enriching user input several existing works enriched specification language interface specifying program behavior adding examples synthesis example enrichment another approach polikarpova ynquid use refinement types instead types encode constraints solution program imposed candidate space constraints mainly semantic unlike gim syntactic predicates embodies ideal passing responsibility user understand code case write code likewise barman suggest interactive extension sketching intended synthesize sketch leveraging user decompose specifications programming example examine results angelic programming leverages programmer knowledge expanded interface synthesizer user user shown synthesized program nondeterministic angelic operation execution traces operation make program correct responsibility identify needed operation replace angelic operator conclusion presented novel granular interaction model gim interacting synthesizer interaction model extends common pbe approaches enables programmer communicate effectively synthesizer first prove using examples insufficient eliminating certain undesired operations program undesired operations easy eliminate using syntactic operations made available gim second show effectiveness gim controlled user study compares gim standard pbe study shows participants strong preference time using granular feedback instead examples able provide granular feedback times faster times faster average acknowledgements research leading results received funding european union seventh framework programme grant agreement authors thank yifat chen solomon indispensable assistance getting user study ground hadas sloin yoav goldberg help analyzing data references aws albarghouthi sumit gulwani zachary kincaid recursive program synthesis international conference computer aided verification springer rajeev alur rastislav bodik garvit juniwal milo martin mukund raghothaman sanjit seshia rishabh singh armando emina torlak abhishek udupa synthesis dependable software systems engineering rajeev alur dana fisman rishabh singh armando results analysis arxiv preprint rajeev alur arjun radhakrishna abhishek udupa scaling enumerative program synthesis via divide conquer international conference tools algorithms construction analysis systems springer tobias anton induction generalizing tree traversal patterns lernen wissensentdeckung und adaptivitt lwa workshops saarbrcken shaon barman rastislav bodik satish chandra emina torlak arka bhattacharya david culler toward tool support interactive synthesis acm international symposium new ideas new paradigms reflections programming software onward acm blaine gilham liu smith westfold planware domainspecific synthesis schedulers proceedings ieee international conference automated software engineering ase ieee computer society washington usa http cfm rastislav bodik satish chandra joel galenson doug kimelman nicholas tung shaon barman casey rodarmor programming angelic nondeterminism acm sigplan notices vol acm rezaul chowdhury pramod ganapathi jesmin jahan tithi charles bachmeier bradley kuszmaul charles leiserson armando yuan tang autogen automatic discovery parallel recursive algorithms solving dynamic programs proceedings acm sigplan symposium principles practice parallel programming ppopp acm new york usa article pages doi http september feng ruben martins yuepeng wang isil dillig thomas synthesis complex apis proceedings annual acm symposium principles programming languages popl john feser swarat chaudhuri isil dillig synthesizing data structure transformations examples acm sigplan notices vol acm joel galenson philip reames rastislav bodik hartmann koushik codehint dynamic interactive synthesis code snippets proceedings international conference software engineering acm sumit gulwani automating string processing spreadsheets using inputoutput examples proceedings annual acm symposium principles programming languages popl acm new york usa doi http sumit gulwani synthesis examples interaction models algorithms symbolic numeric algorithms scientific computing synasc international symposium ieee tihomir gvero viktor kuncak ivan kuraj ruzica piskac complete completion using types weights acm sigplan notices vol acm daqing hou david pletcher evaluation strategies sorting filtering grouping api methods code completion software maintenance icsm ieee international conference ieee shachar itzhaki rohit singh armando kuat yessenov yongquan charles leiserson rezaul chowdhury deriving dynamic programming algorithms using transformations international conference companion object oriented programming systems languages applications appear shachar itzhaky sumit gulwani neil immerman mooly sagiv simple inductive synthesis methodology applications sigplan doi http shachar itzhaky rohit singh armando kuat yessenov yongquan charles leiserson rezaul chowdhury deriving dynamic programming algorithms using transformations proceedings acm sigplan international conference objectoriented programming systems languages applications acm landauer masahito hirakawa visual awk model text processing demonstration tessa lau steven wolfman pedro domingos daniel weld learning repetitive procedures smartedit wish command giving users power instruct software sumit gulwani flashextract framework data extraction examples proceedings conference programming language design implementation michael boyle keshav pingali acm doi http feng liu nayden nedev nedyalko prisadnikov martin vechev eran yahav dynamic synthesis relaxed memory models acm sigplan notices adi omari sharon shoham eran yahav synthesis proceedings international conference software engineering icse austin usa may doi http osera steve zdancewic program synthesis acm sigplan notices vol acm robert paige symbolic finite european symposium programming springer daniel perelman sumit gulwani thomas ball dan grossman typedirected completion partial expressions acm sigplan notices vol acm nadia polikarpova ivan kuraj armando program synthesis polymorphic refinement types proceedings acm sigplan conference programming language design implementation acm oleksandr polozov sumit gulwani flashmeta framework inductive program synthesis acm sigplan notices veselin raychev martin vechev eran yahav code completion statistical language models acm sigplan notices vol acm calvin smith aws albarghouthi mapreduce program synthesis proceedings acm sigplan conference programming language design implementation acm armando program synthesis sketching proquest armando christopher grant jones rastislav bodik sketching concurrent data structures acm sigplan notices vol acm armando liviu tancau rastislav bodik sanjit seshia vijay saraswat combinatorial sketching finite programs acm sigops operating systems review september abhishek udupa arun raghavan jyotirmoy deshmukh sela milo martin rajeev alur transit specifying protocols concolic snippets acm sigplan notices martin vechev eran yahav greta yorsh synthesis synchronization popl acm symposium principles programming languages martin vechev eran yahav david bacon derivation concurrent garbage collection algorithms acm sigplan notices vol acm martin vechev eran yahav david bacon noam rinetzky cgcexplorer search procedure provably correct concurrent collectors acm sigplan notices vol acm chenglong wang alvin cheung rastislav bodik synthesizing highly expressive sql queries examples proceedings acm sigplan conference programming language design implementation acm ian witten dan tels learning text editing tasks examples watch mit press shanchan jerry liu jian fan automatic web content extraction combination learning grouping proceedings international conference world wide web acm hila peleg sharon shoham eran yahav
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coinductive equivalences probabilistic functional programs ugo dal lago davide sangiorgi michele alberti nov abstract study bisimulation context equivalence probabilistic contributions paper threefold firstly show technique proving congruence probabilistic applicative bisimilarity technique follows howe method technicalities quite different relying disentangling properties sets real numbers secondly show bisimilarity general strictly finer context equivalence coincidence two relations attained pure resulting equality induced trees generally accepted finest extensional equivalence pure lazy regime finally derive coinductive characterisation context equivalence whole probabilistic language via extension terms akin distributions may appear redex position another motivation extension operational semantics allows experiment different congruence technique namely logical bisimilarity introduction probabilistic models pervasive formidable tool dealing uncertainty incomplete information sometimes necessity rather option like computational cryptography secure public key encryption schemes need probabilistic nice way deal computationally probabilistic models allow probabilistic choice primitive designing algorithms way switching usual deterministic computation new paradigm called probabilistic computation examples application areas probabilistic computation proved useful include natural language processing robotics computer vision machine learning new form computation course needs available programmers accessible indeed various probabilistic programming languages introduced last years spanning abstract ones concrete ones inspired various programming paradigms like imperative functional even object oriented quite common scheme consists endowing deterministic language one primitives probabilistic choice like binary probabilistic choice primitives distributions one class languages copes well probabilistic computation functional languages indeed viewing algorithms functions allows smooth integration distributions playground nicely reflected level types monads matter fact many existing probabilistic programming languages designed around one incarnations like scheme allows write functions programs take functions inputs produce outputs focus paper operational techniques understanding reasoning program equality probabilistic languages checking computer programs equivalence crucial challenging problem equivalence two programs generally means programs behave manner context specifically two context equivalent convergence behavior terminate possible context finding effective methods context equivalence proofs particularly challenging languages bisimulation emerged powerful operational method proving equivalence programs various kinds languages due associated coinductive proof method useful behavioral relation resulting bisimulation bisimilarity congruence also sound respect context equivalence bisimulation transplanted onto languages abramsky version bisimulation called applicative bisimulation received considerable attention short two functions applicative bisimilar applications applicative bisimilar argument often checking given notion bisimulation congruence languages nontrivial case applicative bisimilarity congruence proofs usually rely howe method forms bisimulation proposed environmental bisimulation logical bisimulation goal relieving burden proof congruence accommodating language extensions work consider pure extended probabilistic choice operator context equivalence two terms means probability convergence contexts objective paper understand context equivalence bisimulation paradigmatic probabilistic language called paper contains three main technical contributions first proof congruence probabilistic applicative bisimilarity along lines howe method technique consists defining every relation terms howe lifting construction essentially definition ensures relation obtained lifting bisimilarity congruence latter proved bisimulation therefore coinciding applicative bisimilarity definitionally probabilistic applicative bisimulation obtained setting labelled markov chain top adapting coinductive scheme introduced larsen skou setting proof congruence construction closely reflects analogous constructions nondeterministic extensions novelties technical details proving resulting relation bisimulation particular proof key lemma essential ingredient howe method relies disentangling properties sets real numbers properties proved modeling problem flow network apply theorem congruence applicative bisimilarity yields soundness respect context equivalence easy corollary completeness however fails applicative bisimilarity proved finer subtle aspect also late early formulation bisimilarity choice operator two versions semantically different construction crucially relies late style second main technical contribution show presence functions probabilistic choice contexts gives context equivalence applicative bisimilarity maximal discriminating power pure proving pure context equivalence applicative bisimilarity coincide tree equality equates terms tree briefly llt llt equality generally accepted finest extensional equivalence pure lazy regime result sharp contrast happens nondeterministic interpretation choice absence choice context equivalence coarser llt equality third main contribution coinductive characterisation probabilistic context equivalence whole language opposed subset pure obtain result setting bisimulation game extension weighted formal sums terms akin distributions may appear redex position thinking distributions sets terms construction reminds reduction nondeterministic deterministic automata technical details however quite different language therefore faced congruence problem bisimulation formal sums may contain infinite number terms proof congruence bisimulation extended language experimented technique logical bisimulation method related method environmental bisimulation clauses applicative bisimulation modified allow standard congruence argument bisimulations firstorder languages bisimulation method exploited establish closure bisimilarity contexts bisimulation logical bisimilarities two key elements first bisimilar functions may tested bisimilar rather identical arguments precisely arguments context closure bisimulation use contexts necessary soundness secondly transition system deterministic least confluent bisimulation game also played internal moves probabilistic setting ordinary logical bisimulation game modified substantially formal sums represent possible evolutions running terms hence appear redex position allowing anywhere would complicate matters considerably also making resulting bisimulation proof technique cumbersome obligation redex position certain terms contrast basic schema logical bisimulation related terms used arguments bisimilar functions therefore end arbitrary positions solve problem moving coupled logical bisimulations bisimulation formed pair relations one terms extended formal sums bisimulation game played relations first relation used assemble input arguments functions another delicate point meaning internal transitions formal sums logical bisimilarity transition system formal sums evolve values finite number steps even number terms composing formal sum infinite satisfy requirements defining transition system extended terms top proof congruence coupled logical bisimilarity also exploits distribution bisimulation proof technique paper adopt evaluation results applicative bisimilarity transported onto contrast transporting results less clear leave future work see section details extended version paper details available related work research probabilistic functional languages far mainly focused either new programming constructs denotational semantics applications underlying operational theory ordinary known rich remained far largely unexplored section give pointers relevant literature probabilistic without hope exhaustive various probabilistic proposed starting pioneering work sahebdjahromi followed advanced studies jones plotkin works mainly focused problem giving denotational semantics probabilistic computation rather studying operational point view recently revamp line work introduction adequate sometimes also denotational models probabilistic variations pcf also another thread research various languages derived given types monadic style allowing way nicely model concrete problems like bayesian inference probability models arising robotics works however attack problem giving operationally based theory program equivalence nondeterministic extensions analysed typed calculi well untyped calculi emphasis works mainly apart cited authors closely follow testing theory modalities may must separately together ong approach inherits testing bisimulation elements definition applicative bisimulation follows larsen skou scheme fullyprobabilistic systems many forms probabilistic bisimulation introduced literature greater complexity usually due presence nondeterministic probabilistic behaviors continuous probability distributions see surveys contextual characterisations llt equality include multiplicities deadlock observable choice parallel composition applications characterisation figure approximation semantics operators contrast contextual derived bisimulation includes clause internal moves observe branching structures behaviours see survey observational characterisations trees preliminaries pure untyped probabilistic lambda calculus let denumerable set variables set term expressions terms defined follows operator probabilistic choice term meant behave either probability general construct computable real number derivable given universality see set free variables term indicated defined usual given finite set variables denotes set terms whose free variables among ones term closed equivalently substitution free occurrences def def denoted sometimes use identity term projector def purely divergent term terms given semantics following term value closed call set values values ranged metavariables like closed terms evaluates top single value partial value distribution function set value distributions distributions necessarily sum model possibility probabilistic divergence given value distribution support subset whose elements values attributes positive probability value distributions ordered pointwise form lower semilattice limits always exist given value distribution sum semantics closed term value distribution defined one ways explained recall though briefly lack space first step consists defining formal system deriving finite lower approximations semantics approximation semantics example derives judgments form term value distribution finite support see figure approximation semantics defined similarly derives judgments form noticeably simulate vice versa second step called semantics set least upper bound distributions obtained either two ways def sup sup notice every set distributions directed thus least upper bound value distribution expone expone exptwo exptwo expthree foldp expthree expone foldp foldp foldp figure three functions def example consider term elsewhere well empty distribution distribution assigns semantics terms satisfies useful equations lemma lemma proof see detailed proofs interested context equivalence probabilistic setting typically qualitative scenario non deterministic one terms considered context equivalent converge diverge need take account quantitative information definition context preorder equivalence expression stands term converges probability context preorder stipulates implies every closing context equivalence induced probabilistic context equivalence denoted remark types open terms results paper stated untyped language adapting language straightforward use integers booleans recursion examples moreover results often stated closed terms generalized open terms expected manner paper context equivalences preorders defined open terms similarities defined closed terms intended extended open terms requiring usual closure substitutions example give basic examples probabilistic programs analyse using coinductive techniques introduce later paper consider functions expone exptwo expthree figure written language extended probabilistic choice also seen terms typed probabilistic integers recursion akin term expone takes function natural number input proceeds tossing fair coin captured binary infix operator depending outcome toss either calls recursively calls fed identity natural number program expone evaluates geometric distribution assigning probability positive natural number similar effect obtained exptwo takes input modifying along evaluation function expthree complicated least apparently understand behavior one first look auxiliary function foldp two natural numbers two functions foldp reduces following expression term expthree works forwarding three arguments foldp fourth argument recursive call expthree however replaced number greater equal chosen according geometric distribution functions expressed using combinators see soon expone exptwo expthree context equivalent whenever natural number probabilistic bisimulation section recall definition basic notions bisimulation labelled markov chains following larsen skou section adapt form bisimilarity probabilistic combining abramsky applicative bisimilarity definition labelled markov chain triple countable set states set labels transition probability matrix function following normalization condition holds usual stands whenever equivalence relation denotes quotient modulo set equivalence classes modulo given binary relation reflexive transitive closure denoted definition given labelled markov chain probabilistic bisimulation equivalence relation implies every every note probabilistic bisimulation definition equivalence relation means principle allowed define probabilistic bisimilarity simply union probabilistic bisimulations matter fact given two equivalence relations necessarily equivalence relation following standard way overcome problem lemma collection probabilistic bisimulations also reflexive transitive closure probabilistic bisimulation def proof let fix fact equivalence relation proved follows reflexivity easy reflexive definition symmetry consequence symmetry relations states every symmetry easily get consequence transitivity easy transitive definition please notice means equivalence class respect union equivalence classes respect suppose states every obtain concludes proof lemma allows define largest probabilistic bisimulation called probabilistic bisimilarity def probabilistic bisimulation indeed lemma probabilistic bisimulation claim inclusion obvious way around follows probabilistic bisimulation hence included union notion probabilistic simulation preorders play role equivalence relations given labelled markov chain probabilistic simulation preorder relation implies every every usual stands namely set lemma adapted probabilistic simulations proposition collection probabilistic simulations also reflexive transitive closure probabilistic simulation def proof fact preorder follows construction probabilistic simulation must satisfy following property implies every every let states every rjn consequence every every holds rjn since definition rjn follows concludes proof def consequence define similarity simply probabilistic simulation symmetric probabilistic simulation probabilistic bisimulation lemma symmetric probabilistic simulation probabilistic bisimulation proof symmetric probabilistic simulation definition also preorder reflexive transitive relation therefore equivalence relation probabilistic bisimulation must also satisfy property srt implies every every fact simulation follows srt every every since class holds latter follows get way around symmetric property implies every label every hence completes proof moreover every probabilistic bisimulation inverse probabilistic simulation lemma probabilistic bisimulation rop probabilistic simulation proof let prove probabilistic simulation first consider set equivalence subclasses module contained formally equivalence class modulo please observe consequence thus result easily follows every every finally rop also probabilistic simulation consequence symmetric property fact proved probabilistic simulation contrary nondeterministic case however simulation equivalence coincides bisimulation proposition coincides proof fact subset straightforward consequence symmetry property fact lemma every probabilistic bisimulation also probabilistic simulation let prove subset former probabilistic bisimulation course equivalence relation preorder consider equivalence class modulo define following two sets states def def observe disjoint set states whose union precisely moreover notice closed respect one hand hand see previous point otherwise would meaning equivalence class modulo thus contradiction consequence given follows similarly thesis technical reasons become apparent soon convenient consider markov chains state space partitioned disjoint sets way comparing states coming different components possible remember disjoint union family sets defineduas set states labelled markov chain disjoint union one wants simulation relations compare elements coming implies case say underlying labelled markov chain multisorted probabilistic applicative bisimulation howe technique section notions similarity bisimilarity introduced spirit abramsky work applicative bisimulation definitionally consists seeing operational semantics labelled markov chain giving larsen skou notion simulation states terms labels two kinds one either evaluate term obtaining distribution values apply term value resulting bisimulation probabilistic applicative bisimulation shown congruence thus included probabilistic context equivalence done generalization howe technique methodology get congruence results presence functions applied probabilistic calculi far formalizing probabilistic applicative bisimulation requires care usual two values defined bisimilar every bisimilar rather want compare two arbitrary closed terms simplest solution consists following larsen skou stipulate every equivalence class modulo bisimulation attributed measure values thus treated two different ways terms values reason corresponds two states underlying markov chain definition seen multisorted labelled markov chain denote labels either closed terms model parameter passing models evaluation please observe states labelled markov chain defined elements disjoint union two distinct states correspond value avoid ambiguities call second one one coming distinguished value want insist fact value distinguished indicate define transition probability matrix follows every term every distinguished value def every term every distinguished value def cases returns terms seen states interact environment performing distinguished values take closed terms parameters simulation bisimulation relations defined labelled markov chain even strictly speaking binary relations often see restrictions formally probabilistic applicative bisimulation pab simply probabilistic bisimulation way one define probabilistic applicative bisimilarity denoted similarly probabilistic applicative simulation pas probabilistic applicative similarity denoted remark early late technically distinction terms values definition means bisimulation late style bisimulations concurrent languages late indicates explicit manipulation functions clause input actions functions chosen first later input value received taken account used contraposition early style order quantifiers exchanged choice functions may depend specific input value received setting adopting early style would mean transitions setting probabilistic bisimulation top resulting transition system leave future work study comparison two styles paper stick late style easier deal especially howe technique previous works applicative bisimulation nondeterministic functions also focus late approach remark defining applicative bisimulation terms multisorted labelled markov chains advantage recasting definition familiar framework importantly formulation useful dealing howe method spell explicit operational details definition probabilistic applicative bisimulation seen equivalence relation whenever equivalence class probability reaching value two terms values say special treatment values clause motivates use multisorted labelled markov chains definition usual one way show two terms bisimilar prove one relation containing pair question pab terms semantics indistinguishable lemma binary relation pab proof fact equivalence easily follows reflexivity symmetry transitivity equality must satisfy following property closed terms every notice clearly every hypothesis moreover must satisfy following property cloned values every close term every hypothesis implies clearly every hypothesis concludes proof please notice previous result yield nice consequence every indeed lemma tells latter terms semantics conversely knowing two terms similar means knowing quite lot convergence probability lemma adequacy moreover bisimulation proof concludes proof example bisimilar terms necessarily semantics one reason using bisimulation proof method basis prove equalities among functions let consider following terms def def semantics differ every value otherwise otherwise nonetheless prove indeed every three terms semantics consider equivalence class distinguished values modulo includes three distinguished values otherwise let prove following technical result moreover stipulate bisimilar distinguished values bisimilar values lemma iff iff proof first double implication obvious matter distinguished values value terms let detail second double implication fact pab implies definition every every suppose contradiction latter means exists according definition iff otherwise since derive implies although distinguished value starting reasoning made still holds get latter form due hypothesis hypothesis equivalence class derive absurd need prove every every supposing holds first let rewrite respectively hypothesis reasoning made every otherwise otherwise proves thesis result holds probabilistic applicative bisimulation congruence section prove probabilistic applicative bisimulation indeed congruence sibling precongruence overall structure proof similar one howe main idea consists defining way turn arbitrary relation possibly open terms another one way satisfies simple conditions pre congruence including key step prove indeed simulation view proposition considering similarity suffices convenient work generalizations relations called sets triples form thus relation pair corresponding include recall applicative similarity extended open terms considering closing substitutions given write said compatible iff four conditions hold pfin pfin pfin pfin often use following technical results establish particular hypothesis lemma let consider properties pfin pfin transitive together imply proof proving means show hypothesis imply using first one steady term follows similarly using second one steady term follows conclude transitivity property lemma let consider properties pfin pfin transitive together imply proof proving means show hypothesis imply using first one steady term follows similarly using second one steady term follows conclude transitivity property notions equivalence relation preorder straightforwardly generalized compatible equivalence relation respectively preorder said congruence respectively precongruence bisimilarity congruence bisimilar whenever context words terms replaced equivalent ones context crucial notion equivalence expected pass proving bisimulation congruence may nontrivial underlying language contains functions also case proving inspecting operational semantics involved terms indeed possible method fails involved contexts contain applications particular proving requires probabilistic applicative bisimilarity stable respect substitution bisimilar terms hence necessarily general called term substitutive pfin note also reflexive implies say closed satisfies way open extension defined closed figure howe lifting unfortunately directly prove enjoy substitutivity property hard thus proceed indirectly defining starting new relation called howe lifting property construction proved equal actually howe lifting relation defined rules figure reader familiar howe method sense indeed precisely definition one finds realm nondeterministic language terms facilitates first part proof indeed one already knows preorder compatible includes since properties already known see depend shape terms operational semantics lemma reflexive compatible proof need prove hold proving means show pfin since reflexive pfin thus conclude formally proving means show pfin since reflexive get moreover hypothesis thus conclude holds formally proving means show pfin since reflexive get moreover hypothesis thus conclude holds formally proving means show pfin since reflexive get moreover hypothesis thus conclude holds formally concludes proof lemma transitive imply proof prove statement inspection last rule used derivation thus structure variable say holds hypothesis last rule used thus get additional hypothesis transitivity deduce conclude latter obtaining formally xrn say holds hypothesis last rule used thus get additional hypothesis transitivity deduce conclude latter obtaining formally application say holds hypothesis last rule used thus get additional hypothesis transitivity deduce conclude latter obtaining formally probabilistic sum say holds hypothesis last rule used thus get additional hypothesis transitivity deduce conclude latter obtaining formally concludes proof lemma reflexive implies proof prove inspection structure variable say holds hypothesis conclude latter obtaining formally say holds hypothesis moreover since reflexive implies compatible reflexive conclude formally application say holds hypothesis reflexivity hence get latter conclude formally probabilistic sum say holds hypothesis reflexivity hence get latter conclude formally concludes proof moreover preorder closed lifted relation substitutive reflexivity implies compatibility lemma follows reflexive hence closed lemma reflexive transitive closed term substitutive hence also closed proof show pfin prove latter induction derivation thus structure variable either latter case suppose hypothesis holds way deduce rule hence fact closed obtain equivalent finally lemma conclude equivalent holds otherwise holds way deduce latter rule hence fact closed obtain equivalent lemma deduce following equivalent thus holds say holds hypothesis way deduce latter rule follows let denote induction hypothesis get moreover fact closed obtain holds deduce following equivalent thus holds application say holds hypothesis way deduce latter rule follows induction hypothesis get moreover fact closed obtain holds deduce following equivalent thus holds probabilistic sum say holds hypothesis way deduce latter rule follows induction hypothesis get moreover fact closed obtain conclude following equivalent thus holds concludes proof something missing however conclude precongruence namely transitivity also follow howe building transitive closure relation defined rules figure easy prove compatible closed lemma compatible proof need prove hold figure transitive closure proving means show pfin since compatible therefore reflexive holds hence follows proving means show pfin prove induction derivation looking last rule used base case last rule thus holds since compatible follows conclude applying latter obtaining otherwise last rule used get hold induction hypothesis conclude applying latter two obtaining proving means show pfin firstly prove following two characterizations particular prove details since similarly provable prove induction derivation looking last rule used base case last rule get holds using compatibility property follows conclude applying latter obtaining otherwise last rule used get hold induction hypothesis along since compatible reflexive holds induction hypothesis along latter get conclude applying obtaining let focus original statement prove induction two derivations name respectively looking last rules used four possible cases four combinations permit conclude observe first three cases addressed hence remains prove last case derivations concluded applying rule according rule definition get two additional hypothesis derivation particular get hold similarly get hold double induction hypothesis firstly secondly get respectively conclude applying latter obtaining proving means show pfin detail proof since boils partial sums play role applications concludes proof lemma closed proof need prove closed pfin prove latter induction derivation looking last rule used base case last rule get holds since closed follows conclude applying latter obtaining otherwise last rule used get hold induction hypothesis conclude applying latter two obtaining important note transitive closure already howe lifted relation preorder starting relation lemma preorder relation proof need show reflexive transitive course transitive closure transitive relation moreover since reflexive lemma reflexive compatible lemma first half story also need prove simulation already know preorder following lemma gives missing bit lemma key lemma every holds proof lemma delicate discussed next section lemma using standard argument derive needed substitutivity results ultimately important result section theorem precongruence relation proof prove result observing precongruence showing first lemma lemma ensure compatible lemma tells preorder consequence precongruence consider inclusion lemma definition transitive closure operator follows show converse proving included relation probabilistic applicative simulation therefore contained largest one particular since closed lemma lemma suffices show latter closed version terms cloned values acts like terms given two cloned values iff since already know preorder thus preorder remain checked following two points every holds let proceed induction structure proof base case last rule get holds particular lemma last rule used obtain hold induction hypothesis get course consequence satisfied every every holds means whenever ultimately hand concludes proof corollary congruence relation proof equivalence relation definition particular symmetric relation since proposition also compatible consequence theorem proof key lemma already said lemma indeed crucial step towards showing probabilistic applicative simulation precongruence proving key lemma turns much difficult deterministic nondeterministic cases particular case application relies another technical lemma going give proved tools linear programming combinatorial problem face proving key lemma actually decontextualized understood independently suppose sets figure disentangling sets whose elements labelled real numbers example could situation like one figure sake simplicity labels indicated fix def three real numbers holds def def routine check every sum labels elements let observe course possible turn three sets three disjoint sets contains copies elements whose labels however obtained splitting ones original elements examples sets figure superpose three sets obtain venn diagram started quite remarkably however examples figure additional property namely every holds show finding sets satisfying properties always possible even arbitrary suppose suppose real number defined every holds said probability assignment always possible disentangle probability assignments answer positive following formulation theorem theorem flow network value maximum flow equal capacity minimum cut def lemma disentangling probability assignments let probability assignment every nonempty every following conditions hold every holds every holds proof every probability assignment let define flow network digraph def def distinguished source target respectively composed three kinds edges every assigned capacity every nonempty assigned capacity every nonempty assigned capacity prove following two lemmas together entail result lemma admits flow summing exist conditions hold def proof let fix idea start flow value input source hypothesis admitted maximum one get split portions going singleton vertices every value afterwards every vertex values flows incoming edges summed distributed outgoing adges one wishes thanks conservation property flow formally flow turned function defined follows def every possibly nonnull component exactly every nonempty soon defined ingoing edges define outgoing ones splitting component want possible course flow ingoing outgoing values def formally let fix indicate def component every set every course similar definition given every nonempty notice way defined guarantees sum components always equal every every nonempty fix ratio component first one hand every nonempty obviously less equal hence condition holds component flow since satisfies capacity conservation constraints moreover structured way component whenever consequence since satisfies capacity constraint every condition holds lemma admits flow summing proof prove result means theorem particular prove def capacity cut must least cut said degenerate easy verify every degenerate cut capacity greater equal thus greater equal consequence concentrate cuts prove def def capacity least given two cuts say iff given call cut canonical def unique please observe definition capacity least forward edges connecting elements going singletons plus edges going sum capacities edges greater equal hypothesis need prove following two lemmas lemma every nondegenerate def def proof let moreover let element def consider verify cut looking indeed obtained hypothesis adding supersets course moreover holds since implies contains supersets well also easy check fact process constructing lose forward edges coming since singleton edge coming element since lemma every def def proof let prove result induction def thesis follows lemma induction hypothesis follows thus two lemmas permit conclude indeed every cut course possibly empty set let consider canonical one hand since lemma hence concludes main proof coming proof lemma widely often implicitly use following technical lemmas denote set distinguished values lemma every proof concludes proof lemma every proof definition therefore lemma remark throughout following proof implicitly use routine result stating implies every property needed latter precisely reason formulated multisorted labelled markov chain consists distinguished values nothing proof lemma equivalent proving every following implication holds induction structure proof course every value proof necessarily ends follows let subset inequality trivially holds contrary consider set terms relation via every hold consequence consequence property see words lemma application obtained follows moreover proof must end follows since induction hypothesis get every holds let take look distribution since finite distribution sum actually sum finitely many summands let support time put form amenable treatment lemma let consider sets term associate probability scope lemma since induction hypothesis know every conclude every real numbers rnu riu conclude riu hqi riu hqi whenever know lemma apply inductive hypothesis derivations hqi obtaining every riu riu riu riu thesis probabilistic sum obtained follows moreover proof must end follows since induction hypothesis get every holds similarly since induction hypothesis get every holds let take look distribution idea prove every holds fact since latter would imply thesis induction hypothesis lemma concludes proof context equivalence formally introduce probabilistic context equivalence prove coarser probabilistic applicative bisimilarity definition context syntax tree unique hole generated follows denote results filling hole def def def def def def also write context resulting replacing occurrence syntax tree tree continue keep track free variables sets variables inductively define subsets contexts following rules use double indexing indicate sets free variables filling hole term two following properties explain idea lemma proof induction derivation rules lemma proof induction derivation rules let recall definition context preorder equivalence definition probabilistic context preorder respect evaluation given iff implies probabilistic context equivalence denoted holds iff lemma context preorder precongruence relation proof proving precongruence relation means prove transitive compatible start proving transitive every pfin every imply definition latter boils prove following hypotheses every implies every implies imply easily apply first hypothesis second hypothesis equal get thesis prove compatible relation starting property trivially valid particular must show every pfin every every definition latter boils prove following hypotheses every implies imply since let consider context lemma context form please note definition therefore second hypothesis rewritten thus follows moreover observe nothing else since proved transitive prove property showing hold fact recall lemma latter two together imply former particular prove must show every pfin every definition latter boils prove following hypothesis every implies imply since let consider context lemma context form please note definition therefore second hypothesis rewritten thus follows moreover observe nothing else detail proof follows reasoning made considering context proving follows pattern resulted fact lemma together imply detail proofs since proceed reasoning made considering appropriate context time concludes proof corollary context equivalence congruence relation proof straightforward consequence definition lemma let compatible proof induction derivation due thus result trivially holds last rule used induction hypothesis holds since compatible relation follows last rule used induction hypothesis holds since compatible relation follows definition means hence result holds case rule holds similar reasoning last rule used induction hypothesis holds since compatible relation follows definition means hence result holds case rule holds similar reasoning concludes proof lemma proof since proposition implies since theorem precongruence hence compatible relation follow lemma theorem pfin every implies proof every follows lemma lemma latter implies means particular iff equivalent definition converse inclusion fails counterexample described following def def example hence prove two terms context equivalent means relation shown coincide context equivalence context lemma proved howe technique see section section supplementary details counterexample context free context equivalence present way treating problem concrete representations contexts right basically work classes contexts let dispense entirely work instead coinductive characterization context preorder equivalence phrased terms definition said adequate every implies let indicate collection compatible adequate let def turns context preorder largest compatible adequate let proceed towards proof latter lemma every proof need show compatible adequate obviously adequate every exists term def note identity relation reflexive particular satisfies compatibility property proving means show hypothesis follows exists term since hence compatible holds latter together imply proving means show hypothesis follows exist two terms one hand hand since hence compatible holds two together imply proceeding fashion one easily prove property lemma adequate proof suffices note property adequate closed taking unions relations indeed adequate relations easy see union every couple either either way implying adequate lemma precongruence proof need show transitive compatible relation lemma implies transitive let prove also compatible note identity relation implies reflexivity hence particular satisfies property clear property closed taking unions relations satisfies true properties lemma respectively lemma suffices show satisfies obvious contrary properties clearly closed taking unions relations concludes proof corollary largest compatible adequate proof straightforward consequence lemma lemma lemma coincide proof definition immediate adequate moreover lemma precongruence therefore implying let prove converse since lemma precongruence hence compatible relation holds every every implies therefore every every implies fact adequate definition concludes proof simpler characterization kind program equivalence interested context equivalence fact prove two notions coincide context equivalence envisages quantification contexts relaxes constraint restricted class contexts without affecting associated notion program equivalence class contexts evaluation contexts particular use different representation evaluation contexts seeing stack evaluation frames definition set frame stacks given following set rules nil set free variables frame stack easily defined union variables occurring free terms embedded given set variables define set frame stacks whose free variables given frame stack term define term follows def enil def define binary relation form pairs form sequences pairs finally define formal system whose judgments form whose rules follows empty nil value term expression stands real number lemma closed frame stacks closed iff particular holds iff nil proof first recall work dal lago zorzi provides various inductive semantics either equivalent result deduced following properties proof induction derivation looking last rule used def empty rule used consider empty distribution observe sen rule value rule used implies nil value say def consider pdistribution observe enil svn rule course term rule used obtained every induction hypothesis exist eti let proceed cases according structure implying def consider distribution observe hence smn rule moreover condef sider distribution observe hence smn rule moreover consider distribution def observe implies moreover concludes proof proof induction derivation looking last rule used refer inductive schema inference rules gave semantics sen rule used every every empty rule course svn rule used value say consider def def nil definition enil value rule nil hence svn rule used every induction hypothesis every exist eti let proceed cases according structure def hence consider nil def definition enil term rule nil nil induction hypothesis result nil latter implies moreover def hence consider nil def definition enil term rule nil hypothesis nil nil induction result nil nil latter implies moreover every hence def def consider nil definition enil term rule nil nil nil induction hypothesis result nil every latter moreover implies concludes proof generally speaking two properties prove following double implication sup sup sup concludes proof given define iff every relation extended relation open terms usual way moreover stipulate ciu ciu iff since preorder proving precongruence boils show following implication indeed converse implication consequence lemma obvious reflexivity relation extend howe construction frame stacks natural way nilrh nil srh lemma every pfin holds proof need show hold since defined open terms taking closing suffices show result close let start prove every close frame stack latter obvious consequence fact reduces let look details distinguishing two cases nil implies proceed similarly similarly prove converse let fix distinguish two cases nil holds empty rule otherwise term term implies proceed similarly concludes proof lemma every proof induction structure proof looking last rule used trivially nil nil since follows latter implies nil thesis otherwise term rule used suppose following situation term let distinguish following cases definition follows form following ciu induction hypothesis conclude observe consequence thesis easily follows given get follows double induction hypothesis follows latter together imply thesis easily follows given case left hence get follows holds substitutivity follow holds induction hypothesis follows thesis easily follows concludes proof theorem pfin iff proof since defined open terms taking closing lemma closed suffices show result closed since reflexive lemma follows compatible hence reflexive taking lemma conclude implies remarked lemma latter entails implies compatible moreover lemma immediately follows also adequate thus contained ciu largest compatible adequate actually lemma follows ciu contained particular latter means implies first please observe since context preorder compatible lemma adequacy property lemma latter implies ultimately holds implies let take account general case open terms compatibility property follows hence fact compatible established part proof lemma every suitable holds corollary coincides proof straightforward consequence theorem proposition coincide proof prove def def easily verified let concentrate prove every let distinguish three cases nil reduced last two pairs reduced consequence proceed similarly observe consequence concludes proof example consider programs example terms expone exptwo differ former performs probabilistic choices natural numbers obtained applying function argument latter choices done functional level argument functions provided later stage consequence two terms applicative bisimilar reason akin inequality terms example contrast bisimilarity expone expthree natural number intuitively holds expone expthree evaluate single term fed function start evolving genuinely probabilistic way second argument provided point two functions evolve different ways semantics sense section lemma bisimulation one use equivalence generated relation expone expthree using body expone expthree respectively discriminating power probabilistic contexts show applicative bisimilarity context equivalence collapse tested terms pure deterministic words probabilistic choices brought terms inputs supplied tested functions applicative bisimilarity context equivalence yield exactly discriminating power show prove pure relations coincide tree equality equates terms tree briefly llt llt lazy variant trees briefly popular tree structure correctly express computational content strong regime fail lazy one instance term unsolvable identical lazy regime would always distinguish hence different llt llt introduced longo developing original idea levy tree coinductively constructed follows def def unsolvable order unsolvable order finally principal head normal form tree root subtrees defined coinductively llt course infinite write iff def example let unsolvable order consider terms def def terms used prove results canonical model lazy abramsky ong show model convergence test definable operator receives argument would return identity function supplied argument convergent would diverge otherwise convergence test distinguish two terms reduces abstraction whereas diverges however pure make distinction two terms also different trees although convergence test operator definable separated using probabilities running context would feed argument whereas example abramsky canonical model coarser llt equality instance def def terms different llt equal abramsky model hence equal context equivalence separated context equivalence def instance using context since whereas already know full applicative bisimilarity implies context equivalence hence prove pure two equivalences collapse llt equality suffices prove pure terms implies implies first implication obtained variation technique powerful methodology separation results often employed proofs local structure characterisation theorems exploit inductive characterisation llt equality via stratification approximants definition key lemma shows difference trees two within level observed suitable context probabilistic write abbreviation term denote term usually called permutator degree permutators play key role technique variant play pivotal role lemma term degree either exists finally function positive integers function degree giving main technical lemma useful auxiliary concepts definitions rely two notions reduction means reduces one step probability matter fact either obtained composing zero times multiplying corresponding real numbers dealing pure abbreviated slight abuse notation also denote lazy reduction relation pure open terms specialised form probabilistic choice thought new syntactic construct thus set pure extended operator derived operator operational rules expected ones restriction applied called following need following lemma lemma let closed terms suppose also proof course integer proof key lemma makes essential use characterization form relation definition open bisimulation relation pure open bisimulation implies exist every conversely reductions open bisimilarity written union open bisimulations open bisimulation advantage easily providing notion approximation definition approximants set def exists exist conversely reductions please observe lemma pure relations coincide ready state prove key technical lemma lemma suppose let free variables integers mxr permutator functions fxr closed terms following holds fxr mxr fxr mxr proof proof proceeds induction least term free variables stand also write sequence occurrences finally term write denote fact converge basic case cases consider symmetric ones analogous case one two terms diverges easy take permutator degree values integers def permutation functions irrelevant set mtf since also nsf since therefore term end head term assume without loss generality take values def integers permutation functions irrelevant set mtf since also nsf since therefore term end head term values integers permutator def def functions irrelevant set empty sequence whereas inductive case two cases look induction variables integers permutator functions mif nif redefine necessary make sure def set msf mif whereas nsf nif depending whether contains case derivations rule used lemma inductive assumption nif derive induction variables integers permutator functions def set term defined except variable left uninstantiated whereas lemma inductive hypothesis derive concludes proof fact technique actually works implies discriminating power probabilistic contexts least strong one llt corollary implies show llt equality included probabilistic applicative bisimilarity proceed follows first define refinement latter essentially one observe probabilistic choices consequence underlying bisimulation game may ignore probabilities obtained notion equivalence strictly finer probabilistic applicative bisimilarity advantage refinement inclusion llt equality refinement inclusion latter probabilistic applicative bisimilarity turn relatively easy prove direct proof inclusion llt equality probabilistic applicative bisimilarity would harder would required extending notion tree reasoning substitution closures trees definition relation strict applicative bisimulation whenever implies converse strict applicative bisimilarity union strict applicative bisimulations two terms llt passing argument produces exactly choice structure intuitively whenever first term finds copy head position also second find lemma strict applicative bisimulation terms strict applicative bisimilar distinguished applicative bisimilarity proper since requirements induced latter less strict ones former imposes lemma strict applicative bisimilarity included applicative bisimilarity since know pure deterministic included lemma lemma included theorem latter included corollary conclude corollary relations coincide coupled logical bisimulation section derive coinductive characterisation probabilistic context equivalence whole language opposed subset section need manipulate formal weighted sums thus work extension weighted sums may appear redex position advantage formal sums transition system extended language deterministic closed term value exactly one possible internal transition make possible pursue logical bisimulation method congruence bisimilarity proved using standard induction argument contexts refinement method handling probabilities called coupled logical bisimulation uses pairs relations need distinguish ordinary terms terms possibly containing formal sums technically proof congruence first prove correspondence transition system extended terms original one derive techniques coupled logical bisimulations needed following proofs finally show coupled logical bisimulations preserved closure first relation context closure second relation evaluation context preferred follow logical bisimulations rather environmental bisimulations former admit simpler definition latter pair terms enriched environment extra set pairs terms moreover unclear environments one also considers formal sums leave future work formal sums tool representing behaviour running terms thus terms formal sums results closed terms interest however characterization contextual equivalence coupled logical bisimulation also holds open terms notation terminology write extension formal sums may appear redex position terms defined follows hmi formal sum hmi countable possibly empty set indices use binary formal sums formal sums ranged metavariables like value abstraction hmi formally summed value values ranged hmi hmj disjoint abbreviates hmr similarly every hmi hhj stands hmi hmi write real number stands hmi set closed terms partial value distribution sense section seen formal sum similarly formal sum hmi mapped distribution indicate reduction defined rules figure rules terms written given top operational semantics defined section invoked premise rule spc value reduction relation deterministic strongly normalizing use reflexive transitive closure lemma shows agreement new reduction relation original one lemma value proof one first show one reasons double induction induction transition induction exploiting determinism spc hmi hdi figure reduction rules context equivalence bisimulation certain terms formal sums may appear redex position ordinary terms terms contrast may appear arbitrary position extending context equivalence therefore distinguish two cases moreover main objective characterisation context equivalence set somewhat constrained context equivalence contexts may contain formal sums thus contexts contexts call simple contexts whereas call general context unconstrained context term hole may appear places term expected including within formal sum later see allowing general contexts affect resulting context equivalence terms possibly containing formal sums tested evaluation contexts contexts form write recall unique given definition context equivalence two context equivafs lent written closing simple contexts iff two equivalent written ufs closing evaluation contexts iff virtue lemma context equivalence coincides context equivalence introduce bisimulation yields coinductive characterisation context equivalence also equivalence coupled relation pair intuitively place pairs terms preserved contexts preserved evaluation contexts coupled relation write union coupled relations defined componentwise coupled relations coupled def def relation relation context closure set closed terms form context definition coupled relation coupled logical bisimulation whenever formally summed value converse coupled logical bisimilarity union coupled logical bisimulations hence union first component coupled logical bisimulations similarly coupled bisimulation bisimulation game played pairs however first relation relevant inputs tested functions built using clause definition actually also pairs tested coupled relations must values produced bisimulation game coupled bisimulation formal sums plain require formal sums appear redex position terms used arguments bisimilar functions therefore end arbitrary positions see another aspect relevance proof technique logical bisimulation allows prove substitutivity bisimilarity arbitrary contexts pairs terms pairs proof technique allows derive preservation evaluation contexts proof congruence coupled logical bisimilarity push many terms possible first relation first relation large possible however proofs bisimilarity concrete terms first relation may small possibly singleton even empty bisimulation clauses become similar applicative bisimulation inputs tested function almost identical summing coupled logical bisimulation use two relations gives flexibility ordinary logical bisimulation depending needs tune size first relation possible aspects coupled logical bisimilarity specific version would require modifications remark coupled logical bisimulation first relation used construct inputs tested functions formally summed values produced bisimulation game second relation therefore first relation may thought global global pair terms bisimulation game played consequence coupled logical bisimulation remains quite different environmental bisimulation environment constructing inputs local pair tested terms coupled logical bisimulation follows ordinary logical bisimulation one global environment ordinary logical bisimulation however global environment coincides set tested terms similarity logical bisimulation also revealed associated functional contrast functional associated environmental bisimulation monotone see remark example use coupled logical bisimulation revisit counterexample completeness applicative bisimilarity respect contextual equivalence example consider terms example show hence also contextual equivalence corollary recall def def def def terms set contain pair interests contain pairs set pairs identical components namely empty formal sum thus coupled logical bisimulation main challenge towards goal relating coupled logical bisimilarity context equivalence substitutivity bisimulation establish latter exploiting techniques bisimulation give definitions techniques omitting statements soundness first technique allows drop bisimulation game silent actions definition bisimulation coupled relation coupled logical bisimulation whenever following holds lemma coupled logical bisimulation coupled logical bisimulation reduction computation performed level formal sums reflected coupled bisimulation application values formal sums following technique allows computation application input values also ordinary terms definition extract formal sum term using function inductively follows def hmi whenever hmi def def definition coupled relation bisimulation formal sums whenever either one bisimulation clauses definition applies one following clauses applies according definition bisimulation game coupled relation given pair either choose follow bisimulation game original definition contain formal sums try one new clauses advantage first new clause allows make split derivatives original terms advantage two new clauses allow directly handle given without using operational rules figure therefore without introducing formal sums understand def def def first clause suppose def sufficient ensure lemma bisimulation formal sums coupled logical bisimulation proof show coupled relation def hhi hki either hmi hni bisimulation apply lemma key point show whenever roughly reason tree whose nodes pairs terms produced bisimulation game root pair proviso node pair values pair one child namely certain paths tree may divergent reach leaf give formal sums produce thus write hzi hyi represent exactly multiset leaves tree together probability path reaching leaf using proof technique prove necessary substitutivity property bisimulation use techniques way bisimulation defined particular presence clause possibility using pairs bisimulation construct inputs functions make possible use standard argument induction contexts lemma bisimulation context closure def def bisimulation formal sums corollary implies implies evaluation contexts using lemma prove inclusion context equivalence corollary moreover converse corollary proved exploiting simple properties ufs transitivity inclusion ufs lemma proof implies ufs hence ufs lemma ufs implies proof follows definition ufs transitivity lemma lemma hmi hni proof suppose hmi hni show hzi similarly hyi result follows theorem proof take coupled relation def def ufs show bisimulation clause one uses lemma transitivity ufs clause consider term ufs definition take arguments sufficient since lemma lemma hence also also holds coupled logical bisimilarity preserved formal sum construct implies hmi hni consequence context equivalence defined general contexts set simple contexts definition remark functional induced coupled logical bisimulation monotone instance pair terms may satisfy bisimulation clauses input functions may taken larger relation recall coupled relations pairs relations hence operations coupled relations union inclusion defined however corollary theorem tell indeed largest bisimulation namely pair logical well environmental bisimulations techniques particularly important relieve burden proving concrete equalities powerful technique languages contexts present form contexts combined version logical bisimilarity relation write rcfs closure relation general closing contexts definition coupled relation coupled logical bisimulation contexts whenever following holds cfs soundness proof first derive soundness context technique whose proof turn similar lemma technique definition already allows context manipulation need technique proof technique example seen terms expone exptwo example applicative bisimilar show context equivalent proving coupled bisimilar sketch proof employ technique definition use def def coupled relation expone exptwo def def expone exptwo coupled logical bisimulation contexts interesting part matching argument terms upon receiving argument yield summed values rcfs beyond reduction far studied problem giving sound sometime complete coinductive methods program equivalence probabilistic endowed reduction one may wonder whether obtained adapted notions reduction particular reduction operational semantics since construction labelled markov chain somehow independent underlying operational semantics defining probabilistic applicative bisimulation effortless proofs congruence bisimilarity soundness paper also transplanted defining multisorted labelled markov chain strict regime one recall functions applied values definition seen multisorted labelled markov chain denote please observe contrary gave definition semantics labels either values model parameter passing models evaluation define transition probability matrix follows every term every distinguished value def every value every distinguished value def cases returns similarly case one define probabilistic applicative simulation bisimulation notions probabilistic simulation bisimulation way one define probabilistic applicative bisimilarity denoted probabilistic applicative similarity denoted proving precongruence follows reasoning outlined lazy regime course one must prove key lemma first lemma every holds statement proof particularly different one provided lemma delicate case obviously application due operational semantics takes account also distribution values parameter reduces anyway one prove implying context equivalence restrict attention pure section strongly relying evaluation llt reflect term equivalence lazy regime leave task generalizing results eager evaluation future work conjecture setting probabilistic choice alone give contexts discriminating power probabilistic bisimulation similarly investigated version coupled logical bisimilarity current proofs rely appearance formal sums redex position constraint would probably lifted comparison nondeterminism syntactically identical eponymous language introduced liguoro piperno semantics present however quantitative course great impact context equivalence nondeterministic setting one observes possibility converging diverging terms different convergence probabilities considered different essential way actually nondeterministic context equivalence probabilistic context equivalence incomparable example terms context equivalent must sense probabilistically take conversely probabilistically equivalent term reduces defined using combinators equivalent must sense since latter diverge divergence irrelevant probabilistically probability zero may context equivalence contrast coarser probabilistic context equivalence despite differences two semantics similarities analogously happens nondeterministic applicative bisimulation context equivalence coincide probabilistic setting least considered counterexamples full abstraction much complicated easily adapted probabilistic setting conclusions first paper bisimulation techniques program equivalence shown applicable probabilistic one hand abramsky idea seeing interaction application shown amenable probabilistic treatment giving rise congruence relation sound context equivalence completeness however fails way probabilistic applicative bisimulation defined allows one distinguish terms context equivalent behave differently choices interactions performed notion coupled logical bisimulation introduced proved precisely characterise context equivalence along way applicative bisimilarity proved coincide context equivalence pure yielding tree equality crucial difference two main bisimulations studied paper style applicative logical rather fact applicative bisimulation insists relating individual terms coupled logical bisimulation flexible allows relate formal sums may think distributions also explains need distinct reduction rules two bisimulations see examples complete applicative bisimulation stands simpler use coupled logical bisimulation moreover natural form bisimulation interesting trying transport techniques handling onto variants extensions language topics future work abound already hinted earlier sections among interesting ones one mention transport applicative bisimulation onto language conjecture resulting relation would coincide coupled logical bisimilarity context equivalence going howe technique seems difficult given infinitary nature formal sums confinement redex positions also interesting would effective notion equivalence even two introduced notions bisimulation avoid universal quantifications possible contexts refer essentially infinitary operational semantics meaning term obtained least upper bound finite approximations would possible define bisimulation terms approximations without getting fine grained bisimulations style logical bisimulation environmental bisimulation known require techniques order avoid tedious equality proofs concrete terms paper introduced techniques coupled logical bisimilarity additional techniques would useful techniques could also developed applicative bisimilarity would like develop sound operational techniques computational indistinguishability key notion modern cryptography computational indistinguishability defined similarly context equivalence context however required work within appropriate resource bounds two terms different observable behaviors although negligible probability see work first step direction complexity bounds yet probabilistic behaviour essential ingredient correctly taken account references abramsky lazy turner editor research topics functional programming pages addison wesley samson abramsky luke ong full abstraction lazy lambda calculus inf egidio astesiano gerardo costa distributive semantics nondeterministic typed theor comput hendrik pieter barendregt lambda calculus syntax semantics volume studies logic foundations mathematics marco bernardo rocco nicola michele loreti uniform framework modeling nondeterministic probabilistic stochastic mixed processes behavioral equivalences inf boudol laneve discriminating power multiplicities inf boudol strict parallel functions inf dorin comaniciu visvanathan ramesh peter meer object tracking ieee trans pattern analysis machine intelligence ugo dal lago davide sangiorgi michele alberti coinductive equivalences probabilistic functional programs long version available http ugo dal lago margherita zorzi probabilistic operational semantics lambda calculus rairo theor inf vincent danos russell harmer probabilistic game semantics acm trans comput rocco nicola matthew hennessy testing equivalences processes theor comput ugo liguoro adolfo piperno non deterministic extensions untyped inf giovannetti bohm theorem observational equivalences informal account electr notes theor comput tiuryn urzyczyn discrimination parallel observers algorithm inf thomas ehrhard michele pagani christine tasson computational meaning probabilistic coherence spaces lics pages shafi goldwasser silvio micali probabilistic encryption comput syst noah goodman principles practice probabilistic programming popl pages andrew gordon bisimilarity theory functional programming electr notes theor comput andrew gordon mihhail aizatulin johannes guillaume claret thore graepel aditya nori sriram rajamani claudio russo pattern bayesian reasoning popl pages matthew hennessy exploring probabilistic bisimulations part formal asp douglas howe proving congruence bisimulation functional programming languages inf radha jagadeesan prakash panangaden model process calculus icalp pages jones gordon plotkin probabilistic powerdomain evaluations lics pages koutavas levy sumii applicative environmental bisimulation electr notes theor comput kim guldstrand larsen arne skou bisimulation probabilistic testing inf lassen relational reasoning functions nondeterminism phd thesis university aarhus lenglet alan schmitt stefani howe method calculi passivation concur pages algebraic interpretation equality models lambda calculus editor lambda calculus computer science theory volume lncs pages giuseppe longo models lambda calculus theories expansions isomorphisms ann pure appl logic christopher manning hinrich foundations statistical natural language processing volume mit press morris lambda calculus models programming languages phd thesis mit luke ong functional setting lics pages prakash panangaden labelled markov processes imperial college press sungwoo park frank pfenning sebastian thrun probabilistic language based sampling functions acm trans program lang judea pearl probabilistic reasoning intelligent systems networks plausible inference morgan kaufmann avi pfeffer ibal probabilistic rational programming language ijcai pages morgan kaufmann pitts howe method languages sangiorgi rutten editors advanced topics bisimulation coinduction pages cambridge university press andrew pitts theories program equivalence semantics logics computation pages cambridge university press norman ramsey avi pfeffer stochastic lambda calculus monads probability distributions popl pages probabilistic lcf mfcs volume lncs pages david sands sos rules proof principles operational metatheory functional languages popl pages sangiorgi lazy lambda calculus concurrency scenario inf davide sangiorgi naoki kobayashi eijiro sumii logical bisimulations functional languages fsen volume lncs pages davide sangiorgi naoki kobayashi eijiro sumii environmental bisimulations languages acm trans program lang davide sangiorgi david walker theory mobile processes cambridge university press kurt sieber nondeterminism tlca volume lncs pages sebastian thrun robotic mapping survey exploring artificial intelligence new millennium pages
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empirical study population dynamics reinforcement learning yaodong lantao yiwei jun weinan ying yong oct university college london shanghai jiao tong university abstract paper conduct empirical study discovering ordered collective dynamics obtained population artificial intelligence agents intention put agents simulated natural context understand induced dynamics population level particular aim verify principles developed real world could also used understanding artificiallycreated intelligent population achieve simulate world laws world designed findings logical equivalence discovered nature endow agents intelligence based deep reinforcement learning scale population size millions results show population dynamics agents driven agent individual self interest reveals ordered pattern similar model studied population biology discover emergent behaviors collective adaptations studying agents grouping behaviors change environmental resources two findings could explained theory nature introduction employing modeling power deep learning singleagent reinforcement learning started display even surpass intelligence wide variety tasks ranging playing games labyrinth mnih atari mnih silver tasks continuous control locomotions lillicrap text generation neural architecture design zoph recently algorithms broadened use demonstrated potentials setting agents incentives economical constraints exist example studies mordatch abbeel lazaridou peysakhovich baroni wang liang manning vries shown different cooperative learning environments compositional language naturally emerges researchers peng first three authors equal contributions correspondence jun wang weinan zhang wnzhang foerster usunier also demonstrated multiple agents trained play combat game starcraft agents mastered collaborative strategies similar experienced human players nonetheless aforementioned systems far limited less tens agents focuses studies rather optimization micro individual level policy studies resulting collective behaviors dynamics emerging large population agents remain untouched yet hand populations exhibit certain orders regularity collective behaviors honey bees use specific waggle dance transmit signals trail ants transfer food leaving chemical marks routes formations bird flocks migration particular sizes fish schools deep ocean even human beings easily show ordered macro dynamics example rhythmical audience applause concerts periodical human waves fanatic football game etc stream research theory ashby explores new approach explaining emergence orders nature fact dynamics appears many disciplines natural sciences bak theory suggests ordered global dynamics matter complex induced repeated interactions local individual parts system initially disordered without external supervisions interventions concept proven important multiple fields nature sciences camazine sumpter kauffman ancient philosopher lucretius said designing intelligence necessary create orders palmer interesting question understand kinds ordered macro dynamics community agents would possess together put natural context paper fill research gap conducting empirical study questions aim understand whether principles theory ashby developed real world could also applied understanding population order achieve argue key study clear methodology introducing intelligence therefore simulate world individual agent endowed intelligence deep reinforcement learning framework population size scaled million level maximize generality laws world designed incorporating natural findings logic equivalence miscellaneous potential dynamics thus studied first study macro dynamics population size predators preys investigate emergence one fundamental collective behavior grouping particular compare statistics dynamics intelligent population theories models biological studies interestingly find artificial ecosystem individual intelligence incorporated reaches ordered pattern dynamics similar model indicates population biology also discover emergence collective adaptations grouping behaviors environment changes two findings well explained based theory results could potentially open interesting research direction understanding population using natural science principles real world related work reinforcement learning sutton barto employs learning scheme reinforces agent maximize cumulative rewards sequentially interacting environment intelligence evolves agent learning past experiences trying perform better future recently deep neural networks succeed marrying algorithms particular show remarkable performance approximating value function mnih policy function lillicrap value policy function mnih increase intelligence traditional methods methods extended settings multiple agents exist interact methods minimax wellman others nash qlearning wellman proposed fundamental question answer different agents communicate reach coherent goal several differentiable communication protocols proposed foerster sukhbaatar fergus others easily embedded error training scheme work peng employed bidirectional recurrent neural networks coordinate groups agents play starcraft combat games achieved micromanagement skills beyond pursuing high performance playing video games researchers recently start shift focus onto studying community agents corresponding attributes concurrent studies mordatch abbeel lazaridou peysakhovich baroni wang liang manning vries conducted different cooperative learning environments emergence compositional language found leibo introduced agents policy learning solving sequential social dilemmas discovered agents behaviors would influenced environmental factors conflicts would emerge competing shared resources nonetheless systems studies consider tens agents thus unfair generalize findings population level macro dynamics large population remain disclosed subject computerized artifacts work also related research conducted natural sciences theory proposed ashby serves fundamental approach understanding emergence orders natural sciences theory believers think global ordered dynamics system originate numerous interactions local individuals initially disordered needs external interventions theory predicts existence ordered dynamics population fact phenomena observed multiple fields natural sciences camazine sumpter kauffman example population biology one important discovery ordered harmonic dynamics population sizes predators preys lynx snowshoe hare gilpin summarized model lotka basically describes fact lag phase space population sizes predators preys even though explained theory models summarized based statistics ecological field studies essentially learning process individual intelligence involved work chose different approach incorporating individual intelligence population dynamics studies agent endowed intelligence make decision rather considered homogeneous intention find whether population still creates ordered dynamics lotkavolterra equations whether dynamics explainable perspective theory multiple disciplines natural sciences spanning zoology psycholog economy sumpter dunbar guillen one fundamental thus important collective behaviors study grouping population units aggregate together collective decisionmaking grouping believed imply emergence sociality induce collective behaviors javarone marinazzo studying grouping behaviors traditional approaches include setting game rigid reductive predefined interactive rules agent conduct simulations based game fryxell niwa inada kawachi rulebased games might work well biological organisms inherit characteristics ancestors however show limits studying heterogeneous agents boccaletti contrast rulebased games learning process involved investigate formation grouping behaviors millionlevel population driven algorithms intention find grouping behaviors population negative feedback negative feedback leads homeostasis helps stabilize collective behaviors produced favor positive feedback going extremes predator prey response threshold response threshold threshold beyond individuals change behaviors response stimulus timestep timestep obstacle health individual variation individual variation essence guarantee continual explorations new solutions problem within population figure illustration world world exist preys predators obstacles predators hunt prey survive starvation predator health bar limited eyesight view predators form group hunt prey chance capturing increase also means captured prey shared among group members multiple group targeting prey largest group within capture radius win example predators form group win prey group predator soon dies starvation emerge change environmental factors food resources collective behaviors emerging based grouping design world paper try understand whether population create ordered patterns population dynamics dynamics collective grouping behaviors predatorprey interaction one fundamental relationship observed nature intend simulate world agents shown fig world deigned easily adaptable incorporate environmental complexity investigate miscellaneous dynamics well collective behaviors population individual agent driven purely self interest axioms natural environments avoid introducing specific rules could harm generality observed results design laws world considering real findings logical equivalence observed natural system regard laws axioms studying population dynamics collective behaviors briefly review axioms accepted refer corresponding natural evidence appendix note axioms treated separately consider instead combination different axioms could produce affect collective dynamics positive feedback positive feedback enhances particular behaviors reinforcement helps spread information meaningful action quickly individuals redundancy redundancy ensures functional continuity whole population even catastrophic event happens synchronization synchronization special kind positive feedback time rather space example would individual unique frequency applause affect frequency crowd concert selfishness individuals always tend maximize utility one behave altruistically others benefit behaving collectively acting alone realization realize world via designing markov game list detailed rules game corresponding axiom refers population dynamics world see fig goal predator species survive ecosystem procreate next generations positions world initialized randomly beginning environment considered infinite horizon population preys predators boomed breeding offsprings however face hazards either hunted preys dying starvation predators realize idea starvation make health status predator decrease time constant factor also restored capturing eating preys predators assumed infinite appetite logical equivalence nature predator normally ability storing food resource future survival predator unique characteristics identity vector eyesight health status unique characteristics agent represents diversity population individual agent make independent decision behave differently even given scenario predators form group increase chance capturing prey group members visible predators within view single agent chooses action join group environment select group within view randomly agent become member group decides leave current group afterwards note single predator may hunt prey alone well hunt group member illustrated fig prey assigned square capture area capture radius reflects difficulty hunted groups predators singles able hunt prey manage stay within capture radius apart capture radius another parameter capture threshold also reflects capturing difficulty prey within capture area meeting threshold group predators become valid candidate multiple valid candidate groups targeting prey group largest group size winner mimics law jungle group wins candidates members group share prey equally pursuit preys grouping encouraged large group help increase probability capturing prey however huge group size also inhibited due less proportion prey group member obtains sharing grouping behaviors considering synchronization selfishness incorporate second type prey captured individual predator alone means set capture threshold species analogy think tigers predators sheep preys whose captures require collaborative grouping predators rabbits preys captured single predator two kinds preys considered abstraction individual reward grouping reward respectively predators make decision either join group hunting sheep conduct hunting rabbit order maximize reward probability survival introduces acting alone collaborating others keep alternating environments feeding two kinds preys one another examine dynamics grouping behaviors emphasize dynamics grouping behaviors also avoid influences systematic preferences grouping result changing population size keep population size predators fixed endowing eternal longevity also considered observation little change predator population size environment optimal strategy agent continuously varies time predator population learn adapt collective strategy correspondingly population built deep reinforcement learning designed world build population deep reinforcement learning setting size population starts million agent plays markov game partial observation local environment agents predators population learn policy capture preys also coexist population formally partially observable markov game denoted tuple denotes set environmental states agent state sit contains observation oti time step well identity sit oti individual agent supposed make independent decisions behave differently based embedding action obs obs updates obs reward action experience buffer obs reward action reward figure system predatorprey world cal observation well embeddings inputs identity embedding vector unique individual observation oti dependent agent current position orientation agent eyesight ranges grids ahead grids sides within horizon five channels observation oti different objects areas map occupy first three channels raw rgb pixels fourth channel indicator whether object group member fifth channel health status object agent otherwise padded zero time step agent take action ait selected action space forward backward left right rotate left rotate right stand still join group leave group considered invalid predator takes join group action group member already takes leave group action single individual also invalid agent tries cross map boarders invalid actions settled time step transition function states deterministic given joint actions states agents reward function agent denotes discount factor reward denotes initial state distribution agent gains intelligence learning policy could maximize expected cumulative reward interacting environment way let denote expected cumulative reward denotes trajectory sampled agent policy initial state distribution deterministic game rules fixed state transition function agent goal learn considering exploration action space apply methods selecting action qit sit ait setting agents share common qvalue function approximated deep neural network sit ait oti ait refer standard setting dqn mnih parameters sharing one way make learning task computationally tractable individual agent however still make independent decision behave differently based local observation unique identity embeddings fact embedding represents agents unique characteristics could help fire different hidden states hand also reasonable let whole population share network biologically speaking individuals species tend inherit characteristics ancestors fryxell niwa inada kawachi therefore safe assume intelligence level predator namely share common network time step agents contribute experienced transitions sit ait rti buffer shown fig system design considering efficiency collect agents experience one time step parallel update shared using time increase utilization gpu based experience coming buffer shared updated sit ait sit ait rti max sit ait worth mentioning experience buffer fig stores experience agents current time step different replay buffer commonly used traditional dqn buffer maintains queue across different time steps unlike setting using replay buffer typically lead issue learning tasks lowe hand mnih introduced replay buffer aiming disrupting consecutive examples setting experiences sampled concurrently millions agents individual agent different states policies therefore naturally strong training examples thirdly unlikely unwanted feedback loops arise since sampled experiences dominated single agent decisions therefore design experience buffer reasonable proves robust experiments findings two sets experiments conducted codes reproducible experimental results provided understanding population dynamics first study population dynamics community predators preys tracking population size species time specifically initialize predators preys randomly scattered map size predators health status set initially decays time step two comparing settings birth rates preys set respectively two hidden layers hidden units interleaved sigmoid layers project outputs one potential action training predators learn code github https reinforcement learning scheme exploratory parameter surprisingly find population reveals ordered pattern measuring population dynamics shown fig population sizes predators preys reach dynamic equilibrium curves present shape lag phase crest one aligned trough underlying logic ordered dynamics could predators population grows learn know hunt efficiently consequence preys captured preys population shrinks later cause predators population also shrinks due lack food supply help less predators population preys recover shrinkage start regrow logic drives contour population sizes see traits row fig harmonic cycles circle patterns become stable increasing level intelligence agents acquire reinforcement learning shown later ablation study enabling individual intelligence key observe ordered patterns population dynamics fact population dynamics possessed agents consistent model studied biology shown orange traits fig population biology model lotka describes hamiltonian system interactions predators preys model population size predators preys change time based following pair nonlinear differential equations preys assumed unlimited food resource thus reproduce exponentially rate meeting predation proportional rate predators prey meet represented predators exponential decay population due natural death denoted meanwhile also boost population hunting prey represented solution equations harmonic function shaped population size predators lagging preys phase phase space plot shows series periodical circle dependent initial conditions words equilibrium cycle reach depends ecosystem starts similar patterns population dynamics might indicate orders population induced logic ecosystem model describes however key difference unlike equations model observed macro dynamics directly start microcosmic point view population driven self interest powered individual agent reaching macroscopic principles test robustness findings perform ablation study three important factors think critical generation ordered figure population dynamics time space row phase space row orange circles theoretical solutions equation red spot equilibrium circles simulation results simulated birth rate preys fitted model simulated birth rate preys fitted model model represents birth rate denote denote namics first analyze whether observed pattern restricted specific settings world expose predator models trained environment birth rate preys fig new environment birth rate preys fig shows period time adjustment predators adapt new environment agents whole manage maintain patterns second break binary relationships introducing second type prey require group hunting shown fig case three species model may find challenging analyze still observe ordered harmonic circles space third investigate role individual intelligence disabling learning function setting fig fig shows population possess ordered dynamics anymore intelligence individual agent disabled whole ecosystem explodes reason predator goes extinct increased birth rate preys leads new distributions states thus observations consequently original optimal policy predators becomes suboptimal new environment given number preys increases exponentially map size limited sheep soon cover blank spaces predators barely aggregate valid groups hunting finally die starvation ing refer group proportion face two kinds preys one requires group hunting predators make decision either join group hunting sheep hunt rabbit alone conduct two experiments predator population size equaling thousands millions map size equaling respectively acting like supplement number preys fixed amount number drops certain threshold supplement alternate types preys feed suppose number species threshold supply species setting study population dynamics preys alternatively fed predator policy needs react correspondingly new environment survive shown fig moment right rabbits fed environment proportion groups drastically drop nearly predators collectively behave selfish rather altruistic group number rabbits captured proportion grouping behaviors increases mildly meets spike soon sheep fed environment reaches another dynamic equilibrium highlyvariable environment population predators show intelligence adapting hunting strategies collectively without external supervisions controls understanding grouping behaviors discussions next investigate dynamics collective grouping behaviors particular intend find relationship environmental food resources proportion predators participate group judging ordered patterns population world reasons agree lucretius designing intelligence necessary create orders nature fact understanding emergence figure population dynamics time space phase space new type prey green line introduced captured single agent population shows ordered dynamics phase space figure population dynamics learning function population disabled simulation follows setting fig ordered dynamics found group proportion figure grouping proportion world two kinds preys fed points time step preys fed tells number prey sheep requires group hunting increases proportion groups population increases adapting grouping becomes collective behaviors vice verse case prey rabbit fed experiment population orders system theory proposed ashby considers global ordered dynamics system spontaneously originate numerous interactions local individuals initially disordered needs external interventions theory predicts existence ordered dynamics numerous local interactions individuals system could potentially explain ordered patterns observed population tested meanwhile according theory created order independent complexity individual involved example dynamics also hold natural systems herbivore plants parasite host even though models based set equations fixed interaction terms findings depend intelligent agents driven consistent learning process generalization resulting dynamics onto population still leads imagine general law could unify artificially created agents population studied natural sciences long time arguably contrast theory reductionist scientists hold different view order created transferring external systems typical example second law thermodynamics boltzmann stating total entropy level disorder always increase time closed system idea widely accepted particularly physics quantitative analysis feasible however argue findings population law agents exceptions simply environment live closed whenever system exchange matter environment entropy decrease system orders emerge still compatible second law discussion entropy life schrodinger certainly goes beyond topic leave future work conclusions conduct empirical study population simulating world individual agent empowered deep reinforcement learning number agents millions find population possesses ordered population dynamics consistent model ecology also discover emergent collective adaptations environmental resources change time importantly findings could well explained theory natural sciences next intend understanding many population dynamics expect findings could enlighten interesting research direction interpreting population using natural science principles developed real world references ashby ashby principles system facets systems science springer bak bak nature works science selforganized criticality springer science business media boccaletti boccaletti latora moreno chavez hwang complex networks structure dynamics physics reports boltzmann boltzmann second law thermodynamics theoretical physics philosophical problems springer camazine camazine biological systems princeton university press vries vries strub chandar pietquin larochelle courville guesswhat visual object discovery dialogue dunbar dunbar group size vocal grooming origins language psychonomic bulletin review foerster foerster assael freitas whiteson learning communicate deep multiagent reinforcement learning advances neural information processing systems foerster foerster nardelli farquhar torr kohli whiteson stabilising experience replay deep reinforcement learning arxiv preprint fryxell fryxell mosser sinclair packer group formation stabilizes dynamics nature gilpin gilpin hares eat lynx american naturalist guillen guillen business groups emerging economies view academy management journal wellman wellman nash stochastic games journal machine learning research nov wellman others wellman multiagent reinforcement learning theoretical framework algorithm icml volume citeseer inada kawachi inada kawachi order flexibility motion fish schools journal theoretical biology javarone marinazzo javarone marinazzo evolutionary dynamics group formation arxiv preprint kauffman kauffman origins order selforganization selection evolution oxford university press usa lazaridou peysakhovich baroni lazaridou peysakhovich baroni cooperation emergence natural language corr leibo leibo zambaldi lanctot marecki graepel reinforcement learning sequential social dilemmas arxiv preprint lillicrap lillicrap hunt pritzel heess erez tassa silver wierstra continuous control deep reinforcement learning arxiv preprint lotka lotka elements physical biology lowe lowe tamar harb abbeel mordatch mixed environments arxiv preprint mnih mnih kavukcuoglu silver graves antonoglou wierstra riedmiller playing atari deep reinforcement learning arxiv preprint mnih mnih kavukcuoglu silver rusu veness bellemare graves riedmiller fidjeland ostrovski control deep reinforcement learning nature mnih mnih badia mirza graves lillicrap harley silver kavukcuoglu asynchronous methods deep reinforcement learning international conference machine learning mordatch abbeel mordatch abbeel emergence grounded compositional language populations niwa niwa dynamic model fish schooling journal theoretical biology palmer palmer reading lucretius renaissance volume harvard university press peng peng yuan wen yang tang long wang multiagent bidirectionallycoordinated nets learning play starcraft combat games arxiv preprint schrodinger schrodinger life university press cambridge silver silver huang maddison guez sifre van den driessche schrittwieser antonoglou panneershelvam lanctot mastering game deep neural networks tree search nature sukhbaatar fergus others sukhbaatar fergus learning multiagent communication backpropagation advances neural information processing systems sumpter sumpter principles collective animal behaviour philosophical transactions royal society london biological sciences sutton barto sutton barto reinforcement learning introduction volume mit press cambridge usunier usunier synnaeve lin chintala episodic exploration deep deterministic policies application starcraft micromanagement tasks arxiv preprint wang liang manning wang liang manning learning language games interaction arxiv preprint zhang wang seqgan sequence generative adversarial nets policy gradient aaai zoph zoph neural architecture search reinforcement learning arxiv preprint appendix table axioms simulated world axiom positive feedback negative feedback individual variation response threshold redundancy synchronisation selfishness description example nature positive feedback enhances one backup nature comes observations ular behaviors ants ant discovers food source ment helps spread particular search trail path food soon tion meaningful action quickly serve trigger positive feedback individuals ants start follow negative feedback leads ant case population size ants limited ostasis helps stabilize increasing number ants forage food tive behaviors produced favor outside distribution ants food sources positive feedback going stable extremes individual variation one evidence comes honey bees social essence guarantee continual insects honey bees evolved highly variable explorations new solutions directional sense response sucrose level focus problem within population food collection order ensure diversification ways food collection otherwise one single food resource depleted quickly response threshold evidence found nature bumble bees old beyond individuals start fan cool hive temperchange behaviors ature inside goes threshold level sponse stimulus redundancy ensures functional kingdom bees community suffers continuity whole population drastic reduction number worker bees younger even catastrophic event bees soon replace positions guarantee pens whole community function well synchronization special kind example would individual unique freof positive feedback time rather quency applause affect frequency crowd space concert empirical evidence includes audience applause often achieved adjustments individuals unique frequency among local average selfishness means agents easily observable nature always maximise utility references edward wilson insect societies insect eric bonabeau guy theraulaz deneubourg serge aron scott camazine social insects trends ecology evolution tanya pankiw robert page response thresholds sucrose predict foraging division labor honeybees behavioral ecology sociobiology robert jeanne interindividual behavioral variability social insects westview press anja control nest climate bumblebee bombus terrestris colonies interindividual variability self reinforcement fanning response behavioral ecology thomas seeley wisdom hive social physiology honey bee colonies harvard university press
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zeros irreducible characters factorised groups mar felipe abstract element finite group said vanishing exists irreducible character case also called zero aim paper obtain structural properties factorised group impose conditions prime power order elements vanishing keywords finite groups products groups irreducible characters conjugacy classes vanishing elements msc introduction within finite group theory close relationship character theory study conjugacy classes widely known regarding last topic several authors investigated connection certain conjugacy class sizes also called indices elements group structure recent results show conjugacy classes elements factors factorised group exert strong impact structure whole group see character theory celebrated burnside result asserts every row character table finite group corresponds complex character zero entry theorem nevertheless conjugacy class column may contain zero fact somehow violates standard duality arising many cases two referred research lines therefore authors introduce next concept element vanishing exists irreducible character first author supported proyecto prometeo generalitat valenciana spain second author supported proyecto mtm ministerio industria competitividad spain proyecto generalitat valenciana spain third author acknowledges predoctoral grant generalitat valenciana spain instituto universitario matemtica pura aplicada universitat camino vera valencia spain mfelipe anamarti vicorso orcid ids literature also called zero otherwise element said immediate consequence cited burnside result get group vanishing elements abelian said various questions concerning vanishing elements studied numerous authors particular appearing references paper therefore natural wonder whether results based conjugacy class sizes remain true restrict focus indices correspond vanishing elements consider vanishing indices spirit researchers recently obtained positive results certain cases instance dolfi pacifici sanus proved prime divide vanishing index group normal abelian sylow theorem brough showed fixed prime vanishing indices divisible soluble theorem moreover vanishing index supersoluble theorem last two results turn vanishing versions theorem theorem respectively besides brough kong also showed hypotheses previous results weakened vanishing indices prime power order elements remark classification finite simple groups cfsg used development paper interested combining novelty research irreducible characters study products groups concretely want analyse information factorised group obtained character table consider conjugacy classes elements factors particular inspired aforementioned investigations deal factorised groups irreducible characters evaluate zero elements factors worthwhile note product two vanishing elements needs vanishing general moreover element normal subgroup vanishing whole group subgroup see example focusing products groups along last decades relations permutability factors considered many authors instance total permutability mutual permutability see see last permutability relations inherited quotients ensure existence minimal normal subgroup contained one factors principally concerned products groups satisfy particular conditions name see definition framework purpose get better understanding vanishing elements factors control structure group moreover also deal arithmetical conditions indices elements paper structured following way firstly defined section properties crucial along paper proved section analyse case group vanishing factors prime see theorem consequence obtain information factorised group prime divisors order considered vanishing prime power order elements factors see corollary later obtain structural properties groups vanishing indices whole group elements factors concretely section study case vanishing indices prime powers theorem corollary next focus section case indices divisible prime see theorem situation indices also handled last section see theorems particular highlight affirmative answer question posed brough given corollary significant mention previous results remain true factors either totally mutually see example remark order avoid repeating arguments previous papers proof runs one known result suitable changes refer corresponding one throughout paper every group assumed finite terminology follows group element call index size conjugacy class set prime divisors order denoted prime element order divisible customary set sylow denoted sylp whilst set hall set primes write irr set irreducible complex characters given group product subgroups subgroup called prefactorised respect factorisation see recall subgroup covers section group remainder notation standard taken mainly particular normal subgroup group denoted symbolically also refer details classes groups definition properties analyse section kind factorisations manage along paper definition let product subgroups say every proper normal subgroup holds exists normal subgroup either either covers note adopt bar convention quotients condition means denotes core group subgroup illustrates given name factorisations remark let state immediate facts either always simple group either take definition exists minimal normal subgroup contained either present examples example let product subgroups let assume satisfy one following permutability properties mutually permutable permutes every subgroup permutes every subgroup every subgroup every subgroup exists permutes iii totally permutable every subgroup permutes every subgroup particular property holds satisfy applying theorem lemma seen also permutability properties clearly inherited quotients thus cases shall see later example group whose factors neither mutually permutable prove quotients inherit property lemma let let proper normal subgroup also proof let use bar convention denote quotients take normal subgroup claim exists normal subgroup covered either normal subgroup either covered either follows analogously valid instead lemma characterisation via normal series lemma let product subgroups following statements pairwise equivalent exists normal series either covered either iii exists chief series either covered either term chief normal series prefactorised also proof implies let either next take desired series previous lemma therefore exists either get series repeating process reach trivial quotient get desired series implies iii refine series chief series get factor exist minimal normal subgroup let see either may assume instance ani thus iii implies show exists normal subgroup covered either let minimum number normal let suppose instance minimality follows claim chief normal series prefactorised work induction case clear since either assume want show also prefactorised may consider fix prefactorised chief normal series like iii showing consider following portion chief normal series let claim satisfies either order apply equivalence assumption instance lemma established point arbitrary prefactorised normal subgroup corefactorisation might next example shows example consider sym hxi sym denotes symmetric group letters sym since minimal normal subgroup neither moreover seen either mutually vanishing elements main objective section prove theorem corollary let state first key ingredients locating vanishing elements given group lemma lemma let normal minimal normal abelian every element vanishing bianchi brough camina pacifici obtained subsequent result lemma corollary let group abelian minimal normal subgroup let chief factor every element vanishing element let prime irr recall zero divide result brauer theorem highlights significance property vanishing elements irreducible character zero every divides order holds following lemma yields elements normal subgroups vanish whole group lemma lemma let normal subgroup group irreducible character zero every element order divisible vanishing element focus vanishing elements simple groups combination results use classification gives following proposition let simple group let either exists irr zero exists irr extends aut vanishes proof either group lie type proposition applies irreducible character zero note case includes groups hence remains consider sporadic simple groups alternating groups firstly virtue lemma sporadic simple group exists always irreducible character extends aut vanishes alternating groups known proposition two irreducible characters vanishes vanishes element order extend aut argument included within proof theorem provides following proposition turns essential remainder section proposition let minimal normal subgroup finite group let exists vanishing proof isomorphic simple group dividing order character zero irr clear also zero let lemma provides vanishing let suppose character zero proposition exists irr extends aut thus proposition follows irr extends moreover result established deal vanishing elements factorised groups next example gives insight occurring phenomena example let sym hxi factorised group example note although vanishing vanishing product hand element vanishing remark claim hypotheses regarding vanishing elements results stated inherited every quotient group indeed let proper normal subgroup note also lemma since exists bijection irr set characters irr containing kernel vanishing prime power order element assume also vanishing prime power order element fact used sequel sometimes reference first significant result analyses vanishing factors remark cfsg needed theorem let let prime every normal sylow proof let counterexample minimal order result take sylp clearly assume proper hence remark minimality may suppose since consider minimal normal subgroup instance let suppose divides order proposition vanishing contradiction divide order particular may assume proper minimality remark obtain normal choose lemma chief series chief factor covered either let minimum number divides minimal normal subgroup abelian follows unique sylow elementary abelian claim every element vanishing note abelian normal also holds addition since lemma yields every element vanishing therefore remains find lying either order get final contradiction since lemma applying lemma affirm unique sylow also prefactorised let pick vanishing hence result established immediate consequence take trivial factorisation theorem obtain theorem proof authors apply lemma centre sylow subgroup order get final contradiction highlight centre subgroup may prefactorised see example reasonings differ another consequence theorem following corollary let let set primes every prime power order nilpotent normal hall proof apply theorem prime note result generalises corollary indeed next corollary extends corollary factorised groups corollary let let every element vanishing order soluble proof denote virtue corollary nilpotent normal hall soluble also soluble consider case hypotheses theorem hold primes follows clearly groups nilpotent actually obtain stronger fact abelian next result essential proof proposition theorem supersoluble every element vanishing particular nilpotent elements vanishing corollary let following statements pairwise equivalent every element every prime power order element abelian proof doubt implications let prove clearly theorem nilpotent since assuming every prime power order element lying proposition provides every sylow subgroup lies thus said burnside result quoted introduction elementary show group abelian vanishing elements indeed enough consider last characterisation prime power order elements directly deduce taking trivial factorisation previous corollary claim also obtained theorem asserts complex character vanishes prime power order element also uses cfsg case proofs emphasize difficulty handling prime power order elements moreover observe theorem imply directly corollary since assure factorised group vanishing prime power order element lies one factors prime power vanishing indices camina camina analysed structure groups groups whose prime power indices given prime next extended study products two arbitrary groups thus stated introduction seems natural address corresponding vanishing problem vanishing indices prime powers particular factorised groups let enunciate first preliminary results subsequent one due wielandt lemma let finite group prime camina camina proved next proposition extends lemma celebrated burnside result groups conjugacy class prime power size proposition theorem elements prime power index finite group lie second term fitting series main result following one proposition let group contains normal prime divide finally lemma elementary lemma let normal subgroup group subgroup let prime divides divides remark note hereafter results stated arithmetical hypotheses indices inherited quotients indeed let suppose element prime power divisible given prime respectively since divides lemma get also prime power squarefree divisible prime respectively ready prove following vanishing versions theorem theorem respectively emphasize techniques used approach valid work zeros irreducible characters theorem let let prime sylp assume every vanishing prime power index considered indices normal normal sylow normal sylow normal sylow proof indices vanishing enough reproduce proof theorem notice contradictions derived lemma let denote may assume show next every vanishing holds applies since remark suppose vanishing assumptions get prime power actually therefore also done let denote let assume statement false theorem exists vanishing remark vanishing power prime follows proposition proposition implies divides divides final contradiction proceed induction order show normal may assume get normal second assertion follows directly remark vanishing analogue theorem true considered vanishing indices powers primes distinct sylow might abelian example let suzuki group degree let normaliser sylow sylow subgroup order sylow subgroup order vanishing nevertheless sylow moreover theorem asserts factor prime power indices whole factorised group unique prime divides considered indices however know vanishing version fact true finally note consider assumptions theorem every prime third statement tells nilpotent fact following result shows abelian group compare corollary corollary let every prime power order element vanishing prime power index abelian particular prime powers actually prime normal sylow abelian hall proof nilpotent theorem let denote let assume exists prime power order element vanishing prime may suppose vanishing assumption prime power since nilpotent proposition follows follows wielandt lemma contradiction thus vanishing prime power order element corollary get abelian second assertion note unique sylow theorem claim abelian hall let denote hence vanishing prime power order element since otherwise elements central assumptions contradiction follows corollary abelian vanishing indices last section focus vanishing indices factorised groups motivated previous developments next theorem treats extreme case vanishing indices divisible fixed prime comment although arguments proof first statement similar theorem include sake comprehensiveness theorem let assume divide every element prime power order vanishing divide every prime power order element vanishing abelian sylow proof assume result false argue minimal counterexample theorem minimality may suppose let minimal normal subgroup instance soluble since divides order follows assume proper since otherwise minimality get hence last fact assumptions produce elements prime power order vanishing corollary applies thus applying arguments second paragraph proof theorem obtained element prime power order vanishing whose conjugacy class size divisible final contradiction let denote unique hall subgroup vanishing prime power order element otherwise hypotheses imply elements central contradiction virtue corollary get abelian note theorem provides vanishing version theorem products two groups even relaxing mutual permutability factors also remark theorem theorem trivial factorisation indeed implies next corollary improves main result considering vanishing indices prime power order elements corollary let group prime divide vanishing index prime power order element abelian sylow regarding vanishing indices first analyse divisible fixed prime next proposition actually vanishing version theorem point result valid arbitrary factorisation proposition let prime number let divide vanishing elementary abelian proof since elements lie centre proposition apply directly theorem order get thesis following lemma essential sequel lemma lemma let prime acting faithfully elementary abelian cyclic author posed following question group vanishing indices divisible prime satisfying must following theorem gives positive answer question even factorised groups see corollary case theorem let let prime suppose divisible every prime power order element vanishing follows soluble sylp elementary abelian proof suppose result false let counterexample minimal order since every group odd order soluble may assume class soluble groups saturated formation suppose exists unique minimal normal subgroup moreover instance enough reproduce arguments proof theorem obtain prime power order element vanishing whose conjugacy class size divisible contradiction assume result true let counterexample least possible order minimality may suppose let minimal normal subgroup thus divides order since soluble abelian moreover class groups saturated formation unique minimal normal subgroup get consider instance take minimal normal subgroup covered either claim element vanishing since abelian prime indeed get follows lemma every element vanishing note sylq elementary abelian take assume conjugation hence divide note divides hand acts coprimely abelian follows observe acts faithfully coprimely previous argument get lemma leads fact cyclic hxi hence normal since minimality obtain order isomorphich subgroup aut isomorphic follows divides final contradiction notice sylp isomorphic previous assertion hence result follows proposition corollary let group let prime assume divide prime power order element vanishing soluble group moreover sylp elementary abelian theorem proved following let product mutually permutable subgroups let fixed prime satisfying prime power order elements divisible elementary abelian sylow point property remain true hypotheses theorem following example shows example let semidirect product cyclic group order acts transitively cyclic group order let prime vanishing elements lying index divisible however elementary abelian sylow highlight arguments used theorem generalised order obtain following general result theorem let let prime suppose every prime power order element vanishing divisible proof sufficient follow proof theorem notice case use lemma minimal normal subgroup lies either thus affirm every element vanishing order apply assumption divisible every prime power order element consider indices primes get theorem let suppose every prime power order element vanishing supersoluble abelian elementary abelian sylow subgroups sylow order prime statements immediate consequences next general result arbitrary factorisation supersoluble group theorem let product subgroups assume supersoluble suppose every prime power order element vanishing abelian elementary abelian sylow subgroups sylow order prime proof adapt proof theorem hypotheses regarding vanishing elements prove either arguing minimal counterexample case assume exists prime sylow since supersoluble proposition yields every vanishing thus apply element class size hypothesis theorem hand following proof theorem need assure prefactorised normal subgroup contains sylow inherits assumptions let see fact also holds case first clearly supersoluble hence take vanishing follows vanishing otherwise get proposition proposition leads fact contradiction moreover vanishing paragraph thus cases divides note particular satisfies hypotheses proposition statement runs proof theorem applying proposition vanishing version theorem proof theorem considering smallest prime divisor theorem conclude soluble hence prime applying theorem get prime supersoluble assertions follow previous theorem theorem author gives supersolubility criterion group every vanishing index prime power order element want highlight following consequence theorem gives information structure group moreover techniques differ used theorem corollary let group let assume prime power order element vanishing supersoluble abelian elementary abelian sylow subgroups sylow order prime references conditional permutability factorized groups ann mat pura appl conditional permutability saturated formations proc edinb math soc cossey mutually permutable products conjugacy classes monatsh math asaad products finite groups vol gruyter expositions mathematics berlin bianchi brough camina pacifici vanishing class sizes finite groups algebra brough elements finite groups algebra brough vanishing criteria control finite group structure algebra brough kong vanishing criteria control finite group structure https camina camina implications conjugacy class size group theory cossey structure mutable product finite groups acta math https mutually perhungar cossey wang remarks length conjugacy classes finite groups commun algebra doerk hawkes finite soluble groups vol gruyter expositions mathematics berlin dolfi pacifici sanus groups whose vanishing class sizes divisible given prime arch math basel dolfi pacifici sanus spiga orders zeros irreducible characters algebra felipe class sizes products groups algebra felipe prime power indices factorised groups mediterr math article isaacs character theory finite groups academic press london isaacs navarro wolf finite group elements irreducible character vanishes algebra malle navarro olson zeros characters finite groups group theory
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nov curiosities algebra bounded dirichlet series raymond mortini amol sasane abstract shown algebra bounded dirichlet series coherent ring infinite bass stable rank corollaries latter result derived infinite topological stable rank infinite krull dimension introduction aim short note make explicit two observations algebraic properties ring bounded dirichlet series particular show coherent ring essentially immediate consequence eric amar proof noncoherence hardy algebra polydisk infinite bass stable rank straightforward adaptation first author proof fact stable rank infinite polydisk algebra infinite corollaries obtain infinite topological stable rank infinite krull dimension giving relevant definitions briefly mention closed banach subalgebra classical hardy algebra consisting bounded holomorphic functions open right half plane striking compare findings corresponding results coherent bass stable rank topological stable rank krull dimension yes see see see see mathematics subject classification primary secondary key words phrases coherent ring hardy algebra dirichlet series bass stable rank topological stable rank krull dimension raymond mortini amol sasane nevertheless results lend support harald bohr idea interpreting dirichlet series functions infinitely many complex variables key theme used proofs main results note recall pertinent definitions algebra set dirichlet series bounded dirichlet series denotes sequence complex numbers holomorphic bounded equipped pointwise operations supremum norm sup unital commutative banach algebra theorem shown banach algebra precisely multiplier space hilbert space dirichlet series importance hilbert space stems fact kernel function related riemann zeta function closed subalgebra consisting dirichlet let series form involving integers generated first primes lemma proof theorem established isometrically banach algebra isomorphic certain algebra functions analytic infinite dimensional polydisk defined plays central role follows give outline based seminal observation made bohr put denotes nth prime view fundamental theorem arithmetic formally dirichlet series considered power series infinitely many variables indeed unique expansion algebra bounded dirichlet series nonnegative obtain formal power series depending whether function let recall kronecker theorem diophantine approximation chapter xxiii proposition map dense range using maximum principle shown norm right hand side norm denotes usual hardy algebra bounded holomorphic functions polydisk endowed supremum norm sup shown result also holds infinite dimensional case order describe result introduce notation let banach space complex sequences tending infinity induced norm let open unit ball banach space thus point set substituting argument given formally obtain function call bohr terminology mte abschnitt said norm functions uniformly bounded denote supremum norms using schwarz lemma polydisk seen max may define lim shown remains true infinite dimensional case may associate proposition exists banach algebra isometric isomorphism raymond mortini amol sasane coherence definition let unital commutative ring let times relation set relations denoted ring said coherent finitely generated property equivalent coherence intersection two finitely generated ideals finitely generated annihilator element finitely generated refer reader article monograph relevance property coherence commutative algebra noetherian rings coherent coherent rings noetherian example polynomial ring noetherian sequence ideals ascending stationary coherent corollary context algebras holomorphic functions unit disk mention shown hardy algebra coherent disk algebra amar showed hardy algebra coherent worth mentioning whether hardy algebra bidisk coherent seems open problem using amar result prove following result theorem coherent stable rank algebraic notion bass stable rank ring introduced order facilitate computations definition let commutative ring identity element denoted element called unimodular exist elements set unimodular elements denoted say reducible exists element bass stable rank least integer every reducible integer say infinite stable rank algebra bounded dirichlet series using idea proposition infinite polydisk algebra infinite bass stable rank show following theorem bass stable rank infinite banach algebras analogue bass stable rank called topological stable rank introduced marc rieffel definition let commutative complex banach algebra unit element least integer dense called topological stable rank say infinite topological stable rank integer exists corollary topological stable rank infinite proof follows inequality bass stable rank commutative unital semisimple complex banach algebra equal topological stable rank see corollary definition krull dimension commutative ring supremum lengths chains distinct proper prime ideals corollary krull dimension infinite proof follows fact ring krull dimension bass stable rank see noncoherence use following fact due amar proof theorem proposition finitely generated module proof theorem main idea proof using isomorphism essentially boil problem working let suppose finitely generated say show section image elements generate contradicting proposition raymond mortini amol sasane applying see exist applying obtain finally taking section obtain follows generate contradiction amar result proposition stable rank proof theorem straightforward adaptation first author proof fact bass stable rank infinite polydisk algebra infinite proposition infinite polydisk algebra uniform closure algebra generated coordinate functions countably infinite polydisk proof theorem fix let given set psn show reducible first let note unimodular indeed expanding product right hand side obtain algebra bounded dirichlet series appropriate suppose reducible exist psn let psn applying obtain let define otherwise continuous map vanishes outside maxn sup implies must exist maps compact convex brouwer fixed point theorem follows exists since zero outside see let obtain know contradicts choice arbitrary follows bass stable rank infinite acknowledgement useful discussions anders olofsson lund university gratefully acknowledged second author raymond mortini amol sasane references amar non certains anneaux fonctions holomorphes illinois journal mathematics bass algebraic benjamin new bohr die bedeutung der unendlich vieler variabeln der theppotenzreihen orie der dirichletscher reihen nachr ges wiss math chase direct products modules transactions american mathematical society glaz commutative coherent rings historical perspective current developments nieuw archief voor wiskunde glaz commutative coherent rings lecture notes mathematics springerverlag berlin hardy wright introduction theory numbers oxford clarendon press hedenmalm lindqvist seip hilbert space dirichlet series systems dilated functions duke mathematical journal heitmann generating ideals domains pacific journal mathematics maurizi remarks algebra bounded dirichlet series journal fourier analysis applications mcvoy rubel coherence rings functions journal functional analysis mortini example subalgebra unit disk whose stable rank finite studia mathematica von renteln primideale der topologischen algebra mathematische zeitschrift rieffel dimension stable rank proceedings london mathematical society seip interpolation dirichlet series linear complex analysis amer math soc transl ser amer math providence trivial gleason parts topological stable rank american journal mathematics treil stable rank algebra equals journal functional analysis lorraine institut cartan lorraine umr ile saulcy metz france address department mathematics london school economics houghton street london address sasane
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vanishing ideals binary hamming spheres feb alessio meneghetti university trento department mathematics trento italy abstract show efficiently obtain algebraic normal form boolean functions vanishing hamming spheres centred zero exploiting symmetry problem obtain formulas particular cases computational method address general case list polynomials corresponding spheres radius provided moreover explicitly provide connection binary transform elementary symmetric functions conclude presenting method based polynomial evaluation compute minimum distance binary linear codes keywords binary polynomials binary transform elementary symmetric functions minimum distance linear codes introduction many computationally hard problems described boolean polynomial systems standard approach computation basis corresponding ideal since quite common scenario restrict ideals containing entire set field equations ease notation work environment therefore quotient ring moreover results depend number variables otherwise specified consider defined infinitely many variables denote set variables work characterise vanishing ideal set binary vectors contained hamming sphere radius characterisation corresponds explicit construction polynomial whose roots exactly set points weight worth mentioning polynomial corresponds algebraic normal form anf preprint submitted elsevier february boolean function vanishes inside hamming sphere see carlet thorough discussion boolean functions direct application work would possibility add generating system ideal therefore force corresponding variety live hamming sphere radius less straightforward application presented section show novel method check whether minimum distance linear code given range believe ideas presented section could eventually starting point new algebraic algorithms computation minimum distance linear codes would however require design dedicated procedures minimize computational complexity algorithms beyond aim paper reader find similar methods guerrini authors proposed technique compute distance distribution systematic codes relying polynomial ideals interesting results obtained however deal particular classes codes find main difference previously known algorithms ideas presented work need rely like methods require computation basis comprehensive work utilisation computational methods address problems algebra geometry see cox everything regarding coding theory refer macwilliams sloane remark possible construct applying binary transform right evaluation vector using standard tools approach would however require restrict oneself specific number variables run one known algorithms computation presently corresponding complexity general case exponential number variables utilisation binary transform compute anf boolean function standard approach found example carlet survey binary transform reader refer pieprzyk zhang paper organised follows section introduce notation provide properties binary symmetric functions section discuss binary transform preliminary results used remaining sections respectively discuss generating polynomials vanishing ideals hamming spheres application putation distance linear codes binary symmetric functions vanishing ideal hamming sphere radius generated single binary polynomial denote precise algebraic normal form map whose zeros binary vectors whose hamming weight less less ones definition depends uniquely weight hence symmetric polynomial therefore written terms elementary symmetric functions esfs set monomials degree recall monomials implies monomial degree multiplication distinct variables case variables section look closely behaviour esfs working quotient properties described section used section derive results definition denote binary representation nonnegative integer even though length usually equal consider equal minimum number bits required context example need perform operation involving binary representation two integers consider length equal definition let two binary vectors say namely support contained support following theorem parity binomial coefficients extensively used remaining part section refer macwilliams sloane applications binary transform theorem lucas theorem mod lemma let symmetric polynomial linear combination elementary symmetric functions proof follows fact polynomial theorem let defined quotient proof let lemma follows product linear combination esfs given degree least pmax deg deg deg deg namely since symmetric polynomials monomial degree appears product observe monomial cardinality set odd number pairs equal first term equal ways choosing variables monomial set variables monomial second term corresponds choosing variables among appear obtaining monomial since second binomial coefficient equation also written product equation odd binomial coefficients odd hence write binary vectors three possible cases first case conditions satisfied implies cases least one binomial coefficients equation even implies lemma assure binary symmetric polynomial written linear combination interesting consequence theorem represent polynomial really need esfs polynomials indeed defined terms corollary set binary symmetric polynomials variables equivalent proof apply lemma theorem equivalently start polynomial compute proposition proof contains monomials degree monomials trivially zero evaluated vector strictly less coordinates proposition let mod proof let support let xji monomial degree exactly monomials degree hence mod corollary let let binary representations proof follows directly proposition theorem proposition let integers vector weight vector multiple proof let binary representations corollary since mod also two inclusions directly imply hence binary transform present section closed formula binary transform next section use describe theorem let boolean function binary transform fractions symbolic since denominators vanish together corresponding term product left provide example proving formula let namely proof binary transform boolean function whose evaluation vector corresponds coefficients precise point identified monomial xbnn mean appear hence definition monomial consider generic monomial observe formula obtain xbnn namely obtain easily checked polynomial assume value evaluated monomial corresponds polynomial zero everywhere else evaluation vector exactly vector coefficients vanishing ideal hamming sphere section provide description theorem allows write formula directly definition let denote polynomial whose coefficients correspond binary vectors weight least let apply binary transform theorem let let proof straightforward application definition observe equation gives formula also remark theorem really depend number variables since always assume set variables obtained restriction larger set combining theorems still obtain somewhat implicit formula since lemma would like write explicitly linear combination esfs explicitly determine need determine appearing equation lemma proof follows proposition lemma proof due lemma linear combination esfs degree least due proposition vector weight follows forces equal lemma let proof symmetric polynomial degree strictly less larger zero vector weight sum implies sum esfs degree strictly larger hence since zero corollary let proof let vector whose weight multiple firstly let observe whenever mod mod lemma proves case let smallest namely value therefore equal parity number equal hence contradicts definition hence lemma let proof prove lemma hold also contradict definition assume contradiction existence let correspond smallest integer property let vector weight equal corollary proves case assume start considering lemma coefficients equal moreover also remark hence value equal since number equal follows mod proved exist follows vector weight saying number odd implies moreover write explicitly evaluation look term defined equation appears evaluation apply corollary finding equal requirement equation follows using equation find already knew looking equation therefore simplify equation obtaining last term would equal would situation equation meaning would equal case instead assumed contradiction implies looking equation gives namely vector weight larger vanishes definition possible corollary proof lemma implies symmetric polynomial degree less appears lemmas say finally lemma allows write following lemma directly implied lucas theorem state useful prove particular property even lemma let two integers equal mod mod proposition let mod mod proof case follows lemma let let proceed induction assume equation holds till coefficient case write recall lemma states linear combination esfs degree larger definition whenever particular weight lemma follows hence summarise results presented point theorems methods compute respectively odd even values theorem let odd integer let let define mod proof lemmas imply completely determined coefficients equation derives definition proposition theorem let odd integer let let sequence coefficients defined theorem proof apply first proposition theorem theorem derive explicit formulas particular cases related corollary corollary let let consequence theorem since use give concise formula stated next corollary corollary let remaining part section use derive another family symmetric polynomials let consider set point weight exactly polynomial vanishing point outside set proposition proof apply definition use corollary let proof apply proposition corollary corollary theorem equal binary transform proof transform exactly polynomial vanishing points whose weight different conclude section following generalisation idea behind derivation related result shown carlet characterise numerical normal form binary symmetric functions theorem bases vector space symmetric boolean functions proof symmetric boolean function assumes value points whose hamming weights completely determine function require evaluation point possible weight case variables need therefore values write denote value point weight formula since define terms also write linear combination application linear codes show way determine minimum distance code using related approach proposed guerrini systematic nonlinear case ideas behind two methods indeed similar even though results section require computation basis denote restriction case variables let code let generator map namely image let minimum distance minimum weight without loss generality assume theorem proof means vector written means boolean function unknowns exactly definition corollary proof theorem definition remark corollary sufficient condition bound since indeed strictly smaller applying theorem times determine precisely would bound however restrict linear codes minimum weight corresponds minimum distance obtain corollary whose proof straightforward application theorem linear case let generator matrix linear code observe generator map case linear map corollary linear case conclusions theoretical point view explicit description allows formulation problems solutions requirements weight even though quite straightforward simply check weight given solution particular cases could advantage add right linear combination generating system ideal theoretical overview several properties polynomials shown obtain either applying binary function algorithm end work reader find list polynomials small numbers variables provide see table finally section shown application results coding theory novel theoretical method check minimum distance linear binary code conclude giving remarks contribution coding theory even though construction dedicated algorithms study complexity procedures beyond purpose work general case deal code structure generator matrix chosen randomly since working length codewords number monomials equal computation requires therefore multiplications one involving linear polynomials however worst case scenario dedicated algorithms could instead take advantage symmetric nature designed compute minimum distance particular classes codes acknowledgements author would like thank massimiliano sala interesting hints polynomial ideals applications coding theory references carlet boolean functions cryptography error correcting codes boolean models methods mathematics computer science engineering cox little shea ideals varieties algorithms edition springer introduction computational algebraic geometry commutative algebra sep algorithm computing minimum distance electronics letters guerrini orsini sala computing distance distribution systematic codes journal algebra applications fossorier eleftheriou june computation minimum distance codes ieee international conference communications ieee cat vol macwilliams sloane theory codes publishing pieprzyk zhang computing transforms boolean functions characterising coincident boolean functions proceedings international conference boolean functions cryptography applications table indices representation equation lemma need values smallest power larger given column marked multiple notation denote presence integers example given indices similarly
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reinterpretation standard deviation concept school geodesy geomatics wuhan university wuhan hubei china key laboratory precision engineering industry surveying state bureau surveying mapping wuhan hubei china abstract existing mathematical theory interprets concept standard deviation dispersion degree therefore measurement theory uncertainty concept precision concept expressed standard deviation times standard deviation also defined dispersion measurement result concept logic tangled comparative analysis standard deviation concept measurement error evaluation principle paper points concept standard deviation actually single error probability interval value instead dispersion degree error regularity evaluated standard deviation corrected mathematical concept gave correction direction measurement concept logic bring global change measurement theory system keywords measurement error standard deviation variance covariance probability theory introduction existing measurement theory precision concept uncertainty concept expressed standard deviation times standard deviation however mathematical theory interprets concept standard deviation dispersion degree measurement concepts precision uncertainty also defined dispersion connotation standard deviation dispersion difference two concepts course lead people understand concept concept controversy took place also dispersion concept actually also justify example state bureau surveying mapping china announced mount everest elevation standard deviation precision ordinary people understanding error measurement result constant deviation precision existence interval error however measurement professionals usually take understanding say precision dispersion degree measurement result random error point single result discrete would say future results repeated measurements discrete strange aspect logic current measurement result interpreted clear also measurements future pulled together actually future dispersion theory also tenable taking measurement conditions current measurement future repeated measurements processes conditions including internal external instruments exactly current measurement every measurement result exactly current result discrete standard deviation every result still source process certainly cause result conditions future repeated measurements different current measurement discussion even pointless current measurement different future measurements relation current measurement dispersion matter needed care current measurement next discussing variance concept error evaluation principle author point concept nature standard deviation dispersion correct mathematical concept specify correction direction measurement theory concept logic understanding standard deviation concept mentioned dispersion concept actually makes measurement issue hardly explain actual standard deviation concept definition variance standard deviation given existing probability theory mathematical expectation random variable defined lim measurement results sequence xmye order correctly understand concept standard deviation clarify several understandings first measurement sequence mutual dispersion process differences forming process measurement result differences process fuzzy uncertain fuzziness uncertainty process finite degree basic method probability theory pay attention fuzzy processes statistical analysis final results infer probability interval unknown numerical value analyzing dispersion many known numerical values purpose probability theory infer unknown value probability interval instead evaluate dispersion known sequence neither logic necessary contrarily numerical value probability interval dispersion future sequence positive proposition prove inverse proposition say although standard deviation dispersion analysis measurement sequence meaning actually probability interval value deviation using final measurement result standard deviation probability interval evaluation value single deviation exi single deviation exi exists within probability interval standard deviation taking normal distribution example standard deviation expresses deviation exi within scope confidence probability standard deviation exi probability interval value instead dispersion strict expression formula must equation refers error exi standard deviation probability interval value error exi error sample directly involved statistic naturally sequence exi formula approximated exi obviously error one member sequence consisting errors logical trouble interpret dispersion interval existing error values probability interval evaluation error contrarily neither logic necessary one error standard deviation dispersion future measurement sequence probability interval value deviation result expectation expressed standard deviation deviation expectation true value answer course yes standard deviation obtained tracing upstream measurements forming deviation error comes measurement say formula expresses deviation result expectation also may express deviation expectation true value even may express deviation result true value using dispersion error sequence error located express probability range error standard deviation meaning besides cases error also may viewed one member existing error sequence obtain different standard deviation value showed standard deviation subjective concept expressing human cognition normal different cognitive method causes different standard deviation evaluation example measure size workpiece caliper error measurement result regarded one member sequence consisting errors whole measurement range caliper also regarded calipers model also regarded calipers result error may get three different standard deviation values obviously standard deviation value error given neither logic necessary explain dispersion future repeated measurement therefore standard deviation probability interval evaluation value single error concept error range probability meaning expresses error possible deviation degree standard deviation precisely associated single deviation say order evaluate probability interval unknown event instead dispersion sample sequence probability theory statistical analysis number known events appropriate interpret standard deviation dispersion degree precisely correct general public understand mount everest elevation precision error possible range seen formula minimum standard deviation minimum basic idea least squares principle say concept variance gives method evaluating error probability interval also gives least square principle getting best measurement result covariance covariance propagation law definition variance actually use errors group error located evaluate probability range one error another problem may two different error groups therefore considering case multiple errors definition variance extended variance error error error meaning assumed errors uncorrelated standard deviations respectively two errors therefore variance respectively according definition covariance errors uncorrelated equation becomes seen mathematical meaning covariance variance communal error component among two errors long communal error component among different errors must covariance example two measurement result errors measured instrument two instrument errors made manufacturer existing measurement theory covariance propagation law proved dispersion concept actually aims deviation measurement result mathematical expectation variance covariance extended errors naturally covariance propagation law extended errors proof process also become simple example linear functions ktn take total differential equation therefore covariance matrix error sequence standard deviation regular error standard deviation probability interval value error first concept change brought regular error also evaluated standard deviation past noticed error sample sequence random discrete noise voltage discrete error sample sequence random regularity example electronic noise error electronic instruments random function time shown fig however also makes people form serious misconception error random regularity fig random distribution noise error evaluated standard deviation error certain regularity front said probability theory study error probability interval statistic analysis results sequence ignoring process various regularities process actually artificially ignored mean objective process must random regularity error certain regularity also shows random regularity certain regularity ignored random regularity certain regularity different ways observe things measurement practice common fact various regular errors incorporated statistical models process example cyclic error phase photoelectric distance meter disc eccentricity error theodolite sine regularity function model asin phase unknown considered equal probability distribution error inevitably exist within probability interval equal probability distribution see fig derived distribution density function standard deviation fig random distribution sine error seen relating error phase observe error shows sine regularity phase viewed arbitrary sine cycle error also follow random distribution standard deviation error another example rounding error cycle error sawtooth regularity shown fig relating error actual value observe error shows sawtooth cycle regularity actual value viewed arbitrary error also follow random distribution distribution density function standard deviation another example certain fig random distribution sawtooth error regularity frequency error quartz crystal temperature however temperature viewed arbitrary temperature value within probability interval corresponding error also exists within certain probability interval see fig fig random distribution quartz crystal frequency error standard deviation evaluation value probability interval error error evaluated standard deviation incorrect think regular error standard deviation statistical method standard deviation measurement error model exi formula becomes probability interval value error first measurement errors sequence final result given least square method every error independent every error variance according law variance propagation next separate error two parts according formula must key question comes error unknown however according equation already know thus take mathematical expectation sides equation aware get therefore famous bessel formula finally formula standard deviation probability interval evaluation value error visible critical step reasoning process according existing dispersion concept may possible explain dispersion sequence explaining dispersion unique final measurement result illogical equation established respectively interpreted probability interval values error exi logic fluent error model exi eai formula becomes eai probability interval value error eai next error model final result given least square method variance propagation relation therefore derivation process evolves eai equation transformed according formula must take mathematical expectation sides equation substitute equation equation form bessel formula still remains unchanged finally formula standard deviation probability interval evaluation value error similarly critical step reasoning process equation according existing dispersion concept explaining dispersion unique final measurement result also illogical also according dispersion sequence mainly depends dispersion sequence unable equal naturally impossible explain dispersion sequence example error equation solution equation obtained obviously although sequence discrete dispersion actually equal standard deviation dispersion sequence two different things therefore respectively interpreted probability interval values error exi logic fluent multivariate error model aij aij exi aij formula becomes aij probability interval value error aij next error equation general adjustment principle according principle least squares normal equation measurement results according covariance propagation law covariance matrix final measurement results covariance matrix therefore called factor matrix replace measurement results error equation obtain numerical value residual error please note residual relative final measurement result instead mathematical expectation guarantee make aey aey make ant take squares sum sides equation according formula equation evolves take mathematical expectation sides equation aware get make att att let see matrix please note att att symmetric matrix ait ait however according formula make equal substitution equation equation ait ait seen equation elements main diagonal equal ajt replace equation equation ajt therefore case unequal weights transformed equal weights adjustment similar formula obtained derivation process bessel formula multivariate adjustment finally covariance matrix final measurement result errors obtained formula similarly critical step reasoning process formula according existing dispersion concept people thinking still fall trap single value final measurement result dispersion equation illogical also aij according aij dispersion sequence mainly depends aij unable equal naturally impossible explain dispersion sequence therefore standard deviation interpreted error probability interval value logic fluent total standard deviation final measurement result section standard deviation obtained bessel formula variance propagation law actually probability interval value constant deviation final measurement result mathematical expectation use express constant deviation use express standard deviation final measurement result difference mathematical expectation true value also constant deviation expressed different come dispersion sequence error model final result given least square method every measurement result contains error final measurement result added error component dispersion sequence affected error model final result given least square method measurement result contains error respectively final measurement result added error component dispersion sequence affected error model aij final results given least square method measurement result contains error aij respectively final measurement result added error component respectively dispersion sequence affected importantly standard deviation deviation also obtained seeking upstream measurement historical measurement forming error essential difference upstream measurement current measurement obtain error standard deviation long tracing source way final measurement result total error according variance propagation law total standard deviation equation constant deviation generated measurement respective variance difference nature naturally actually error classification issue systematic error random error therefore existing measurement theory conceptual thinking interpreting random error random discrete interpreting systematic error follow random distribution standard deviation corresponding precision trueness concept logic naturally abolished equation total standard deviation total error probability interval evaluation value final measurement result say long consider upstream measurements current measurement whole error final measurement result true value evaluated total standard deviation naturally existing measurement error evaluation concept redefined according concept connotation standard deviation conclusion concept standard deviation actually probability interval evaluation value error deviation based concept quantitative evaluation method unknown error realized error probability interval evaluated standard deviation conceptual thinking based error classification philosophy abandoned existing measurement error evaluation concept redefined according concept connotation standard deviation bring global change measurement theory system international vocabulary metrology basic general concepts associated terms vim jcgm guide expression uncertainty measurement international organization standard first edition corrected general terms metrology definitions evaluation expression uncertainty measurement basic terms surveying mapping schmidt gum anmerkungen zur normdefinition der messunsicherheit und verzerrten elementarfehlermodellen http distance meter edm instruments iso optics optical instruments field procedures testing geodetic surveying instruments part distance meters edm measurements reflectors optical theodolites iso optics optical instruments field procedures testing geodetic surveying instruments part theodolites xiao shi ling new concepts measurement error theory measurement volume april pages errors classification philosophy critique proceedings national doctoral forum surveying mapping ling zhou qiang wang xiao new philosophical view measurement error theory acta metrologica sinica liu ling xiao new concepts measurement error regularities effect characteristics
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distributional representation model collaborative filtering zhang junlin cai heng huang tongwen xue huiping zhangjlh caiheng huangtw xuehp abstract paper propose concise deep learning approach collaborative filtering jointly models distributional representation users items proposed framework obtains better performance compared current algorithms made distributional representation model promising direction research collaborative filtering introduction recommender systems best known usage websites bringing much extra profit website better recommendation algorithms attracted attention industry academic community collaborative filtering one popular approaches among recommendation algorithms utilizes user feedback infer relations users items ultimately relate users items like side recent years witnessed breakthrough applying deep learning algorithms object recognition speech recognition nlp another filed deep learning widely used inspired successful application deep learning nlp tasks especially distributional representation method want explore distributional representation users items collaborative filtering paper proposed framework combines neutral network distributional representation users items collaborative filtering explicitly encoding features vectors explore complex nonlinearity interdependencies features neutral network though seems simple method proved effective recommendation domain experiment results main contributions work summarized following propose distributional representation approach recommender system best knowledge first study introduce word embedding concept collaborative filtering experiment results show promising direction research section describes distributional representation framework collaborative filtering section present experiment results indicate proposed method outperforms many commonly used algorithms research field section presents brief overview related work final section conclusion paper distributional representation model recommendation output layer hidden layer input layer input features lookup table concatenation user item fig neutral network structure distributional representation model collaborative filtering one popular approaches building recommendation systems mostly relies past user behavior previous transactions product ratings convenience call transaction product item following part paper order identify new associations analyzes relationships users interdependencies among items proposed model explicitly transforms user item vectors encode latent features uses combined vectors neutral network input explore complex nonlinearity interdependencies features regard regression problem proposed model figure shows main structure distributional representation model transforming user item vectors user embedding space checking lookup table parameter matrix needs learned training column vector size defined hand item also represented vector mapping lookup table parameter matrix column user item given input recommendation system predict score item user concatenate user vector item vector longer vector applying neutral network structure proposed neutral network three layers input layer hidden layer output layer mentioned input layer concatenation vector node hidden layer full connection nodes input layer transforms features encoded user vector item vector number nonlinearity function hyperbolic tangent tanh function used nonlinearity function following tanh tanh function rescaled version sigmoid output range instead linear function input vector edge weight parameter connect nodes input layer hidden layer output hidden layer used features logistic regression classier output layer return probability means predicted scores item user sigmoid function used nonlinearity function scales output range bigger score obviously means preference however applications always prefer score range say rescale output neutral network right range multiplying result factor see section following parameters need trained edge weight nodes output layer hidden layer rating records users used training set training data takes form triplet rating user item full learning objective takes following form structural risk minimization tries minimize prediction error training predicting function distributional representation model sequentially consists tanh function sigmoid function use standard regularization parameters weighted general used train model taking derivatives respect four groups parameters use optimization converges local optimum objective function experiment datasets evaluating proposed model use movielens eachmovie datasets movielens dataset contains ratings approximately movies made movielens user eachmovie contains ratings entered user different movies experiments ninety percentage rating data randomly chosen training rest used test set experiment results rmse commonly used evaluation standard recommendation system use experiments order compare performance distributional representation model model algorithms use mahout test bed commonly used recommendation algorithms classical knn based model slopeone als improved knn based model proposed koren elaborately tuned get good performance experiment results listed table best run model following parameters length user vector item vector number nodes hidden layer results indicate consistently good performance model datasets made distributional representation model promising direction research collaborative filtering table results movielens eachmovie datasets model rmse movielens dataset rmse eachmovie dataset knn knn slopeone als koren knn model related works many popular algorithms proposed recent years among improved knn proposed koren latent factor shows great performance advantages latent factor models explain ratings characterizing items users terms factors inferred pattern ratings one successful realizations latent factor models based matrix factorization svd proposed distributional representation model categorized latent factor explicitly encodes latent features users items word embedding vectors compared matrix factorization distributional representation model directly combine latent factor vectors neutral network structure explore complex nonlinearity interdependencies features framework neutral network method deep leaning approach rbm wang model show different network structures different optimization target compared proposed model conclusion present paper concise distributional representation model collaborative filtering best knowledge first study use word embedding recommendation system conclude experiment results model outperforms algorithms many cases made distributional representation model promising direction research collaborative filtering introduce tensor model natural regard model special case model explore general deep leaning model future work references rifai dauphin vincent bengio muller manifold tangent classifier nips krizhevsky sutskever hinton imagenetclassification deep convolutional neural networks inadvances neural information processing systems nips dahl deng acero context dependent deep neural networks large vocabulary speech recognition ieee transactions audio speech language processing mohamed dahl hinton acoustic modeling using deep belief networks ieee trans audio speech language processing ronan collobert jason weston unified architecture natural language processing deep neural networks multitask learning proceedings international conference onmachine learning pages acm ronan collobert jason weston michaelkarlen koraykavukcuoglu pavel language processing almost journal machine learning turian ratinov bengio word representations simple general method learning proceedings acl pages bengio ducharme vincent neural probabilistic language march huang socher manning improving word representations via global context multiple word prototypes movielens dataset http eachmovie dataset http apache mahout http bell koren scalable collaborative filtering jointly derived neighborhood interpolation weights ieee international conference data mining koren factorization meets neighborhood multifaceted collaborative filtering model inkdd proceeding acm sigkdd international conference knowledge discovery data mining pages new york usa acm koren bell volinsky matrix factorization techniques recommender systems ieee ruslan salakhutdinov andriy mnih geoffrey hinton restricted boltzmann machines collaborative filtering proceedings international conference machine learning salakhutdinov boltzmann aistats pages hao wang naiyan wang yeung collaborative deep learning recommender
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mar virtual network embedding approximations leveraging randomized rounding matthias rost stefan schmid berlin germany email mrost university vienna austria email stefan schmid virtual network embedding problem vnep captures essence many resource allocation problems today cloud providers offer physical computation networking resources customers customers request resources form virtual networks directed graph specifying computational requirements nodes bandwidth requirements edges embedding virtual network shared physical infrastructure joint mapping virtual nodes suitable physical servers together mapping virtual edges onto paths physical network connecting respective servers study offline setting vnep multiple requests given task find profitable set requests embed exceeding physical resource capacities paper initiates study approximation algorithms vnep employing randomized rounding linear programming solutions show standard linear programming formulation exhibits inherent structural deficit yielding large even infinite integrality gaps turn focusing class cactus graphs virtual networks devise novel linear programming formulation together algorithm decompose fractional solutions convex combinations valid embeddings applying randomized rounding obtain first approximation algorithm classic resource augmentation model ntroduction cloud applications usually consist multiple distributed components virtual machines containers results substantial communication requirements provider fails ensure communication requirements met performance suffer dramatically consequently last years several proposals made jointly provision computational functionality virtual machines together appropriate network resources virtual network embedding problem captures core problem given directed graph specifying computational requirements nodes bandwidth requirements edges virtual network must embedded physical network computational network requirements met figure illustrates two incarnations virtual networks service chains virtual clusters paper study objective maximizing profit deciding requests embed embed selected requests capacities physical substrate network violated note results also extended objective minimizing overall cost embedding requests customer nat cache internet fig examples virtual networks wild left graph shows use case service chains mobile operators particular example used route parts traffic cache using optimize user experience furthermore firewall translation used security concerns right graph depicts virtual cluster abstraction used provisioning virtual machines data centers abstraction provides connectivity guarantees via virtualized switch center virtual machine connects formal problem statement light recent interest service chaining extend vnep general definition considering different types computational nodes accordingly physical network offering set computational types considered set types may contain firewall server etc refer physical network substrate network type set denotes substrate nodes host functionality type denoting node resources rsv substrate resources rsv capacity nodes edges denoted set request denoted request directed graph given refer nodes graphs virtual request nodes edges virtual request edges using indicate virtual node types mapping virtual node restricted set vsr mapping virtual edge restricted esr virtual node edge attributed resource demand respectively virtual nodes edges mapped substrate nodes edges sufficient capacity vsr request attributed benefit denote dmax maximal demand request may impose resource dmax rsv dmax max following notions valid mappings respecting mapping constraints feasible embeddings respecting resource constraints introduced formalize vnep definition valid mapping valid mapping rev quest tuple mvr functions following holds virtual nodes mapped allowed substrate nodes mvr vsr holds mapping virtual edge connecting mvr mvr using allowed edges holds denote set valid mappings request definition allocations valid mapping allocation induced mapping resource defined follows maximal allocation valid mapping request may impose substrate resource denoted amax maxmkr definition feasible embedding feasible embedding subset requests collection valid mappings cumulative allocations nodes edges exceed substrate capacities holds resources definition virtual network embedding problem vnep asks finding feasible embedding subset requests maximizing profit related work last decade vnep attracted much attention due many applications survey already lists different algorithms many variations vnep known strongly phard amaldi recently shown approximated within factor unless holds based hardness vnep works consider heuristics without performance guarantee works proposed exact methods integer constraint programming coming cost exponential runtime acknowledging hardness general vnep diversity applications several subproblems vnep studied recently considering restricted graph classes virtual networks substrate graph example virtual clusters uniform demands studied line requests studied tree requests studied considering approximation algorithms even employed randomized rounding obtain constant approximation embedding line requests arbitrary substrate graphs approximation guarantee however comes price strong assumptions benefits capacities interesting work bansal give time log algorithm minimizing load embedding trees based strong relaxation inspired hierarchy best knowledge approximation algorithms known arbitrary substrate graphs classes virtual networks beyond trees lastly work closely related unsplittable flow problems usfp exist several approximation results usfp similar inapproximability result vnep usfp approximated within factor proven baveja srinivasan gave approximation algorithm matching bound shown guruswami usfp approximated randomized rounding within factor log high probability assumption demands polynomially bounded high probability refers success probability parameter solution found almost certainly setting large enough preliminary version work published technical report work essentially obtain results using much simpler formulation furthermore paper focus approximating profit variant vnep technical report also details approximations cost variant outline randomized rounding vnep shortly revisit concept randomized rounding given integer program certain problem randomized rounding works computing solution linear relaxation decomposing solution convex combinations elementary solutions iii probabilistically choosing elementary solutions based weight denoting set valid mappings request convex combination valid mappings frk mkr frk must recovered linear programming solution request profit convex combinations equals profit achieved linear program convex combinations violate substrate capacities results organization paper initiates study approximation algorithms vnep general substrates general virtual networks beyond lines trees specifically employ randomized rounding obtain first constant factor approximation algorithm class cactus graph requests resource augmentation model studying classic flow mcf formulation vnep section show tree requests linear solutions decomposed convex combinations valid mappings hence allowing application randomized rounding contrast result tree requests show requests containing cycles general decomposed valid mappings result ramifications beyond inability apply randomized rounding prove mcf formulation exhibits large even infinite integrality gaps investigating root cause surprising result devise novel linear programming formulation convex combinations valid mappings recovered class cactus graph requests see section iii given ability decompose linear programming solutions present analyze performance randomized rounding algorithm section obtaining first approximation algorithm virtual network embedding problem lassic ulti ommodity ormulation vnep imits section study relaxation standard flow mcf formulation solving vnep first show positive result formulation sufficient decompose virtual networks trees convex combinations valid mappings subsequently show formulation fails allow decomposition cyclic requests prohibits randomized rounding approach yields integrality gap best classic formulation formulation presents classic mcf formulation vnep first describe integer variant computes single valid mapping request using binary variables linear programming variant obtained relaxing binary variables domain variable indicates whether request embedded variable indicates whether virtual node mapped substrate node similarly flow variable indicates whether substrate edge used realize virtual edge variable denotes cumulative allocations embedding request induces resource constraint virtual node request must placed suitable substrate nodes vsr iff holds constraint forbids mapping nodes may host respective virtual node constraint induces unsplittable unit flow virtual edge substrate location onto mapped substrate location onto mapped constraint virtual edges may mapped allowed substrate edges constraints compute cumulative allocations nodes edges respectively constraint guarantees capacities substrate resources respected state following lemma formalizing connectivity property enforced constraints lemma local connectivity property formulation mcf formulation safeguards following connectivity property virtual edge substrate node vsr exists path flow along edge respect variables greater path computed polynomial time proof first note constraint holds hence virtual nodes must mapped equal extent suitable substrate nodes fix substrate node vsr also partially mapped holds result follows directly connects using allowing empty path hand holds formulation classic mcf formulation vnep max esr rsv right hand constraint induces flow value side constraint may attain negative values nodes vsr holds flow emitted node must eventually reach node vsr hence result follows path constructed simple search considers edges holds decomposing solutions tree requests given lemma present algorithm decompose solutions formulation convex combinations valid mappings underlying undirected graph tree note formulation binary variables relaxed take value interval given virtual network algorithm processes virtual edges according acyclic representation era undirected interpretation rooted consider tree requests arborescence computed simple breadthfirst search underlying undirected graph starting denote edges request whose orientations reversed process computing algorithm decomposition algorithm mcf solutions tree requests input tree request solution formulation acyclic reorientation output convex combination set set mkr mvr set set mvr choose vsr choose set foreach era compute connecting according lemma set mvr else let compute connecting mvr vsr according lemma set reverse set mvr lemma given virtual network request whose underlying undirected graph tree algorithm decomposes solution formulation valid mappings mkr frk following holds decomposition complete holds decomposition resource allocations bounded holds resource set limits classic mcf formulation set set frk min set frk set add drk frk mkr set return algorithm extracts mappings mkr value frk iteratively long holds initially iteration none virtual nodes edges mapped holds constraint must exist node vsr algorithm sets mvr given initial fixing algorithm iteratively extracts nodes queue already mapped considers outgoing virtual edges era outgoing edge contained lemma readily applied obtain mapping edge node edge orientation reversed iff holds apply lemma reversing flow tion concretely flow variables interpreted reversed substrate edges introduced lemma allows construct path reversed substrate mvr holds reversing path path obtained correctly connects mapped node mvr mvr according original edge orientation note mapping virtual node virtual edge valid construction furthermore arborescence edge node eventually mapped hence mkr valid mapping mapping value frk computed minimum mapping variables used constructing mkr reducing values mapping variables together allocation variables constraints continue hold decomposition process continues long holds step least one variable value set easy check algorithm terminates complete decomposition drk frk holds algorithm polynomial runtime number variables request bounded lastly observations note following lemma relating mcf solution decomposition computed algorithm lemma shown constructively relaxations classic mcf formulation decomposed convex combinations valid mappings underlying graph tree show case anymore virtual networks contain cycles theorem fractional solutions standard multicommodity flow formulation general decomposed request solution substrate partial decomposition fig example showing linear relaxations integer program general decomposable convex combinations valid mappings request simple cyclic graph shall mapped substrate graph assume vsr vsr vsr holds solution depicted follows substrate nodes annotated mapping virtual nodes hence node indicates virtual node mapped substrate node substrate edges dashed according dash style virtual links mapped onto virtual links also mapped using flow values dash style substrate edge therefore implies holds convex combinations valid mappings virtual networks contain cycles proof figure visually depict example solution formulation single valid mapping extracted validity depicted solution follows fact virtual node mappings sum virtual node connects neighboring node half unit flow assume sake contradiction depicted solution decomposed virtual node mapped onto substrate node neighboring node hosts must exist mapping mvr mvr similarly mvr must hold however flow virtual edge leaving leads hence virtual node must mapped possible argument holds considering mapping onto valid mapping extracted solutions also yields high unbounded integrality gap proven theorem integrality gap mcf formulation unbounded considering edge mapping restrictions even holds capacities substrate infinite proof consider example figure introduce following restrictions mapping virtual links esr esr esr note solution depicted figure still feasible hence attain objective hand exist valid mapping request optimal solution achieves profit hence integrality gap unbounded theorem integrality gap mcf formulation lies considering node mapping restrictions proof consider following instance substrate cycle even number nodes edges consider unit edge capacities set consider request edges unit bandwidth demands assume mapped substrate nodes uneven index mapped substrate nodes even index clearly valid mapping request use edge resources consider following mcf solution nodes together respective edges mapped alternating fashion holds vsr similarly set edges originating uneven even nodes respectively mapping induces allocations edge hence many copies request may embedded mcf solution optimal solution may embed single request integrality gap therefore lies iii ecomposable ormulation mcf formulation suffices compute decomposable solutions tree requests lemma consider requests may contain cycles concretely consider virtual networks whose undirected interpretation cactus graph two undirected cycles intersect single virtual node partitioning cactus requests let request whose underlying undirected interpretation cactus graph consider acyclic reorientation rooted following easily seen lemma graph cactus request uniquely partitioned set cyclic subgraphs corresponding graphs forest corresponding graph edge era contained exactly one graphs edge set era cycle partitioned two branches end branches start node node note gra subgraphs may contain edges whose orientation reversed original edge set denote respective subgraphs agree edge orientation illustrate partitioning consider partition request figure era partition consists single cyclic subgraph furthermore note holds shorten notation denote potential mapping locations target cycle set vsr novel formulation cactus requests novel formulation uses priori partition lemma set cycles forest construct partial embedding formulations respective subgraphs request corresponding forest mcf formulation employed compute mappings subgraph constraint refer associated variables etc similarly request cycle potential target node mapping mcf formulation constructed constraint use square brackets reference specifc cycle target node variable belongs bind together first independent formulations mappings respective subgraphs reuse variables introduced already mcf formulation refer variables defined outside formulations global variables mark using square brackets note variables used formulation continuous binding together different mapping subgraphs done follows similar constraint formulation employ constraint enforce setting global node mapping variables constraints node mappings mapping subgraphs must agree global node mapping variables exist many copies embedding cyclic subgraph sum node mappings formulations must agree global one constraint crucial importance decomposability considering cycle target node enforces target node cycle may mapped lastly allocation variables computed follows node mappings global node mapping variables considered constraint global edge allocations set sum edge allocations formulations decomposing solutions novel formulation show adapt decomposition algorithm tree requests algorithm also enable decomposition cactus requests first note construction arbitrary acyclic interpretation used requests decompose request exactly graph needs handed decomposition algorithm second formulation contain global edge mapping variables edge mapping variables used lines algorithm must substituted edge mapping variables respective formulations concretely edge request covered exactly clear whether virtual edge part cyclic subgraph holds edge mapping variables used hand edge covered cyclic subgraph exist many choose respective edge mapping variables ensure decomposability proceed follows edge first edge mapped iteration employ mapping variables belonging arbitrary target node holds sublp must exist constraint another edge cycle already mapped iteration commodity formulation decomposable formulation cactus requests max variables variables vrf vsr vrck vsr rsv chosen variables commodity employed minor additions valid mappings cactus requests indeed recovered construction algorithm edge mappings valid extracted valid hence need check mapping nodes agrees previous node mappings considering virtual target nodes cycles mapped mapping decisions target node cycles anticipated deciding variables consider branches cycle always lead substrate node lastly note steps taken lines algorithm must slightly adapted also decrement variables lemma still holds respect obtained decompositions cactus requests state following without proof theorem given solution novel formulation cactus request solution decomposed convex combination valid mappings drk mkr frk following holds decomposition complete holds decomposition resource allocations bounded holds resource pproximation via andomized rounding shown optimal convex combinations vnep computed cactus requests turn approximation algorithm vnep whose profit presented algorithm algorithm first performs preprocessing lines removing requests fully fractionally embedded absence requests requests fully embedded respective requests never part feasible solution hence removed lines randomized rounding scheme applied solution formulation computed decomposed rounded rounding procedure iterated long constructed solution sufficient quality number maximal rounding tries exceeded concretely seek solutions achieve least factor times optimal profit exceed node edge capacities factors respectively following discuss parameters solutions found high probability note algorithm indeed algorithm novel formulation times larger mcf formulation hence solved ellipsoid algorithm furthermore decomposition constructing solution using rounding performed probabilistic guarantee profit bounding profit achieved randomized rounding scheme recast profit achieved terms random variables discrete random variable models profit achieved potentially embedding request probability request embedded profit obtained equals accordingly drk profit achieved overall hence equals decomposition complete lemma blp blp denotes profit optimal solution bound probability achieving fraction profit optimal solution make use continuous random variables theorem let sum independent random exp holds preprocessing requests removing requests embedded absence requests know attain least maximal profit requests lemma blp acknowledging objective value linear program upper bound objective integral embedding applying obtain following algorithm randomized rounding algorithm vnep foreach preprocess requests compute solution formulation request set maximizing remove request set holds compute solution formulation request set maximizing decompose solution convex combinations drk frk mkr perform randomized rounding construct solution choosing mapping mkr probability frk solution maximal rounding tries exceeded theorem probability achieving less profit optimal solution upper bounded exp proof let denote maximum benefit requests consider random variables holds let denote total profit achieved scaling profits blp holds lemma choosing theorem obtain exp plugging minimal value equation obtain exp accordingly exp mentioned blp denoting optimal profit integer program optimal solution bip observing bip blp holds solution contained solution space bip accordingly obtain bip exp completing proof probabilistic guarantees capacities following analyze probability randomly rounded solution violates substrate capacities certain factor employ hoeffding inequality theorem hoeffding inequality let indep pendent random variables exp holds vsr esr contain substrate elements actually support mapping virtual node virtual edge terms resources respectively lemma dmax model allocation resource request random variable amax frk furthermore denote drk random variable describing overall allocations resource rspafter rounding solution drk holds definition using theorem obtain holds resources using apply hoeffding inequality bound node edge resource violations lemma consider node resource rsv choose dmax holds let max max log following holds proof choose log apply hoeffding log exp amax log exp amax log exp second inequality amax amax used follows dmax next step factored denominator plugged definition lastly utilize expected load upper bounded resource capacity obtain lemma statement noted network functions unique within request equals number requests since case amax dmax holds virtual nodes share type may bepmapped substrate nodes bounded edge resource violations essentially result obtained lemma consider single edge choose dmax holds let amax log probability capacity exceeded factor log bounded main result applying union bound joint probability either achieve enough profit less fraction violate resource capacity factor nodes edges bounded hence probability return solution within rounds bounded obtain following theorem theorem assume let chosen minimally dmax holds algorithm approximation algorithm vnep finds solution high probabiliy least fraction optimal profit cumulative allocations within factors nodes edges original capacities rsv log log defined max max dmax proof apply union bound argument first note defined using function maximum taken values occurring lemmas accordingly probability cumulative allocations node resource rsv larger less lemma given maximally many network resources overall probability exceeds capacity factor less similarly lemma probability cumulative allocations edge surpass less edges probability allocations edges exceeds upper bounded lastly theorem probability finding solution optimal objective less equal exp probability find suitable solution satisfying objective capacity criteria within single round therefore upper bounded exp holds probability find suitable solution within many rounds hence lower bounded hence randomized rounding scheme yields solution high probability rsv pand bounded factors respectively letting denote maximal size request contained obtain realistic assumption number considered types constant holds obtain resource violation bounds log respectively hence obtain following corollary corollary let denote maximal size request graph assumption number node types constant holds log holds resources request randomized rounding yields approximation vnep high probability assuming furthermore maximal size virtual network request bounded constant holds randomized rounding yields constant approximation profit resource violations high probability onclusion paper initiated study approximation algorithms virtual network embedding problem supporting arbitrary substrate graphs supporting virtual networks containing cycles obtain approximation extended classic formulation surprisingly decomposed convex combinations enable decomposability cactus requests novel linear programming formulation relies cactus nature requests analysis performance randomized rounding scheme independent convex combinations computed hence hope extend presented formulation corresponding decomposition algorithm obtain approximations even larger graph classes future acknowledgements work partially supported aalborg university prelytics project well german bmbf software campus grant authors would like thank elias alexander elvers parts paper contributing significantly implementation found https eferences edoardo amaldi stefano coniglio arie mca koster martin tieves computational complexity virtual network embedding problem electronic notes discrete mathematics hitesh ballani paolo costa thomas karagiannis ant rowstron towards predictable datacenter networks acm sigcomm computer communication review volume pages acm nikhil bansal lee viswanath nagarajan murtaza zafer minimum congestion mapping cloud proc acm podc alok baveja aravind srinivasan approximation algorithms disjoint paths related routing packing problems mathematics operations research chowdhury rahman boutaba virtual network embedding coordinated node link mapping proc ieee infocom devdatt dubhashi alessandro panconesi concentration measure analysis randomized algorithms cambridge university press guy even moti medina boaz online path computation function placement sdns proc international symposium stabilization safety security distributed systems sss guy even matthias rost stefan schmid approximation algorithm path computation function placement sdns proc sirocco fischer botero till beck meer hesselbach virtual network embedding survey comm surveys tutorials ieee venkatesan guruswami sanjeev khanna rajmohan rajaraman bruce shepherd mihalis yannakakis hardness results approximation algorithms paths related problems journal computer system sciences jon kleinberg decision algorithms unsplittable flow paths problem proc annual acm symposium theory computing stoc pages tamas lukovszki stefan schmid online admission control embedding service chains proc international colloquium structural information communication complexity sirocco sevil mehraghdam matthias keller holger karl specifying placing chains virtual network functions proc ieee cloudnet october jeffrey mogul lucian popa talk talk cloud network performance acm sigcomm ccr napper haeffner stiemerling lopez uttaro service function chaining use cases mobile networks april url https hartert declarative expressive approach control forwarding paths networks sigcomm matthias rost carlo fuerst stefan schmid beyond stars revisiting virtual cluster embeddings proc acm sigcomm computer communication review ccr matthias rost stefan schmid service chain virtual network embeddings approximations using randomized rounding corr url http matthias rost stefan schmid anja feldmann time optimal virtual network embeddings temporal flexibilities proc ieee ipdps pages xie ning ding charlie ramana kompella constant change incorporating network reservations data centers proc acm sigcomm minlan yung jennifer rexford mung chiang rethinking virtual network embedding substrate support path splitting migration sigcomm comput commun
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squaring operation commutative rings oct amnon yekutieli abstract let homomorphism commutative rings squaring operation functor derived category complexes operation needed definition rigid complexes sense van den bergh turn leads new approach grothendieck duality rings schemes even stacks paper zhang introduced squaring operation explored properties unfortunately proofs paper severe gaps present paper reproduce construction squaring operation done general context first paper consider homomorphism commutative rings first main result square independent resolutions used present second main result trace functoriality squaring operation give precise statements complete correct proofs subsequent paper reproduce remaining parts paper require fixing allow proceed papers mentioned bibliography rigid approach grothendieck duality proofs main results require substantial amount foundational work commutative noncommutative rings including study rings lifting properties homotopies part paper could independent interest contents introduction facts rings modules resolutions modules central pairs rings resolutions ring homotopies pairs modules compound resolutions rectangles rectangle operation squaring operation references introduction background rigid dualizing complexes concept rigid dualizing complex introduced van den bergh influential paper vdb done noncommutative ring base field let recall date october key words phrases rings modules derived categories derived functors resolutions mathematics subject classification primary secondary supported israel science foundation grant amnon yekutieli definition simplify matters interested commutative situation shall state commutative ring let field commutative denote derived category given complex square complex structure comes first argument rhom note single module hhq hochschild cohomology assume finitely generated hence noetherian rigid dualizing complex relative pair dualizing complex sense isomorphism van den bergh proved rigid dualizing complex exists unique isomorphism work direction done zhang author series papers see references papers dealt noncommutative situation significantly complicated current paper interested commutative rings relative situation instead base field homomorphism commutative rings want produce useful theory squaring rigidity homomorphism flat pretty easy way generalize follows given complex may define however flat formula meaningless since way interpret object paper zhang proposed solve flatness problem replacing ring flat ring resolution work rigid dualizing complexes commutative rings done paper outlined survey paper work progress rigid dualizing schemes stacks indeed rigid approach grothendieck duality allows first time state prove global duality proper map stacks ideas outlined lecture notes unfortunately serious flaws proofs explained subsection introduction discovery flaws forced back repair foundations thus present paper provide comprehensive correct treatment squaring operation using ring method companion paper destined repair problems enhance results repairs foundations done intend proceed geometric application rigid dualizing complexes namely papers progress rings resolutions let ring usual short differential graded texts would call unital associative algebra see remark discussion nomenclature say nonpositive ring called strongly commutative odd squaring operation call commutative ring nonpositive strongly commutative term appearing title paper denote category commutative rings observe category commutative rings full subcategory since ring seen ring concentrated degree rings appearing introduction commutative exception subsections however paper must deal noncommutative rings explained subsection even though motivation squaring operation commutative rings develop squaring operation generally commutative rings way shall present results introduction reason twofold first added difficulty working commutative rings compared commutative rings second presentation cleaner working totally commutative framework let homomorphism commutative rings morphism pair commutative rings notation refer pairs form category given another pair morphism pairs commutative diagram resulting category denoted pdgrsc consider pair pdgrsc resolution morphism surjective see section review various kinds module resolutions including ones surj let morphism resolution commutative diagram vertical arrows resolutions also say morphism resolutions case identity automorphism pair say morphism resolutions according proposition resolutions pairs morphisms pairs exist amnon yekutieli let homomorphisms refer data triple commutative rings resolution triple commutative diagram resolution pair resolution pair first triple viewed morphism pairs viewed resolution resolutions exist proposition squaring operation recall rings commutative ring associate category derived category additive functor identity objects sends isomorphisms sections recall facts modules resolutions related derived functors let homomorphism rings namely object given resolution let proposition definition provide explicit presentation terms compound resolutions see remark regarding symmetric compound resolutions consider morphism pairs module module morphism clarification actually morphism forw forw forgetful functor restriction scalars corresponding ring homomorphism however time shall suppress forgetful functors sake clarity given resolution morphism sqb morphism constructed using compound resolutions see proposition definition construction functorial arguments mean another morphism resolution morphism morphisms sqb see proposition key technical result paper repeated theorem body paper proved brief discussion proof found subsection introduction squaring operation theorem homotopy invariance let morphism let let let phism suppose resolutions respectively morphisms resolutions morphisms sqb equal first main theorem paper theorem existence squares let homomorphism commutative rings let unique unique isomorphism together isomorphism resolution satisfying following condition morphism resolutions diagram idm isomorphisms commutative module called square relative theorem repeated theorem body paper already mentioned assertion rings rings appeared theorem proof large gap discussion comparison ailn theorem see subsection second main theorem paper theorem trace functoriality let homomorphisms commutative rings let let let morphism unique morphism satisfying condition resolution triple diagram commutative amnon yekutieli theorem repeated theorem statement already appeared theorem rings rings proof loc cit also incorrect known functoriality operation section implies assignments sqidb functor called squaring operation relative functor linear fact quadratic functor following sense given morphism element morphisms see theorem slightly general statement proofs theorems rather easy prove know theorem existence resolutions however proof theorem quite long difficult name suggests involves homotopy homomorphisms way know produce homotopy ring noncommutative equality see definition theorem thus forced deal noncommutative rings instead pairs commutative rings discussed relevant noncommutative object central pair rings namely central homomorphism commutative source see definitions let central pair rings resolution central pair rings together quasiisomorphism diagram bimodule namely module enveloping ring must replace single module pair consisting left right situation instead square rectangle object lives derived category center graded sense see definition noncommutative situation homotopy theorem noncommutative version theorem theorem proved two stages first important special case noncommutative done lemma proof lemma relies theorem uses cylinder ring general case reduced special case theorem existence rectangles noncommutative version theorem rectangle operation interesting even commutative pair rings due connection monoidal operation see remark discussion related papers early versions theorems already appeared paper zhang theorems respectively however great embarrassment proofs results severe gaps gapsi text one clear error homomorphism middle page proof theorem make sense unless order give squaring operation correct treatment necessary work noncommutative ring resolutions explained subsection mistake proof theorem discovered avramov iyengar lipman nayak ailn also found way fix ailn theorem generalization theorem indeed ailn theorem establishes rectangle operation case central pair rings proof ailn theorem relies quillen model structure category dgr noncommutative rings central commutative base ring following aspect found remark note theorem noncommutative version theorem stronger ailn theorem allow central pair nonpositive rings goals different ailn hence adopt different strategy relative situation homomorphism commutative rings crucial must consider triples commutative rings reason stick commutative ring resolutions relying theorem know whether methods ailn adapted yield trace functoriality squaring operation theorem acknowledgments wish thank james zhang bernhard keller vladimir hinich liran shaul rishi vyas asaf yekutieli sharon hollander help writing paper facts rings modules section review known facts rings modules talk central homomorphisms rings finally introduce cylinder construction rings modules differential graded ring ring short graded ring together additive endomorphism degree called differential differential satisfies graded derivation namely satisfies graded leibniz rule texts would call associative unital algebra definition denote dgr category rings morphisms graded ring homomorphisms commute differentials consider rings rings concentrated degree thus category rings becomes full subcategory dgr definition let ring ring pair ring ring homomorphism suppose rings homomorphism rings homomorphism homomorphism rings denote dgr category rings thus dgr usual coslice category except say instead see remark regarding direction arrows obvious forgetful functor dgr dgr morphism dgr shown amnon yekutieli commutative diagram remark probably justify use ring instead traditional algebra maintain generally whenever homomorphism category rings ought called explanation consider commutative base ring given ring homomorphism target commutative simply say commutative noncommutative context things delicate one says associative unital meaning ring homomorphism central namely image center suggest instead use central context one says associative unital image lies inside subring degree central cocycles propose use central instead definition notice name ring allows drop adjectives associative unital also separates rings lie algebras etc present paper actually need consider ring homomorphisms source genuine ring cases even noncommutative thus sometimes center subring cocycles say say homomorphism might central nonetheless definition consider graded ring homogeneous elements define graded commutator extends using bilinearity namely aik ajl often notation used graded commutator definition let ring center subring ace ring called weakly commutative ace words homomorphism dgr called central ace case resulting ring homomorphism ace denoted uce denote dgr full subcategory dgr consisting pairs central squaring operation implicit item definition easy check fact ace indeed subring ring ace weakly commutative item morphism category dgr required central centrality condition central homomorphisms dgr also central thus get subcategory dgrce dgr consisting objects central homomorphisms letting dgrwc full subcategory dgr weakly commutative rings get functor dgrce dgrwc letl ring recall left left graded differential degree satisfies graded leibniz rule equation instead right defined similarly default modules paper left modules let ring let left let degree additive homomorphism say sign convention abelian group homomorphisms degree denoted homa let homa homa graded abelian group homa differential given homa particular get ring enda homa suppose ring abelian group see structure ring homomorphism endz right left usual tensor product graded abelian group subgroup generated tensors graded abelian group differential satisfying let rings tensor product ring multiplication aik differential like let ring opposite ring abelian group denote elements bop multiplication bop note centers satisfy iff weakly commutative amnon yekutieli right seen left action bop observation working left modules limitation shall switch notations according convenience consider namely left action right action commute viewed left action bop thus structure abelian group homomorphism rings endz let weakly commutative ring dgr abimodule dgr obvious ring structure moreover dgr given tensor product module action definition let ring denote dgmod category left morphisms homomorphisms degree commute differentials since name long mostly use abbreviation dgmod category abelian set also category structure set morphisms abelian group homa relation structures homm homa group ofl let graded abelian group recall shift twist translation suspension graded abelian group defined follows graded component degree note ungraded abelian groups gradings different thus identity automorphism becomes degree invertible homomorphism graded abelian groups shall usually represent elements following structure differential action right module bimodule structure determined concretely squaring operation note sign conventions shift canonical bijection isomorphism power denoted corresponding degree homomorphism graded abelian groups identity underlying ungraded abelian group note automorphism category usual write recall cone construction looks using operator let ring let homomorphism cone cone whose differential express module column left multiplication matrix degree abelian group homomorphisms learned concept cylinder ring defined keller appears without name consider matrix ring cyl element degree multiplication differential dcyl graded commutator element explicitly dcyl ring homomorphisms cyl formulas easy see idz identity automorphism definition let ring cylinder ring cyl cyl multiplication induced differential induced using formulas convenient denote elements cyl matrices rather tensors note element central degree proposition let ring consider induced ring phisms cyl homomorphisms ring homomorphisms central moreover induced homomorphisms ace cyl cyl ace bijective amnon yekutieli proof since cyl free graded via forgetting differential direct consequences corresponding assertions clear definition let ring cylinder cyl cyl cyl cyl multiplication induced differential induced definition relates also right modules bimodules explained proposition let ring let induced module homomorphisms cyl cyl quasiisomorphisms considering homomorphism cyl view cyl via ring homomorphism cyl likewise considering homomorphism cyl view cyl module via ring homomorphism cyl proof proposition immediate fact cyl cyl cylinder operation rings modules following functoriality ring homomorphism induced ring homomorphism cyl cyl cyl cyl idcyl explicitly formula cyl likewise modules remark abstract categorical sense ring cyl plays role dual cylinder perhaps called path object indeed precisely path object described page scsh decided adhere name cylinder two reasons first name used second reason dealing rings happens algebraic geometry arrows tend reversed reversal occurs definition talk rings rather rings categorical convention would dictate resolutions modules let ring already mentioned dgmod category equipped translation automorphism homotopy category dgmod dgmod definition dgmod homa triangulated category derived category dgmod gotten dgmod inverting localization functor identity objects dgmod dgmod ring module category mod dgmod mod dgmod mod dgmod mod squaring operation references derived category modules section section section another reference course notes sections category associated abelian category replaced category dgmod definition shall use following abbreviations triangulated categories related ring dgmod dgmod let recall facts resolutions module called acyclic resp called resp acyclic homa resp homa also acyclic module called every acyclic aop acyclic classical situation implies definitions introduced section shown property property every admits resolutions resolutions see also afh chapter modules discussed section recall diagram called commutative homotopy homomorphisms homotopic words corresponding diagram commutative next proposition essentially standard fact state play key role especially section proposition let assume either kinjective additive homomorphism homm homd surjective kernel group homa homomorphisms let morphism exists homomorphism unique homotopy let homomorphisms homomorphisms homotopic iff words item says given morphism first diagram exists homomorphism making diagram commutative unique homotopy item says second diagram commutative homotopy iff third diagram commutative amnon yekutieli proof according propositions additive homomorphism homk homa homd bijective note ring definitions proofs pass without change case ring immediate ring homomorphism induces forgetful restriction scalars functor foru exact hence gives rise triangulated functor foru usually mention functor foru explicitly unless important discussion hand foru equivalence aop rhoma isomorphisms rhomb proposition result used lot paper restriction functors suppressed easy proof left reader proposition let homomorphisms dgr let let homomorphisms respectively abelian group aop action function formula homomorphism aop equality say also homomorphisms respectively homotopies respectively homotopy aop item proposition illustrated proposition let rings let aop let let quasiisomorphisms squaring operation aop homv homa proof implicit proof proposition detailed proof symmetry situation assume either respective rings begin reducing second case first case choose resolution suffices prove first case assume consider homomorphism ida ida claim see let look first commutative diagram homomorphism idp since homomorphism idp isomorphism therefore ida idp idp idp ida ida next look second commutative diagram homomorphism defined isomorphism homomorphism idp ida idq ida aop therefore choose resolutions resolutions respectively proposition lift diagrams commutative homotopy get diagram homa homid homa homv homid homv amnon yekutieli commutative homotopy let examine commutative diagram homa homid idi homa homid homid idq homa homid homid homa homomorphisms homid homid quasiisomorphisms least one homomorphisms homid idq homid idi follows diagonal homomorphism homid similar argument shows homid quasiisomorphism going back diagram see enough prove homv let homomorphism fits commutative diagram homid canonical isomorphism quasiisomorphisms see finally consider commutative diagram homa homv homid homa homid adj adj adjunction isomorphism since quasiisomorphisms follows homv central pairs rings resolutions section introduce several special kinds rings homomorphisms recall dgr category rings definition ring called nonpositive full subcategory dgr consisting nonpositive rings denoted definition ring called strongly commutative satisfies two conditions weakly commutative definition namely odd denote dgrsc full subcategory dgr strongly commutative rings squaring operation definition ring called commutative ring nonpositive strongly commutative full subcategory dgr consisting commutative rings denoted words dgrsc dgr example commutative ring concentrated degree commutative ring remark name strongly commutative ring suggested palmieri earlier version paper used name strictly commutative ring following algebra name weakly commutative ring commutative algebra algebra ailn course number invertible contains condition definition implies condition distinction weak strong commutativity disappears see also remark central homomorphisms introduced definition definition central pair rings central homomorphism dgr ring commutative usually denote pair keeping implicit suppose central pairs rings morphism central pairs consists homomorphisms dgr resulting category called category central pairs rings denoted pdgr morphism pdgr shown diagram require homomorphism central homomorphism automatically central commutative definition nonpositive central pair central pair nonpositive rings full subcategory nonpositive central pairs denoted commutative central pair central pair ring also commutative full subcategory commutative central pairs denoted morphism pdgr called central homomorphism central subcategory pdgr objects central homomorphisms denoted pdgrce morphism pdgr called homomorphisms quasiisomorphisms definition let central pair rings resolution morphism pdgr surjective resolution called strict ida identity automorphism see diagram illustration amnon yekutieli definition let morphism pdgr let resolution let resolution morphism pdgr said morphism resolutions pdgr case also say resolution identity automorphism say morphism resolutions morphism shown diagram graded set mean set partition elements said degree say nonpositive graded set filtered graded set graded set ascending filtration graded subsets let ring denote graded ring gotten forgetting differential modules talking free modules assume ring nonzero avoid nonsense variable degree let shift element denote element note consider graded set variables free free isomorphic graded set variables image called basis let filtration issan ascending filtration submodules grf free say admits filtration see afh module hence also let filtration choose graded subset whose image grj basis ofsthis free graded set filtered filtered graded set called note every isomorphism graded element image grj satisfies hence see given graded set variables form noncommutative polynomial ring zhxi free set monomials elements obvious multiplication grading next definition standard first occurrence seems definition let graded set variables strongly commutative polynomial ring quotient zhxi ideal generated elements odd definition consider nonpositive central pair rings say commutative ring commutative ring homomorphism commutative pair rings isomorphism graded squaring operation nonpositive graded set strongly commutative polynomial ring graded set called set commutative ring generators say noncommutative ring noncommutative ring homomorphism noncommutative pair rings isomorphism graded zhxi nonpositive graded set zhxi noncommutative polynomial ring graded set called set noncommutative ring generators say ring ring homomorphism pair rings left right matter proposition let nonpositive central pair rings either commutative noncommutative proof noncommutative case let set noncommutative ring generators monomials using fact graded set nonpositive commutative case let set commutative ring generators choose ordering set monomials odd using fact graded set nonpositive see cases thus example let commutative ring let sequence elements koszul complex associated sequence commutative ring set commutative ring generators degree remark strongly commutative polynomial ring free graded makes proposition work weakly commutative polynomial ring quotient zhxi ideal generated elements flat indeed odd element definition let central pair rings let resolution definition assume commutative pair say commutative resolution pair commutative definition say noncommutative resolution pair noncommutative definition say resolution pair definition example ring resolution terminal among ring resolutions example let surjective homomorphism commutative rings let sequence elements generates ideal ker assume sequence regular define koszul complex strict commutative ring resolution amnon yekutieli remark require surjective omission significant mostly concerned case ring proposition nonzero negative components surjectivity needed proof theorem next results section enhanced versions propositions category introduced definition lemma let commutative ring let assume either two conditions holds commutative commutative commutative ring generating set noncommutative noncommutative ring generating set let degree function function extends uniquely graded homomorphism graded ring homomorphism extends ring homomorphism iff proof easy exercise lemma let nonpositive graded set let commutative ring consider graded ring zhxi let degree function function extends uniquely degree derivation differential derivation differential iff proof take elements consider monomial resp zhxi let commutative case nothing check noncommutative case extending additively obtain degree homomorphism resp zhxi extends degree homomorphism satisfies graded leibniz rule easy exercise ring denote set cocycles set coboundaries kernel image respectively subring zero differential course graded ideal theorem let nonpositive central pair rings commutative exists strict commutative ring resolution exists strict noncommutative ring resolution squaring operation proof looking commutative quasiisomorphism shall construct increasing sequence commutative differential denoted time shall construct increasing sequence graded sets compatible sequence isomorphisms shall also construct compatiblessequence ring homomorphisms homomorphism desired properties construction recursion moreover every following conditions hold homomorphisms surjective degrees homomorphism bijective degrees start choosing set degree elements function set generates let set degree elements bijection let unique function choose set degree elements function set generates define graded set graded ring setting gives degree function according lemma get differential graded ring becomes lemma says function extends uniquely homomorphism condition holds condition holds trivially take assume already defined satisfies conditions choose graded set degree elements function cohomology classes elements generate let consider choose graded set degree elements function cohomology classes elements generate ker exists let choose graded set degree elements function set generates let set degree elements bijection let unique function define lastly choose graded set degree elements function letting amnon yekutieli set generates since surjective exists define define graded set commutative graded ring degree function according lemma function induces differential get ring likewise degree function lemma induced homomorphism satisfies looking noncommutative quasiisomorphism proof part except replace zhfi everywhere theorem let central pair nonpositive rings suppose given two factorizations surjective either two conditions holds ring commutative ring commutative semifree ring noncommutative exists homomorphism situation shown commutative diagram dgr surj proof proof similar proposition let set commutative resp noncommutative generators define let generated set resp zhfi construct consistent sequence homomorphisms satisfying construction recursion required properties start take since surjective exists let resulting function extends uniquely homomorphism lemma squaring operation next consider assume homomorphism defined satisfying required conditions take element since surjective exists cocycle cocycle let also cocycle thus cohomology class satisfies bijective conclude hence define way obtain function extends according lemma function extends uniquely homomorphism graded equation lemma imply fact homomorphism rings corollary object admits nonpositive resolution morphism admits nonpositive resolution proof let noncommutative ring resolution see theorem strict nonpositive resolution let noncommutative ring resolution let commutative ring resolution next let noncommutative ring resolution resolution resolution also homomorphism finally according theorem find homomorphism respects homomorphisms ring homotopies section introduce concept homotopy ring homomorphisms many ideas section communicated privately keller course responsibility correctness definition suppose homomorphisms rings additive homomorphism degree called derivation satisfies twisted graded leibniz formula aki ring homotopy degree satisfying homotopy formula homotopy results valid noncommutative rings definition recall notions commutative ring central homomorphism category definitions lemma let commutative ring let homomorphisms assume noncommutative set noncommutative ring generators let degree function amnon yekutieli unique degree extends homomorphism ring homotopy iff proof define additive homomorphism zhxi degree letting elements need noncommutative work unless extend elements using isomorphism zhxi easy calculation using induction shows theorem let commutative ring let central momorphism suppose given two factorizations surjective noncommutative let homomorphisms satisfy ring homotopy satisfies situation depicted commutative diagrams surj proof choose set noncommutative ring generators graded rings let zhxi let generated thus zhfk define linear homomorphism degree ring homotopy recursively take since follows isomorphism graded rings see let construct use fact elements cocycles take define since cohomology class isomorphism see therefore consider element namely cocycle using fact bijective find cocycle exists squaring operation surjectivity says define way get function lemma shows function extends uniquely homomorphism ring homotopy consider assume already take note claim element cocycle indeed since ring homotopy claimed proof like case see therefore cohomology class using fact isomorphism conclude hence cocycle using fact bijective find cocycle exists surjectivity says define way get function extend function defining lemma shows function extends uniquely homomorphism ring homotopy ring homotopies expressed using cylinder construction see definition next result similar theorem proposition let ring homomorphisms let homomorphism degree following conditions equivalent homomorphism ring homotopy homomorphism ucyl cyl formula ucyl ring homomorphism proof straightforward calculation amnon yekutieli remark let commutative ring ring categories dgr admit quillen model structures weak equivalences noncommutative rings cofibrant see theorem think quite plausible true even ring nontrivial negative part evidence provided theorems hand know whether categories dgrsc admit similar quillen model structures negative indication theorem seem hold commutative rings another negative indication even ring known except contains see recall ailn theorem discussed subsection introduction deals commutative base ring proof theorem hinges quillen model structure produced model structure exist general case commutative ring predict would likely imply theorem proof similar ailn however even model structures categories would probably sufficient imply theorem proof theorem actually proof noncommutative variant theorem requires delicate treatment central morphisms central pairs see setup definition something quillen model structure rather coarse structure appear able provide pairs modules compound resolutions rectangles recall pdgr category central pairs rings see definition resolutions pdgr defined definition category central dgr defined definition definition let object pdgr enveloping ring ring dgr let morphism pdgr induced ring homomorphism dgr way obtain functor pdgr dgr lemma let pdgr let resolution proof homomorphisms also forget rings structures view modules respectively proposition applies definition let object pdgr let object namely left right define module squaring operation let morphism pdgr let let morphism define morphism remark omitted reference forgetful functors definition thus homomorphism actually homomorphism forw homomorphism actually homomorphism forwop shall continue omissions sake clarity forgetful functors made explicit one given homomorphism two rings hope shortcut cause confusion definition let object pdgr let object let resolution compound resolution following data resolution resolution resolution denote compound resolution easy see compound resolutions exist consider resolution pdgr left right actions commute commute action graded sense thus module hence make next definition spirit proposition definition situation definition define module homb next setup shall used rest section setup given central pair pdgr resolution pdgr pair modules mkl mkr bkop given central morphism pdgr morphism pdgrce see definition morphism mkl mkr bkop morphism consider index morphisms redundant hence suppress amnon yekutieli definition consider setup let compound resolution mkl mkr compound morphism following data resolutions respectively homomorphisms respectively respectively homomorphism morphisms denote compound morphism also say compound resolution definition illustrated next diagrams commutative diagrams categories respectively vertical arrows diagrams namely whose names contain letters isomorphisms note letter denotes localization functor confused modules etc composition rule compound morphisms taken care following lemmas lemma situation definition compound resolutions morphism exist proof homomorphisms exist respective categories homomorphism exists see proposition next definition crucial homomorphism central guaranteed setup demand morphism pdgrce since central get induced ring homomorphism proposition applies squaring operation definition situation definition let morphism homwen lemma morphism independent compound resolution namely another compound resolution equality morphisms proof say modules homotopy equivalent equivalence homomorphisms homotopic likewise therefore modules homotopy equivalent equivalence homomorphisms homotopic implies homomorphisms homotopic lemma assume morphisms identities also assume compound morphism morphism identity automorphism proof clear lemma consider setup given compound resolution mkl mkr given compound morphism also given compound morphism equality rect morphisms proof choosing homomorphisms lift respectively concoct composed morphism homotopy whose component thus rect hand lemma know rect rect amnon yekutieli lemma pdgr isomorphism isomorphism proof according proposition homwen proposition let pair pdgr resolution let pair object unique unique isomorphism together isomorphism rect every compound resolution pair satisfying condition let compound resolutions pair let compound morphism idm idm diagram rect rect isomorphisms commutative proof uniqueness clear existence let fix resolution define isomorphism rect identity given resolution let morphism identity define isomorphism rect lemma depend choice verify condition let morphism identity according lemma proposition situation setup unique morphism satisfying condition squaring operation let compound resolution mkl mkr let compound morphism diagram rect rect morphisms commutative proof choose compound resolution choose compound morphism superscript stands basic done lemma let unique morphism condition holds respect choices prove condition holds arbitrary choice compound resolutions compound morphism let choose compound morphisms idm idm consider following diagram rect rect rect rect top square commutative definition bottom square commutative lemmas commutative condition proposition therefore outer paths equal prove proposition situation setup diagram rect morphisms commutative morphisms identity automorphisms identity automorphism amnon yekutieli proof triangular diagram commutative lemmas condition proposition assertion identity automorphisms true lemma remark could defined rectangle object directly composition three functors words proposition would become definition however approach would made hard make precise sense morphism prove functoriality result proposition rectangle operation section continue material section slightly different notation recall pdgr category central pairs rings full subcategory nonpositive central pairs see definitions main result theorem throughout section work following setup setup given central morphism namely morphism see definition resolutions morphisms see definitions object object morphism ring input depicted following commutative diagram category four central homomorphisms belonging four central pairs lemma inclusive also assume next condition condition homomorphism noncommutative definition squaring operation homomorphisms equal view item condition may write homomorphism lying ring input setup condition summarized following commutative diagrams category recall cylinder ring definition lemma condition homomorphism cyl dgr diagram commutative cyl cyl cyl proof know noncommutative surjective theorem applies obtain ring homotopy proposition deduce existence ring homomorphism cyl formula shows commutativity diagram next element commutes homomorphism cyl deduce commutativity first square diagram second square trivially commutative central homomorphisms cyl cyl thus two central pairs cyl cyl central pairs fit commutative diagram amnon yekutieli cyl cyl cyl category pdgr moreover morphisms pairs lower row central namely pdgrce pairs upper row vertical morphisms consider module cyl cyl define homomorphisms cyl cyl cyl homomorphism cyl next define homomorphism cyl cyl homomorphism done consider next commutative diagram category cyl cyl lemma condition equality morphisms rectb proof step assume morphisms identity automorphisms equality also assume morphism identity automorphism may also assume important functoriality proposition cyl squaring operation compare diagrams lemma three morphisms isomorphisms caution needed phism actually category cyl forgetful functor cyl hidden know see proposition ring homomorphism cyl induces isomorphism cyl therefore functor equivalence since formula leads conclude isomorphisms cyl rectcyl cyl cyl cyl step special assumptions choose homomorphism lemma functoriality equalities formula part applies depend deduce holds key technical result paper noncommutative version theorem homotopy invariance situation setup equality morphisms rectb proof choose following resolutions strict resolution strict noncommutative resolution strict noncommutative resolution exist theorem homomorphism homomorphism homomorphisms exist theorem resolutions homomorphisms two paragraphs fit following commutative diagram central pairs resolutions central pairs resolutions amnon yekutieli let introduce temporary abbreviations rectb etc applying functoriality proposition commutative diagram obtain commutative diagram morphisms whose names contain letter isomorphisms note get commutative diagram condition satisfied morphisms resolutions lemma tells since isomorphism conclude going back diagram using fact isomorphisms see theorem existence rectangles let commutative ring let central homomorphism nonpositive rings given pair module unique unique isomorphism together isomorphism every resolution following condition holds squaring operation let morphism resolutions diagram idm idm isomorphisms commutative proof choose commutative resolution choose noncommutative resolution possible theorem subscript stands universal note resolution define consider resolution according theorem morphism resolutions define idm idm theorem tells isomorphism depend choice remains verify condition suppose another resolution morphism resolutions choose morphism resolutions diagram isomorphisms proposition says finally theorem says diagram condition indeed commutative amnon yekutieli remark let regular finite dimensional noetherian commutative ring let cohomologically commutative ring see section suppose ring homomorphism essentially finite type let full subcategory consisting modules whose cohomology bounded whose cohomology modules finite let write gaitsgory states field characteristic operation makes monoidal category proof statement given assume ring noetherian commutative ring essentially finite type according corrections rigid dualizing complex relative functor rhoma duality category exchanges recently shaul showed bifunctorial isomorphism dbf proof relies reduction formula hochschild cohomology ailn theorem isomorphism implies operation monoidal structure subcategory dbf fid dbf consisting complexes finite injective dimension monoidal unit work progress shaul indicates result hold greater generality first paragraph remark objects squaring operation section rings commutative namely work inside category see definition category pdgrsc commutative pairs rings introduced definition full subcategory category pdgr central pairs rings definition let object resolution resolution sense definition pair commutative definition morphisms resolutions resolutions morphisms pdgrsc definition pairs commutative proposition object morphism admits resolution proof like proof corollary using commutative resolutions commutative ring course used following definitions definition let pair commutative rings let given resolution let see proposition squaring operation definition let morphism let let let morphism given resolution define sqb morphism proposition remark explained remark write however morphism definition standard use explicit approach compound resolutions done section still simplify matters using symmetric compound resolutions symmetric compound resolution data consisting resolution resolution likewise morphism resolved symmetric compound morphism consists resolution homomorphism lifting homomorphism lifting key technical result paper theorem introduction actually special case theorem theorem homotopy invariance let morphism let let let phism suppose resolutions respectively morphisms resolutions morphisms sqb equal proof take theorem brings first main theorem paper theorem introduction theorem existence squares let homomorphism commutative rings let module unique unique isomorphism together isomorphism resolution satisfying following condition amnon yekutieli let morphism resolutions equality idm isomorphisms proof proof similar theorem fix universal resolution commutative resolution let commutative resolution let commutative resolution possible theorem define given resolution theorem says morphism resolutions define isomorphism idm verification condition proof theorem except use theorem instead theorem definition triple data homomorphisms dgrsc refer triple commutative resolution triple together homomorphisms surjective see diagram introduction commutative resolution viewed follows resolutions morphism resolutions morphism pairs proposition commutative resolutions triples exist proof follows theorems compare proof corollary second main theorem paper theorem introduction theorem trace functoriality let homomorphisms commutative rings let let let morphism unique morphism satisfying condition squaring operation commutative resolution triple equality morphisms proof begin choosing universal resolution triple choose order commutative resolutions run sun tun way obtain resolving triple define let verify condition given arbitrary commutative resolution triple according theorem find homomorphisms diagram commutative sun tun let look corresponding diagram squaring operation diagram top square commutative proposition applied twice two commutative condition theorem outer paths equal definition vertical arrows isomorphisms conclude bottom square commutative prove corollary let pair commutative rings assignments sqidb functor amnon yekutieli proof consider morphisms combination proposition condition theorem shows lemma idm definition object theorem called square relative functor corollary called squaring operation relative following result says squaring operation also functorial ring proposition given homomorphisms commutative rings modules morphisms morphisms proof choose resolution obvious sense generalizes definition according condition theorem suffices prove morphisms true proposition proposition let commutative ring action proposition makes category proof center see linear category see formula next formula see category hence localization category given homomorphism commutative rings objects set homd action coming action object proposition end paper next result theorem situation theorem let morphisms proof choose resolution choose strict commutative resolution putting together obtain resolution commutative let symmetric compound resolution let symmetric compound resolution see remark definition find homomorphism lifts obtain symmetric compound morphism squaring operation specifically homomorphism lift proposition represented homven homc homb remind choose represents cohomology class lifts lifts get symmetric compound morphism resolving homomorphism homven homven homven represents theorem shows functor quadratic functor references afh avramov foxby halperin differential graded homological algebra preprint dated june available authors ailn avramov iyengar lipman nayak reduction derived hochschild functors commutative algebras schemes advances mathematics ail avramov iyengar lipman reflexivity rigidity complexes commutative rings algebra number theory bernstein lunts equivariant sheaves functors lecture notes mathematics springer bokstedt neeman homotopy limits triangulated categories compositio math baues pirashvili shukla cohomology additive track theories eprint arxiv gaitsgory sheaves eprint hinich homological algebra homotopy algebras communications algebra erratum arxiv keller deriving categories ann sci norm sup keller cyclic homology exact categories journal pure applied algebra correction author web page keller algebras modules functor categories eprint arxiv milne cohomology princeton university press maclane homology reprint hartshorne residues duality lecture notes math berlin shaul hochschild category commutative algebras via twisting eprint appear comm algebra shaul reduction hochschild cohomology algebras finite center pure applied algebra scsh schwede shipley algebras modules monoidal model categories proc london math soc spaltenstein resolutions unbounded complexes compositio math stacks project online reference jong editor http vdb van den bergh existence theorems dualizing complexes graded filtered ring algebra yekutieli rigid dualizing complexes via differential graded algebras survey triangulated categories lms lecture note series amnon yekutieli yekutieli duality tilting commutative rings eprint yekutieli residues duality schemes stacks lecture notes http yekutieli course derived categories http yekutieli rigid complexes commutative rings revisited preparation yekutieli rigidity residues duality schemes preparation yekutieli rigidity residues duality stacks preparation yekutieli zhang dualizing complexes perverse sheaves noncommutative ringed schemes selecta math yekutieli zhang rigid complexes via algebras trans ams yekutieli zhang rigid dualizing complexes commutative rings algebras representation theory number department mathematics ben gurion university sheva israel address amyekut
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popperian falsification lighthill argument defended steven meyer smeyer february abstract area computation called artificial intelligence falsified describing previous falsification british applied mathematician james lighthill explained lighthill arguments continue apply current argued use popperian scientific method duty every scientist attempt falsify theories theories falsified replace modify paper describes popperian method detail discusses paul nurse application method cell biology also involves questions mechanism behavior arguments used lighthill original report falsifed discussed lighthill arguments shown apply current argument uses recent scholarship explain lighthill assumptions show arguments based assumptions continue falsify modern iimportant focus argument involves hilbert philosophical programme defined knowledge truth provable formal sentences current takes hilbert programme dogma beyond criticism lighthill mid century applied mathematician abandoned paper uses recent scholarship explain john von neumann criticism claim assumed lighthill paper discusses computer chess programs show lighthill combinatorial explosion still applies humans argument showing turing machines correct description computation given paper concludes advocating studying computation peter naur dataology introduction paper applies method falsification discovered karl popper show artificial intelligence programs intelligent fact normal computer programs programmers express ideas writing computer code meaningless metaphysics popperian sense metaphysics based number incorrect assumptions dogmas falsified james lighthill evaluation british science funding agency lighthill paper defends lighthill century falsification explains applies current paper presents material author developed encouraged criticize stanford university undergraduate talk given paul feyerabend philosophy science seminar author computer science student berkeley order understand lighthill criticism falsifies arguments still apply second decade century spite vast improvements computer speed capacity necessary understand development modern computers primarily physicists wwii paper uses recent historical scholarship explain lighthill background assumptions shows background knowledge also falsifies current popperian falsification falsification method discovered karl popper argues general statements scientific merit singular statements popper calls basic statements simple structure meaning statements disproven either scientific experiments logic popper popper major contribution philosophy science insist duty every scientist criticizes one theories fullest extent possible false theories modified replaced popperians believe scientific method consists numerous bold conjectures tested falsified eliminated modified popper method calls bold conjecture followed stringent criticism popper original falsification theory developed late early called naive falsification lakatos theory improved generalized popper colleagues century using term popperian philosophy sense includes modifications improvement popper theory mostly carried london school economics popper also imre lakatos paul feyerabend thomas kuhn aspects popperian methodology clearly expressed imre lakatos methodology scientific research programmes msrp lakatos disagreements among popperians questions emphasis methodology importance rationality science james lighthill holder lucasian chair applied mathematics cambridge university familiar part milieu developed popperian theory falsification theory philosophy science usually discussed terms physics developers trained physicists physics possibly good fit study methodology mechanism functional explanation involved attempting understand physical reality describe fields particle interactions example connection cell biology attempts understand utilize mechanisms cell behavior closer paul nurse popper memorial lecture discusses importance bold conjectures diligent attempts eliminate incorrect theory falsification nurse nurse also discussed data analysis cell biology readers unfamiliar popperian falsification nurse lecture provides excellent introduction falsification important claimed computational intelligence successful discussions ethical issues involving inferior humans deal superior intellect robots required author believes primary obligation scientists eliminate false theories lighthill falsification lighthill falsification quite simple lighthill claim continues apply spite changes mostly vastly faster computers execute machine instructions parallel new names algorithms deep learning replaces alpha beta heuristics improve logic resolution algorithms implement intelligence lighthill argues described using language human intelligence views computers computation tools expressing people ideas lighthill divides three areas category automation feedback control engineering category computer based studies central nervous system category bridge area supposedly going provide magic synergy allow creation intelligent robots example current deep learning would fall areas falls category involves automatic logical deduction without need person program ideas algorithm also category looks beyond conventional data processing problems involved data banking retrieval think lighthill arguing studies normal computer science rephrases problems terms human attributes paragraph according lighthill control engineering matter engineering accomplished lighthill writes section discussing category nevertheless must looked natural extension previous work automation human activities judged essentially criteria paragraph years computer development programmable digital computers usually best choice control engineering modern terms current feedback control engineering based improvements camera technology allowing precise location measurements complex feedback advances cost reductions computer storage technology allow large amounts data processed faster lower cost criticizing approach area since obviously makes sense study neurophysiology lighthill distinguishes syntactic automation advocated currently versus conceptual automation asks device mimics human function somehow assists studying making theory function central nervous system paragraph lighthill criticizes use mathematical logic arguing practical use runs combinatorial explosion paragraph argues difficulties storing axioms favored logicians versus heuristic knowledge favored paragraph view crucial falsifier namely although lighthill attempting provide neutral assessment believe hilbert programme central tenet lighthill also discusses organization problems methodology questions claims robots better humans probably replace lighthill applied mathematician also discusses combinatorial explosion humans solve solved formal algorithms understanding lighthill falsification modern terms lighthill falsified showing individual claims false arguing unified subject rather normal problems computation involving computer applications study data researchers convinced time think lighthill make popperian view science clear remainder paper discusses scientific background knowledge especially physics applied mathematics areas falsifies current methods discussion possible recent scholarship especially areas hilbert philosophical programme study john von neumann thinking development digital computers skepticism toward hilbert programme truth formal proof mathematician david hilbert conjectured knowledge truth consists solely sentences proven axioms hilbert original conjecture mathematical problem however interpreted philosophical theory truth became formal proof axioms paradigmatic example birkhoff von neumann formalization quantum mechanics axiomatized logic birkhoff popper attempted falsify hilbert programme basic assumption knowledge world expresses formal sentences knowledge expressed formulas derived using logic usually predicate calculus sentences world true addition belief knowledge formal sentences foundation belief church thesis copeland true namely nothing exist outside formally proven sentences proven axioms community dogma beyond criticism however philosophical hilbert programme abandoned starting various reasons reason often given goedel incompleteness results showed hilbert programme could succeed hilbert programme still believed logic area seems grasping straw attempts mitigate goedel disproof finding practice areas goedel results apply zach stanford encyclopedia philosophy article discusses attempts mitigate goedel results see detlefsen skeptical view hilbert programme number reasons hilbert philosophical programme rejected reasons explain argument since people intelligence computer programs also intelligence view problem building faster computers developing better algorithms computers discover learn formal sentences people heads fact reasons hilbert programme abandoned show lighthill falsification correct meaningless metaphysics von neumann argument automata neural networks useless high levels complexity second half century john von neumann work computers computations widely accepted publication von neumann work computing occur years lighthill falsification written particular aspray neumann kohler however lighthill applied mathematician certainly familiar von neumann work john von neumann studied automata neural networks developing von neumann computer architecture von neumann combined skepticism toward linguistics automata sources algorithms discussing problems formal neural networks wrote insight formal neuron network anything describe words important insight simplifies matters enormously low complication levels means certain simplification high complication levels perfectly possible high complication levels value theorem reverse direction namely express logics terms efforts converse may true von neumann quoted aspray note von neumann also considered rejected current methodology developed von neumann computer architecture paper herman goldstine design digital computer von neumann believed sort intuition built programs instead using brute force searching aspray edward kohler kohler describes von neumann discovery developing modern computer architecture article von neumann rejected carnap duality information concepts readers tempted regard claim trivial automata simulate arbitrarily complex behavior assuming described exactly enough fact describing behavior exactly first place constitutes genuine scientific creativity prima facie superficial task von neumann achieved famous explication von neumann machine regarded standard architecture post computers problem context area operations research solution space searching influenced von neumann lighthill pre computer algorithmic operations research experience see budiansky detailed story understanding limitations combinatorial explosion arises naturally experience skepticism toward linguistics formal languages computing starting ludwig wittgenstein late skepticism toward linguistics especially formal languages become prevalent wittgenstein claim mathematical language nothing pointing wittgenstein popperians english science general receptive wittgenstein pointing philosophy mathematics popperians avoid linguistic philosophy viewed creating problems solved read lighthill falsification assuming attitude toward language modern still claims knowledge truth limited provable formal sentences physicist skepticism towards mathematics axiomatized logic view important reason rejection hilbert programme physicists always skeptical toward axiomatized mathematics albert einstein lecture geometry expresses skepticism einstein believed formal mathematics incomplete disconnected physical reality einstein stated view axioms advocated modern axiomatics purges mathematics extraneous elements expurgated exposition mathematics makes also evident mathematics predicate anything objects intuition real objects einstein niels bohr argued first comes conceptual theory calculation finsler rejection axiomatics general inconsistency result addition skepticism toward axiomatics also skepticism toward set theory core claim sentences derivable axioms zermelo fraenkel probably exist swiss mathematician paul finsler believed mathematics exists outside language formal sentences finsler claimed shown incompleteness formal systems goedel proof superior tied russell logic goedel see restoration failed paul finsler theory sets breger discussion finsler result undecidability formal proofs history also finsler finsler chess elite human players response chess programs superiority chess programs even best human chess players cited evidence future robots superior areas involving intelligence fact situation complicated response world best chess players shows lighthill claims even formal sentenced based toy world combinatorial explosion limits problem solving ability algorithms study chess playing programs evaluation efficacy show problems recent claims successes general deep blue chess program defeated world champion gary kasperov since world best chess players adjusted computer chess programs december financial times newspaper chess column leonard barton referring champion fabiano caruana writes champion world unleashed brilliant opening novelty incidentally showed limitations powerful computers barton taken two decades caruana five years old kasperov lost deep blue appears computer algorithms run combinatorial explosion problems best players defeat computers possibly interesting claims show problems scientific methodology emphasize lack diligent attempts falsify theory first financial incentive structure challenge meant kasperov made money losing rather winning kasperov viewpoint could win back collecting meager chess tournament prize money lose collect large appearance fee plus receiving numerous appearance fees marketing representative many claims success involving human competition computers follow pattern minimum tests type need use double blind protocols better method determining computers defeat best human players would use double blind tournaments opponents may humans computers participants officials allowed know even better would system chess player natural competitiveness utilized losing lower rated human player would result large deduction rating points finally progress chess playing computer programs shows chess programs normal data processing applications lighthill sense human knowledge chess expressed amplified injecting computer writing computer program turing machine incorrect model computation central argument based church turing thesis namely turing machines universal anything involves intelligence calculated tms applying lighthill combinatorial explosion arguments seems tms wrong model computation instead different computational model called mrams random access machines unit multiply bounded number unbounded size memory cells better model computation meyer von neumann understood need random access memory design von neumann architecture ibid mram machines deterministic non deterministic computations solvable polynomial bound time least problems class combinatorial explosion mitigated suggests algorithms studied normal data processing assumption heuristics guessing somehow improve algorithms problematic conclusion suggestion replace naur dataology problem paper people trained perform advanced computational research primarily physicists imagine content people trained became formalized object oriented programming computer programs verified correctness proofs axiomatized proofs algorithm efficiency imagine anything computation formalized logic computation researchers trained unable imagine alternatives dogmas suggestion adopt ideas danish computer scientist trained astronomer peter naur naur argued computation studied dataology dataology theory neutral term studying data naur wrote mental life twentieth century become entirely misguided ideological position discussions adopt computer inspired form accepted naur peter naur one founders computer science realized become much formal mathematics separated reality naur advocated importance programmer specific program development use preconceptions would put computation allows people express ideas writing computer programs clearest explanation naur method appears book conversations pluralism software engineering naur books amplifies program development method naur described turing award lecture naur naur page interviewer asks basically say foundations thing computer science must formalize sake formalization alone naur answers sure see way see techniques tools applicable cases definitely basic sense naur continues programmer realize alternatives choose one suits understanding best nothing formal proofs dataology without preconceptions predictions imminent replacement human intelligence robots would improve scientific study computation next step advocates would try falsify naur dataology references aspray barton birkhoff breger budiansky copeland aspray john von neumann origins modern computing mit press barton chess column financial times games page weekend life style section edition birkhoff von neumann logic quantum mechanics annals math breger restoration failed paul finsler theory sets gillies revolutions mathematics oxford budiansky blackett war men defeated nazi brought sciene art warfare knopf copeland thesis stanford encyclopedia philosophy summer edition url detlefsen detlefsen encyclopedia hilbert programme philosophy einstein formalism routledge url hilbert einstein geometry experience lecture prussian academy sciences berlin january url finsler finsler kohler lakatos lakatos lighthill finsler ueber die unabhaengigkeit der continuumshypothese dialectica finsler finsler set theory platonism circularity booth ziegler eds birkhauser kohler von neumann rejected carnap duality information concepts redei stoltzner eds john von neumann foundations quantum physics vienna circle institute yearbook kluwer lakatos falsification methodology scientific research programmes lakatos musgrave eds criticism growth cambridge press lakatos feyerabend method motterlini university chicago press lighthill artificial intelligence general survey artificial intelligence paper symposium science research council url meyer meyer naur naur meyer adding methodological testing naur iacap proceedings college park maryland meyer philosophical solution equal also solution page naur knowing mystique logic rules kluwer academic naur computing science anatomy human mental life publishing appendix url naur naur neumann nurse naur computing versus human thinking comm acm naur conversations pluralism software engineering daylight belgium lonely scholar publishing von neumann redei john von neumann selected letters history mathematics series vol american mathematical society nurse philosophy drives discovery scientists view popper popper memorial lecture london school economics podcast url http popper popper wittgenstein zach popper logic scientific discovery harper row original german popper birkhoff von neumann interpretation quantum mechanics nature wittgenstein wittgenstein lectures foundations mathematics cambridge diamond university chicago press zach hilberts program stanford encyclopedia philosophy spring edition url https
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mis congested clique model log log christian konrad feb department computer science centre discrete mathematics applications dimap university warwick coventry abstract give maximal independent set mis algorithm runs log log rounds congested clique model maximum degree input graph improves log upon log log log rounds algorithm ghaffari podc log number vertices input graph first stage algorithm simulate first polynlog iterations sequential random order greedy algorithm mis congested clique model log log rounds thins input graph relatively quickly stage maximum degree residual graph second stage run mis algorithm ghaffari podc residual graph completes log log rounds graphs degree introduction local congest models local congest models studied computational models distributed graph algorithms models communication network represented graph also constitutes input computational graph problem vertex network node hosts computational unit identified unique log initially besides every vertex knows neighbors ids network nodes simultaneously commence execution distributed algorithm algorithm proceeds synchronous rounds round consists two phases computation phase every vertex may execute unlimited computations followed communication phase vertices may exchange individual messages neighbors message lengths unbounded local model congest model every message length log goal design algorithms employ communication rounds possible output typically distributed independent set problems focus paper upon termination algorithm every vertex knows whether participates independent set local model provides abstraction allows study locality distributed problem far network nodes need able look network order complete certain task addition locality constraint congest model also addresses issue congestion example local model network nodes learn neighborhoods rounds generally possible congest model due limitation message sizes model recent years model variant congest model received significant attention differs congest model every pair vertices opposed every pair adjacent vertices exchange messages sizes log communication phase focus model thus solely lies issue congestion since message exchanges possible model least powerful congest model many problems computing minimum spanning tree computing size maximum konrad supported centre discrete mathematics applications dimap warwick university epsrc award matching fact solved much faster congest model ghaffari asks whether classic local problems maximal independent set mis maximal matching solved much faster model congest model maximum degree log log input graph ghaffari made progress question gave log log rounds mis algorithm model best known congest model algorithm runs log log log rounds algorithm separates two models regards mis problem since known min rounds required mis congest model log log log result ghaffari gave roughly quadratic improvement best congest model mis algorithm paper show exponential improvement possible main result follows theorem main result let graph maximum degree randomized algorithm model operates deterministic log log rounds outputs maximal independent set high probability techniques ghaffari gave variant mis algorithm runs log log rounds graphs maximum degree polylog lemma achieve runtime log log rounds even graphs arbitrarily large maximum degree give log log rounds algorithm computes independent set residual graph denotes inclusive neighborhood maximum degree run ghaffari algorithm residual graph complete independent set computation algorithm implementation sequential greedy algorithm mis model greedy processes vertices input graph arbitrary order adds current vertex initially empty independent set non neighbors previously added key idea simulate multiple iterations greedy rounds model simulation iterations rounds done follows let arbitrary ordering vertices ids observe subgraph induced first vertices edges lenzen gave routing protocol used collect edges one distinguished vertex rounds vertex simulates first iterations greedy locally observe knowledge sufficient notifies nodes chosen independent set selection presented simulation used obtain rounds mis algorithm model reduce number rounds log log identify residual sparsity property greedy algorithm greedy processes vertices uniform random order maximum degree residual graph processed kth vertex log high probability lemma make use property thus first compute uniform random ordering vertices processed first vertices maximum degree residual graph allows increase block size simulate next iterations rounds using fact maximum degree residual graph hard see lower bound even holds local model variant works fact graphs maximum degree bounded log sufficiently small constant degree bound sufficient purposes use notation equals usual notation factors ignored subgraph induced next random vertices maximum degree high probability thus contains edges pursuing approach process vertices ith block since residual sparsity lemma maximum degree ith residual graph hence processed log log blocks maximum degree becomes section give slightly involved arguments show log log iterations opposed log log iterations fact enough residual sparsity property greedy author aware work exploits mentions residual sparsity property random order greedy algorithm mis context correlation clustering data streaming model similar property greedy clustering algorithm used lemma lemma fact strong enough give version required paper since provide proof residual sparsity property central functioning algorithm give proof follows main idea adapted needs related work maximal independent set problem one classic symmetry breaking problems distributed computing without communication luby independently alon gave log rounds distributed algorithms years barenboim improved certain ranges gave algorithm currently fastest algorithm ghaffari runs log log rounds log log rounds log mis algorithm designed model previously mentioned algorithm ghaffari ghaffari shows multiple rounds congest model algorithm simulated much fewer rounds model similar approach taken paper however algorithm simulation multiple iterations sequential greedy algorithm performed one distinguished node every node participates simulation congest model algorithm ghaffari algorithm outline proceed follows first give necessary definitions notation state known results employ paper section give proof residual sparsity property sequential greedy algorithm section log log rounds mis algorithm subsequently presented section followed brief conclusion section preliminaries assume simple unweighted graph node write denote exclusive neighborhood write degg inclusive neighborhood defined inclusive neighborhoods extended subsets given subset vertices subgraph induced denoted independent sets independent set subset vertices independent set maximal every independent set given independent set call graph residual graph respect clear context may simple call residual graph say vertex uncovered respect adjacent vertex clear context simply say uncovered without specifying explicitly ghaffari gave following result reuse paper authors kindly shared extended version paper theorem ghaffari let graph poly log distributed algorithm runs model computes mis log log rounds routing subroutine algorithm needs solve following simple routing task let arbitrary vertex suppose every vertex holds messages size log wants deliver guaranteed lenzen proved model deterministic routing scheme achieves task rounds following refer scheme lenzen routing scheme concentration bound dependent variables analysis algorithm require chernoff bound dependent variables see example theorem chernoff bound dependent variables let random variables inequality holds let every last say event occurs high probability probability event occuring sequential random order greedy algorithm mis greedy algorithm maximal independent set processes vertices input graph arbitrary order adds current vertex consideration initially empty independent set none neighbors already algorithm progressively thins input graph rate graph loses edges depends heavily order vertices considered vertices processed uniform random order algorithm number edges residual graph decreases relatively quickly variant next lemma proved context correlation clustering streaming model lemma let integer let set beginning iteration algorithm probability least following holds proof fix arbitrary index prove either vertex neighbors probability least result follows union bound error probabilities vertices consider following process random order vertices determined first reveal reveal vertices iteration algorithm let input graph let uniform random ordering set uncovered elements return algorithm random order greedy algorithm mis set neighbors uncovered beginning iteration let every following holds since one yet revealed vertices distinguish two cases first suppose result follows immediately since construction sequence decreasing suppose next prove high probability one iteration neighbor considered algorithm turn implies mis algorithm congest clique model algorithm mis algorithm depicted algorithm consists three parts first vertices agree uniform random order follows vertex smallest choses uniform random order locally informs vertices positions within order vertices broadcast positions vertices result vertices know entire order let order next simulate greedy maximum degree residual graph bound chosen convenience number equally suitable end iteration first determine number function maximum degree current residual graph subgraph induced yet uncovered vertices edges see lemma using lenzen routing protocol edges collected vertex continues simulation greedy iteration informs chosen vertices selection turn inform neighbors selection vertices compute new residual graph maximum degree proceed next iteration prove input graph maximum degree set parameter nodes agree random order vertices exchange ids one round let vertex smallest vertex choses uniform random order informs every vertex position within order every vertex broadcasts vertices result vertices know order let resulting order simulate sequential greedy every vertex sets true indicating uncovered let every vertex broadcasts vertices every vertex knows let every vertex true sends incident edges true using lenzen routing protocol rounds vertex knows subgraph uncovered vertices true continues simulation greedy iteration using let vertices selected independent set vertex informs nodes selection one round nodes inform neighbors selection one round every node sets alse let true every vertex broadcasts vertices every vertex computes locally broadcasts vertices result every vertex knows end run ghaffari algorithm run ghaffari mis algorithm log log rounds algorithm log log rounds mis algorithm model lemma log log iterations necessary drops last run ghaffari algorithm completes maximal independent set computation analysis let denote graph beginning iteration notice let let value iteration observe executed hence log holds every iteration let graph iteration establish runtime algorithm need bound number iterations end next lemma bound every conclude log log lemma probability least every maximum degree bounded follows proof prove statement induction observe statement thus trivially true suppose statement holds index recall residual graph obtained running greedy vertices hence applying lemma following holds probability resolving recursion obtain observe invoked times lemma thus union bound result holds probability corollary log log establish correctness algorithm need ensure apply lenzen routing protocol collect edges vertex feasible need prove every contains edges high probability lemma probability least graph edges proof let vertex set set uncovered vertices beginning iteration prove probability least every following holds since vertex set subset vertices result follows applying union bound error probabilities every vertex prove inequality observe graph solely determined vertices execution algorithm far affected outcome random variables thus principle deferred decision every vertex seen uniform random vertex chosen let indicator variable event observe furthermore observe every inequality holds using bound follows inequality since worst case implies choices left least possibilities thus use chernoff bound dependent variables stated theorem order bound probability deviates expectation distinguish two cases first suppose log theorem setting log exp thus using inequality high probability since suppose log theorem setting log log calculation since completes proof inequality theorem restated algorithm operates log log rounds congestedclique model outputs maximal independent set high probability proof concerning runtime step algorithm requires communication rounds observe every iteration requires rounds terminates log log rounds high probability corollary since ghaffari algorithm requires log log log log rounds maximum degree residual computed last iteration case overall runtime bounded log log concerning correctness algorithm step collection graph vertex achieved using lenzen routing protocol used since proved lemma graph vertices high probability conclusion paper gave log log rounds mis algorithm runs congestedclique model simulated sequential random order greedy algorithm exploiting residual sparsity property greedy conceivable round complexity reduced lower bounds known mis model results problems minimum weight spanning tree problem log log rounds algorithm lotker subsequently improved log log log rounds rounds finally rounds give hope similar improvements may possible mis well simulate centralized greedy algorithms rounds congestedclique model acknowledgements author thanks amit chakrabarti anthony wirth graham cormode discussions residual sparsity property clustering algorithm given references ahn cormode guha mcgregor wirth correlation clustering data streams proceedings international conference international conference machine learning volume icml http alon babai itai fast simple randomized parallel algorithm maximal independent set problem algorithms dec http barenboim elkin pettie schneider locality distributed symmetry breaking acm jun http kaski korhonen lenzen paz suomela algebraic methods congested clique proceedings acm symposium principles distributed computing podc acm new york usa http drucker kuhn oshman power congested clique model proceedings acm symposium principles distributed computing podc acm new york usa http kesselheim improved algorithms latency minimization wireless networks theor comput sci may http ghaffari improved distributed algorithm maximal independent set proceedings annual symposium discrete algorithms soda society industrial applied mathematics philadelphia usa http ghaffari distributed mis via communication proceedings acm symposium principles distributed computing podc acm new york usa http ghaffari parter mst rounds congested clique proceedings acm symposium principles distributed computing podc acm new york usa http hegeman pandurangan pemmaraju sardeshmukh scquizzato toward optimal bounds congested clique graph connectivity mst proceedings acm symposium principles distributed computing podc acm new york usa http hegeman pemmaraju lessons congested clique applied mapreduce structural information communication complexity springer international publishing cham hegeman pemmaraju sardeshmukh distributed algorithms congested clique kuhn distributed computing springer berlin heidelberg berlin heidelberg jurdzinski nowicki mst rounds congested clique proceedings annual symposium discrete algorithms soda new orleans usa january https korhonen suomela brief announcement towards complexity theory congested clique international symposium distributed computing disc october vienna austria https kuhn moscibroda wattenhofer local computation lower upper bounds acm mar http kuhn moscibroda wattenhofer computed locally proceedings twentythird annual acm symposium principles distributed computing podc acm new york usa http gall algebraic algorithms congested clique model applications problems gavoille ilcinkas eds distributed computing springer berlin heidelberg berlin heidelberg lenzen optimal deterministic routing sorting congested clique proceedings acm symposium principles distributed computing podc acm new york usa http linial distributive graph solutions local data annual symposium foundations computer science los angeles california usa october https lotker pavlov peleg spanning tree construction log log communication rounds siam comput jul https lotker pavlov peleg mst construction log log communication rounds proceedings fifteenth annual acm symposium parallel algorithms architectures spaa acm new york usa http luby simple parallel algorithm maximal independent set problem proceedings seventeenth annual acm symposium theory computing stoc acm new york usa http peleg distributed computing approach society industrial applied mathematics philadelphia usa
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modules self tor vanishing mar olgur celikbas henrik holm bstract auslander reiten conjecture states finitely generated module algebra extai extai must projective inspired work avramov buchweitz jorgensen others consider paper possible counterpart conjecture commutative local rings terms vanishing tor main result shows class local rings satisfy counterpart closed standard procedures ring theory ntroduction throughout denotes commutative noetherian local ring unique maximal ideal residue field purpose establish result also give observations examples might useful studies following open problem question finitely generated torir must finite projective dimension question true rings may considered homology counterpart celebrated auslander reiten conjecture commutative rings question previously studied several people various partial results recently obtained special cases see example likely complete solution question yield new perspective homological commutative algebra question appears implicitly paper paragraph preceding thm thm loc cit positive answer given case gorenstein admits exact zero divisor classes rings question known true include complete intersection rings cor see also thm thm golod rings thm study question consider following condition every finitely generated satisfying torir finite projective dimension pdr best knowledge approach question different papers literature instead determining specific conditions give affirmative answer question show property preserved standard procedures local algebra main result stated follows theorem following conditions equivalent iii satisfies satisfies satisfies satisfies mathematics subject classification key words phrases projective dimension vanishing tor olgur celikbas henrik holm one use theorem conjunction lemma construct new examples rings satisfying see example point work motivated result similar one proved auslander condition however arguments quite different since techniques used loc cit work setting see remark cor section prove theorem show construct examples local rings satisfying furthermore give way obtain certain kinds regular sequences power series rings might independent interest section consider slightly weaker version condition call condition related gorenstein dimension prove result similar theorem show results section strengthened new setting results lemma let local homomorphism commutative noetherian local rings satisfies finite flat dimension satisfies proof assume satisfies let finitely generated torir torir flat dimension replacing sufficiently high syzygy dimension shifting assume torir torir every case isomorphism derived category yields complex homologically bounded homology even concentrated degree zero since finite flat dimension side homologically bounded hence side toris satisfies follows finite projective dimension follows pdr finite proposition let commutative noetherian local ring let sequence satisfies satisfies converse true holds every proof first statement special case lemma prove partial converse assumption non maximal ideal since induction suffices consider case assume satisfies let non see satisfies let finitely generated see also lem long exact sequence torir therefore torir since satisfies get pdr finite follows finite see example prop remark would interesting know last assertion proposition holds without assumption property preserved passing quotient ideal generated regular sequence proposition modules self tor vanishing remark sequence regular belong follows proposition safisfies safisfies proposition used construct new examples rings satisfying known examples see example however useful concrete way constructing regular sequences property mentioned lemma give one construction commutative ring element happen perhaps surprisingly see example however noetherian situation much nicer let commutative noetherian ring consider elemement follows thm coefficient unit non lemma let commutative noetherian local ring consider power series ring write unique maximal ideal let integers let elements every following conditions hold xmi unit element regular sequence proof first note condition implies power series coefficient unit coefficient indeed next show regular sequence condition says coefficient unit non next show non write vmi hvmi xmi xnn xmi xmi xmi xmi isomorphism xmi xmi particular image identified element vmi xmi xnn side image xmi hence show non suffices argue one coefficients unit know xmi coefficient unit means one elements hvmi unit consequently hvmi hvmi xmi unit xmi image also unit desired next show holds suppose contradiction olgur celikbas henrik holm unit follows assumption identity already mentioned side unit contradicts side belongs indeed furthermore depend variables every zero example following less arbitrarily chosen sequence corresponding satisfies assumptions lemma proposition indeed clear holds since implies satisfies note fiber product ring artinian gorenstein thm satisfies hence also holds following ring chosen proof theorem equivalence iii noted remark let set elements generate thm sequence clearly satisfies assumptions lemma equivalence follows note equivalence isomorphic completions isomorphic iii follows already established equivalence onnections orenstein dimension section give remarks observations pertaining aulander self tor vanishing commutative noetherian local ring consider following variant condition every finitely generated satisfying torir finite general weaker see prop two conditions equivalent maximal ideal decomposable see thm testing finiteness via vanishing tor form idea pursued number papers example thm proved finitely generated module commutative noetherian ring finite vanishes every furthermore finitely stable homology tor generated modules testing finiteness via vanishing absolute homology tor also examined property following stronger version proposition proposition let commutative noetherian local ring let sequence satisfies satisfies modules self tor vanishing proof part proceed proof lemma note replaced sufficiently high syzygy sequence becomes regular standard see also lem finiteness infer finiteness cor part proceed proof proposition finiteness one always gets finiteness assumption needed thm arguments proof theorem applies give following theorem let commutative noetherian local ring following conditions equivalent iii satisfies satisfies satisfies satisfies acknowledgments part work completed holm visited west virginia university march grateful kind hospitality wvu department mathematics eferences maurice auslander anneaux gorenstein torsion commutative paris commutative par pierre samuel texte des maurice auslander par marquerite mangeney christian peskine lucien szpiro normale jeunes filles available http maurice auslander idun reiten generalized version nakayama conjecture proc amer math soc luchezar avramov infinite free resolutions six lectures commutative algebra bellaterra progr vol basel luchezar avramov buchweitz support varieties cohomology complete intersections invent math luchezar avramov foxby ring homomorphisms finite gorenstein dimension proc london math soc luchezar avramov srikanth iyengar saeed nasseh sean homology trivial extensions commutative algebras olgur celikbas lars winther christensen liang greg piepmeyer stable homology associative rings trans amer math soc olgur celikbas sean testing gorenstein property collect math lars winther christensen gorenstein dimensions lecture notes vol berlin lars winther christensen henrik holm vanishing cohomology rings manuscripta math david fields zero divisors nilpotent elements power series rings proc amer math soc robert gilmer anne grams tom parker zero divisors power series rings reine angew math craig huneke roger wiegand tensor products modules rigidity tor math ann tensor products modules rigidity local cohomology math scand david jorgensen tor torsion complete intersection algebra olgur celikbas henrik holm generalization formula pure appl algebra hideyuki matsumura commutative ring theory second cambridge stud adv vol cambridge university press cambridge translated japanese reid saeed nasseh sean vanishing ext tor fiber products proc amer math soc saeed nasseh ryo takahashi structure irreducible homomorphisms free modules appear algebr represent theory local rings maximal ideal preprint joseph rotman introduction homological algebra pure applied mathematics vol academic press harcourt brace jovanovich publishers new york liana freeness commutative artinian rings pure appl algebra epartment athematics est irginia niversity organtown address epartment athematical ciences niversitetsparken niversity openhagen openhagen enmark address holm url http
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oct dwell time computation methods switched linear systems karabacak september abstract analytical computation methods proposed evaluating minimum dwell time average dwell time guaranteeing asymptotic stability switched linear system whose switchings assumed respect given directed graph minimum average dwell time found using graph governs switchings associated weights approach used previous work systems subsystems adapted switched systems generalized allow defective subsystems moreover present method improve dwell time estimation case bimodal switched systems method scaling algorithms minimize condition number used give better minimum dwell time average dwell time estimates keywords switched systems minimum dwell time average dwell time optimum cycle ratio asymptotic stability switching graph introduction problems regarding stability switched linear systems attracted interest many researchers last two decades approach stability switched linear systems impose constraints set switching signals guarantee stability mainly two kinds constraints considered minimum dwell time constraint average dwell time constraint former intervals two consecutive switchings assumed larger equal number called minimum dwell time whereas latter intervals assumed average larger equal number called average dwell time literature many methods find minimum dwell time average dwell time guarantee stability given switched system however general method gives smallest possible minimum average dwell time terms subsystem properties efficient methods used approximate minimum average dwell time based linear matrix inequalities therefore give insight relationship subsystem properties minimum average dwell time switched systems subsystem matrices estimate minimum average dwell time explicitly depending subsystem properties derived minimum dwell time found function subsystem eigenvalues eigenvectors method based viewing switching signal walk complete directed graph digraph whose nodes correspond subsystems whose arcs correspond switching events general problem considered namely finding minimum dwell time switched systems whose switchings governed digraph called switching graph note special case problem switching graph complete corresponds standard dwell time problem literature hand general problem important switched systems whose switchings governed digraph encountered control engineering applications theoretically systems first considered switched system literature best knowledge stability conditions reduced conditions strongly connected components digraph later second author paper presents sufficient conditions stability constrained switched systems using properties digraph governs switchings method improved conference paper continuoustime systems cover systems defective subsystem matrices recently stabilization switched systems whose switchings governed digraph considered digraphs also used multiple lyapunov function method generalized paper apply method proposed switched linear systems waive condition considering jordan form subsystem matrices addition improve minimum dwell time estimate bimodal systems applying scaling algorithm minimizes condition number matrices sequel first explain switching graph arises naturally considering bound norm solution linear switched system section define switching graph used estimate minimum average dwell time based switching graph methods minimum dwell time average dwell time computation given section section respectively finally discuss possible future research area section notation denote set real numbers set positive integers respectively denotes vectors spectral norm matrices switched systems switching graph consider switched linear systems form state system finite set schur stable subsystem matrices number subsystems dimension state space index set subsystems denotes set admissible switching signals assume switching instants denote index active subsystem time time let denote number switchings time interval two different sets switching signals considered smin smin consists switching signals interval two consecutive switchings always larger whereas save set switching signals property time number past switchings satisfies average dwell time condition given initial condition solution switched linear system written save let consider jordan matrix decomposition generalized eigenvector matrix jordan matrix use fact diagonal jordan matrix conventional replaced constant explain let assume jordan form consists one jordan block matrices jordan form consists one jordan block treated similarly changing generalized eigenvector matrix matrix written jordan form place known kdk kdk denotes diagonal part denotes nilpotent part schur stable matrix kdk choose kdk one gets matrix whose jordan form consists one jordan blocks let choose sufficiently small satisfying kdk diagonal part jordan block since maxk kjk obtain moreover since kdk maxk kdk condition equivalent kdk subsystem matrix choose generalized jordan decomposition sufficiently small satisfying kdi diagonal part kji following drop superscript matrices simplicity notation unless necessary indicate dependence note subsystem matrix jordan form diagonal schur stability implies kji kdi substituting using norm inequalities tkk let define kji note diagonal hence equal spectral radius writing term parentheses inequality exponential form maxi kpi since subsystem schur stable kji using smin write following show function seen weight walk length doubly weighted digraph called switching graph definition switching graph switched systems transitions subsystems restricted digraph whose nodes represent subsystems whose directed edges represent admissible transitions subsystems consequence idea switching signal viewed walk graph easily seen transition subsystem subsystem gives contribution function function ordered pair namely values assigned weights directed edges admissible transitions subsystems consider gain transition subsystem sub system loss dwelling subsystem per unit time doubly weighted graph follows switching graph switched linear system doubly weighted digraph set nodes isomorphic index set subsystems set directed edges represent admissible transitions subsystems set given case restriction imposed transitions subsystems namely fully connected switching graph weight functions defined set switched system four subsystems switching graph shown figure assume switchings switched system consideration respect directed graph case set admissible switching signals restricted directed edges switching graph figure switching graph switched system four subsystems edge shown respectively denote min smin set smin restricted switching graph set switching signals contains switching signals respect given digraph satisfies minimum dwell time property similarly ave save maximum cycle ratio consider weighted digraph weight function defined walk length defined path walk distinct cycle path weight walk defined consider doubly weighted graph ratio cycle defined maximum cycle ratio defined max max max denotes set cycles length cycle similarly mean cycle defined cycle mean defined max max maximum optimum minimum maximum cycle ratio also known ratio optimum cycle mean considered graph theory literature many applications different areas scheduling problems performance analysis digital systems many algorithms used find optimum cycle ratio optimum cycle mean given doubly weighted digraph see terms practical complexity one fastest algorithms given sequel use algorithm code available ali dasdan personal web page minimum dwell time computation section consider switched linear system whose switchings governed digraph time interval consecutive switching instants larger equal minimum dwell time special case bimodal case discussed subsection theorem let family schur stable matrices let switching graph switched linear system given min asymptotically stable maximum cycle ratio found using weights given parameters satisfying proof note switching signal represented walk switching graph length walk finite last subsystem stays active forever thus guaranteeing asymptotic stability switched linear system hence consider walks infinite length represent switching signals infinitely many switchings using seen weight walk weight function using fact walk digraph nodes decomposed union cycles path length decomposed weight path sum weights note assumption implies cycle namely weights cycles negative since finite bounded therefore valid min hence switched linear system asymptotically stable remark subsystem matrices theorem reduces version theorem bimodal case theorem enhanced bimodal switched systems namely switched systems two subsystems since one cycle graph associated bimodal case maximum cycle ratio denotes condition number spectral norm namely therefore using theorem bimodal switched system stable known eigenvectors scaled nonzero scalar eigenvector matrix multiplied right nonsingular diagonal matrix also eigenvector matrix let denote set nonsingular diagonal matrices consider eigenvector matrices let new eigenvector matrices obtained scaling columns using respectively hence condition theorem replaced stronger condition mindl analytical method minimizing condition number spectral norm scaling rows columns algorithmic methods available corollary switched linear system two schur stable subsystems asymptotically stable mindl one use inequality condition implies diagonal hence kjkp denotes spectral radius denotes using fact one similarly obtain condition kakp analytical method minimizing condition number norms scaling rows columns according method min adr denotes matrix whose elements absolute value corresponding elements denotes spectral radius hence following result corollary switched linear system asymptotically stable average dwell time computation section average dwell time problem considered namely finding smallest possible value switched system asymptotically stable average dwell time set special case bimodal case discussed subsection theorem let family schur stable matrices let switching graph switched linear system given ave asymptotically stable maxi maximum cycle mean found using weights given parameters satisfying proof inequality written consider walk associated switching signal seen walk equal since walk decomposed union cycles path length less number nodes path length less say sum cycles length defining varies possible paths get consider maximum cycle mean switching graph namely denotes length cycle set cycles obtained since cycle weights get substituting defining obtain since assumption conclude remark subsystem matrices theorem reduces version theorem bimodal case similarly minimum dwell time case average dwell time method improved bimodal switched systems one cycle bimodal system average dwell time satisfies hence method subsection applied computation average dwell time illustrative examples apply obtained minimum dwell time computation method two illustrative examples compare results two different methods literature method given morse finds minimum dwell time guaranteeing subsystem contractive method given geromel colaneri uses linear matrix inequalities based multiple lyapunov function technique skip comparison obtained average dwell time computation methods literature since either require specified convergence rate refer modedependent form namely subsystem certain average dwell time condition imposed example consider switched system consisting four linear subsystems whose matrices given figure ring switching graph considered example edge shown respectively figure ring switching graph considered example edge shown respectively cos sin cos sin assume switchings respect one switching graphs fully connected fig ring fig ring fig different switching graphs minimum dwell time values computed using theorem table seen results better results obtained method given morse comparing results method given geromel colaneri seen switching graph theorem gives worse result let consider bimodal system use corollary equilibration method compute better value minimum dwell time one obtained theorem example let given subsystem matrices switched system calculated using condition given theorem however applying corollary calculated equal value found linear matrix inequalities method use two sided equilibration method based idea condition number reduced making norms rows well norms columns equal row scaling matrix column scaling matrix calculated conclusion method computation minimum dwell time guarantees asymptotic stability switched system presented method applicable systems whose switchings governed digraph graphtheoretical nature method allows fast computation estimate minimum dwell time using maximum cycle ratio algorithms graph theory note many problems considered switched systems whose switchings governed digraphs role nature switching digraph plays dynamics switched system considered shown average dwell time computed using minimum cycle mean switching graph approach improved two different ways firstly one introduce average dwell time try find sufficient conditions average dwell times given switching graph secondly one consider preassumed convergence rate calculate average dwell time given switching graph less conservative method acknowledgement research supported scientific technological research council turkey project table minimum dwell time values computed example minimum dwell time switching geromel graph theorem morse colaneri references hespanha uniform stability switched linear systems extensions lasalle invariance principle ieee trans auto cont morse supervisory control families linear exact matching ieee trans auto cont geromel colaneri stability stabilization switched linear systems siam contr optim guo wang stability analysis class switched linear systems asian control cai mijanovic stability analysis linear hybrid systems application switched control refrigeration process asian control shorten wirth mason wulff king stability criteria switched hybrid systems siam review jan available http graziano colaneri geromel middleton shorten nonconservative lmi condition stability switched systems guaranteed dwell time ieee trans auto cont haimovich braslavsky felicioni feedback stabilization switching systems via techniques ieee trans auto cont karabacak dwell time approach stability switched linear systems based distance eigenvector sets int system science karabacak dwell time average dwell time methods based cycle ratio switching graph syst control lett hou dong wang stability analysis switched linear systems locally overlapped switching law guid control dyn wang hou stability analysis stabilisation fullenvelope networked flight control systems switched system approach iet control theory zhang hou liu zhou switched control voltage source pwm rectifier rapidly varying active load ieee trans power electron garcia sontag wang uniform stability properties switched systems switchings governed digraphs nonlinear karabacak ilhan explicit sufficient stability conditions dwell time linear switched systems proc conf decision control los angeles california kundu balachandran chatterjee algorithmic synthesis stabilizing switching signals switched linear systems available http ahmadi jungers parrilo roozbehani analysis joint spectral radius via lyapunov functions graphs siam contr optim karp characterization minimum cycle mean digraph discrete math golitschek optimal cycles doubly weighted graphs approximation bivariate functions univariate ones numer math dasdan gupta faster maximum minimum mean cycle algorithms analysis ieee comut aid dasdan experimental analysis fastest optimum cycle ratio mean algorithms acm trans des autom electron syst bouyer brinksma larsen staying alive cheaply possible alur pappas editor hybrid systems computation control proceedings vol lect notes comput leus herroelen scheduling stability production systems sched kats levner cyclic routing algorithms graphs performance analysis applications robot scheduling comput ind eng koh performance analysis systems ieee comput aid dasdan personal webpage http braatz morari minimizing euclidean condition number siam contr optim liu equilibration method reduce condition number linear system model eng bauer optimally scaled matrices numer math geromel colaneri stability stabilization discrete time switched systems int control zhang stability analysis switched timedelay systems automatica available http zhang han zhu huang stability stabilization positive switched systems average dwell time nonlinear analysis hybrid systems available http
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modified soft brood crossover genetic programming hardik parekh vipul dabhi information technology department dharmsinh desai university nadiad india information technology department dharmsinh desai university nadiad india abstract premature convergence one important issues using genetic programming data modeling avoided improving population diversity intelligent genetic operators help improve population diversity crossover important operator genetic programming analyzed number intelligent crossover operators proposed algorithm modification soft brood crossover operator help improve population diversity reduce premature convergence performed experiments three different symbolic regression problems made performance comparison proposed crossover modified soft brood crossover existing soft brood crossover subtree crossover operators index terms intelligent crossover genetic programming soft brood crossover introduction genetic programming model programming uses ideas biological evolution handle complex problem number possible solutions effective solutions survive compete solutions way reach closer needed solution premature convergence one important issues using genetic programming data modeling premature convergence leads evolution solutions locally optimal premature convergence avoided improving population diversity population diversity improved using intelligent crossover research aim improve population diversity using intelligent crossover analyzed different toolkits available found jclec java class library evolutionary computation useful research work jclec open source platform independent implemented java convergence leads evolution solutions locally optimal evolve globally optimal solutions avoidance premature convergence required aim avoid premature convergence run hence improve population diversity intelligent crossover operator useful improve population diversity analyzed intelligent crossover operators like context aware crossover cac semantic aware crossover sac semantic similarity based crossover ssc soft brood crossover approximating geometric crossover selective crossover size fair crossover comparison different crossover operators bases different criteria specified table observed soft brood crossover operator useful improve population diversity table comparison intelligent crossover operators soft brood crossover sbc intelligent crossover operators crossover important operator genetic programming standard crossover may produce offspring parents standard crossover intelligence avoid problem generating offspring parents intelligent crossover combines parents way generate offspring better fitness parents premature soft brood crossover differs crossover operators number crossover performs pair parents performed operations generate number offspring offspring evaluated based fitness generated offspring two best fittest offspring passes next generation rest discarded modified soft brood crossover msbc proposed crossover operator modifies existing soft brood crossover operator help prevent premature convergence improve population diversity presents pseudo code proposed crossover operator detail algorithm modified soft brood crossover select parent crossover random crossover operations performed generate brood children generation total generation fitness children calculated two dissimilar based fitness children copied next generation rest discarded else evaluated children sorted based fitness two best fittest children copied next generation discarded modified soft brood crossover operator generates number offspring pair parent first half generation passing two dissimilar offspring next generation based fitness rest half generation passing two best fittest offspring next generation table results different ratio generations problem cos cos sin tan problem sextic polynomial problem iii toolkit analyzed different toolkits available found jclec java class library evolutionary computation useful research work jclec open source platform independent implemented java need specify parameters configuration file jclec toolkit xml file format run experiments configuration parameters first select algorithm solve problem selected sge simple generational elitist algorithm available jclec specifically genetic programming process sge standard uses tree representation represent individual also used tree representation represent individual thus package must used establishing minimum tree size maximum tree size list terminal symbols functions present set tree size terminals functions species type species terminal class function function function population randomly initialized using expression trees class provider type performed runs problem using different percentage generations passing first second half crossover operator obtained results represented table found passing generations first half generations second half gives best results need specify max generation stopping criterion selection parents set using package tournament selection gives better performance selected tournament selector parent selector type tournamentselector fitness function calculated using evaluator declaration evaluator type mandatory use symbolic regression problem specified symregevaluator evaluator type evaluator packages used experiment package contains implementations ifitness interface jclec support soft brood crossover operator implemented subtree crossover tree crossover operators available jclec genetic programming subtree crossover operators performs crossover branches tree tree crossover performs crossover whole tree implementation soft brood crossover operator modified subtreecrossover class exprtreerecombinator class files modified subtreecrossover class file contains logic crossover point selection helpful implement proposed crossover modification exprtreerecombinator class file required contains method called genetic operator set configuration file xml format several package implementations several selection methods boltzmann selector random selector roulette selector stochastic remaining selection universal stochastic selection range selection tournament selection available selectors package contains exprtreeindividual defines type individual package also contains exprtreeindividualspecies class defines structure individuals operators manipulate continuously subtree crossover tree crossover allnodesmutator demotemutator growmutator onenodemutator promotemutator promotemutator truncmutator available operators jclec implementation details symbolic regression problem solving facility available jclec implemented modifying symregevaluator class file three functions available addition multiplication subtraction implemented following functions division sin cosine tan square root exponential log jclec experiments use newly created terminals functions need set configuration file modified seed generator class file pass current time seed rather static seed comparing performance proposed crossover operator standard subtree crossover soft brood crossover generate graphs fitness versus generation modified populationreporter class file generates file contains generation fitness experiments performed experiments three different symbolic regression problems problem cos cos sin tan problem sextic polynomial problem table iii parameters problems parameters value population size maximum generation min tree size max tree size terminal set problem terminal set problem terminal set problem constants function set problem function set problem problem parent selector sqrt sin cos tan tournament selector size crossover probability mutation probability problem set parameters shown table iii prepared results runs problem using subtree crossover soft brood crossover modified soft brood crossover operators fig plot generations fitness using subtreecrossover fig plot generations fitness using subtreecrossover fig plot generations fitness problem using soft brood crossover fig plot generations fitness using soft brood crossover fig plot generations fitness using modified soft brood crossover fig plot generations fitness using modified soft brood crossover figure shows best fitness obtained generation using subtree crossover figure shows best fitness obtained generation using soft brood crossover figure represents best fitness obtained generation using modified soft brood crossover problem proposed crossover gives best fitness less number generations compare subtree crossover soft brood crossover operators figure shows best fitness obtained generation using subtree crossover figure shows best fitness obtained generation using soft brood crossover figure represents modified soft brood crossover obtained best fitness generation obtained results problem say proposed crossover obtains best fitness less number generations compare soft brood crossover subtree crossover operators conclusions fig plot generations fitness using subtreecrossover proposed new crossover operator genetic programming modifies existing soft brood crossover operator implemented soft brood crossover proposed crossover modified soft brood crossover jclec toolkit performed experiments three different symbolic regression problems high dimension sextic polynomial symbolic regression constants using subtree crossover soft brood crossover modified soft brood crossover operators obtained results three different problems conclude proposed crossover modified soft brood crossover gives good performance existing soft brood crossover subtree crossover operators references nguyen nguyen neill semantic aware crossover genetic programming case function regression tackett recombination selection genetic construction computer programs phd thesis university southern california department electrical engineering systems ventura romero zafra delgado jclec java framework evolutionary computation soft computing vol apr fig plot generations fitness using soft brood crossover hammad majeed conor ryan less destructive crossover opera tor proceedings european conference genetic programming pages lecture notes computer science springer april neill hoai mckay galv semantic similarity based crossover function proceedings international conference artificial evolution krawiec lichocki approximating geometric crossover semantic space proceedings annual conference genetic evolutionary computation gecco langdon size fair homologous tree genetic programming crossovers known time programs within hengpraprohm chongstitvatana selective crossover genetic programming proceedings iscit international symposium communications information technologies pages november fig plot generations fitness using modified soft brood crossover figure shows subtree crossover obtained best fitness generation figure shows soft brood crossover obtained best fitness generation figure represents modified soft brood crossover obtained best fitness generation case problem modified soft brood crossover obtained fitness soft brood crossover subtree crossover modified soft brood crossover obtained fitness less number generations two crossover operators holland langton wilson holland genetic programming programming computers means natural selection mit press cambridge usa quang nguyen hoai galv crossover genetic programming application symbolic regression journal genetic programming evolvable machines volume issue june neill vanneschi gustafson banzhaf open issues genetic programming genetic programming evolvable machines hien hoai brief overview population diversity measures genetic programming proceedings third asianpacific workshop genetic programming military technical academy hanoi vietnam
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jun minimum ber precoding massive mimo systems hela jedda josef nossek amine mezghani institute circuit theory signal processing technische munich germany email university california irvine irvine usa email amezghan dacs converters adcs gaining interest massive mimo systems economical computational efficiency present new precoding technique mitigate iui channel distortions downlink mumiso system qpsk symbols transmit signal vector optimized taking account quantization develop sort mapping based table lut input signal transmit signal lut updated channel realization simulation results show significant gain terms uncoded ber compared existing linear precoding techniques ntroduction massive mimo systems seen promising technology next generation wireless communication systems huge increase number antennas base station improve spectral efficiency energy efficiency reliability large number antennas say antennas serves simultaneously much smaller number users price pay massive mimo systems increased complexity hardware number radio frequency chains signal processing resulting increased energy consumption transmitter several approaches considered literature decrease power consumption spatial modulation use parasitic antennas use transceivers one attractive solution overcome issues high complexity high energy consumption associated massive mimo use low resolution adcs dacs power consumption adc dac one devices reduced exponentially decreasing resolution quantization drastically simplify amplifiers mixers therefore focus massive mimo systems resolution dacs adcs restricted bit massive mimo systems knowledge csi csit large spatial dof massive mimo systems exploited significantly increase spatial gain using precoding literature linear precoders designed massive mimo systems based criterion mmse mitigate iui distortions due coarse quantization however mmse criterion may optimal since desired receive signals restricted discrete qpsk points corectly detected belong respective quadrants thus aim changing design criterion minimun ber mber goal get receive signal desired quandrant far possible decision thresholds contribution design precoder transmit signal vector design sort mapping based lut input signal vector transmit signal vector lut updated channel entries transmit signal vector belong square formed qpsk constellation points minimize quantization distortions transmitter paper organized follows section introduce downlink system model section iii give overview mapping idea mber criterion illustrated explained section section formulate optimization problem based mber criterion show derivations corresponding solution give two linear precoder designs section aim comparing sections vii viii interpret simulation results summarize work notation bold letters indicate vectors matrices nonbold letters express scalars operators adj stand complex conjugation transposition hermitian transposition adjugate expectation respectively identity matrix denoted zeros ones matrix rows columns defined vector represents zero vector position define sign sign sign additionally diag denotes diagonal matrix containing diagonal elements ystem odel fig downlink system model qpsk symbols consider downlink scenario depicted fig antennas serves singleantenna users signal vector contains data symbols users represents set qpsk constellation assume system deploy quantization transmitter well receiver use quantizer transmitter delivers signal mitigate iui distortions due coarse quantization input signal vector mapped unquantized transmit signal vector prior dac mapping based lut size generated beginning coherence slot etx transmit signal gets scaled etx available power transmitter received decoded signal vector users reads etx channel hxq desired quadrant need get receive signal far possible quantization thresholds make less sensitive noise illustration consider fig red points designate qpsk constellation solution set mber criterion represented four squares however mmse criterion tries receive signal close possible desired signal get green circles thus mmse solution set restricted subset mber solution set massive mimo employed signals get larger magnitude prohibited mmse preserved mber matrix entries zero mean unit variance awg noise vector iii apping work design precoder design transmit vector signal given input signal vector depending channel assume full csit depicted fig first optimization problem solved possible input vectors find optimal transmit vectors used optimization problem introduced section second solutions stored lut size since restricted qpsk modulation get possible input vectors third map given input vector signal vector according lut updated channel aim optimization problem jointly minimize iui quantization distortions optimization criterion minimun ber mber constraint constraint leads linear behavior quantizer transmitter thus quantization distortions transmitter omitted lut optimization problem fig mmse mber criterion order explore mber criterion formulate appropriate mathematical optimization problem end refer fig illustration mentioned aim making receive signal belong safe red area one maximize minimize fortunately mathematical expression enable maximizing given max max cos solution requires cos positive achieved mapping fig processing steps channel mber criterion single user scenario minimize ber case qpsk symbols need get receive signal quadrant desired signal since receive signal gets distorted additive gaussian noise may remove fig illustration optimization problem multi user scenario multi user scenario make use optimization problem apply user max max cos cost functions jointly expressed following matrix diag rsh diag hxsh diag diag diag end cost functions jointly maximized single transmit vector cost functions combined maximize together question arises shall maximize sum minimal contribution product ptimization roblem optimization problem nconvex since solution set optimal solution found exhaustive search however complexity exhaustive search increases exponentially number antennas decrease complexity problem make solvable linear methods relax constraint since matrix function real imaginary parts stacked constraint reformulated maximizing sum may lead maximizing expression user highest value cost users maximizing product seems fairer method since product maximized values users contribute considerably thus relaxed optimization problem reads max det optimization problem resort gradient projection algorithm fulfill constraint end need find derivative expression cost function respect gradient given det adj adj gradient projection algorithm used algorithm summarized algorithm initial value depends choice chosen precoder hhh algorithm gradient projection algorithm iteration step tolerable error initialization repeat det det det det det iteration step start value denoted iteration step large cost function decreases instead increasing elements become negative step size reduced order ensure algorithm convergence iteration step optimization performed step xisting linear precoders precoder precoder design introduced based mmse criterion given wwf fwf etx fwf etx etx transmit vector reads wwf fulfill power constraint transmit vector scaled etx factor ensures equal power allocation antennas wfq precoder precoder design consists two stages analog precoder pdigital precoder whwfq dwfq diag wwfq wwfq analog precoder diagonal matrix assign antenna desired amount power optimize quantization levels end wwfq multiplied dwfq transmitting leads unequal power allocation antennas uncoded ber wfq unq simulation results averaged channel realizations used modulation scheme qpsk transmit symbols per channel use compare proposed design existing linear precoding techniques wfq terms uncoded ber mutual information calculated numerically based toolbox proposed ideal case precoder used quantization performed denoted fig see proposed mapping method outperforms existing linear precoders terms uncoded ber uncoded ber achieve gain compared wfq wfq design unequal power allocation antennas performed requires number power amplifiers pas equal number antennas adjust power antenna proposed method power antenna equal allows run saturation region thus efficiently use energy fig different precoder designs plotted function transmit power gain less significant compared uncoded ber means proposed method requires less perfomant codes achieve capacity additionally complexity method studied terms average number iterations needed get one optimal solution table drawn table required number iterations decreases larger tolerable error around iterations per algorithm run without degrading much uncoded ber fig ber performance system wfq unq vii imulation esults etx precoder design presented mmse precoder takes account quantization effects based linear covariance approximation precoder expressed nondiag fwfq etx fwfq etx ttr nondiag etx etx fig performance system table omplexity performance method average iterations snr ber bpcu bpcu bpcu viii onclusion presented novel precoding technique based mber criterion instead designing precoder design transmit output vector fulfills relaxed constraint qpsk set minimize distortions due quantization transmitter based mber criterion method gives promising results compared existing linear precoding techniques although equal power allocation antennas performed achieve significant gain ber compared precoders allow unequal power allocation antennas furthermore run saturation region get energy efficient systems however advantages achieved higher complexity running nonlinear optimization problem input however lut based implementation possible systems small number users eferences marzetta noncooperative cellular wireless unlimited numbers base station antennas wireless communications ieee transactions vol november bjornson kountouris debbah massive mimo small cells improving energy efficiency optimal coordination telecommunications ict international conference may rusek persson lau larsson marzetta edfors tufvesson scaling mimo opportunities challenges large arrays signal processing magazine ieee vol jan renzo haas ghrayeb sugiura hanzo spatial modulation generalized mimo challenges opportunities implementation proceedings ieee vol jan muller sedaghat fischer load modulated massive mimo signal information processing globalsip ieee global conference dec kalis kanatas papadias parasitic antenna arrays wireless mimo systems springer sedaghat mueller mueller fischer novel transmitter massive mimo smart antennas wsa international itg workshop march bjornson matthaiou debbah massive mimo nonideal arbitrary arrays hardware scaling laws design wireless communications ieee transactions vol svensson andersson bogner power consumption analog digital converters norchip conference nov peel hochwald swindlehurst technique multiantenna multiuser channel inversion regularization communications ieee transactions vol jan gershman sidiropoulos shahbazpanahi bengtsson ottersten convex beamforming signal processing magazine ieee vol may mezghani ghiat nossek transmit processing low resolution electronics circuits systems icecs ieee international conference dec usman jedda mezghani nossek mmse precoder massive mimo using quantization acoustics speech signal processing icassp ieee international conference bertsekas tsitsiklis parallel distributed computation numerical methods joham utschick nossek linear transmit processing mimo communications systems signal processing ieee transactions vol aug brown pocock zhao conditional likelihood maximisation unifying framework information theoretic feature selection journal machine learning research vol jan
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theoretical assessment solution quality evolutionary algorithms knapsack problem jun boris mitavskiy yuren zhou apr abstract evolutionary algorithms well suited solving knapsack problem empirical studies claim evolutionary algorithms produce good solutions knapsack problem nonetheless rigorous investigations address quality solutions evolutionary algorithms may produce knapsack problem current paper focuses theoretical investigation three types evolutionary algorithms exploit bitwise mutation truncation selection plus different repair methods knapsack problem assesses solution quality terms approximation ratio work indicates solution produced pure strategy mixed strategy evolutionary algorithms arbitrarily bad nevertheless evolutionary algorithm using helper objectives may produce solutions knapsack problem index terms evolutionary algorithm approximation algorithm knapsack problem solution quality ntroduction knapsack problem combinatorial optimisation problem includes variety problems knapsack problem knapsack problem last two decades evolutionary algorithms eas especially genetic algorithms gas tackling knapsack problem problem received particular interest evolutionary computation community following two reasons first reason binary vector representation candidate solutions natural encoding knapsack problem search space thereby provides ideal setting applications genetic algorithms hand knapsack problem natural optimization problem often taken test problem studying optimization evolutionary algorithms moeas number empirical results literature see instance assert eas produce good solutions knapsack problem naturally arising question measure goodness solutions eas may produce address question popular approach compare quality solutions generated eas via computer experiments example solution quality measured best solution found within generations comparison may help compare performance different eas yet seldom provides information regarding proximity solutions produced eas optimum viewpoint algorithm analysis important assess good solution terms notion approximation ratio see several effective approximation algorithms solving knapsack problem example fully polynomial time approximation scheme knapsack problem presented nonetheless rigorous investigations addressing approximation ratio eas knapsack problem exist recast knapsack problem knapsack problem two conflicting objectives maximizing profits minimizing weights set knapsack problem introduced optimization problem moea called restricted evolutionary multiobjective optimizer designed obtain set pioneering contribution rigorous runtime analysis proposed moea current paper focuses investigating approximation ratio three types eas combining bitwise mutation truncation section diverse repair mechanisms knapsack problem first type several pure strategy eas single repair method exploited eas second type several mixed strategy eas choose repair method repair method pool randomly third type using helper objectives simplified version remainder paper organized follows knapsack problem introduced section section iii analyse pure strategy eas section analyse mixed strategy eas section devoted analysing moea using helper objectives section concludes article jun boris mitavskiy department computer science aberystwyth university aberystwyth yuren zhou school computer science engineering south china university technology guangzhou china napsack roblem pproximation olution knapsack problem important knapsack problem one intensively studied combinatorial optimisation problems given instance knapsack problem set weights profits capacity knapsack task find binary vector subject item selected knapsack item selected knapsack feasible solution knapsack represented binary vector satisfies constraint infeasible one violates constraint vector represents null knapsack last two decades evolutionary algorithms especially genetic algorithms gas well adopted tackling knapsack problem order assess quality solutions eas follow classical algorithm see detailed exposition define evolutionary approximation algorithm follows definition say algorithm optimization problem instances problem produce solution within polynomial runtime value within factor value optimal solution regardless initialization runtime measured expected number function evaluations instance case knapsack problem evolutionary algorithm always find solution value least half optimal value within polynomial runtime iii ure trategy volutionary lgorithms section analyze pure strategy eas knapsack problem pure strategy refers employs single repair method genetic operators used eas bitwise mutation truncation selection bitwise mutation flip bit probability truncation selection select best individuals parent population child number diverse methods available handle constraints eas empirical results indicate repair methods efficient penalty function methods knapsack problem thus repair methods investigated current paper repair procedure explained follows input infeasible infeasible select item knapsack setp feasible end end end output several select methods available repair procedure repair repair random repair methods repair sort items according decreasing order corresponding profits select item smallest profit remove knapsack repair sort items according decreasing order corresponding ratios select item smallest ratio remove knapsack random repair select item knapsack random remove knapsack thanks repair method infeasible solutions repaired feasible ones fitness function feasible solution first let consider pure strategy using repair solving knapsack problem described follows input instance knapsack problem initialize population considering individuals mutate one individual generate child child infeasible solution repair feasible solution using repair end select individuals parent population child using truncation selection end output maximum fitness function following proposition reveals using repair produce good solution knapsack problem within polynomial runtime proposition constant using repair algorithm knapsack problem proof according definition suffices consider following instance knapsack problem item profit weight capacity without loss generality suppose large positive integer sufficiently large global optimum instance described local optimum ratio fitness local optimum global optimum suppose starts local optimum highest fitness truncation selection combined repair prevents mutant solution entering next generation unless mutant individual global optimum thus arrives global optimum bits flipped ones bit flipped bits remain unchanged probability event happening thus deduce expected runtime exponential completes argument let constant towards proposition tells solution produced using repair polynomial runtime may arbitrarily bad next consider another pure strategy uses repair tackle knapsack problem described follows input instance knapsack problem initialize population considering individuals mutate one individual generate child child infeasible solution repair feasible solution using repair end select individuals parent population child using truncation selection end output maximum fitness function similarly may prove produce good solution knapsack problem within polynomial runtime using instance proposition proposition constant using random repair algorithm knapsack problem proposition tells solution produced using random repair arbitrary bad finally investigate pure strategy using repair solving knapsack problem described follows input instance knapsack problem initialize population considering individuals mutate one individual generate child child infeasible solution repair feasible solution using repair end select individuals parent population child using truncation selection end output maximum fitness function proposition constant using repair algorithm knapsack problem proof let consider following instance item profit weight capacity without loss generality suppose large positive integer sufficiently large local optimum global optimum fitness ratio local optimum global optimum suppose starts local optimum let investigate following mutually exclusive exhaustive events infeasible solution generated case infeasible solution repaired back repair feasible solution fitness smaller generated case truncation selection prevent new feasible solution accepted feasible solution generated fitness smaller way truncation selection preserve new mutant solution nonetheless event happens first bit individual initial population flipped least bits individual flipped probability event follows immediately starts local optimum expected runtime produce better solution exponential desired conclusion follows immediately definition proposition tells solution produced using repair may arbitrarily bad well summary demonstrated none three pure strategy eas algorithm knapsack problem given constant ixed trategy volutionary lgorithm section analyse mixed strategy evolutionary algorithm combines several repair methods together mixed strategy refers employing two repairing methods selected respect probability distribution set repairing methods may worth noting types mixed strategy eas considered literature example mixed strategy employs four mutation operators naturally want know whether mixed strategy combining two repair methods together may produce approximation solution guarantee knapsack problem mixed strategy solving knapsack problem described follows combines repair methods together input instance knapsack problem initialize population considering individuals mutate one individual generate child child infeasible solution select either repair repair method uniformly random repair feasible solution end select individuals parent population child using truncation selection end output maximum fitness function unfortunately quality solutions mixed strategy still guarantee proposition given constant mixed strategy using repair repair algorithm knapsack problem proof consider instance proof proposition item profit weight capacity local optimum global optimum fitness ratio local optimum global optimum suppose starts local optimum let analyse following mutually exclusive exhaustive events occur upon completion mutation feasible solution generated fitness smaller case truncation selection prevent new feasible solution entering next generation feasible solution generated fitness smaller truncation selection may allow new feasible solution enter next generation event happens first bit flipped least bits flipped probability event infeasible solution generated fewer bits flipped bits case either infeasible solution repaired repair repaired feasible solution fewer bits among rest bits repair later case fitness new feasible solution smaller therefore accepted truncation selection infeasible solution generated fewer bits flipped bits event happens least bits flipped bits probability event afterwards positive probability repaired feasible solution fewer bits among rest bits repair later case fitness new feasible solution smaller therefore prevented entering next generation truncation selection summarizing four cases described see starts local optimum possible generate better solution probability know expected runtime produce better solution exponential conclusion proposition follows proposition tells solutions produced mixed strategy exploiting repair repair may arbitrarily bad furthermore prove even mixed strategy combining repair repair together algorithm knapsack problem proof practically identical proposition summary demonstrated mixed strategy eas algorithms knapsack problem given constant ulti bjective volutionary lgorithm far established several negative results eas knapsack problem naturally arising important question construct evolutionary approximation algorithm straightforward approach apply approximation algorithm first produce good solution afterwards run seek global optimum solution nonetheless eas sometimes get trapped absorbing area local optimum less efficient seeking global optimum analyse using helper objectives denoted moea short similar presented small changes made helper objectives sake analysis experiment results shown moea using helper objectives performs better simple combination approximation algorithm moea designed using technique optimisation problems transferred optimisation problems decomposing original objective several components adding helper objectives may bring positive negative effects approach used solving several combinatorial optimisation problems example knapsack problem vertex cover problem minimum label spanning tree problem describe moea using helper objectives similar original single objective optimization problem recast optimization problem using three helper objectives first let look following instance item profit weight capacity global optimum instance optimal solution average profit packed items largest thus first helper objective maximize average profit items knapsack use original value profits instead use ranking value profits assume profit item kth smallest let ranking value example instance helper objective function defined next consider another instance item profit weight capacity global optimum instance optimal solution average ratio packed items largest however average profit items largest second helper objective maximize average ratio items knapsack use original value instead ranking value assume item kth smallest let ranking value example instance helper objective function defined finally let see following instance item profit weight capacity global optimum instance optimal solution neither average profit packed items average ratio largest instead number packed items largest average weight smallest thus third helper objectives maximize number items knapsack objective functions consider optimization problem subject optimisation problem solved using bitwise mutation truncation selection plus mixed strategy two repair methods input instance knapsack problem initialize population considering individuals mutate one individual generate child child infeasible solution select either repair repair method uniformly random repair feasible solution end select individuals parent population child using truncation selection end output maximum fitness function novel truncation selection operator adopted since target maximise several objectives simultaneously select individuals higher function values respect objective function selection described follows input parent population child merge parent population child temporary population consists individuals sort individuals temporary population descending order denote select individuals left right denote satisfy number selected individuals greater truncate individuals end add selected individuals next generation population resort individuals temporary population descending order still denote select individuals left right still denote satisfy number selected individuals greater truncate individuals end add selected individuals next generation population resort individuals temporary population descending order still denote select individuals left right still denote satisfy number selected individuals greater truncate individuals end add selected individuals next generation population next generation population size less randomly choose individual parent population child add next generation population end output new population algorithm steps selecting individuals higher values order preserve diversity choose individuals different values similarly steps selecting individuals higher value choose individuals different values maintaining diversity steps selecting individuals higher value choose individuals different values preserving diversity explicitly select individuals based instead implicitly steps steps using helper objectives truncation selection brings benefit searching along several directions implicitly hence moea may arrive local optimum quickly time get trapped absorbing area local optimum experiment results demonstrate moea using helper objectives outperform simplified combination approximation algorithm analysis based fact derived analysis greedy algorithm knapsack problem see section consider following algorithm let feasible solutions largest profit item resort items via ratio profits corresponding weights greedily add items order knapsack long adding item knapsack exceeding capacity knapsack denote solution fitness smaller fitness optimal solution based fact prove following result theorem moea produce feasible solution worse within runtime proof without loss generality let first item profitable one first suffices prove generate feasible solution fitting holland schema usual stands care symbol could replaced either within polynomial runtime suppose value individuals population smaller fit holland schema let individual chosen mutation mutation flipped probability child feasible arrive desired individual denote child infeasible probability first item kept thanks repair feasible solution generated denote shown generate feasible solution includes profitable item probability least thus generate feasible solution fitting holland schema within expected runtime without loss generality let let demonstrate reach within polynomial runtime via objectives first prove reach within polynomial runtime exploit drift analysis tool establish result binary vector define distance function population distance function min according definition distance function suppose none individuals current population let individual value whose distance smallest current population individual belongs one two cases case fits holland schema least one bit takes value case fits holland schema individual chosen mutation probability analyse mutation event related two cases analysis case one first bit flipped bits changed event happen probability let establish value increases mutation denote bits objective value without loss generality bit flipped mutation bits becomes objective value thus value increases equivalently value decreases thanks truncation selection value always increases negative drift therefore drift case analysis case first bit flipped bits changed analysis identical case drift case case recall distance function applying drift theorem theorem deduce expected runtime reach included population kept ever according truncation selection next prove reach within polynomial runtime starting suppose current population includes individual individual individual may chosen mutation probability mutated probability individual second largest value thus according truncation selection kept next generation population hence expected runtime reach individual similarly prove reach within runtime within runtime expected runtime reach combining discussions together see expected runtime produce solution worse change helper objective functions used proof work need new proof obtaining conclusion furthermore mentioned none three objectives removed otherwise moea produce solution guaranteed approximation ratio side performance might better adding objectives example min onclusions work assessed solution quality three types eas exploit bitwise mutation truncation selection solving knapsack problem proven pure strategy eas using single repair method mixed strategy combing two repair methods algorithm constant words solution quality eas may arbitrarily bad nevertheless shown using helper objectives algorithm runtime work demonstrates using helper objectives good approach design evolutionary approximation algorithms advantages using helper objectives search along several directions also preserve population diversity eas using strategies preserving diversity niching methods investigated paper extension work eas future research another work future study solution quality moeas knapsack problem acknowledgements work supported epsrc grant nsfc grant eferences martello toth knapsack problems chichester john wiley sons michalewicz arabas genetic algorithms knapsack problem methodologies intelligent systems springer khuri multiple knapsack problem genetic algorithms proceedings acm symposium applied computing acm chu beasley genetic algorithm multidimensional knapsack problem journal heuristics vol raidl improved genetic algorithm multiconstrained knapsack problem proceedings ieee world congress computational intelligence ieee michalewicz genetic algorithms data structures evolution programs new york springer verlag zitzler thiele multiobjective evolutionary algorithms comparative case study strength pareto approach ieee transactions evolutionary computation vol jaszkiewicz performance genetic local search knapsack comparative experiment ieee transactions evolutionary computation vol captivo figueira martins luis santos solving bicriteria knapsack problems using labeling algorithm computers operations research vol case study memetic algorithms constraint optimization soft computing vol kumar singh assessing solution quality biobjective knapsack problem using evolutionary heuristic algorithms applied soft computing vol rohlfshagen bullinaria nature inspired genetic algorithms hard packing problems annals operations research vol williamson shmoys design approximation algorithms cambridge university press ibarra kim fast approximation algorithms knapsack sum subset problems journal acm vol kumar banerjee analysis multiobjective evolutionary algorithm knapsack problem theoretical computer science vol dong novel genetic algorithm using helper objectives knapsack problem arxiv vol coello carlos theoretical numerical techniques used evolutionary algorithms survey state art computer methods applied mechanics engineering vol zhou comparison gas using penalizing infeasible solutions repairing infeasible solutions proceedings international symposium intelligence computation applications wuhan china springer hou dong mixed strategy may outperform pure strategy initial study proceedings ieee congress evolutionary computation ieee knowles watson corne reducing local optima problems evolutionary multicriterion optimization springer jensen using evolutionary algorithms optimisation journal mathematical modelling algorithms vol handl lovell knowles multiobjectivization decomposition scalar cost functions parallel problem solving springer brockhoff friedrich hebbinghaus klein neumann zitzler effects adding objectives plateau functions ieee transactions evolutionary computation vol lochtefeld ciarallo optimization strategies scheduling problem applied soft computing vol friedrich hebbinghaus neumann witt approximating covering problems randomized search heuristics using models evolutionary computation vol lai zhou zhang performance analysis evolutionary algorithms minimum label spanning tree problem ieee transactions evolutionary computation accpeted online yao drift analysis average time complexity evolutionary algorithms artificial intelligence vol
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shafahi wang haghani school bus routing maximizing trip compatibility ali shafahi phd candidate department civil environmental engineering university maryland college park email ashafahi zhongxiang wang graduate student department civil environmental engineering university maryland college park email ali haghani professor department civil environmental engineering glenn martin hall university maryland college park phone fax email haghani word count words text figures words words submission date july shafahi wang haghani abstract school bus planning usually divided routing scheduling due complexity solving concurrently however separation two steps may lead worse solutions higher overall costs solving together finding minimal number trips routing problem neglecting importance trip compatibility may increase number buses actually needed scheduling problem paper proposes new formulation homogeneous fleet routing problem maximizes trip compatibility minimizing total travel time incorporates trip compatibility scheduling problem routing problem since problem inherently routing problem finding good solution cumbersome compare performance model traditional routing problems generate eight data sets importing generated trips routing problems bus scheduling blocking problem shown proposed model uses fewer buses common traditional routing models keywords school bus routing trip compatibility school bus scheduling bus blocking shafahi wang haghani introduction school bus routing scheduling problem sbrsp traditionally broken two pieces due extra complexity added solved together first piece school bus routing sbrp trip building second piece scheduling blocking sbsp sbrp goal build trip consists several stops trips usually start schools drop students different bus stop locations trips reverse sbsp blocking hand ties trips together assigns grouped trips routes blocks blocking problem possible buses serve trips different schools school public school county department pupil transportation authority charge operation design school bus routes monetary point view interested safe reliable transportation services keeping cost services low possible since major driving factor cost number buses blocks benefit buses mixed trips private school setting school owns operates school buses trips blocks school setting solving routing problem scheduling problem independently cause loss terms funds separation problem thus beneficiary improves running time required search optimal solution public school setting however separation problems could result financial losses overall requirement school buses number buses required county operation output blocking scheduling problem blocking problem compatibility trips main influencing factor helps reduce overall number buses say trip compatible trip trip served bus enough time drive trip initial stop trips compatible put block however compatible assigned separate blocks potentially increases required number blocks compatibility trips generated routing problem trips essential inputs blocking problem therefore important school bus routing problems somehow generate compatible trips majority school bus routing problems either minimizing number total trips minimizing total travel time objective objectives somewhat treating routing problem blocking problem independent problems consider example depicted figure example three stops school bus stops school population school exceeds capacity one bus therefore need least two trips school addition school ends times school without considering blocking minimum number trips needed school minimum total duration achieved trip trip soonest one trips reach school time past end time school therefore neither trip trip compatible trips school consider case trip trip second trip stop stop passes reach stop case total travel time however trip could reach school therefore compatible trips built school shafahi wang haghani figure example illustrates importance treating school bus routing scheduling problems related problems existing research solve either sbrp sbsp solve together becomes computationally burdensome find good solutions however choice good objective school bus routing problem considers sbsp bus blocking potentially reduce cost increase efficiency paper structure next section review school bus routing problem presented new mathematical model explained next section computational tests conducted evaluate model performance finally performance proposed model applications limitations future research directions presented literature review whole process school bus routing scheduling problem involves five detailed steps data preparation bus stop selection bus route generation school bell time adjustment route scheduling first two steps incorporated solved lar strategy arl strategy difference application two strategies referred last two steps solved scheduling problem time windows school bus routing problem sbrp always regarded variation vehicle routing problem vrp one difference sbrp classic vrp travel time depot first pickup vertex trips insignificant well travel time last vertex depot trips school bus service providers mainly concerned ride time students opposed total travel time buses makes sbrp become similar open vehicle routing problem ovrp specifically many consider capacitated distance travel time constrained ovrp main difference ovrp general vrp former tries find set hamiltonian paths rather hamiltonian cycles classical vrp however minimum hamiltonian path problem still problem transferred minimum hamiltonian cycle problem problem shafahi wang haghani paper focuses sbrp similar vrp vrp papers also listed literature review characteristics sbrp school bus routing problem divided various ways different categories divide problem based number schools problem solved another category based area school fall type fleet another mean categorization common public school settings routing scheduling realistic however quite lot papers focus problem due simplicity similarity classic vrp dealing problems many papers divide problem problems assuming bus trip exclusive one school still papers consider mixed load service problem braca solved sbrp multischool bus routing new york city location based heuristic method forms routes inserting vertex minimal insertion cost among vertices repeating procedure starting random vertices pick best solution among iterations another classification urban school rural area school bus routing problem urban areas tends students stop bus capacity usually binding constraint stops may need served therefore maximum ride time constraint relaxed certain conditions rural areas stop tends small number students distance adjacent stops longer maximum ride time usually critical constraint moreover vans smaller vehicles rather buses might economical thus vehicle type selection mix fleet preferred homogeneous heterogeneous fleet becomes important affects bus capacity degree crowding allowance standing etc national association state directors pupil transportation services regulates maximum three young students lower third grade sit typical school bus seat applied efficiency effectiveness equity criteria proposed savas sbrp order balance total ride time student afternoon trips could replicated sequence except school morning trip thus may monetarily temporally efficient solve routing problem replicate trips balance maximal load maximal ride time another equity concern considered objective formulation objective common objectives sbrp minimizing total travel distance travel time minimizing total number trips minimizing total cost including bus purchase fixed cost bus operation cost latter total cost combination first two objectives considering bus purchase cost proportional number trips without scheduling trip assumed served one bus bus operation cost equivalent total travel distance incorporated students walking distance model tried minimize number buses minimizing maximum ride time students two objectives conflicting less number buses would require longer route lengths another factor objective function could penalty penalties generally added hard constraint relaxed allow violation constraints penalty could various forms including time window violation penalty capacity maximum ride time constraints violation penalty shafahi wang haghani constraints logistic degree constraints basic vrp constraints formulate trips regulate conservation flow assuring vehicle enters vertex node also exits vertex addition trip starts depot goes back depot vertex served exactly cases allowed cases papers consider scenario although sbrp ovrp discussed easier generate round trips first discard first trips last trips arc trip capacity constraints also quite common papers sbrp papers vrp vehicle capacity constraints maximum ride time constraint also considered especially rural areas russell morrel used minutes maximum ride time chen limited ride time less equal minutes elimination constraints another common set constraints prevent formation illegal trips connected depot adopted way enforce connectivity set appropriate lower bound number vehicles required visit subset vertices another common method using artificial commodity flow introduction flow variables adopted miller tucker zemlin subtour elimination constraint solve sbrp chain barring constraints seen scenarios constraints aim eliminate illegal trips start one school end another school constraints enforced implicitly simply allowing trips assigned school stops assigned school thus chain barring constraints considered implicitly paper note assumption implied constraint school trip mixed students different schools yet huge shortcoming reality mixed students popular among school agencies time windows widely incorporated scheduling rather route generation applied time window constraints school buses must arrive school earlier minutes later minutes start school proposed time window constraints prevent formation time solving routing scheduling together modified several routing heuristic algorithms saving algorithm insertion algorithm nearest neighbor algorithm sweep algorithm solve routing scheduling problem however optimality heuristics proven evaluated due lack optimal solutions exact algorithms paper propose new mathematical model new objective function proposed model maximizes trip compatibility minimizing total travel time solve routing scheduling problem simultaneously hence finding good solutions problem computationally burdensome however provides routing solution set trips high compatibility result routes served one bus likely generated number buses needed entire school operations could reduced methodology present mixed integer programming mip model aims maximize compatibility among trips hope minimizing overall number buses result presented mip novel model solves routing problem considering bus blocking problem table summarizes variables parameters used mathematical model shafahi wang haghani table notation summary variables parameters variables school bus routing variable name parameter name description binary decision variable equals stop assigned trip binary decision variable equals trip assigned school portion capacity bus trip filled stop binary variable equals trip bus goes directly stop stop travel time duration trip start time trip end time trip binary variable equals trips served trip compatible travel deadhead duration last stop trip first stop trip binary variable equals last stop trip stop variable used subtour elimination units artificial commodity shipped stop using trip variables bus blocking binary variable equals trip follows trip within block tour variable created compatible pair trips binary variable equals trip belongs tour size trip goes binary variable equals trip trip served also another trip served afterwards binary variable equal trip served first trip tour size greater parameters school bus routing bus blocking description set schools set possible trips dedicated school set trips set outputted trips school bus routing problems set stops students school come set stops capacity bus number students stop school origin stop school duration drive stop plus dwell time required stops time school closes allowable buffer picking students schools cases solved due trips trips flexibilities large value penalty total number trips cases solved coefficient compatibility cases solved number allowable trips minimum required trips school explained earlier main purpose objective function simultaneously shafahi wang haghani maximize number compatible trips minimizing number overall trips required travel time done assigning penalties positive weights total number trips used assigning benefits negative weights compatible pair trips min constraints listed charge building trips school trip built various stops shafahi wang haghani constraints prevent assignment stops trips assigned respective schools trips ones use constraints trip capacity constraints constraints ensure students served constraints disallow assignment students trip pass stop students constraints assure stop assigned trip trip visits node traversing arcs lead node constraints enforces trip start school trips conservation flow expressed constraints constraints calculate travel duration trips trips start school end node last stop stop going back school constraints calculate end time trips using start time travel time start time constrained time schools close constraints constraints used identifying compatible trips deadhead pairs trips used compatibility calculations computed using constraints constraints used identifying last stops trips last stop last actual stop stop right school goes back start point complete fictional closed loop trip constraints eliminating symmetries note one trip assigned school trip could assume many values constraints prevent higher trip ids occur prior lower ones speed search good solutions constraints flow subtour elimination constraints finally constraints limiting number trips assigned school minimum number needed based population stop discussed urban school setting bus capacity binding constraint maximum ride time constraint relaxed sake improving model efficiency compare trip outcomes model versus traditional models input trips models bus blocking problem summarized constraints make sure trip selected assigned school either served another trip pair served alone constraints identify middle trips blocks objective blocking problem minimizing total number blocks could achieved maximizing number middle trips minimizing number alone trips alone trips trips sole trips tour block size one shafahi wang haghani min important note way objective function written may putting much importance total trip compatibility consider scenario trip compatible trips trips incompatible case one bus serve either trip trip trip bus serve total trip compatibility equal two one bus saved implies trip compatibility equal number buses saved one approach hybrid model routing blocking however mentioned earlier problem complex due additional variables constraints problem almost impossible solve problems toy problems using exact algorithms another approach look compatibility trips schools looked approach however upon preliminary investigations find great gain proposed model computational result order evaluate model eight set randomly generated problems generated tested typical routing models found literatures minimizing number trips minimizing number buses without scheduling minimizing total travel time distance parameters distributions used randomly generating cases mostly based real world cases detailed information scenario results different models shown table scenario many parameters total number stops stops indicator problem size number vertices problem number schools also affects size problem since school certain number students require bus transport total number trips school needs bound ratio total number students capacity bus school average number students school require bus transportation also different scenario note school different population maximum number stops assigned school another parameter influences problem complexity finally bus start range scenario time difference earliest latest school dismissal time dismissal times schools follow discrete uniform distribution minutes time intervals within range scenario four objectives tested proposed objective maximizing trip compatibility minimizing total travel time maximizing trip compatibility maxcom deleting travel time proposed objective minimizing total number trips minn minimizing total travel time mintt problems solved commercial solver fico mosel xpress five computers feature cpu ghz ram due slow rate reduction gap solution processes test problems terminated certain running times minutes hours relative performance measure used models total number blocks output blocking problem input trips solutions respective model scenario illustrating impact additional allowed trips parameter constraints complexity problem solution time perform sensitivity analysis run scenario using four different values parameter recall school allowed use two trips minimum number requires minimum number trips school requires easily calculated based school population capacity bus seen additional trips allowed shafahi wang haghani running time required however better solutions may found example buses additional trips hrs current best solution comparison additional trips short running time minutes solutions additional trips cases keeping mind recently mentioned note sake saving running time paper limits remaining scenarios additional trips noted real applications additional trips allowed try find best potential solution simply allowing models run longer real applications running time sbrsp less important finding optimal solution even one bus saving could significantly beneficial table summary computational result scenario stops schools avg student school max stops school bus service start time range hrs running time minute gap unit number trips trip nob number buses bus additional trips gap nob gap nob gap nob gap nob max com max com min min scenario number stops increased model presented paper save buses comparison solution mintt buses saved minn figure considering higher gap proposed model running time minutes result potential improved scenario similar scenario except increasing average number students school students students shafahi wang haghani based result buses saved compared mintt minn respectively scenario expansion scenario scenario average students school relatively large case saving maxcom significant buses bused saved scenario number stops number school increased scenario mintt need buses maxcom needs buses reason might bigger gap maxcom seen general uses less buses compared traditional models scenario similar scenario except expands bus service start time range minutes minutes purpose change see model performs comparison traditional models many trips going easily compatible anyway trips easily compatible increasing start time range bus saving might less smaller time range scenarios result figure proves hypothesis one bus saved scenario scenario similar scenario change bus service start time range range increased minutes scenario similar results scenario additional improvement made proposed models scenario another experiment decreases bus service start time range minutes scenario one bus saved maxcom conditions bus service start time range small trips compatible maxcom would take whatever need make compatible leads extremely long travel time trips figure cross examination reveals application limits proposed objective proposed objective extremely beneficial comparison traditional models scenarios bus service time range falls range either big trips compatible anyway small trips compatible range like school bus service provided county department pupil transportation proposed objective greatly outperforms traditional objectives another concern sbrsp maximum ride time trip since maxcom model maximum ride time constraint incentive minimizing total travel time maxcom focuses maximizing trip compatibility even expense long travel time trips could minutes scenario long school bus trip perspective makes sense figure travel time distribution frequency calculated minutes interval marked beginning interval instance frequency interval minutes plotted travel time horizontal axis equals minutes results figure show tends short travel time trips mostly less minutes maxcom figure total travel time different objectives shows scenarios results mintt minimal total travel time maxcom minn long travel times much less total travel time maxcom minn total travel times slightly higher mintt general minn worst result model would stop right finding solution uses minimal number trips since objective improved furthermore optimal model simply ignores decreasing travel time increasing trip compatibility makes harder group block tends routes figure routes exist less buses needed overall generates best result use least buses trips relatively short travel times shafahi wang haghani figure summary school bus blocking result shafahi wang haghani figure travel time distribution shafahi wang haghani figure total travel time comparison worth noting gain savings number buses needed serving trips come free figure shows tradeoff reduction number buses increase travel time per bus comparison made best results proposed models maxcom best results traditional models mintt minn seen buses saved expense additional travel time minutes per bus scenario buses saved financial point view savings gained needing fewer buses could easily justify additional travel times note scenario means travel time per bus solution new objectives minutes less traditional objectives figure tradeoff decrease number buses travel time increase best new models best traditional models shafahi wang haghani summary conclusion paper propose new mathematical model optimizes new objective solve school bus routing problem new objective component maximizing trip compatibility potentially decrease total number buses required operation eight problems solved test performance illustrate applications limitations proposed method shown proposed model outperforms traditional models requiring fewer buses model significance greater school dismissal times within minutes recommend using model cases research opens venue many future studies one sensitivity analysis parameters proposed objective another one double counting compatibility problem mentioned end methodology section larger size problem also applied test performance efficiency proposed model heuristic algorithm also developed solve model larger problems needed references desrosiers ferland rousseau lapalme chapleau overview school busing system jaiswal scientific management transport systems amsterdam park kim school bus routing problem review european journal operational research solving school bus routing problems integer programming journal operational research society gendreau potvin hasle metaheuristics vehicle routing problem extensions categorized bibliography vehicle routing problem latest advances new challenges springer syslo deo kowalik discrete optimization algorithms pascal programs new jersey laguna marti heuristic solutions problem routing school buses multiple objectives journal operational research society braca bramel posner computerized approach new york city school bus routing problem bowerman hall calamai optimization approach urban school bus routing formulation solution method transportation research part policy practice ripplinger rural school vehicle routing problem transportation research record journal transportation research board national association state directors pupil transportation school bus seat capacity savas equity providing public services mangagment school bus routing problem case study journal operational research society russell morrel routing school buses interfaces chen kallsen chen tseng bus routing system rural school districts computers industrial engineering shafahi wang haghani thangiah nygard march school bus routing using genetic algorithms aerospace sensing international society optics photonics spada bierlaire liebling methodology school bus routing scheduling problem transportation science schittekat sevaux mathematical formulation school bus routing problem proceedings ieee international conference service systems service management troyes france november vertical transfer algorithm school bus routing problem innature biologically inspired computing nabic fourth world congress ieee taillard badeau gendreau guertin potvin tabu search heuristic vehicle routing problem soft time science gendreau hertz laporte tabu search heuristic vehicle routing problem management science tillman multiple terminal delivery problem probabilistic demands transportation science held karp problem minimum spanning trees operations research wren holliday computer scheduling vehicles one depots number delivery points journal operational research society gillett miller heuristic algorithm problem operations research pacheco tabu search routing problem journal operational research society laporte nobert arpin exact algorithm solving capacitated problem annals operations research laporte nobert taillefer solving family vehicle routing problems transportation science desrosiers soumis lagrangian relaxation methods solving minimum fleet size multiple traveling salesman problem time windows management science laporte vehicle routing problem overview exact approximate algorithms european journal operational research miller tucker zemlin integer programming formulations traveling man problems assoc comput mach solomon algorithms vehicle routing scheduling problems time window constraints operations research
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statistical science vol doi institute mathematical statistics enhancing advanced compilation tools methods sep duncan temple lang abstract describe approach compiling common idioms code directly native machine code illustrate several examples yield significant performance gains allows use new approaches computing importantly compilation requires changes done entirely via packages allows others experiment different compilation strategies even define new languages within use virtual machine llvm compiler toolkit create native code perform sophisticated optimizations code adopting widely used software within leverage ability generate code different platforms cpus gpus continue benefit ongoing development approach potentially allows develop code also fast compiled work different data representations sources could even run outside approach aims provide compiler limited subset language also enable programmers write compilers another approach help write descriptions want compute key words phrases programming language efficient computation compilation extensible compiler toolkit ploiting coming terms technologies parallel computing including shared nonshared computing data interesting processors gpus graphics processriod present significant implications ing units challenge innovate sigfor choose forward computnificantly enhance existing computing platforms ing platforms education statistics related develop new languages systems fields simultaneously leveraging able meet tomorrow needs level interpreted languages matlab next decade python recently julia dealing increasstatisticians play important role big ing volume complexity data iii data surge therefore must pay attention logistical performance details statistical comduncan temple lang associate professor putations could previously ignore need department statistics university california think best meet computdavis math sciences building davis california ing needs near future also best usa duncan able participate efforts require serious computing involving statistical ideas electronic reprint original article published institute mathematical statistics methods best served computing platform core team statistical science vol need system afford luxreprint differs original pagination typographic detail ury system given limited resources background motivation temple lang field develop maintain innovate system alternatively would better reimplementing environment system developed supported larger communities example python java leave others build new fast computing environment leverage within field statistics would right give opportunity escape legacy current systems position innovation would developing new system splinter already small community reduce effectiveness disseminating new statistical methods effectively excellent package mechanism wrestled questions decade believe simple answer best way proceed much social issue technical one community amazing valuable phenomenon large code base packages scripts widespread use important science even new systems emerge replace take several years need significant improvements performance make competitive systems least near future must improve performance allow continue deal larger complex data problems paper discuss one direct approach improve performance code extensible enables many people improve essence approach suggesting conceptually quite simple emerged numerous languages platforms around time first started implementing idea compile code directly native machine instructions run cpu gpu device target instead insisting code evaluated one interpreter may generate code perform equivalent computations quite different way dynamically compile fast native code combining information code data representation processed code available computing hardware cpu gpu location different sources data whether need handle missing values nas form jit compilation leverages additional knowledge context code run maps code machine instructions rather evaluated interpreter approach presented quite different programmers typically improve performance code manually implement slow computationally intensive parts call routines call programming around instead trying compile around approach statisticians use familiar idioms express desired computations compiler infrastructure generates optimized instructions realize intended computations efficiently possible hardware platform use input statistician analyst process code code good humans easily understand debug adapt extend code written furthermore compiler understand intended code optimize quite different ways also allows code optimized different ways future code says left compiler makes approach feasible practical availability virtual machine compiler infrastructure llvm lattner adve llvm winner association computing machinery system software award award conferred language highly extensible compiler toolkit llvm library integrate languages generate native code various different cpus gpus output targets ability integrate adaptable extensible tool languages game changer developing compilation tools strategies use technology continue evolve developed domain experts adapt purposes domain knowledge within extensible package rather interpreter leaves compilation infrastructure user space allowing development new compilation strategies shared without changes interpreter contrasts compiler interpreter part fixed executable rllvm temple lang package provides bindings llvm api use generate arbitrary native code rllvmcompile temple lang buffalo enhancing advanced compilation tools methods package provides simple compiler attempts translate code native code mapping code instructions llvm leaving llvm optimize code generate native machine code explore examples use llvm improve performance change computational strategies certain types problems worth thinking little potential implications fast code alternative data models practical level compile scalar nonvectorized functions almost fast code use process individual observations streaming updating manner means escape data memory approaches strongly rewards exporting code alternative execution environments write code export run different systems example databases python javascript web browsers map code llvm intermediate representation use emscripten zakai compile directly javascript code systems share code use llvm bindings compile native code particular hardware richer data structures provides small number primitive data types example vectors lists functions currently use create new aggregate composite data structures however introduce new different data structures linked lists suffix trees opaque data types programmed native code compile code native instructions also opportunity new code use different data structures represent data differently code merged descriptions new data types yield quite different native instructions better suited particular problems templating concepts ability create native code code allows think functions expressions differently descriptions done without specifics compiler rewrite generate code behave differently interpreter give results hopefully functions compiler use knowledge particular representation data functions process generate native code intelligent manner example code may access elements data columns two different representations data frames matrices individual elements accessed different compiler generate specialized code might even change order computations improve efficiency cache coherency representations function tied particular data representation summary compiling programs llvm yields novel computational potentials directly relevant improving statistical learning communication big data era compiling highlevel code native code used many systems julia interesting modern project numpy jones python another several years ago ross ihaka explored using lisp ihaka temple lang platform new statistical computing platform ideas used many years performance gains impressive important premise underlying approach paper project large code base important least another years users likely immediately change new system even technically superior reason important improve performance also important allow developers outside core development team contribute effort avoid many projects need extensible system within one requires continual changes centralized source code addition focusing immediate future improving also need exploring new language computing paradigms within field statistics julia interesting modern project need foster experiments new ideas emerge also need increase quantity quality computing within curricula section constitutes majority paper explore different examples computing compiling code make computations efficient simply obtaining faster temple lang execution speed also allowing change approach problem section discuss additional general strategies exploit improve computations future briefly discuss exciting research projects improve efficiency contrast llvm approach outline section road map ongoing work llvm approach related projects part future activities aim illustrate feasibility entire approach paper focus reasonably standard code approaches improved generating native code semantics generated code original code approach also allows develop new languages semantics within compilation framework explore different computational models however examples discussed paper stay within existing computational model order anchor discussion avoid many degrees freedom becoming distraction hope explore new semantics language features within via compilation enhancing advanced compilation tools methods section explore examples write code compile machine code explore different strategies illustrate approach computations differently option compile code rather interpret chosen problems several reasons reasonably simple state illustrate potential benefits compilation like benchmarks examples may reflect typical use cases would things however problems concrete practical represent ways would like able program performance issues way examples illustrate continue use additional computational model overcome interpreter performance issues still using essentially code note following subsections present absolute comparative timings different approaches implementations different tasks timings performed three different machines used macbook pro running ghz intel core mhz also two linux machines first linux machines older ghz amd opteron second much recent faster ghz intel core ram additionally different machines different compilers used compile may impact timings used gcc gcc first linux machine gcc clang second linux box cases compiled optimization level flag absolute times quite different across machines expect withinmachine relative performance llvm generated code native code differs linux however results similar across linux machines finally current steps optimize native code generate llvm quite simple expect improve near future included time compile code measurements two steps compiling code intermediate representation compiling native code former done latter session done llvm code several reasons omitting steps timings first focus tasks take long time run example many hours days compilation time order minutes compilation time negligible second expect compiled code reused multiple calls overhead compiling amortized across calls also ignored time functions compile install code used packages fibonacci sequence fibonacci sequence interesting mathematical sequence integers defined relation implement function easy obvious manner fib function else fib fib enhancing advanced compilation tools methods simplicity verify nonnegative assuming caller provide meaningful inputs maps mathematical description sequence computational form essentially manner good thing makes code computations easy understand debug maintain however scalar function vectorized meaning computes value fibonacci sequence single integer value rather vector inputs makes slow want compute multiple values sequence example apply element vector instead implementing function natural form gain performance would look implementations since sequence described recurrence relationship simple closedform formula computing nth value sequence easily implemented vectorized function alternatively might use memoization remember results computed previous calls function avoid repeating computations might even use lookup table precomputed results common input values combination approaches key get good performance think problem quite differently instead explore make simple implementation perform better hope avoid change entire approach rely vectorized operations use function compilefunction rllvmcompile package create native compiled version fib function compilefunction fib list specify type return value also type input case function return type input regular integer values corresponding integer type could use integer using type wanted deal larger numbers fact create two separate different versions function different types two calls compilefunction simple illustration easy adapt code different situations create different compiled routines different characteristics compilefunction return function invoke directly however default currently returns object representing compiled routine llvm invoke routine using object function analogous functions calls compiled routine returns value unlike functions function knows expected types routine parameters coerces inputs types expected routine case converts value numeric value integer verifying routine gives correct results explore performance code recursive function computationally intensive calculating example fib calculate fib twice fib fib similarly compute fib multiple times repetition one reasons code slow compare time evaluate fib using three different versions fib function original interpreted function llvm routine version fib compiled llvm routine fastest table shows elapsed times table timings computing fibonacci sequence values interpreted code code code linux linux time speedup time speedup time speedup timings call fib using regular function version llvm version improve accuracy timings calculate duration replications two slower functions replications llvm routine divided duration llvm version clearly much faster temple lang ratio time function relative time interpreted function convey relative speedup factor see macbook pro llvm routine times faster interpreter linux machine speedup bit smaller still significant factor attempted reduce variability timings observed different speedups ranging linux timings report recent rather best simple example necessarily representative would calculate fibonacci sequence production code however ability express algorithm natural mathematical form makes easier program verify extend would much like able write code manner without sacrificing good performance random walk ross ihaka developed instructive example writing straightforward code compared clever highly vectorized code means illustrate profiling make code efficient task simulating random walk natural write loop iterations corresponding steps walk step toss coin determine whether move horizontally vertically given choice toss another coin determine whether move left right calculate store new location call approach code shown figure several refinements based profiling nontrivial knowledge ihaka defines efficient implementation random walk shown figure removes explicit loop samples steps one call sample determines positions using two calls cumsum function makes good use several vectorized functions implemented code therefore fast manually compiled implementation using rllvm similarly used create two compiled versions function simulated million step random walk using original function compiled function fully vectorized version llvm version table shows relative speedups see manually vectorized function times faster function xpos ypos numeric truefalse true false decide whether moving horizontally vertically sample truefalse xpos xpos sample ypos ypos else xpos xpos ypos ypos sample list xpos ypos fig implementation random walk sample directions separately function xsteps ysteps dir sample replace true xpos cumsum xsteps dir ypos cumsum ysteps dir list xpos ypos fig fast vectorized implementation random walk implementation illustrating important vectorization make code efficient however also see compiling implementation llvm outperforms even vectorized version taking time vectorized version probably due compiled code using single loop vectorized version two calls cumsum hence least one additional loop steps sampling text file suppose one large value csv files example download airline traffic delay data year approximately megabyte csv file research innovative technology administration rita enhancing advanced compilation tools methods table timings simulating random walk interpeted code byte compiled code vectorized code code time speedup time linux speedup linux time speedup generate million steps approach compare manually vectorized implementation code version written llvm version function vectorized version times faster regular function however llvm version outperforms vectorized version likely removing one loop part bureau transportation http rather working entire data set might choose take random sample observations concern appropriateness simple random sample also assume know number observations csv file efficiently extract sample lines file could use unix shell tools difficult randomly generate specify lines sample sampling indices something want passing shell command awkward best alternatively could entire sampling could read entire file memory via readlines function subset ones want however requires significant amount memory first store lines make copy ones want discard larger vector may feasible may enough memory may simply slow think different strategies one first identify indices lines want sample read file chunks get line sample store line continue read next line sample make work need able continue read currently file use file connection first step generate vector line numbers want sample using example linenum sort sample samplesize number lines csv file sorted line numbers read sample lines file sequentially next step determine many lines successive lines sample compute lineskips diff linenum gives vector pairwise difference successive elements example suppose first two lines want sample first two elements lineskips read first two lines sample con file readlines con readlines con element lineskips tells many lines read get next line sample next need function read many lines return last following function readto function numlines con readlines con numlines numlines final step obtaining entire sample call readto element lineskips example readselectedlines function lineskip file sapply lineskip readto file obtain sample call readselectedlines passing variable lineskips open connection con file sample readselectedlines lineskips con temple lang functions concise efficient since sapply essentially implemented loop within interpreter using connection readlines read blocks lines readto efficient uses code within unfortunately involve reading allocating storing subsetting discarding potentially large character vector returned call readlines however want single line end vector call call involves significantly fewer lines reading entire file allocating large character vector still slows computations extensively involves memory manager different approach avoid memory issue change readto function reads line individually returns last one could change readto function numlines con ans numlines ans readlines con ans straightforward easy understand unfortunately extremely slow looping almost every line file idea reading one line time would work well could avoid overhead loop mechanism compile new version readto native code almost need equivalent readlines read single line file exactly standard routine fgets similar connection pass fgets pointer opaque file data structure puts contents next line reads location memory also provide simplicity exposition define function fgets proxy call fgets fgets function file fgets ptr file code assumes function named fgets ptr somehow address array memory character elements space long string run code variables ptr file fgets actually exist instead allocate llvm compiled native routine generate fgets compile fgets function llvm module collection routines variables using compilefunction also define global variable ptr pointer array want creating actual array characters another global variable compiling fgets also need tell compiler signature external fgets routine make call fgets correctly via mod module filetype pointertype declarefunction list stringtype stringtype filetype fgets mod done explicitly could automate step using rcindex temple lang package obtain signature programmatically also need tell llvm engine locate fgets routine llvmaddsymbol fgets note fgets function assumed longest line less characters specify different length knew suspected otherwise similarly provide error checking whether reached end file assuming caller knows total number lines sampling number example shortcuts make compiling code particular situation writing general robust code used many different situations could also tell compiler add tests wanted avoid extra computations know redundant obtain instance file data type pass compiled fgets routine use routine fopen write function mimics compile however rllvmcompiler package function automate creation proxy function know signature routine interest example illustrates dynamically create bindings existing compiled routines different libraries case file enhancing advanced compilation tools methods table timings sampling csv file interpreted loop readlines loop fgets code fastcsvsample time speedup time linux speedup time linux speedup use vectorized code read blocks data extract final line block llvm approach compiles simple functions read one line time fastcsvsample thing manually written code compiled approaches avoid memory usage related readlines see nontrivial speedup fastcsvsample package outperforms llvm version approaches outperform approach using connections readlines functionality also implemented code also use existing function cfile rcurl temple lang package read single line open file object via compiled fgets routine redefine readto function readto function numlines con ans numlines ans fgets con ans almost identical original function replaces call readlines con fgets con compile native code via compilefunction resulting code quite fast fast replacement reading next line sample last step make readselectedlines fast recall implemented simply sapply lineskip readto file compile returning character vector compiler recognizes sapply call converts loop native code populates returns new character vector summary compiled three functions fgets readto readselectedlines allow read one line time use minimal amount memory collect lines sample using two loops native code rather compare performance approach using readlines consume chunks lines compiled version reads one line time addition two approaches also manual implementation essentially equivalent llvm approach fastcsvsample package temple lang timings based extracting sample one hundred thousand lines uniformly csv file contains one hundred million lines approach elapsed times given table see compiled approach reading one line time around twenty times faster collecting many unnecessary lines readlines looping even sapply difference llvm native approaches may inherent also possibly due different optimization techniques may able enhance llvm short outperform native vectorized code compiling relatively straightforward code exposition example may make seem complicated essentially want efficiently read one line file time order get next line sample compiled fgets function compiled two functions perform loops number lines important implications example sidestep memory management get control computations using dynamically generated routines use existing native routines data structures fgets file code compiled could already dynamically call native routines directly using example rdyncall adler rffi temple lang important also compiling iterations temple lang fusing loops consider computing given vector observations density function say dnorm write efficiently sum log dnorm sigma indeed could reduce sum dnorm sigma log true purpose example consider general sequence calls vectorized functions functions dnorm log sum implemented code two use efficient mechanism result code seems fast true given way interprets expression one time however two ways make efficient compiling expression first reducing number loops three one generally vectorized functions reduce loops one second typically eliminate least one allocation potentially large vector third way might speed computations use parallel capabilities gpu multiple cores cpus discuss conceptually quite straightforward generally compiling code dynamically indeed ability programmatically combine particular function general parallel strategy makes expedient writing evaluate expression uses three separate loops ignoring pedantic details essentially evaluates call dnorm loops elements computes density values stores values newly allocated vector returns becomes input call log iterates elements vector computes log individual value case may recognize need create new vector return results reuse input vector since essentially anonymous final step overall expression call sum iterates elements vector receives returns single scalar value importantly three loops three vectors length allocate one new large vector could use single loop avoid allocating intermediate vector rewriting code normalloglik function sigma ans val ans ans log dnorm val sigma ans instead vectorized calls put scalar function calls inside single loop combined calls dnorm log together took result element added cumulative sum combination operations called loop fusion large vectors yield significant performance improvements new scalar version faster avoiding loop allocation course evaluated much slower could write would specific normal density generally would write implementations various sequences calls example different density functions sum log pdf expressions involving functions prod dchisq practical however given ability dynamically generate native code compile expression original expression sum log dnorm sigma native equivalent scalar code compile normalloglik function need able call scalar versions log dnorm routines log function available ubiquitous math library libm refer normal density function standard arrange native code invoke dnorm function scalar value vector awkward inefficient instead write version dnorm directly would slow invoke many times compile dnorm normalloglik functions together single module fast another possible approach case take advantage good design modularity rmath library provides routine regular native routine unconnected data types etc invoke log routine reasons quite clear present machine version takes enhancing advanced compilation tools methods table times relative performance fusing loops linux linux fused loops interpreted vectorized functions regular time first two rows show times fusing loops compiling llvm using sequence calls vectorized functions final row shows ratio two times within machine fusing loops slower faster linux longer code million observations longer million observations suspect able improve llvm code exploring optimization facilities however linux machines see speedup even million observations llvm code runs time code timings relative performances given table regardless exact numbers results indicate compiling code competitive manually writing vectorized routines outperform routines difference two approaches uses mechanism rather standard function call via function however reduce three loops one also avoid dealing missing values nas additional parameters base log function even better access multiple cores gpus may able execute code much efficiently simply via parallel execution fusing loops operations together also avoid three separate transitions host gpu transferring memory two systems times need explicitly wrote normalloglik function show fuse loops could also written original expression sum log dnorm sigma reduce map log map dnorm moreargs list sigma explicitly using functional programming concepts easy see fuse loops rewrite code loop rllvmcompile package recognize expression compile instructions either require programmers order gain performance native code try make compiler recognize vectorized nested function call idiom form computing distances observations distances pairs observations important common statistical techniques clustering scaling support vector machines many methods use kernel trick smola provides dist function allows compute distance pairs observations matrix data frame using six different metrics core computations implemented fast however issues rigidities dist function insists data passed code represented matrix make copy data data frame given caller large data sets significant issue essentially two copies data memory also dist function accepts single data set computes distances pairs observations within contrast reasonably common situation start two separate data want compute distance observation observation distances pairs observations within within risk three copies data memory two separate data frames two combined one data frame converted matrix dist function also perform many unnecessary computations observations discard two data sets observations respectively dist function computes distances temple lang interested diverge number unnecessary computations increases especially burdensome number variables observation large another rigidity choice distance metric fixed wanted introduce new distance metric would useful able reuse code underlying dist could function pointer code dist would need modified support accordingly want introduce new metric copy entire code code underlying dist use parallel capabilities openmp detected compiled use gpus change parallel strategy within session without rewriting code result would like able express computations select different strategy parallelizing computations short useful dist would like much flexible want able compute distances two sets observations within single data set use data frame matrix perhaps data representation without making copy data introduce new metrics within infrastructure use different parallel computing approaches current dist function help meet goals essentially code package pdist wong provides way compute pairwise distances two data sets avoids redundant computations unfortunately supports euclidean metric also insists matrices passed code also support parallel computing could write basics dist function make fast could address enhancements listed well make code comprehensible accessible users basic approach computing distance pair observations two data sets expressed following quite function written aid compiling dist function nrow nrow ncol ans numeric ctr total posx posy total total posx posy posx posx posy posy ans ctr sqrt total ctr ctr ans basic steps loop observation first data set loop observation data set pair observations compute distance via third nested loop could made simpler general using vectorized expression calling function final loop however inlined computations directly reason suppose written part computation unfortunately would cause create two new intermediate vectors one specific rows two data sets row data set simple vector containing elements interest pass subtraction function instead arrange data row matrix data frame new vector contiguous values convenient inefficient happen code builtin dist routine uses matrices knows access elements individually rather creating new temporary vector use approach loop also could allocate vectors row values reuse observation still populate different observation avoid intermediate vectors code explicitly accesses individual elements directly matrix merely vector elements matrix arranged sequentially column order therefore first element observation position vector enhancing advanced compilation tools methods table timings computing distances code dist function calling native code speedup factor linux linux shows total elapsed time distance computations variables observations two data sets approach dist function extra memory allocation also distances computed discarded outperform native implementation platforms second element ith observation position nrow compute distance two observations loop variables present observations compute difference code illustrates computations euclidean distance could easily change implement distance metrics could changing code either manually programmatically replacing expression posx posy example abs posx posy rewriting code programmatically powerful feature allows treat code template compile nested loop code via rllvmcompile native instructions compiler currently works primarily primitive data types limited support working directly objects example knowing dimensions matrix accordingly arrange pass matrices dimensions routine currently explicitly specify signature distc compilefunction dist realsxptype list doubleptrtype doubleptrtype future allow caller specify two data sets however making representation matrices explicit valuable information compiler native code compare using code computes distances combining two data sets calls dist converts result interest comparison favors code since form inputs expected form output however quite reasonable timed functions compute distances two data sets size observations variables case distances computed using dist function irrelevant discarded table shows results illustrates fewer computations indeed outperform native code platforms used data sets similar numbers observations results would less dramatic however observations llvm native code still three times faster linux slower comparing results similar native code pdist package timings show native code pdist outperforms llvm compiled code faster one machine times faster another illustrates room significant improvement llvm compilation however fact outperform native approach encouraging readily adapt different purposes different computational strategies indicates significant opportunities potential final note could remove third loop insert call function compute distance two variables example euclidean compiler could recognize matrices arrange compiled version euclidean function access elements displayed without computing intermediate vector row tell compiler data frames would generate different code access elements avoid intermediate vectors since compiler temple lang opportunity compile code main loop metric function together knows representations inputs create better code wrote separately rigidly possible compilation enhancement strategies examples previous section explored different ways could change way compute new facilities generating native code considered compiling code native routines reusing existing native routines within generated routines changing computational strategies employ within embrace new approaches many simple examples could consider improve performance code one ability write functions focus scalar operations create vectorized versions automatically given scalar function write vectorized version sapply mapply compiler turn native loop indeed many performance gains achieved making looping faster also potentially reduce necessity use vectorized code hopefully make programming intuitive new users addition handling loops several aspects evaluation model might able improve choosing different compilation strategies different contexts idea user compiling code may information computations data representation available computing resources compiler examining code extra information important programmer may able give hints compiler choose different compiler altogether control code understood native instructions generated following reasonably obvious general improvements might able infer make certain situations guided user different compilers may yield different code even different semantics code omitting checks values many routines loop elements vector must check element see missing value code general purpose code test fixed part almost every computation involving routine however dynamically generate code may know missing values data set run code omit code perform additional redundant tests similarly example sampling csv file knew number lines file knew call fgets routine would succeed result check return status call reading end file also assumed largest line less characters validate iteration applies accessing elements vector whether first need check index within extent array bounds checking verify conceptually within loop vector declaration user omit checks tests typically simple computationally expensive however become significant instructions invoked often example loop elements large vector memory allocation example discussing loop fusion saw could reduce number overall iterations computation also could reduce memory usage avoided creating vector result call dnorm log potential opportunities reduce memory usage uses concept calls functions theory makes copy argument call function lazy evaluation means arguments never evaluated copied however interpreter smarter copies object modified part another object compiling code want able determine object modified avoid copying analyzing code detect whether parameters considered reduce memory consumption cases verify safe avoid copying object identify within regular code however would modify interpreter make use information generating native code enhancing advanced compilation tools methods example compilefunction make use information dynamically bypassing interpreter another example reduce memory footprint code reuse memory different computation example consider simple bootstrap computation something like following data sample replace true ans computational model allocate new data frame bootstrap sample unnecessary reuse memory sample sample structure differs values cell analyzing sequence commands rather executing one separately without knowledege others take advantage opportunity reuse memory also reuse vector store result repeated calls sample reasonably clear would wrote code reusing data structure instances however possible within individual computations connected code dynamically generate native code utilize information similarly scripts create large object perform several computations move tasks code analysis allow identify object longer used insert calls remove object however may able recognize object longer needed subsequent tasks reuse data format representation case reuse memory least parts data representations small number fundamental data types makes computational reasoning quite simple use implement course choice data type structure important many computations sequences example seq along common code represented explicit vectors containing values sequence seen avoid creating sequence vector populating used loop counter similarly represent regular sequence start end stride increment elements generating code access elements sequence using appropriate calculations specialized sequence type many cases simple data types cause use integer need byte even bits represent possible snpstats clayton package successfully using bytes reduce memory footprint large genomic data operations subset data different format need modified default excessively slow however generate native code free use different ways access individual elements idea important specify code precisely generating code combine code information represent data generate different code strategies realizations somewhat similar template functions dynamic due runtime contextual information several aspects code compile example matching named arguments compile time rather run time overview generating code llvm section briefly describe basic ideas generate code llvm rllvm rllvmcompile little technical low level examples readers need understand material understand main ideas paper use compiler compiled code describing illustrate programmers readily experiment tools generate code different ways use fibonacci sequence fib function example illustrates different aspects generating code fib function expects integer value returns integer body consists single expression contains condition test two blocks code one evaluated depending outcome condition map code llvm concepts need create temple lang moduleid fib define fib entry icmp slt label label ret preds entry preds entry sub call fib sub call fib fib fib add ret fib fib fig intermediate representation compiled fib routine create function block objects create insert llvm instruction objects corresponding expressions function result description intermediate form llvm optimize compile native code different targets example cpu gpu different blocks label correspond different parts statement possibly parts loop generally different instruction blocks contains one instructions call routine evaluation starts first instruction block executes instructions sequentially end instruction block terminator identifies next block jump returns routine jumping blocks allows implement conditional branching loops etc fib function start entry block might create local variables computations function block simply contains code evaluate condition depending value test instruction branch one two blocks corresponding expressions else parts block less add single instruction return value variable block corresponding else part add several lowlevel instructions start computing call fib value store result local variable calculate call fib store result add two local intermediate results store result finally return result figure shows code called intermediate representation form llvm uses illustrates lowlevel computations code function reasonably simple many details involved generating native code defining routine parameters creating instruction blocks loading storing values creating instructions perform subtraction call fib function return value llvm api application programming interface provides numerous classes methods allow create instances conceptual items functions block many different types instructions rllvm package provides interface classes methods allows create manipulate objects directly within example following code shows define function entry instruction block generate call fib mod module function fib list mod start block irbuilder start parms getparameters binop sub parms createconstant createcall want write code manually although rllvm enables instead want programmatically transform code fib function create llvm objects rllvmcompile package since functions regular objects query manipulate directly traverse expressions body function analyze one perform translation concepts llvm concepts basic way compilefunction generates code using customizable handler functions different types expressions recognize calls functions accessing global variables arithmetic operations statements loops use functions rllvm create corresponding llvm objects instructions compiler finished defining instructions routine llvm description want form blocks instructions description intermediate representation look code look something similar enhancing advanced compilation tools methods shown figure code shows somewhat details blocks instructions described see three blocks labeled entry important understand details able use compiled routine translated function show illustrate different steps compilation process indicate programmer chose change steps next instruct llvm verify optimize code point call new routine via function rllvm corresponds method llvm api first time code used llvm generates native code form enhancements patches enthusiasm therefore approach tends work one person limited resources jit compiler extension associated jit package another approach using jit compilation focuses compiling loops arithmetic expressions loops like compiler requires almost change existing code call function jit evaluating code performance gain problems apparently high factor see http unfortunately longer maintained cran primary central repository packages approach suffers fact requires modified version contrasts related research terpreter compiled source several projects exploring places burden author continto improve performance code discuss ually update changes also requires users trust take time build relesome section one visible projects vant binary installations mentioned previously important motivatthe compiler developed luke tierney ing goals work avoid modifying tierney consists package provides compiler support core self allow people build adapt interpreter execute resulting tools directly leverage ongoing work code compiler maps code instructions domain experts compiler technology intein spirit llvm instructions grating tools perform compilation termediate representation instructions approach differs llvm specific respects java virtual machine several typical speedup provided projects working developing implementation factor much larger using java programming language virspeedups problems may tual machine one fastr https ficient obviate need writing code developed collabc probably need see factor oration researchers purdue oracle inria another renjin https closer common tasks compiler written run java virtual others adapt extend however machine offers several benefits many intails resulting generates teresting projects implemented java evaluated tightly embedded example apache hadoop mahout inteplementation engine means one grating code projects funcwants change interpreter one tionality would much tighter effective modify interpreter one platform share private version one make computational engine importantly would benethese changes available others without also fit passively acquiring features java compiling modified version words libraries example security threads one interpreter extensible interesting development researchers oratime regular users furthermore core cle collaborating developers fastr development team always greet suggested creating tool graal http temple lang compiling code ordinarily interpreted virtual machine example fastr could yield performance gains seek passively leveraging general work others merely using widespread technologies contrasts ongoing development relatively small community actively manually import new technologies features ideas languages systems communities translating code another approach translate code code attractive would give similar speedup get llvm potentially produces code allows leverage standard tools languages compilers linkers importantly debuggers also potentially reuse generated code outside simon urbanek package urbanek example exploring translation code code rcpp eddelbuettel inline package widely used improve performance packages provide way include code within code compile call within code code uses syntax make relatively easier write code valuable addition obtain performant code approaches compile code directly preferable get performance first reason programmer program also harder programmers read code understand second reason code essentially opaque code analysis compiler manage generally compile code effectively implement automated parallel computing strategy code code easily part example map code run gpu kernel many cores easily combine code take advantage cores parallel speedup several interesting projects aimed improving performance exclusively running code parallel important sense orthogonal compilation code speed computations single cpu speedup benefit running code cpu however also want compile code take advantage multiple hope able tegrate ideas projects compilation strategies unfortunately longer active projects example taskpr illustrates one aspect observed community researchers implement ideas sometimes phd thesis move projects one terrific aspects ongoing commitment support community probably significant reason widespread use important consideration developing new environments languages one forces motivating continued work within even developing new system would intellectually stimulating important aspect work recognize many positive ways make faster efficient one approaches may dominate others future important pursue comparative approaches continue motivate work much learned different approaches improve others future work compiling subsets code domain specific languages dsls within using llvm promising approach certainly worth vigorously pursuing near term work currently started summer recently returned almost hiatus due projects people however foundations many important components place rllvm package basics extensible adaptable compiler mechanism rllvmcompile allow others make relatively quick progress programming almost entirely develop compilation strategies however many tasks make transparent reliable many related projects make powerful convenient one immediate tasks undertake program rich examples explicitly code implementing code versions recursive partitioning trees random forests boosting also plan explore compiling code expectation maximization algorithm particle filters run gpus aim share sample projects researchers investigating different compilation strategies compare approaches substantive real tasks want program pure code enhancing advanced compilation tools methods plan add functionality available llvm yet bindings rllvm includes topics different optimization passes adding meta data instructions also developed initial infrastructure compile code kernel routines used gpus ptx parallel thread execution code able generate kernel functions code along existing bindings manage memory launch kernels host device allows program gpus directly within code contrasts code developed existing packages target gpus example gputools buckner rgpu kempenaar dijkstra also exploring different approaches compiling code run parallel distributed settings think able use information distribution data code important minimize movement data keep busy actual computations rather transferring inputs outputs computations able write code directly calls routines powerful saw relation fgets routine section need specify signature routines want call preferable able programmatically identify signatures rather require programmers explicitly specify rcindex package interface libclang carruth parsing facilities clang compiler already allows read code identify different elements contains allows determine signatures routines also discover different data structures enumerated constants etc also understand routines manipulate arguments whether perform memory management leave caller saw examples information types parameter local variable necessity able compile using llvm currently programmer must specify information function compiling also functions calls want make transparent least require programmer specify information ambiguity end working type inference package starts known set fundamental functions signatures determine signatures many higher level calls always deal many features language nonstandard evaluation likely get much type information need programmatically since types flexible different return types based types inputs also content inputs need flexible way specify types perhaps existing typeinfo package temple lang gentleman package help analyze code type information variable dependencies build upon codedepends temple lang peng nolan codetools tierney packages related activities envisage working also encourage others collaborate work independently using llvm optionally rllvm rllvmcompile community ends better tools conclusion described one approach making parts language fast leverage compiler toolkit infrastructure llvm generate native code allows incorporate technical knowledge another community future generate code cpus gpus targets dynamically specialize functions different computational approaches data representations sources contextual knowledge giving new flexible approach thinking computing developing simple extensible customizable compiler translate code native code make code run fast also allows compute quite different ways interpret code usual way even outperform native code way work considered general compiler language many aspects language yet dealt considered vectorized subsetting recycling lazy evaluation nonstandard evaluation examples others add facilities compiler support make sense feasible initial results simple approach encouraging important implication temple lang efforts make code efficient benefit writing code describes compute use smart interpreters compilers generate efficient code simultaneously freeing programmers concentrate tasks leveraging domain expertise executing code hope others able use basic building blocks improve matters also explore quite different approaches new languages within environment acknowledgments vincent buffalo made valuable contributions designing developing rllvmcompile package initial work vincent carey provided important ideas insights advice motivation grateful organizing collection papers session joint statistical meetings also appreciate useful comments initial draft paper three reviewers also john chambers supplementary material code examples paper along timing results available https rllvmtimings git repository versions rllvm rllvmcompile packages involved timings also retrieved respective git repositories specific code used associated git tag statscipaper references adler rdyncall improved foreign function interface ffi dynamic bindings libraries package version buckner wilson seligman athey watson meng gputools package enables gpu computing bioinformatics available http carruth christopher gregor korobeynikov kremenek mccall rosier smith libclang translation unit parser library available http clayton snpstats snpmatrix xsnpmatrix classes methods package version eddelbuettel rcpp seamless integration journal statistical software ihaka temple lang back future lisp base statistical computing system proceedings computational statistics springer heidelberg jones oliphant peterson scipy open source scientific tools python kempenaar dijkstra using graphics processing unit speedup bioinformatics analysis available https lattner adve llvm compilation framework lifelong program analysis transformation proc international symposium code generation optimization cgo san jose usa ieee computer society washington available http core team language environment statistical computing foundation statistical computing vienna austria smola learning kernels mit press cambridge temple lang rcurl general network client interface package version temple lang rcindex interface clang parser api available https temple lang rllvm interface lowlevel virtual machine api available https temple lang rffi interface libffi dynamically invoke arbitrary compiled routines without compiled bindings package version temple lang fastcsvsample package sample lines text file available https temple lang buffalo rllvmcompile simple compiler code available https temple lang gentleman typeinfo optional type specification prototype package version temple lang peng nolan codedepends analysis code reproducible research code comprehension available https tierney compiling preliminary report proceedings international workshop distributed statistical computing march hornik leisch technische wien vienna austria tierney codetools code analysis tools package version urbanek compiler available http wong pdist partitioned distance function package version zakai emscripten compiler available https
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international journal computer applications volume november effective evolutionary clustering algorithm hepatitis case study marghny rasha abd ahmed taloba computer science department faculty computer information assiut university egypt computer science department faculty science assiut university egypt computer science department faculty computer information assiut university egypt abstract clustering analysis plays important role scientific research commercial application algorithm widely used partition method clustering however known algorithm may get stuck suboptimal solutions depending choice initial cluster centers article propose technique handle large scale data select initial clustering center purposefully using genetic algorithms gas reduce sensitivity isolated point avoid dissevering big cluster overcome deflexion data degree caused disproportion data partitioning owing adoption applied method public datasets show advantages proposed approach example hepatitis dataset taken machine learning warehouse university california aim evaluate hepatitis dataset order evaluate dataset preprocessing operation reason preprocessing summarize data best suitable way algorithm missing values instances adjusted using local mean method general terms data mining keywords genetic algorithms clustering algorithm squarederror criterion virus hcv introduction data mining supported host models capture character data several different ways clustering one models clustering find groups different whose members similar clustering process group unlabeled patterns partitioned number sets similar patterns assigned cluster dissimilar patterns assigned different clusters two goals clustering algorithms determining good clusters efficiently popular category clustering algorithms dealing kcenter clustering problems clustering conventional algorithm classifies data minimizing mse objective function however clustering procedure assumes data must presented known number clusters turn allows simple implementation clustering however classical suffers several flaws first algorithm sensitive choice initial cluster centers second deal massive data nowadays people affected hepatitis virus found world liver swelling redness without indicating particular reason referred hepatitis currently estimated approximately million people worldwide roughly present global population infected hcv global burden disease attributable chronic liver diseases substantial paper present improved genetic algorithm igk handle large scale data select initial clustering center purposefully using genetic algorithms gas gas based clustering technique provides optimal clustering respect clustering metric considered genetic algorithms gas procedure used find approximate solutions search problems application principles evolutionary biology genetic algorithms use biologically inspired techniques genetic inheritance natural selection mutation sexual reproduction recombination crossover excellent survey gas along programming structure used found genetic algorithms applied many classification performance tuning applications domain knowledge discovery databases kdd clustering cluster analysis important technology data mining effective method analyzing discovering useful information numerous data cluster algorithm groups data classes clusters objects within cluster high similarity comparison one another dissimilar objects clusters dissimilarities assessed based attribute values describing objects often distance measures used branch statistics example unsupervised learning clustering provides exact subtle analysis tool mathematic view international journal computer applications volume november algorithm far popular clustering tool used scientific industrial applications proceeds follows first randomly selects objects initially represents cluster mean center remaining objects object assigned cluster similar based distance object cluster mean computes new mean cluster process iterates criterion function converges typically criterion used defined section briefly describe original algorithm popularclustering algorithmbut suffers drawbacks shown steps algorithm follows algorithm original input number clusters dataset containing objects output set clusters minimize criterion original clustering sum objects database number clusters number objects cluster point space representing given object mean cluster adopting criterion works well clusters compact clouds rather well separated one another difficulty detecting natural clusters clusters widely different sizes densities shapes attempting minimize criterion divide objects one cluster two clusters addition applying criterion evaluate clustering results optimal cluster corresponds extremum since objective function many local minimal values results initialization exactly near local minimal point algorithm terminate local optimum random selecting initial cluster center easy get local optimum entire optimal overcoming criterion hard distinguish big difference among clusters one technique developed based representative technique besides various approaches solving problem performance algorithm heavily depends initial starting conditions simplest one repetition different random selections algorithms also employ simulation anneal technique avoid getting local optimal algorithm improved kmeans algorithm proposed solve clustering problem algorithms used gas solve clustering problem literature bradley fayyad present iterative refinement approach sampling dataset many times clustering twice get optimal initial values cluster center idea multiple drawn dataset clustered independently solutions clustered respectively refined initial center chosen solution minimal distortion solutions aiming dependency initial conditions limitation algorithm applies criterion measure quality clustering paper presents new improved genetic algorithm based effective techniques search optimal initial values cluster centers experimental results demonstrate new algorithm obtain better stability excel original clustering results begin initialize prototypes chooses objects initial centers repeat begin compute center cluster min end assign object cluster based mean calculate mean value objectsfor cluster error function end algorithm algorithm input number clusters dataset containing objects output set clusters minimize begin initialize population compute fitness termination criterion achieved select crossover mutate output best stop end international journal computer applications volume november criterion improved algorithm algorithm improved input number clusters dataset containing objects output set clusters minimize criterion begin multiple produce clusters groups compute choose min refined initial points chosen initial producing repeat combining two near clusters one cluster recalculate new center generated two centers merged number clusters reduces end proposed algorithm original algorithm choose points initial clustering centers different points may obtain different solutions order diminish sensitivity initial point choice employ mediod centrally located object cluster obtain better initial centers demand stochastic sampling naturally bias sample nearly represent original dataset say samples drawn dataset cause distortion reflect original data distribution example original data set depicted left fig sampling results shown right fig producing group mediods respectively finally comparing solutions choosing one group minimal value function refined initial points avoid dividing one big cluster two ones adopting criterion assume number clustering depends balance clustering quality time general bigger expand searching area solution space reduce situation initial values near extremum subsequently reclustering dataset genetic chosen initial conditions would produce mediods merging clusters nearest clusters number clusters reduced mean steps proposed algorithm summarized follows algorithm improved genetic input number clusters dataset containing objects output set clusters minimize criterion begin multiple genetic genetic kmeans produce clusters groups compute choose min refined initial points genetic genetic chosen initial producing repeat combining two near clusters one cluster recalculate new center generated two centers merged number clusters reduces end improved algorithm works small samples compared whole dataset needs significantly less iteration experimental results fig dataset comparing two solutions generated clustering sample drawn original dataset using respectively location clustering centroids two almost similar method applicable refine initial conditions order lessen influence sample choosing initial starting points following procedures employed first drawing multiple say original dataset size capability memory sum size subsamples close possible size original dataset second use genetic run experiments three data sets vowel data data consists indian telugu vowel sounds uttered context three male speakers age group years value therefore chosen data iris data data represents different categories irises four feature values four feature values represent sepal length sepal width petal length petal width centimeters three classes overlap classes samples per class value therefore chosen data crude oil data overlapping data data points features classes hence value chosen data set also run experiments three synthetic datasets denoted shown fig international journal computer applications volume november genetic paramet ers used experim ental average population size selection roulette introduced section crossover single point crossover demonstrated section probability crossover mutation occasional random alteration character clustering algorithm probability mutation initial improved clustering igk results implementation algorithm gaclustering algorithm improved algorithm improved genetic algorithm shown respectively table vowel dataset table iris dataset table crude oil dataset table dataset table dataset table dataset algorithms run iterations purpose demonstration five different initial configurations algorithm improved kmeans algorithm five different initial populations gaclustering algorithm improved genetic algorithm shown tables initial clustering improved table show calculated error algorithm algorithm improved algorithm improved genetic algorithm obviously obtained improved genetic algorithm data set vowel iris crude oil better calculated vowel iris crude oil algorithm vowel iris crude oil improved algorithm vowel iris crude oil table iterations improved igk clustering initial igk average table iterations average clustering improved igk average initial fig datasets left center right table vowel iterations table iris iterations initial average improved clustering table iterations igk initial improved igk clustering table crude oil iterations international journal computer applications volume november average table show calculated error algorithm algorithm improved algorithm improved genetic algorithm obviously obtained improved genetic algorithm data set better calculated algorithm improved algorithm conclusion along fast development database network data scale clustering tasks involved becomes large algorithm popular partition algorithm cluster analysis limitations restrictions computing resources time especially huge size dataset genetic algorithm used search cluster centers minimize clustering error improved genetic algorithm presented paper solution handle large scale data select initial clustering center purposefully using gas reduce sensitivity isolated point avoid dissevering big cluster overcome deflexion data degree caused disproportion data partitioning owing adoption multisampling also run experiments dataset hepatitis dataset contains fields one output field output shows whether patients hepatitis alive dead purpose dataset forecast presence absence hepatitis virus given results various medical tests carried patient database holds attributes hepatitis dataset contains samples belonging two different target classes value therefore chosen data features binary attributes discrete values references proceeding model fitting applied datareduction techniques order trim dimensionsbecause data problem dimensionality applied principal component analysis nineteenindependent variables realized first sixprincipal components cover filho treleaven andalippic genetic algorithm programmingenvironments ieee comput initial improved clustering yasin jilanit danish classification using data mining techniques international journal computer jilanit yasin yasin classification patients international journal computer wang encyclopedia data warehousing mining idea group publishing maulik bandyopadhyay genetic pattern recognition igk average totalvariability continuous data space therefore acquired six independent variables afterapplying data reduction approaches six autonomousvariables age bilirubin alk phosphate serum glutamicoxaloacetic transaminase sgot albumin prothrombintime protime results implementation hepatitis dataset shown table table hepatitis iterations proposed approach perform clustering time series data using gas proposed approach takes advantage global search capacity gas uses local fine tuning anderberg cluster analysis application academic press new york hartigan clustering algorithms wiley new york devijver kittler pattern recognition statistical approach london jain dubes algorithms clustering data prenticehall englewood cliffs tou gonzalez pattern recognition principles addison wesley reading zhang mao xiong efficient clustering conference machine learning cybernetics han kamber data mining concepts techniques morgan kaufmann forgy cluster analysis multivariate data efficiency interpretability classifications biometrics mcqueen methods classification analysis multivariate observations computer chemistry tang yang wang employing genetic algorithm toimprove algorithm international journal computer applications volume november clustering application probability cheng wang ning wang design realization using representation point valid clustering algorithm patten recognition artificial intelligence vol duda hart patten classification scene york john wiley sons selim alsultan simulated annealing algorithm theclustering problem patten recognition krishna narasimha genetic algorithm systems ieee transactions man cybernetics part murthy chowdhury search optimal clusters usinggenetic lett tang yang wang improve algorithm cluster method mathematical statistics applied probability wang clustering based genetic engineering optimal clustering algorithm based onhybrid genetic technique journal east china university scienceand technology natural science edition chittu sumathi modified genetic algorithm initializing clustering global journal computer science technology kumar chhabra kumar initializing cluster center using biogeography based optimization advances computing communication control reddy mishra jana cluster initialization advances computing communication control min andsiqing improved clustering based genetic algorithm ieee computer application system modeling zhang wang kmeans clustering algorithm combined genetic algorithm ieee digital content multimedia technology applications fayyad reina bradley initialization iterative refinement clustering algorithms proceedings international conference knowledge discovery data mining pal majumder fuzzy sets decision making approaches vowel speaker recognition ieee trans systems mancybernet smc fisher use multiple measurements taxonomic eugenics bandyopadhyay maulik evolutionary technique based algorithm optimal clustering information sciences johnson wichern applied multivariate statisticalanalysis englewood cliffs yang optimization study value system engineering theory application virmajoki pairwise nearest neighbor method revisited phd thesis university ofjoensuu joensuu finland
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stochastic interchange scheduling electricity market jan yuting tongxin zheng senior member ieee lang tong fellow ieee problem interchange scheduling presence stochastic generation load considered new interchange scheduling technique based stochastic minimization overall expected operating cost proposed directly solving stochastic optimization intractable equivalent problem maximizes expected social welfare formulated proposed technique leverages operator capability forecasting locational marginal prices lmps obtains optimal interchange schedule without iterations among operators index interchange scheduling multiarea economic dispatch seams issue ntroduction since restructuring electric power industry independent system operators isos regional transmission organizations rtos faced seams issue characterized inefficient transfer power neighboring regions inefficiency caused incompatible market designs independently controlled operating regions inconsistencies scheduling protocols different pricing models economic loss due seams new york new england customers estimated level million annually recent effort addressing seams issue optimizing interchange flows across different regions particular new interchange scheduling technique referred tie optimization proposed minimize overall operating cost federal energy regulatory commission ferc recently approved coordinated transaction scheduling cts allows market participants participation implementations various versions cts carried several system operators one main challenges eliminating seams inherent delay interchange scheduling actual power delivery across regions caused lack information necessary scheduling operation constraints example information used cts interchange scheduling minutes prior actual power delivery increasing integration renewables interchange scheduling needs cognizant uncertainty arises time interchange scheduling power transfer work supported part doe certs program national science foundation grant part work appeared tong school electrical computer engineering cornell university ithaca usa ltong zheng iso new england holyoke usa tzheng goal paper obtain optimal interchange schedule presence system operation uncertainty end propose stochastic optimization formulation aimed minimizing expected overall system cost proposed optimization framework takes account random fluctuations load renewable generations systems directly solving stochastic optimization intractable paper presents approach transfer stochastic optimization problem equivalent deterministic problem maximizes expected economic surplus transformation allows generalize deterministic solution intersecting expected demand supply functions therefore avoiding costly iterative computation operators related work extensive studies seams issue paper consider inefficiencies arise market designs focus instead optimizing interchange schedule highlight approaches relevant technique developed broadly related work see references therein mathematically optimal interchange scheduling obtained optimal power flow opf problem decentralized optimization power flow solved using various decomposition techniques general approach based principle lagrangian relaxation decomposes original problem smaller subproblems earliest approaches include pioneer work kim baldick conejo aguado predate broad deregulation electricity market opf problems explicitly involve multiple isos widely studied general decentralized opf based techniques typically require iterations isos one control center uses intermediate solutions solves dispatch problem although convergence techniques often guaranteed formulation number iterations large practical cost communications computations substantial note recent marginal decomposition technique shown converge finite unknown number iterations growth renewable integration brought new attention uncertainty seams stochastic optimization robust optimization approaches considered recently particular conejo cherkaoui formulate stochastic market clearing model energy reserve dispatch problem solution stochastic optimization obtained based scenario enumerations requires prohibitively high computation effort scheduling considered unit commitment problem wind generation uncertainty specifically adaptive robust optimization problem formulated goal minimizing cost wind scenario solved generation algorithm present paper complements existing results focusing interchange scheduling develop tractable stochastic optimization technique pragmatic approach seams problem one adopted practice incorporate external market participants use proxy buses representing interface neighboring regions technique presented falls category among existing prior work work chen coordinated interchange scheduling scheme proposed energy ancillary services technique based augmented involving iterations among neighboring control centers work closest technique presented work ilic lang underlying principle based economics argument supply demand functions exchanged neighboring operators scalar net interchange functions succinctly characterized exchange needs made need iterations among control centers eliminated approach also based economics argument innovation incorporating system operation uncertainty note type approaches solve opf problem except special case single tie line connecting two operating regions eterministic nterchange cheduling proxy bus representation practice coordination neighboring control regions markets typically use proxy bus mechanism pointed proxy bus models location marginal changes generation assumed occur response changes transactions proxy bus mechanism utilized existing lmp based markets representing valuing interchange power paper consider power system consisting two independently operated subsystems illustrated figure operator selects proxy bus represent location import export neighboring region specifically shown figure operator region assumes withdrawal proxy bus operator region assumes injection quantity proxy bus interchange scheduling determine value region region figure system interface region region figure single proxy bus representation net minimizes overall operating cost subject generation transmission constraints note except single tie line connects two regions proxy representation approximation optimal interchange scheduling based proxy representation provide optimal interchange original system general optimal interchange via proxy representation strictly suboptimal compared opf solutions optimal interchange scheduling interchange scheduling problem proxy bus model formulated minimizing generation costs regions respect power balance transmission internal interface generator constraints simplicity make following assumptions throughout paper system lossless cost function quadratic form matrix positive definite single proxy bus system net interchange modeled explicitly additional scalar variable optimization problem follows min subject net interchange two neighboring regions total amount power flowing one operating region another generation offer function region vector forecasted load renewable generation region net interchange region region region region otherwise vector dispatches region vector transmission limits region interface limit generator constraints region sij shift factor matrix buses region transmission lines region sqi shift factor vector buses region interface shadow price power balance constraint region shadow prices transmission constraints region shadow price net interchange constraint problem centralized formulation determining optimal interchange region optimization problem requires coordinator full access related information regions unsuitable present deregulated electricity markets centralized problem written hierarchical form decentralized optimization follows min subject optimal dispatch region given interchange level regional dispatch problem region specified min subject region min subject note optimization problem involves outer problem optimize interchange level inner problem naturally decomposed two regional problems parameterized words optimizer associated lagrangian multipliers functions tie optimization key idea determine interchange schedule intersecting demand supply curves interchange curve mean incremental cost regional dispatch interface essentially lmp proxy bus given interchange level lmp proxy bus region defined qto figure illustration note function iso incremental dispatch cost interface net interchange serves supply curve exporting iso demand curve importing iso use graphical representation illustrate basic principle shown figure represents generation supply curve region drawn descending cost order example direction interface region region serve supply demand curve respectively optimal schedule qto set intersection two curves note quantity exceeds interface capacity schedule set maximum capacity instead interface transmission constraint case becomes binding price separation happens markets also noted import export transactions settled lmp calculated proxy bus delivery according interchange schedule optimal solution well intuitive argument manifestation deeper connection social welfare optimization illustrated figure cost minimization defined follows exploit connection presence uncertainty iii tochastic nterchange cheduling sqi lagrangian multipliers asand sociated optimal solution far described interchange scheduling deterministic system setting focus incorporation random load generation scheduling scheme stochastic programming formulation stochastic optimization common framework model optimization problems involving uncertainty consider case load stochastic generation treated negative load random interchange scheduling formulated stochastic optimization problem first stage involves optimizing net interchange minimize expected overall cost min subject second stage solves regional optimal dispatch problem given interchange level realization direction interface flow determined comparing prices power flows region otherwise direction interface flow opposite interface constraint simply intersection expected demand supply curves general interchange maximizes expected social welfare given consumer surplus producer surplus qsto figure illustration sto random load specified note optimal dispatch associated lagrangian multipliers parameterized two factors interchange level load realization lmp proxy bus function directly solving problem requires distribution regional cost function interchange level coordinator determine optimal schedule neither achievable present deregulated electricity markets general stochastic optimization problem intractable especially load renewable generation forecast follows continuous distribution proposed scheduling technique hand solve problem without increasing computation complexity deterministic details provided next two subsections social welfare optimization main idea solving exploit connection cost minimization social welfare optimization uncertainty randomness present second stage obvious optimization problem transformed corresponding form social welfare optimization turns optimal solution obtained solving deterministic problem using expected demand supply functions present optimization problem importexport perspective taking account import export regions must agree forward interchange quantity presence future demand supply uncertainty interchange quantity fixed ahead actual power delivery region may rely internal resources compensate uncertainty real time end reasonable export region maximize expected producer surplus import region maximize expected consumer surplus without loss generality let region exporter fixed interchange let random lmp proxy bus function interchange expected supply curve averaged internal randomness similarly expected demand curve averaged internal randomness region time delivery shown figure optimal interchange quantity qsto maximizes expected social welfare absence max subject solve operator needs compute interchange quantity expected lmp proxy bus computation requires conditional expectation future lmp time delivery conditional expectation obtained probabilistic lmp forecast using models load generation see example also conceivable conditional expectations also approximated via regression analysis expected demand supply functions obtained solving optimal interchange quantity becomes search stochastic tie optimization section establish formally equivalence solution solves stochastic optimization problem theorem optimal dual solutions unique equivalent sense optimizer theorem provides new way call stochastic tie optimization sto solve intractable problem result significant optimal interchange obtained deterministic optimization problem requires information expected supply demand curves since price functions nonconfidential information solved one operators operator shares price curve way operators need iteratively update exchange information within scheduling procedure property contrast decomposition methods subproblems resolved intermediate results exchanged iteration information exchange sufficient optimal schedule operators need repeatedly solve regional opf computationally expensive sufficiently large systems within scheduling procedure property significantly reduces computation costs real time thereby providing potential higher scheduling frequency provide proof theorem proof theorem first show differentiability objective functions follows immediately well known results multiparametric quadratic programming summarized lemma lemma dual problem degenerate optimizer associated vector lagrangian multipliers continuous piecewise affine affine critical region optimal objective continuous convex piecewise quadratic quadratic critical region lemma objective function denoted differentiable derivative second equality holds envelope theorem lemma also implies continuous functions objective function denoted differentiable derivative derive connection optimal solutions first order conditions optimal solution associated lagrangian multiplier satisfy first order condition similarly optimal solution associated lagrangian multiplier satisfy first order condition exactly finally show prove need monotonicity price function fixed defined summarized following lemma whose proof provided appendix lemma dual problem unique optimal solution monotonically increasing monotonically decreasing show following cases either case impossible statement trivially true case interface constraint binding either problem implies lemma preservation monotonicity expectation operation unique solution therefore construct solution using associated optimal functions defined zero note solution satisfies first order conditions optimal however contradicts uniqueness optimizer therefore case impossible case also impossible proof follows logic case sum equivalent sense share optimal solution qscts figure illustration scts tochastic cts section incorporate external market participants sto generalizes cts proposal currently implementation generalization call stochastic coordinated transaction scheduling scts simply replacing supply demand curves used cts expected values cts market participants allowed submit requests buy sell power simultaneously side interface request called interface bid includes price indicating minimum expected price difference two regions participant willing accept transaction quantity direction use similar graphical representation sto illustrate scheduling procedure scts shown figure expected supply curve region adjusted curve subtracting aggregated interface bids scts schedule set intersection interface bids left qscts accepted settled lmp difference scheduling clearing procedure described summarized follows share expected lmp functions determine direction interchange flow comparing construct aggregated interface bid curve stack interface bids direction determined step increasing order submitted price difference calculate optimal scts schedule following optimization problem max subject note difference sto scts inclusion interface bids components identical implies information exchange sufficient iteration operators necessary scheduling procedure one operator submits expected generation supply curve executes scheduling clearing procedure region table comparison sto region scenario valuation section compare performance proposed sto two systems system ieee system particular focus two common symptoms seams interface transmission presence counterintuitive flows high cost region low cost region examples uses certainty equivalent forecast stochastic generation mean value sto uses probabilistic forecast distribution various scenarios studied two examples example system consider system depicted figure generator incremental cost functions capacity limits load levels default values presented figure lines identical except maximum capacities tie lines line line internal transmission lines region maximum capacities internal lines region maximum capacities system randomness comes wind generator bus region entire network model shift factor matrix assumed known isos default chose bus proxy bus represent network region bus represent network region impact location proxy buses investigated baseline first tested baseline probabilistic wind forecast distribution two levels load chosen illustrate two symptoms inefficiency schedule first load level example counter intuitive flow occurrence second load level shows case interface utilization results presented figure table figure shows generation supply curves region region sto two examples respectively incremental cost region deliver power proxy bus using forecasted mean wind production expected incremental cost using forecast distribution since randomness region supply curves region sto examples interchange level sto schedule expected overall system cost minimized cases shown figure expected prices two proxy buses converge cases shown figure price proxy bus figure system interchange interchange figure generation supply curves table expected price difference level schedule first example means expected price importing region region lower exporting region implies interchange scheduled high cost region low cost region counter intuitive hand second example expected price difference interchange level schedule marginal price importing region higher exporting region price difference increasing interchange level reduce expected overall cost implies interchange capacity utilized interchange level sto schedule optimal design schedule optimal level cause counter intuitive flow schedule less lead interface utilization impact forecast uncertainty impact forecast uncertainty level investigated varying standard deviation probabilistic wind production forecast loads set default values given figure results presented figure interchange level schedule change since uses mean value wind production forecast sto hand captures uncertainty level probabilistic forecast adjusts interchange schedule accordingly expected overall cost increases expected overall cost price proxy bus cost expected overall cost method sto sto interchange interchange figure expected overall cost marked blue square sto red circle interchange scenario interchange figure expected price difference marked blue square sto red circle sto forecast uncertainty level interchange interchange forecast uncertainty level example system divided standard ieee bus two regions region includes bus region bus generator incremental cost functions capacity limits load levels default values given matpower imposed maximum capacity line interface transmission limited default impact interface constraint studied bus selected proxy buses represent adjacent region network introduce randomness system assumed three wind generators located bus produce power according discrete distribution specifically denote wind production probabilistic forecast consists probability mass function two levels wind considered three scenarios high wind scenario medium wind scenario low wind scenario uses mean value respective scenario baseline case verified optimality sto schedule presence discrete randomness three wind scenarios tested results shown figure table since generation cost functions quadratic price functions continuous piecewise affine performances sto schedule similar system example expected overall cost minimized sto schedule three cases indicated red bus branch indices referred interchange expected overall cost figure high wind scenario expected overall cost forecast uncertainty observed sto uncertainty schedules sto costs generation supply curves figure impact forecast uncertainty cost circles figure prices converge schedule sto shown table schedules counter intuitive flows observed high wind medium wind scenario interface utilization happens low wind scenario impact interface congestion investigate impact interface congestion tested three wind scenarios setting except interface capacity set case results shown table iii presence interface constraint influences performances high wind scenario price separation happens sto binding interface constraint prevents economic interface flow price proxy bus sto sto interchange generation supply curves interchange expected overall cost figure medium wind scenario price proxy bus interchange expected overall cost price proxy bus method sto sto sto expected overall cost table comparison sto expected overall cost expected overall cost expected price difference expected price difference interchange generation supply curves interchange expected overall cost figure low wind scenario table iii impact interface congestion scenario method sto sto sto cost lemma affine critical region derivative exists addition quadratic implies second derivative derivative respect positive therefore monotonically increasing monotonically decreasing within critical region lemma indicates continuous monotonicity preserved eferences impact proxy bus location finally tested impact proxy bus location medium wind scenario tie line bus internal bus wind bus selected proxy buses results presented table impact proxy bus location internal bus tie line bus proxy bus sto cost sto table observe interchange schedule associated expected cost sensitive location proxy bus sto different selections proxy bus direction interchange schedule different example although several considerations guide choice proxy bus location theoretical results show universal selection rule onclusion paper presents stochastic interchange scheduling technique incorporates load renewable generation uncertainties using forecast expected lmp proxy bus proposed approach obtains optimal interchange schedule deterministic optimization problem maximizes expected economic surplus essence technique providing way reduce stochastic optimization problem deterministic optimization problem decision addition proposed technique require iteration operators scheduling procedure information exchange sufficient optimal scheduling ppendix proof lemma denote lagrangian function lemma convex quadratic critical region derivative exists envelope theorem tong stochastic coordinated transaction scheduling proc ieee pes general meeting iso new england new york iso interchange scheduling iris analysis options online available http white ferc approves coordinated transaction scheduling new york iso iso new england http iso nyiso ferc approves coordinated transaction scheduling pjm nyiso http conejo castillo minguez decomposition techniques mathematical programming engineering science applications springer science business media zhao litvinov zheng marginal equivalent decomposition method application optimal power flow problems ieee transactions power systems vol baldick chatterjee coordinated dispatch regional transmission organizations theory example computers operations research vol shahidehpour zhang adaptive robust tieline scheduling considering wind power uncertainty interconnected power systems ieee transactions power systems chen thorp mount coordinated interchange scheduling opportunity cost payment market proposal seams issues proc annual hawaii international conference system sciences ilic lang methods selecting desired net interchange dni across areas demonstration seams solution npcc online available http sessiona kim baldick distributed optimal power flow ieee transactions power systems vol conejo aguado coordinated decentralized optimal power flow ieee transactions power systems vol cadwalader harvey pope hogan market coordination transmission loading relief across multiple regions cambridge center business government harvard university conejo cherkaoui energy reserve dispatch wind uncertainty equipment failures ieee transactions power systems vol harvey proxy buses seams markets draft http tong thomas probabilistic forecast lmp network congestion arxiv preprint borrelli bemporad morari predictive control linear hybrid systems online available http zimmerman matpower matlab power system simulation package online available http
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parametric closure problem david eppstein sep computer science department univ california irvine usa abstract define parametric closure problem input partially ordered set whose elements linearly varying weights goal compute sequence downsets partial order weights vary give polynomial time solutions many important special cases problem including semiorders reachability orders graphs partial orders bounded width partial orders result orders provides significant generalization previous result carlson eppstein bicriterion subtree problems introduction parametric optimization problems variation classical combinatorial optimization problems shortest paths minimum spanning trees input weights fixed numbers vary functions parameter different parameter settings give different weights different optimal solutions goal list solutions intervals parameter values within optimal simple example consider maintaining minimum input values change parameter controlling values changes parametric minimum problem formalized problem constructing lower envelope functions map parameter value input value linear functions problem constructing lower envelope lines equivalent projective duality planar convex hull lower envelope linear complexity constructed time log well obvious applications parametric optimization problems predictable data route planning parametric optimization problems another class applications bicriterion optimization bicriterion problems input element two numbers associated solution value obtained summing first number selected element separately summing second number selected element evaluating nonlinear combination two sums instance element two numbers might interpreted coordinates point plane associated element might wish find solution whose summed coordinates give point close origin possible two numbers might represent mean variance normal distribution might wish optimize function summed distribution two numbers element might represent initial investment cost expected profit business opportunity might wish find feasible combination opportunities maximizes return investment two numbers might represent cost failure communications link might wish find communications network high probability success many natural bicriterion optimization problems type expressed finding maximum quasiconvex function two sums function whose lower level sets convex sets equivalently finding minimum quasiconcave function two sums case optimal solution always obtained one solutions parametric problem one variable defined two numbers associated input element slope linear function gives weight element single number function parameter way algorithm solving parametric optimization problem also used solve bicriterion versions type optimization problem even though bicriterion problem might combine two numbers nonlinear way corresponding parametric problem uses linearly varying edge weights paper formulate provide first study parametric closure problem natural parametric variant classical optimization problem maximum closure problem closures parametric closures closure directed graph subset vertices edges vertex subset another vertex subset maximum closure problem problem finding closure graph equivalently seek downset weighted partial order downset subset elements partial order order belongs subset also belongs subset partial order converted equivalent directed graph considering element order vertex edge whenever order note reversed edge direction usual conventions direction given directed graph one may obtain equivalent partial order strongly connected components graph component less component partial order whenever path vertex vertex graph one classical applications problem involves open pit mining vertices directed graph represent blocks ore covering material must removed reach ore edges represent ordering constraints removal blocks weight vertex represents net profit loss made removing block applications problem include military attack planning freight depot placement scheduling precedence constraints image segmentation stable marriage maximum satisfaction treemap construction information visualization maximum closures found polynomial time reduction maximum flow direct algorithms parametric closure problem assign weights vertices directed graph elements partial order vary linearly functions parameter seek closures downsets maximum weight possible value parameter figure instance open pit mining problem profit loss block ore likely vary function current price refined commodity produced ore parametric version open pit mining problem determine range optimal mining strategies depending future commodity prices vary described algorithm parametric closures also solve bicriterion closure problems maximizing quasiconvex function minimizing quasiconcave function two sums values although able resolve complexity parametric closure problem general case prove polynomial complexity several important special cases problem related work previous work parametric optimization considered parametric versions two standard network optimization problems minimum spanning tree problem shortest path problem parametric minimum spanning tree problem linear edge weights polynomially many solutions constructed polynomial time contrast parametric shortest path problem polynomial least output must represented explicit list paths number solutions running time exponential graphs know previous work general parametric closure problem two previous papers seen retrospect solving special cases abcd abd fig partially ordered set weights vary linearly function parameter different choices lead different downsets lawler studied scheduling minimize weighted completion time precedence constraints sought closure maximizes ratio priority processing time job set jobs used closure decompose instances problem smaller subproblems instead using reduction bicriterion parametric problems lawler showed optimal closure found polynomial time binary search step involves solution weighted closure problem replacing binary search megiddo parametric search would make algorithm strongly polynomial search methods depend specific properties ratio function however extended bicriterion problems carlson eppstein consider bicriterion versions problem finding best subtree containing root given rooted tree weighted edges show many problems solved time log although carlson eppstein formulate problem closure problem subtrees seen closures directed version tree edge directed towards root reachability ordering directed tree example partial order greatly generalize results carlson eppstein new results parametric closures arbitrary partial orders parametric optimization implicit convex hull problem parametric optimization problems formulated dually problems computing convex hulls implicitly defined point sets figure suppose given parametric optimization problem weight element linear function parameter abcd abc abcd abd abc abd fig instance parametric closure problem left hasse diagram partially ordered set four elements weight varies linearly parameter center distributive lattice downsets right point set project convex hull upper hull dashed gives order sequence six distinct closures parameter varies continuously weight candidate solution subset elements constrained specific optimization problem question sum functions solution value also linear function whose coefficients sums element coefficients instead interpreting numbers coefficients linear functions may two numbers coordinates respectively points euclidean plane way family candidate solutions determines planar point set set corresponds point given sum elements coefficients call point set project sets thought vertices hypercube whose dimension number input elements project determines linear projection vertices euclidean plane let hull project denote convex hull projected planar point set parameter value set minimizing maximizing parameterized weight corresponds projective duality vertex hull true maximizer quasiconvex function two sums coefficients thus parametric optimization reformulated problem constructing convex hull bicriterion optimization solved choosing best hull vertex new results arbitrary partially ordered set define family downsets let parametrically weighted project defined convenient abbreviation define polygon hull project convex polygon represents solution parametric closure problem given weights consider following classes partially ordered sets partial order one classes prove polynomial bounds complexity polygon time constructing polygon results imply time bounds parametric optimization maximizing quasiconvex function bicriterion optimization semiorders class partial orders introduced model human preferences element associated numerical value pairs elements whose values within fixed margin error incomparable pairs ordered numerical values equivalently interval orders intervals unit length proper interval orders interval orders interval contains another orderings give section bound log complexity polygon show constructed time using algorithm based quadtree data structure partial orders orders formed recursively smaller orders type two operations series compositions elements one order placed earlier combined ordering elements order parallel compositions pairs one element ordering incomparable orderings applied instance scheduling applications lawler orderings sets form polygon corresponding recursive construction two operations convex hull union two convex polygons minkowski sum two convex polygons follows polygon complexity construction immediately lead fast construction algorithm section adapt splay tree data structure construct polygon time log previous results optimal subtrees follow special case result bounded treewidth suppose partial order elements transitive reduction covering graph forms directed acyclic graph whose undirected version treewidth prior work treewidth partial orders see prior work parametric optimization graphs bounded treewidth see show section polygon polynomially many vertices exponent constructed polynomial time incidence posets incidence poset graph vertices edges elements order relation whenever endpoint one initial applications closure problem concerned design freight delivery systems certain profit could expected set routes system cost setting depots endpoint routes modeled incidence poset graph vertex depot location edge potential route since profits costs different timeframes reasonable combine nonlinear way giving bicriterion closure problem transitive reduction incidence poset subdivision treewidth technique partial orders bounded treewidth also applies incidence posets graphs bounded treewidth fences generalized fences polytrees fences zigzag posets partial orders whose transitive reduction path alternating edge orientations generalized fence may either oriented path poset oriented tree polytree polynomial bounds complexity construction time polygon classes partial orders follow result treewidth however cases simplify construction provide tighter bounds quantities theorem theorem bounded width width partial order maximum number elements antichain set elements partial orders arise instance edit histories version control repositories treewidth partial order less twice partial orders width downsets tighter bound would obtained using treewidth use quadtrees show strongly section obtain width partial order width consider linear extension partial order partition partial order downset upset position linear extension fig two convex polygons left convex hull union center minkowski sum right case polygon log vertices constructed time within logarithmic factor bound unable obtain example family partial orders nonlinear lower bound complexity polygon able obtain nontrivial upper bound hull complexity unrestricted partial orders additionally unable obtain polynomial bounds hull complexity types partial orders one parameter weights dimension higher two also know computational complexity bounds parametric closure problem class partial orders finite dimension leave problems open future research preliminaries minkowski sums hulls unions results complexity convex polygons polygon associated partial order hinge decomposing polygons recursively combinations simpler polygons use two natural geometric operations combine pairs convex polygons produce complex convex polygons definition define convex polygon convex hull nonempty finite set points euclidean plane vertex polygon point obtained intersection polygon closed halfplane edge polygon line segment obtained intersection polygon closed halfplane definition requires include two degenerate special cases consider single point degenerate convex polygon one vertex edges consider line segment degenerate convex polygon two vertices one edge definition two convex polygons let denote minkowski sum set points vector sum point point let denote convex hull union figure lemma folklore convex polygons vertices respectively vertices constructed time form sequence sets union maxima downset minima complementary upset proof complexity bound follows fact set edge orientations union sets edge orientations therefore many edges sum numbers edges result follows fact convex polygon numbers vertices edges equal except degenerate special cases one vertex edges compute may merge two lists edge orientations two polygons linear time use sorted list trace boundary combined polygon similarly complexity bound follows fact vertex set subset union vertices vertices therefore number vertices sum numbers vertices compute linear time convenient partition convex hull lower hull upper hull two monotone polygonal chains splitting leftmost rightmost vertex hull may sort vertices upper hulls single merge linear time apply graham scan sorted list lower hulls corollary suppose convex polygon described formula combines set points plane single polygon using sequence operations suppose addition written expression tree formula height vertices may constructed formula time complex data structures reduce time log see section higher dimensions convex hull points minkowski sum line segments polynomial complexity exponent depends linearly dimension however know analogous bound complexity convex sets formed mixing minkowski sum operations bound held could extend results parametric closures corresponding higher dimensional problems semiorders semiorder type partial order defined luce model human preferences element order associated numerical value application preference modeling utility element person whose preferences modeled pairs items whose utilities sufficiently far ordering two items semiorder numerical ordering utilities however items whose utilities within global margin error incomparable semiorder formally definition let collection items given numerical utilities together global threshold information determines partial order whenever partial orders obtained way called semiorders call margin error semiorder note authors use irreflexive binary relations instead partial orders define semiorders distinction important similar concepts comparisons numerical values margins error give rise semiorders many areas science statistics efficient computations semiorders assume utility values element part input algorithm margin error normalized one instance semiorder figure represented semiorder utilities respectively information hand comparison two elements determined constant time ordering given numerical utility values constructed time concept downset particularly natural semiorder set elements whose utility values could lie sharp numerical threshold perturbing utility value half margin error way closure problem problem finding maximum weight downset alternatively interpreted problem finding maximum possible discrepancy weighted point set location point known imprecisely thus problem related several recent works geometric computations imprecise points see semiorders may exponentially many downsets instance items utilities within one unit sets downsets nevertheless show section semiorder complexity polygon number downsets optimal parameter value log mapping downsets grid parametrically weighted semiorder may write sorted order utility values elements may write elements order scaling utility values may assume without loss generality threshold used define semiorder values set padding items fixed zero weight utility smaller elements margin error may additionally assume without loss generality power two without changing values parametric closure problem convenient parameterize downsets pairs integers follows definition let arbitrary downset let largest index element let smallest index element belong element exists define extremes pair integers thus extremes maps family integer grid mapping potentially many downsets may mapped grid point however every grid point image particular point image downset element defining first coordinate extremes must index smaller element defining second coordinate additionally point utility values beyond margin error semiorder also image downset case semiorder every downset includes also includes thus image extremes lies orthogonally convex subset grid bounded main diagonal monotone curve figure definition let square subset integer grid define subproblem subset semiorder consisting elements whose indices among rows columns define free unordered set elements belong subproblem whose indices pairs indices belong subproblem define set elements whose indices smaller elements subproblem see figure left example decomposition grid squares definitions allow decompose downsets mapped extremes given square beyond margin error fre indices order fig grid two regions part image extremes left image shows square subproblem free right image shows quadtree decomposition grid used prove theorem lemma given square suppose subfamily mapped extremes nonempty set disjoint union three sets nonempty downset subproblem set arbitrary subset free proof downset must remain downset subset particular intersection subproblem must also downset additionally possible set omit member include element outside subproblem free would mapped extremes similarly condition row index largest element subproblem must met extremes would map given set outside therefore every set form described conversely let set formed disjoint union nonempty downset subproblem arbitrary subset free set necessarily downset although might mapped extremes depending choice downset subproblem disjoint union decompose convex hull projections downsets contribution subproblem another contribution free set contribution free set simple structure based minkowski sums lemma square let powerset free family subsets free let weight function define projection project families sets point sets project powerset free minkowski sum sets free convex hull centrally symmetric convex polygon hull project powerset free minkowski sum corresponding line segments free sides fewer line segments collinear constructed time log proof fact project powerset free minkowski sum line segments proved induction number members free particular element subfamilies downsets include exclude project translates minkowski sum induction hypothesis convex hull union two translates thing minkowski sum one line segment bounds number sides construction time follow immediately corollary using fact minkowski sum segments expressed tree binary minkowski sums logarithmic height log time bound lemma reduced algorithm described using balanced tree binary minkowski sums interpreted performing merge sort slopes line segments whose minkowski sum desired hull slopes segments already sorted minkowski sum could constructed linear time case possible preprocess input contiguous subsequence elements sort corresponding line segments slopes quickly using integer sorting algorithms however omit speedup adds complication overall algorithm without improving total time proof used lemma also allows quickly compute polygon subproblem square entirely within margin error using fact family downsets subproblem power set lemma let square grid defined given semiorder entirely within margin error semiorder side length polygon subproblem polygon sides computed time log computing polygon subproblem remaining squares trickier may performed decomposing polygon polygons type four smaller squares lemma let square grid containing even number grid points subdivide four congruent smaller squares let polygon subproblem vertices define way let side length polygon subproblem constructed corresponding hulls smaller squares time log proof smaller square define polygon polygon subproblem project powerset free subproblem translated adding weights elements subproblem vertex polygon represents downset subproblem lemma viewing subproblem subproblem every downset subproblem mapped extremes included polygon may also include downsets mapped problematic therefore construct polygon subproblem polygon subproblem thus found formula expresses polygon subproblem using constant number minkowski sums unions hulls corresponding hulls smaller squares together free subproblems total size result follows using lemma construct polygons free subproblems corollary combine resulting polygons single polygon semiorder algorithm putting together use observations decompose polygon combination polygons smaller grid squares recursively decomposed even smaller squares leads algorithm performs decomposition uses order construct polygons larger subproblems smaller ones eventually producing solution whole problem theorem semiorder elements specified utility values system weights polygon complexity log constructed time proof sort utility values pad next larger power two necessary form quadtree decomposition grid shown figure right square quadtree associate convex polygon empty set polygon subproblem computed according following cases subset grid points downsets mapped extremes associate square empty set subset grid points every two elements subproblem incomparable case associate square polygon hull project powerset subproblem computed according lemma otherwise split four smaller squares construct polygon associated using lemma combine polygons associated children follows induction total complexity polygon constructed square quadtree total time constructing log side length ith square quadtree sum ranges descendants log contribution time bound includes terms form log appearing lemma lemma also amount time spent combining polygon square polygons log higher levels recursive subdivision base case induction square containing single grid point associated subproblem one element one downset maps grid point degenerate convex polygon single vertex polygon constructed root quadtree desired output follows combinatorial complexity time complexity form sum ranging quadtree squares conditions define two monotone curves grid split quadtree square crossed one two curves follows squares side length subdivided part algorithm form two monotone chains number squares side length results theorem follow summing contributions polygon complexity time complexity log different possible values partial orders partial orders considered context scheduling problem lawler include special case tree orderings previously studied work bicriterion optimization recursive decomposition although possible characterize partial orders orders four elements form figure convenient define terms natural recursive decomposition take advantage algorithms definition partial orders partial orders constructed partial orders repeatedly applying following two operations series composition parallel composition fig partial order redrawn wikipedia illustration author series composition given two partial orders form order disjoint union every element less every element parallel composition given two partial orders form order disjoint union order relations see figure example composition operations correspond naturally two geometric operations convex polygons already using observation series composition polygon convex hull union polygon translate sum weights elements polygon parallel composition polygon minkowski sum polygon polygon recursively continuing decomposition gives formula polygon terms operations corollary immediately obtain corollary partial order elements polygon vertices however depth formula polygon may linear using corollary construct polygon could inefficient describe faster algorithm key idea follow formula build polygon represent intermediate result convex polygon data structure allows operations performed quickly pairs polygons unbalanced sizes data structure fast unbalanced polygon merges note minkowski sum operation polygon high complexity polygon bounded complexity change constant fraction vertex coordinates allow fast minkowski sums representation store coordinates explicitly lemma possible store convex polygons data structure destructively merging representations two polygons vertices respectively operation performed time log proof store lower upper hulls separately binary search tree data structure node represents vertex polygon inorder traversal tree gives order vertices node root tree stores cartesian coordinates vertex node stores vector difference coordinates parents coordinates additionally node stores vector difference clockwise neighbor around polygon boundary way traverse path tree adding stored vector difference determine coordinates vertex encountered along path may also perform rotation tree update stored vector differences constant time per rotation keep tree balanced amortized sense using splay tree balancing strategy whenever follow search path tree immediately perform splay operation sequence double rotations moves endpoint path root tree dynamic finger property splay trees sequence accesses sequential order splay tree size take time log compute hull union operation insert vertex smaller polygon number vertices order larger polygon insert vertex search larger polygon find edges use edges check whether belongs lower hull upper hull neither belongs one two hulls search larger polygon find two neighbors hull performing splay neighbors rotated root binary tree cutting tree points may remove vertices new neighbors tree without consider vertices create new node add two neighbors left right child compute minkowski sum operation must simply merge two sequences edges two polygons slopes search edge slope smaller polygon position found splay vertex node split position root tree split tree left right subtrees copy root node translate vertices one side split vector difference inserted edge adding vector root tree rejoin trees proof dynamic finger property splay trees complicated lemma would follow using dynamic binary search data structure finger property also allows additional operation removing large consecutive sequences elements time proportional logarithm length removed sequence instance trees finger property proven time bound deletions structure form need additional analysis would needed make applicable problem algorithm data structure hand ready prove main result section fig fence left generalized fence center polytree right theorem partial order represented decomposition tree polygon complexity may constructed time log proof follow formula constructing polygon operations using data structure lemma charge merge operation partial order elements smaller side merge partial order element belongs subproblems sizes nqwhere height element time charged log log log bounded section consider partial orders whose underlying graphs bounded treewidth underlying graph undirected graph obtained forgetting edge directions covering graph partial order partial order defined reachability directed acyclic graphs underlying graph reachability relation obtained forgetting edge directions transitive reduction minimal directed acyclic graph reachability relation given one fences polytrees first consider case underlying undirected graph treewidth one tree path bound number solutions parametric closure problem time listing solutions using approach later generalize larger values treewidth definition fence zigzag poset partially ordered set whose transitive reduction path alternating edge orientations polytree directed graph formed orienting edges undirected tree oriented path poset directed graph formed orienting edges undirected simple path generalized fence partially ordered set whose transitive reduction oriented path see figure figure depicts example fence definition generalized fence follows munarini distinguished authors define generalized fences reachability orders polytrees theorem let generalized fence polygon complexity constructed time log proof let denote maximum number vertices polygon ranges generalized fences bound let generalized fence maximum complexity let middle element oriented path defined let generalized fence determined half path left removing elements half path let generalized fence determined left removing elements define way right half path downsets contain correspond downsets correspondence obtained adding back elements downsets contain exactly downsets therefore formula polygon polygon polygon polygon polygon suitable translation applied performing operation since four elements use formula lemma derive recurrence complexity base case recurrence standard form familiar algorithms whose solution using formula polygon together lemma construct polygon gives another recurrence running time log may apply method construct solutions parametric closure problem oriented tree always able obtain perfect split two subproblems size consequence method becomes somewhat complex although asymptotic time bound remain pick splitting vertex use following separator theorem trees appears already work jordan lemma undirected tree vertex removal splits remaining tree components vertices proof let vertex tree chosen minimize number vertices largest component exceeds suppose contradiction zero let component vertices let unique neighbor removing produces components strict subsets together component containing vertices therefore components fewer vertices contradicting choice minimizer contradiction implies must zero therefore removing splits remaining tree components vertices theorem let reachability order polytree polygon complexity constructed time log proof theorem remove component resulting forest define partial orders removing elements respectively theorem use decomposition derive formula polygon polygon polygon polygon polygon lemma follows total number vertices polygon total number subproblems obtained recursively performing decomposition size remaining subtrees goes factor two level recursion levels recursion level recursion also doubles number subproblems remaining tree vertex belongs vertex belongs component remaining tree vertices also included two subproblems therefore contribution one tree vertex total complexity polygon total complexity polygon use lemma directly evaluate formula lemma allows binary combinations however using lemma instead allows formula operations size evaluated time log applying method recursive formula derived gives total time log following sections apply method general graphs bounded treewidth tree decompositions definition undirected graph tree whose vertices called bags associated set vertices following two properties vertex belongs set bags induces connected subtree edge exists bag containing endpoints edge width one less maximum number vertices bag treewidth minimum width extension define treewidth directed graph equal treewidth undirected graph obtained forgetting orientations edges treewidth optimal found amount time linear number vertices given graph exponential width definition width minimal two adjacent bags union elements every graph treewidth minimal decomposition width given decomposition minimal two adjacent bags small union edge connecting two bags could contracted bags merged decreasing number bags decomposition without increasing width decomposition property allows control number bags decomposition lemma minimal width graph bags proof consider bags order given traversal starting bag cardinality first bag vertices successive bag traversal must least one new vertex seen previous bags otherwise would subset bag parent traversal decomposition would minimal therefore number vertices least equal number bags plus equivalently number bags partial orders low treewidth lemma let partial order elements whose underlying graph treewidth polygon represented formula operations total size depth log proof fix minimal width use lemma recursively partition decomposition repeatedly removing bag whose removal splits remaining part decomposition connected components size number bags half size previous level partition elements partitions two subsets downset complementary upset use partitions specify elements part eventual downset part eventual downset let one connected components formed removal one partitions downset upset downset whole problem consistent partition elements least one element must also included downset elements least one element must also included upper set define cij subproblem closure problem consisting remaining elements ones belong bags neither member member notation express polygon formula polygon polygon cij term outer translated appropriately according weights additional elements belong downsets term subproblems included term levels recursive partition causes number subproblems vertex participates increase factor therefore whole recursive partition total number subproblems single vertex participates since vertices partition gives rise formula total complexity applying lemma formula lemma gives following result theorem let partial order elements whose underlying graph treewidth polygon complexity constructed time log note larger linear factor bounds theorem theorem know whether dependence exponent complexity bound necessary unless improved acts obstacle existence algorithms parameterized treewidth construction polygon incidence posets partial orders defined reachability graphs low treewidth include particular incidence posets undirected graphs bounded treewidth incidence poset undirected graph reachability poset graph obtained subdividing undirected edge orienting two resulting new edges outward subdivision point figure replacement increase treewidth following folklore lemma shows fig undirected graph left directed graph obtained replacing undirected edge two oppositelyordered directed edges center partial order simultaneously incidence poset undirected graph reachability poset directed graph right lemma let obtained replacing one edges paths treewidth higher proof may assume obtained replacing single edge path instance lemma may obtained multiple replacements type tree also tree treewidth otherwise treewidth least optimal must contain least one bag contains endpoints width may obtained attaching another bag contains three vertices endpoints subdivision point replacement path one important applications closure problem transportation planning uses incidence posets graphs whose edges represent collection truck routes methods applied parametric bicriterion versions problem graph small treewidth following result immediate corollary theorem corollary let incidence poset graph treewidth polygon complexity constructed time log bounded width width partially ordered set size largest antichain partially ordered sets bounded width arise instance version histories distributed version control repository controlled small set developers assumption developer maintains single branch version history application may many elements partially ordered set versions repository width may bounded number developers downset application set versions could possibly describe simultaneous states developers past moment history repository downsets partially ordered set correspond independent sets downset uniquely determined independent set set maximal elements downset therefore partially ordered set width elements downsets precisely dilworth theorem every partial order width partitioned subsets chains number downsets product numbers downsets chains product takes maximum value chains equal size show however even tighter bound complexity polygon fig hasse diagram partial order partitioned two chains left subset integer lattice formed downsets right consider first case general case bounded width follow similar reasoning thus let partially ordered set elements width two dilworth theorem decomposed two chains every downset described pair integer coordinates number elements belong first chain number elements second chain observation partial order described set pairs coordinates describing downsets forms orthogonally convex subset integer grid edges points one unit apart set grid points form covering graph distributive lattice downsets figure depicts example use following definition observation partition parametric closure problem smaller subproblems type definition let grid rectangle subproblem family downsets whose grid points lie observation every grid rectangle set subproblem represented uniquely disjoint union set elements whose single coordinate value left downset restriction elements whose coordinate value within decomposition set forms subset every set subproblem thus observation shows contribution parametric closure problem rectangle obtained solving smaller parametric closure problem restriction partial order elements within translating results single vector determined fixed set observation suppose every integer point grid rectangle corresponds downset restriction elements whose coordinate value within partial order form parallel composition two chains special case partial order follows corollary hull project subproblem complexity proportional perimeter theorem partial order width two polygon complexity log constructed covering relation time proof covering relation determine partition two chains trace boundaries orthogonal grid polygon describing downsets linear time use quadtree partition downsets squares side length power two grid point within square corresponds one downsets squares size form two monotone chains within grid total perimeter squares log also use projection quadtree partition two grid coordinates recursively subintervals whose sizes powers two subinterval compute convex hull downsets corresponding chain whose grid points lie within hull union two polygons two halves computation takes time log overall also compute sequence vectors sums weights prefix coordinates time compute polygon squares partition downsets observation polygon constructed translating polygon grid coordinates within square set observation translation vector determined smaller grid coordinates set observation observation polygon set computed time proportional perimeter square minkowski sum polygons two defining subintervals translation vector set found sum two vectors computed two corresponding prefixes two grid coordinates overall solution seek polygon computed applying operation combine polygons computed square partition total perimeter squares log generates polygon complexity proportional perimeter time proportional perimeter total time compute polygons log total complexity log convex hull log points time perform final operation combine polygons compute polygon higher widths idea works using octree three dimensions etc total complexity polygon proportional sum side lengths octree squares construction time within logarithmic factor bound conclusions introduced parametric closure problem given polynomial complexity bounds algorithms several important classes partial orders bounds general problem nontrivial lower bounds problem remain open research acknowledgements research supported part nsf grant onr grant references agarwal eppstein guibas henzinger parametric kinetic minimum spanning trees proc ieee symp foundations computer science focs ahmed chowdhury gibson islam sherrette maximum weight objects decomposable based rectilinear convex objects proc int symp algorithms data structures wads springer lect notes comput sci avery algorithmic proof semiorders representable algorithms balinski selection problem manag sci bannister devanny eppstein small superpatterns dominance drawing proc analytic algorithmics combinatorics bodlaender linear time algorithm finding small treewidth siam comput brown tarjan design analysis data structure representing sorted lists siam comput buchin eppstein silveira spatial treemaps proc int symp algorithms data structures wads springer lect notes comput sci carlson eppstein weighted subtree bicriterion subtree problems proc scand worksh algorithm theory swat springer lect notes comput sci carstensen parametric cost shortest path problems unpublished memo bellcore chang edmonds poset scheduling problem order cole dynamic finger conjecture splay trees proof siam comput cole mishra schmidt siegel dynamic finger conjecture splay trees splay sorting log sequences siam comput dilworth decomposition theorem partially ordered sets ann math eppstein geometric lower bounds parametric matroid optimization discrete comput geom faaland kim schmitt new algorithm computing maximal closure graph manag sci slutzki optimal parametric search graphs bounded algorithms slutzki eppstein using sparsification parametric minimum spanning tree problems nordic comput gibson han sonka maximum weight digital regions decomposable digital regions proc int symp algorithms computation isaac springer lect notes comput sci hochbaum algorithm closure graphs networks hochbaum anniversary article selection provisioning shared fixed costs maximum closure implications algorithmic methods today manag sci irving leather gusfield efficient algorithm optimal stable marriage acm jordan sur les assemblages lignes journal die reine und angewandte mathematik http joret micek milans trotter walczak wang dimension unpublished manuscript katoh bicriteria network optimization problems ieice trans fundamentals electronics communications computer sciences lawler sequencing jobs minimize total weighted completion time subject precedence constraints ann discrete math lerchs grossmann optimum design mines trans canad inst mining metallurgy van kreveld largest smallest convex hulls imprecise points algorithmica van kreveld largest bounding box smallest diameter related problems imprecise points comput geom theory appl luce semiorders theory utility discrimination econometrica lower envelopes lectures discrete geometry springer graduate texts mathematics megiddo applying parallel computation algorithms design serial algorithms acm munarini combinatorial interpretation chebyshev polynomials siam discrete math orlin optimal weapons allocation layered defenses nav res logist picard maximal closure graph applications combinatorial problems manag sci pirlot vincke semiorders properties representations applications springer rhys selection problem shared fixed costs network flows manag sci ruskey transposition generation alternating permutations order sleator tarjan binary search trees acm
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polymorphic type inference machine code matthew noonan alexey loginov david cok mar grammatech ithaca usa mnoonan alexey dcok abstract many compiled languages types erased early compilation process result compiler passes may convert source machine code idioms original source optimizations mean type information stripped binary essentially nonexistent problem recovering types performing type inference stripped machine code called type reconstruction offers useful capability support reverse engineering decompilation paper motivate develop novel type system algorithm type inference features type system developed surveying wide collection common idioms building catalog challenging cases type reconstruction found idioms place sophisticated set requirements type system inducing features polymorphic types many features identify often seen expressive powerful type systems used functional languages using features guideline developed retypd novel static algorithm machine code supports recursive types polymorphism subtyping retypd yields accurate inferred types existing algorithms also enabling new capabilities research developed funding defense advanced research projects agency darpa views opinions findings contained material authors interpreted representing official views policies department defense government istribution approved public release distribution unlimited reconstruction pointer const annotations recall retypd operate weaker program representations current state art removing need highquality information may impractical compute categories subject descriptors logics meanings programs semantics programming languages software engineering distribution maintenance enhancement programming languages language constructs features mathematical logic formal languages formal languages keywords reverse engineering type systems polymorphism static analysis binary analysis pushdown automata introduction paper introduce retypd typeinference tool finds regular types using pushdown systems retypd includes several novel features targeted improved types reverse engineering decompilation program analyses features include inference type schemes inference recursive structure types figure sound analysis pointer subtyping tracking customizable information purposes typedef names inference type qualifiers const dependence data accurate recovery types retypd continues tradition secondwrite tie introducing principled static algorithm applicable stripped binaries diverging previous work type reconstruction use rich type system supports polymorphism mutable references recursive types principled phase followed second phase uses heuristics downgrade inferred types types display factoring type inference two phases sequester unsound heuristics quirks type systems sound core engine adds degree freedom design space may leverage relatively complex type system type analysis yet still emit familiar types benefit reverse engineer retypd operates intermediate representation recovered automatically disassembling binary using grammatech static analysis tool binaries generating type constraints abstract interpreter retypd operate uniformly binaries platform supported codesurfer including arm development retypd carried extensive investigation common idioms compiled code create challenges existing methods challenging case identified requirements type system could correctly type idiomatic code results investigation appear type system used retypd specifically designed satisfy requirements common idioms pushed far richer type system first expected including features like recursively constrained type schemes previously applied machinecode type inference due space limitations details proofs algorithms appear appendices available online version paper scripts data sets used evaluation also appear challenges many challenges carrying type inference machine code many common idioms lead sophisticated demands feature set type system section describe several challenges seen development retypd led particular combination features optimizations type erasure since type erasure typically happens early compilation process many compiler optimizations may take machine code produce functionally equivalent illtyped results found three common optimization techniques required special care use variable syntactic constant early returns along error paths stack slots constants suppose function signature void int char invoked null usually compiled machine code similar xor push push call eax eax eax eax null represents optimization since push eax encoded one byte instead five bytes needed push immediate value must careful type variables unified eax void null return null return call test eax eax push eax call add esp ret figure common fortuitous known value used like syntactic constant dynamic value typed fortuitous values related situation appears common pattern represented snippet corresponding machine code figure note procedure exit return value eax may come either return value return value null situation detected see false relationship incompatible return types stack slots function uses two variables size disjoint scopes need allocate two separate stack slots variables often optimizer reuse stack slot variable dropped scope true even new variable different type optimization even applies stack slots used store parameters figure function argument longer needed optimizer overwrite local variable incompatible type generally assume map program variables physical locations even make weaker assumption program variables inhabiting single physical location different times belong single type handle issues combination typesystem features subtyping instead unification program analyses reaching definitions stack variables trace partitioning polymorphic functions discovered although directly supported type system programs define make use functions effectively polymorphic example malloc return value expected immediately cast type call malloc may thought returning pointer different type type malloc effectively void rather problem polymorphic malloc could mitigated treating call site call distinct function mallocp may distinct return type unfortunately sufficient treat handful special functions like malloc way common see binaries include stdlib struct struct next int handle int struct list list next null list list next return close list handle push ebp mov ebp esp sub esp mov edx dword ebp jmp mov edx eax mov eax dword edx test eax eax jnz mov eax dword edx mov dword ebp eax leave jmp close int filedescriptor int successz typedef struct int filedescriptor int successz const figure example code compiled gcc linux flags disassembly type scheme inferred machine code reconstructed type tags filedescriptor successz encode inferred purposes use allocators wrappers malloc functions would also need accurately identified duplicated callsite similar problem exists functions like free polymorphic lone parameter even complex functions like memcpy polymorphic first two parameters return type though three types independent furthermore polymorphic type signatures malloc free void memcpy strictly informative reverse engineer standard signatures else could one know void returned malloc meant opaque handle rather cast pointer type compiled binaries polymorphic functions even common example class member function must potentially accept types foster noted using bounded polymorphic type schemes libc functions increased precision inference level source code advance state art type recovery believe important also embrace polymorphic functions natural common feature machine code significant improvements static type monomorphic require capability infer polymorphic types nontrivial complexity recursive types relevance recursive types decompilation recently discussed schwartz lack recursive type system machine code cited important source imprecision since recursive data structures relatively common desirable typeinference scheme machine code able represent infer recursive types natively offset reinterpreted pointers unlike source code syntactic distinction machine code example type struct char file platform possible infer safely passed fclose conversely passed fclose may need infer points structure offset contains file affects typing local structures well structure stack may manipulated using pointer starting address manipulating members directly frame pointer idioms along casts derived base fall general class physical subtyping retypd model forms subtyping using type scheme specialization additional hints extent local variables found using datadelineation analysis disassembly failures problem producing correct disassembly stripped binaries equivalent halting problem result never assume reconstructed program representation perfectly correct even sound analyses built top unsound program representation may exhibit inconsistencies quirks thus must careful incorrect disassembly analysis results one part binary influence correct type results may gathered rest binary type systems model value assignments type unifications vulnerable issues caused bad since unification bad constraints one part binary degrade type results another instance problem arises use register parameters although cdecl calling convention uses stack parameter passing optimized binaries include many functions pass parameters registers speed often functions conform standard calling convention although work hard ensure true register parameters reported conservativeness demands occasional false positive methods based unification generally sensitive precision loss due register parameters common case push ecx idiom reserves space single local variable stack frame function ecx incorrectly viewed register parameter scheme whatever type variables bound ecx callsite mistakenly unified early experiments found overunifications persistent source imprecision early experiments mitigation heuristics overunification quickly ballooned disproportionately large unprincipled component type analysis designed retypd constraint system avoid need prophylactics overunification table example field labels type capabilities label variance capability function input location function output location readable pointer writable pointer field offset incomplete information degradation accuracy large programs identified source loss systems algorithm provide types even absence information precision improved increasing knowledge via analyses vsa good results already attained analysis beyond simpler problem tracking stack pointer subtyping reducing storage requirements list programs may define type hierarchy via typedefs idiom appears windows api variety handle types defined typedefs void handle types used subtypes handles example gdi handle hgdi generic handle used represent one specific hbrush hpen etc cases typedef may indicate supertype lparam dword although typedefs int intended semantics generic type different contexts may used pointer integer flag set accurately track hierarchies requires type system based around subtyping rather unification models common api type hierarchies useful still better ability end user define adjust initial type hierarchy run time support feature parameterizing main type representation uninterpreted lattice described directly manipulating bit representation another bit twiddling even level source code already many idioms common use idioms operate directly manipulating bit representation value either encode additional information perform computations possible using type usual interface common examples include hashing values treating untyped bit blocks stealing unused bits pointer tag information whether thunk evaluated next prev pointers type inverse square root trick idioms important scheme continues produce useful results even presence apparently contradictory constraints handle situation three ways separating phases constraint entailment solving consistency checking modeling types sketches carry information types using unions combine types otherwise incompatible capabilities type system type system used retypd based around inference recursively constrained type schemes solutions constraint sets modeled sketches sketch associated value consists record capabilities value holds whether stored called accessed certain offset sketches also include markings drawn customizable lattice used propagate information typedef names purposes type inference retypd also supports recursively constrained type schemes abstract set types subject constraint set language type constraints used retypd weak enough constraint set satisfiability subtyping derived type variable formation eft var ight var var efl var nherit var var nherit var var refix var rans var var ield var var ointer figure deduction rules type system represent derived type variables represents label reduced cubic time checking set scalar constraints constants belonging thanks reduction constraint satisfiability scalar constraint checking omit expensive satisfiability checks type inference instead delay check final stage internal types converted types display providing natural place instantiate union types resolve inconsistencies since compiler optimizations idioms original source frequently lead program fragments unsatisfiable type constraints trait particularly desirable syntax constraint type system throughout section fix set type variables alphabet field labels function denoting variance definition label require set finite retypd makes use large set labels simplicity focus table within assume distinguished set type constants type constants symbolic representations elements belonging lattice otherwise uninterpreted usually sufficient think type constants type names semantic tags definition derived type variable expression form definition variance label encodes subtype relationship subtype formalized rules ield figure variance function extended defining hxwi hxi hwi sign monoid word called covariant hwi contravariant hwi definition let set base type variables constraint expression form var existence derived type variable subtype derived type variables constraint set finite collection constraints type variables constraint either type constants members say entails denoted derived constraints using duction rules figure also allow projections given constraint set free variable projection binds internal variable constraint set see figure example treatment constraint projection see field labels used form derived type variables meant represent capabilities type example constraint var means readable pointer derived type variable represents type memory region obtained loading let briefly see operations original program translate type constraints using pseudocode clarity full conversion disassembly type constraints described appendix value copies value moved program variables assignment like make conservative assumption type may upcast supertype generate constraint form loads stores suppose pointer type value loaded assignment generate constraint form similarly store results constraint examples paper omit final access simplify presentation function calls suppose function invoked generate constraints reflecting flow actuals formals note define two constraints equivalent rules figure encodes fact called function type must least specific type used callsite one primary goals engine associate procedure type scheme definition type scheme expression form quantification set type variables constraint set type schemes provide way encoding postconditions function places types calling context without constraint sets would able represent conditions form input must subtype output must supertype constraint set used encode interesting type relations inputs outputs case memcpy example function returns second element struct may type scheme figure two programs mediating copy pair aliased pointers deduction rules deduction rules type system appear figure rules interpretation definition require additional motivation ield rules ensure field labels act type operators generating subtype relations derived type variables subtype relations original variables nherit nherit rule nherit uncontroversial since subtype capabilities supertype rule nherit unusual since moves capabilities direction taken together rules require two types subtype relation must exactly set capabilities form structural typing ensuring comparable types shape structural typing appears odds need cast capable objects less capable ones described indeed nherit eliminates possibility forgetting capabilities value assignments still maintain capability procedure invocations due use polymorphic type schemes explanation instantiation enables forget fields object appears details rules ensure retypd perform iterative variable recovery lack iterative variable recovery cited creators phoenix decompiler common cause incorrect decompilation using tie type recovery ointer rule consistency condition ensuring type loaded pointer supertype type stored pointer without rule pointers would provide channel subverting type system example rule used practice appears deduction rules figure simple enough proof may reduced normal form see theorem encoding normal forms transition sequences modified pushdown system used provide compact representation entailment closure pushdown system modeling queried manipulated provide interesting functionality outline functionality appears modeling pointers model pointers soundly presence subtyping found initial approach suffered unexpected difficulties combined subtyping following type system seemed natural model pointers introducing injective unary type constructor ptr ptr type pointers type system approach works expected presence subtyping new issue arises consider two programs figure since type variables associated seen pointers begin writing ptr ptr first program generate constraint set ptr ptr second generates ptr ptr since program effect copying value constraint sets satisfy pointer subtype constraint must entail constraint one assume ptr covariant ptr ptr entails hand make ptr contravariant seems recourse make subtyping degenerate type equality ptr forced declare ptr ptr course means ptr ptr already catastrophe subtyping used machine code since many natural subtype relations mediated pointers example unary ptr constructor handle simplest kind class subtyping derived class physically extends base class appending new member variables root cause difficulty seems conflating two capabilities pointers ability written ability read retypd two capabilities modeled using different field labels label contravariant label covariant see separation pointer capabilities avoids loss precision suffered ptr revisit two example programs first generates constraint set rule nherit may conclude also field type ointer infer finally since contravariant ield says also putting parts together gives subtype chain second program generates constraint set splitting pointer achieve sound account pointer subtyping degenerate type equality note importance consistency condition ointer rule ensures writing pointer reading result subvert type system need separate handling writecapabilities mutable reference rediscovered multiple times instance covariance array type constructor java cause runtime type errors array mutated languages read capabilities soundly modeled sacrificing soundness write capabilities subtyping nherit noted rule nherit leads system form structural typing two types subtype relation must capabilities superficially seems problematic modeling typecasts forget fields cast derived base derived additional fields missing piece allows effectively forget capabilities instantiation callee type schemes callsite demonstrate polymorphism enables forgetfulness consider example type scheme figure function invoked providing type particular fields required simply select capable type type variable effect used specialization polymorphic types model subtyping idioms subtyping used model structural subtyping idioms restricts introduction subtypes points type scheme instantiated call site sta eax since field conclude field well next ointer requires since covariant implies gives subtype chain int successz int filedescriptor semantics poset sketches simple type system defined deduction rules figure defines syntax legal derivations type system constraint solver designed find simple representation conclusions derived load load figure sketch instantiating type scheme figure set type constraints yet notion type inherent deduction rules figure defined rules game equipment played found introducing entities level constraints types resulted much loss precision working challenging examples described consequently developed notion sketch kind regular tree labeled elements auxiliary lattice sketches related recursive types studied amadio cardelli kozen depend priori knowledge ranked alphabet type constructors definition sketch possibly infinite tree edges labeled elements nodes marked elements lattice finitely many subtrees labeled isomorphism collapsing isomorphic subtrees represent sketches deterministic finite state automata state labeled element set sketches admits lattice structure operations described figure lattice sketches serves model interpret type constraints interpretation constraint var sketch admits path root label sequence interpreted sketch obtained traversing label sequence subsketch lattice order sketch obtained traversing sequence main utility sketches nearly free tree model constraint language constraint set satisfiable lattice sketches long prove impossible subtype relation auxiliary lattice particular always solve fragment reference constants stated operationally always recover tree structure sketches potentially solve observation formalized following theorem theorem suppose constraint set variables exist sketches var proof idea symmetrize using algorithm similar spirit steensgaard method pointer analysis begin forming graph one node derived type variable appearing along prefixes add labeled edge derived type variable form graph quotient equivalence relation defined whenever edges either construction exists path label sequence starting equivalence class var regular set paths yields tree structure working lattice elements label trickier problem basic idea use automaton constructed constraint simplification theorem answer queries type constants upper lower bounds given derived type variable full algorithm listed retypd use large auxiliary lattice containing hundreds elements includes collection standard type names common typedefs popular apis semantic classes filedescriptor figure lattice helps model subtyping preserve semantic type names discussed note sketches one many possible models deduction rules could proposed general approach fix poset types interpret interpret field labels monotone resp antimonotone functions separation syntax semantics allows simple way parameterize engine model types choosing model symmetric relation type system similar secondwrite generated hand forming lattice type intervals interval inclusion would obtain type system similar tie outputs upper lower bounds type variable analysis framework reconstruction retypd built top grammatech tool codesurfer binaries codesurfer carries common program analyses binaries multiple cpu architectures including arm codesurfer used recover raw machine code type constraints generated directly resolved types applied back become visible gui later analysis phases codesurfer achieves platform independence tsl language defining processor concrete semantics terms concrete numeric types mapping types model flag register memory banks interpreters given abstract domain automatically created concrete semantics simply specifying abstract domain interpretation concrete numeric mapping types retypd uses codesurfer recovered determine number location inputs outputs procedure well program call graph graphs abstract interpreter generates sets type constraints concrete tsl instruction semantics detailed account abstract semantics constraint generation appears appendix approach type resolution initial recovered type inference proceeds two stages first sets generated fashion components callgraph type schemes externally linked functions may inserted stage constraint set simplified eliminating type variables belong scc interface simplification algorithm outlined type schemes available callgraph traversed assigning sketches type variables outlined stage type schemes specialized based calling contexts function appendix lists full algorithms constraint simplification solving translation types final phase type resolution converts inferred sketches types presentation user since types sketches directly comparable resolution phase necessarily involves application heuristic conversion policies restricting heuristic policies single phase provides flexibility generate types maintaining soundness generality type reconstruction example simple example involves generation const annotations pointers decided policy introduced const annotations function parameters annotating parameter location constraint set procedure satisfies var var retypd appears first machinecode system infer const annotations comparison recovered annotations original source code appears example complex policy used decide union types generic types incompatible scalar constraints must resolved retypd merges comparable scalar constraints form antichains elements antichains used resulting union type example initial stage results types general possible often means types found general strictly helpful human observer policy applied specializes type schemes specific scheme compatible uses example object may include getter function highly polymorphic type scheme since could operate equally well structure field correct type correct offset expect every calling context getter called specific object type perhaps derived types specialize getter type choosing least polymorphic specialization compatible observed uses specializing function signature presenting final type user trade generality types likely match original source simplification algorithm section sketch outline simplification algorithm core constraint solver complete algorithm appears appendix inferring type scheme goal simplification algorithm take inferred type scheme procedure create smaller constraint set constraint implied also implied let denote constraint set generated abstract interpretation procedure analyzed let set free type variables could already use constraint set procedure type scheme since input output types used valid invocation tautologically satisfy yet practical matter use constraint set directly since would result constraint sets many useless free variables high growth rate nested procedures instead seek generate simplified constraint set interesting constraint well makes constraint interesting definition type variable constraint called interesting one following forms capability constraint form var recursive subtype constraint form subtype constraint form type constant call constraint set simplification every interesting constraint since entail set constraints valid replace valid type scheme simplification heuristics systems studied aiken simplification algorithm encompasses heuristics unconstrained pushdown systems algorithm works constraint set building pushdown system whose transition sequences represent valid derivations subtyping judgements briefly review pushdown systems necessary generalizations definition unconstrained pushdown system triple set control locations set stack symbols possibly infinite set transition rules denote transition rule define set configurations configuration called control state stack state note require neither set stack symbols set transition rules finite freedom required model derivation ointer figure corresponds infinite set transition rules definition unconstrained pushdown system determines transition relation set configurations suffix rule transitive closure denoted definition state primary theorem behind simplification algorithm theorem let constraint set set base type variables define subset regular set automaton recognize constructed time proof basic idea treat transition rule pushdown system addition add control states tart transitions tart moment assume labels covariant rule ointer ignored construction tart theorem ensures two control states standard unconstrained pushdown system set pairs regular language caucal gives saturation algorithm constructs automaton recognize language full proof add two novelties first support contravariant stack symbols encoding variance data control states transition rules second novelty involves rule ointer rule problematic since natural encoding would result infinitely many transition rules extend caucal construction lazily instantiate necessary applications ointer saturation details see appendix since usually entail infinite number constraints theorem particularly useful tells full set constraints entailed finite encoding automaton manipulations constraint closure efficient minimization carried restricting transitions tart algorithm used eliminate type variables producing desired constraint simplifications overall complexity inference saturation algorithm used perform simplification construction worst case cubic number subtype constraints simplify since pointer analysis methods also cubic complexity andersen reasonable wonder retypd free analysis really offers benefit system built top analysis data understand retypd efficiencies found first consider retypd core saturation algorithm cubic number subtype constraints due simplicity instructions roughly one subtype constraint generated per instruction furthermore retypd applies constraint simplification procedure isolation eliminate type variables resulting constraint sets relate procedure formalins globals type constants practice simplified constraint sets small since procedure constraint set simplified independently factor controlled largest procedure size overall size binary contrast sourcecode analysis andersen generally cubic overall number pointer variables exponential duplication variables depending depth used context sensitivity situation even difficult analyses vsa since syntactic difference scalar pointer machine code effect every program variable must treated potential pointer benchmark suite programs found execution time retypd scales slightly number program instructions following calculation heuristically explain much disparity theoretical complexity measured complexity benchmark suite maximum procedure size grew roughly like could expect analysis would perform worst program partitioned procedures size program perprocedure analysis may expected behave like analysis overall particular cubic analysis like retypd could expected scale like global analysis remaining differences observed versus theoretical execution time explained facts constraint graphs tend exercise simplification algorithm behavior distribution procedure sizes heavily weighted towards small procedures wparam wparam lparam lparam int wparam accounts getcount protoaccount lparam accounts getarray return figure types declared original source necessarily reflect program semantics example evaluation implementation retypd implemented module within codesurfer binaries leveraging disassembly capabilities codesurfer operate arm code performed evaluation using minimal analysis settings disabling analysis vsa computing affine relations stack frame pointers enabling additional codesurfer phases vsa greatly improve reconstructed expense increased analysis time existing algorithms tie secondwrite require modified form vsa resolve data approach shows types recovered absence information allowing type inference proceed even computing data unreliable expensive evaluation setup benchmark suite consists binaries linux windows compiled variety gcc microsoft visual versions benchmark suite includes mix executables static libraries dlls suite includes coreutils benchmarks used evaluate rewards tie secondwrite additional benchmarks came standard suite programs used precision performance testing codesurfer binaries binaries built optimizations enabled debug information disabled ground truth provided separate copies binaries built settings debug information included dwarf linux pdb windows used idapro read debug information allowed use scripts collecting types dwarf pdb data benchmarks evaluated ghz intel xeon cpu running single logical core ram utilization codesurfer retypd combined capped benchmark description instructions codesurfer benchmarks libidn zlib ogg distributor glut pngtest freeglut miranda xmail yasm tinycad shareaza domain name translator tutorial compression library multimedia library ultravnc repeater bzip library dll library test libpng library irc client email server modular assembler python quake design file sharing benchmarks lattice boltzmann method vehicle scheduling quantum computation compression chess quantum field theory speech recognition protein sequence analysis video compression gnu perl core compiler figure benchmarks used evaluation binaries compiled source using optimized release configurations benchmarks chosen match benchmarks used evaluate secondwrite benchmark suite includes individual binaries figure well collections related binaries shown figure found programs single collection tended share large amount common code leading highly correlated benchmark results example even though coreutils benchmarks include many tools disparate purposes tools make use large common set statically linked utility routines section tail consists routines yes number common code specific idioms appearing coreutils make particularly lowvariance benchmark suite order avoid program collections results treated collections clusters data set cluster computed average metric cluster inserted average single data point final data set retypd performs well many clusters averaging procedure tends reduce overall precision conservativeness measurements still believe gives less biased depiction algorithm expected behavior average benchmarks sources imprecision although retypd built around sound core constraint simplification solving several ways imprecision occur described disassembly failures lead unsound constraint generation second heuristics converting sketches types lossy necessity finally treat source types ground truth leading failures whenever retypd recovers accurate type match original common situation source code representative example last source imprecision appears figure source code belongs irc client uses architecture functionality implemented service functions fixed signature int serviceproc wparam lparam types wparam lparam used certain windows apis generic values two parameters immediately cast types body service functions figure const correctness separately modeling capabilities retypd easily able recover information pointer parameters used input output take account converting sketches types function sketch includes annotate parameter const figure figure retypd appears first system infer const annotations directly benchmark suite found parameter const annotations original source code recovered retypd furthermore retypd inferred const annotations many parameters unfortunately since code use const every possible situation straightforward way detect many retypd additional const annotations correct manual inspection missed const annotations shows instances due imprecision analyzing one two common statically linked library functions imprecision propagates outward callers leading decreased const correctness overall still believe recovery rate shows retypd offers useful approach const inference distance source type coreutils interval size pointer accuracy coreutils rit dynamic rit conservativeness dynamic figure distance types size interval inferred upper lower bounds smaller distances represent accurate types smaller interval sizes represent increased confidence figure conservativeness pointer accuracy metric perfect type reconstruction would conservative match pointer levels note axis begins pointer accuracy elwazeer also introduced rate attempts quantify many levels pointers correctly inferred secondwrite benchmark suite retypd attained mean pointer accuracy compared secondwrite reported across benchmarks retypd averages pointer accuracy conservativeness best type system would high conservativeness rate unsound decisions coupled low interval size tightly specified results low distance inferred types close types metrics retypd performs well better existing approaches retypd mean conservativeness rate compared tie note tie evaluated coreutils cluster retypd conservativeness secondwrite overall conservativeness measured subset benchmarks retypd attained slightly lower subset interesting note retypd conservativeness rate coreutils comparable rewards even though rewards use dynamic execution traces suggests would conservative static analysis virtue generating feasible type constraints comparisons tools gathered results several metrics used evaluate secondwrite tie rewards metrics defined lee briefly reviewed tie infers upper lower bounds type variable bounds belonging lattice types lattice naturally stratified levels distance two comparable types roughly difference levels lattice maximum distance recursive formula computing distances pointer structural types also used tie also determines policy selects upper lower bounds type variable final displayed type tie considers three metrics based lattice conservativeness rate interval size distance type interval conservative interval bounds overapproximate declared type variable interval size lattice distance upper lower bound type variable distance measures lattice distance final displayed type type rewards secondwrite use algorithms evaluated using tie metrics evaluation rewards using tie metrics appears lee distance interval size retypd shows substantial improvements approaches distance intervalsize metrics indicating generates accurate types less uncertainty mean distance type retypd compared dynamic tie rewards static tie secondwrite mean interval size shrunk retypd compared secondwrite tie performance although core simplification algorithm retypd cubic complexity needs applied basis suggests scaling behavior depend distribution procedure sizes size cluster count description coreutils putty freeglut samples gnu coreutils vpx decoders vpx encoders speech recognition ssh utilities instructions retypd reported retypd without clustering distance interval conserv ptr acc const figure clusters benchmark suite metric average cluster given cluster average worse retypd overall average certain metric box drawn around entry type inference memory usage type inference time seconds program size number cfg nodes program size number cfg nodes figure time benchmarks line indicates exponential demonstrating slightly superlinear scaling behavior coefficient determination figure memory usage benchmarks line indicates exponential coefficient determination practice figure suggests retypd gives nearly linear performance benchmark suite ranges size instructions measure retypd performance used numerical regression find model relating execution time program size results relation coefficient determination suggesting nearly variation performance data explained model words programs retypd demonstrates nearly linear scaling execution time cubic behavior constraint solver translate cubic behavior overall similarly found model explains memory usage retypd note regressions performed numerically fitting exponential models space rather analytically fitting linear models space models minimize error predicted values rather minimizing errors log log linear regression space results lesspredicative models workload retypd would straightforward parallelize directed acyclic graph components callgraph reducing scaling constant reduction memory usage required retypd could swap constraint sets disk current implementation keeps working sets ram related work type recovery reverse engineering tool idapro early example type reconstruction via static analysis exact algorithm proprietary appears idapro propagates types unification library functions known signature halting propagation type conflict appears idapro reconstructed relatively sparse type propagation fails produce useful information many common cases falling back default int type however analysis fast secondwrite interesting approach static reconstruction particular emphasis scalability authors combine vsa variant pointsto analysis engine accurate types secondwrite depend pointsto data authors note cause type accuracy suffer larger programs contrast retypd dependent data type recovery makes use subtyping rather unification increased precision tie static tool used part carnegie mellon university platform bap tie first system track subtype constraints explicitly maintain upper lower bounds type variable abstraction type system tie type lattice relatively simple missing features recursive types later identified authors important target future research howard rewards take dynamic approach generating type constraints execution traces comparison howard creators tie showed static type analysis produce types dynamic type analysis though small penalty must paid conservativeness generation tie also showed type systems designed static analysis easily modified work dynamic traces expect true retypd though yet performed experiments previous work type recovery including tie secondwrite either disallows recursive types supports recursive types combining typeinference results oracle example infer type struct struct approach like secondwrite first must resolved points memory region admits abstract location offset type unified type pointer analysis failed compute explicit memory region pointed possible determine type correctly complex interplay type inference analysis delineation leads relatively fragile method inferring recursive types contrast type system infer recursive types even facts completely absent robbins developed smt solver equipped theory rational trees applied type reconstruction although allows recursive types lack subtyping performance smt solver make difficult scale approach binaries except test cases order instructions precision recovered types assessed related type systems type system used retypd related recursively constrained types types eifrig smith trifonov retypd generalizes type system building types using flexible records even constructor taken fundamental type system decomposed record fields allows retypd operate without knowledge fixed signature type constructors drawn essential analysis stripped machine code use cfl reachability perform polymorphic subtyping first appeared rehof extending previous work relating simpler type systems graph reachability retypd continues adding handling pointers simplification algorithm allows compactly represent type scheme function cfl reachability also used extend type system java support additional type qualifiers reconstructed const annotations seen instance idea although qualifier inference separated type inference best knowledge prior work applied polymorphic type systems subtyping machine code future work one interesting avenue future research could come application dependent type systems type inference although rapidly approach frontier type inference undecidable natural example dependent types appearing machine code malloc could typed malloc denotes common supertype types key feature value parameter determines type result types needed properly model functions accept pointers polymorphic functions parameters functions entirely uncommon example function parameterized custom polymorphic allocator rank retypd implemented inference phase runs codesurfer main analysis loop expect moving retypd codesurfer analysis loop opportunity interesting interactions generation type reconstruction conclusion examining diverse corpus optimized binaries identified number common idioms stumbling blocks type inference idioms identified feature could enable difficult code properly typed gathered features type system implemented inference algorithm tool retypd despite removing requirement data retypd able accurately conservatively type wide variety binaries assert retypd demonstrates utility type systems reverse engineering binary analysis lee avgerinos brumley tie principled reverse engineering types binary programs network distributed system security symposium pages acknowledgments lim tsl system generating abstract interpreters application analysis acm transactions programming languages systems authors would like thank vineeth kashyap anonymous reviewers many useful comments manuscript john phillips david ciarletta tim clark help test automation references technical report library extensions agesen type inference parametric polymorphism static analysis symposium pages amadio cardelli subtyping recursive types acm transactions programming languages systems andersen program analysis specialization programming language phd thesis university cophenhagen balakrishnan gruian reps teitelbaum platform analyzing executables compiler construction pages balakrishnan analyzing memory accesses executables compiler construction pages carayol hague saturation algorithms modelchecking pushdown systems international conference automata formal languages pages caucal regular structure prefix rewriting theoretical computer science eifrig smith trifonov sound polymorphic type inference objects programming systems languages applications pages elwazeer anand kotha smithson barua scalable variable data type detection binary rewriter programming language design implementation pages foster johnson kodumal aiken flowinsensitive type qualifiers acm transactions programming languages systems aiken making program analyses scale workshop set constraints gopan driscoll nguyen naydich loginov melski software binaries application discovery international conference software engineering pages greenfieldboyce foster type qualifier inference java programming systems languages applications pages idapro http kozen palsberg schwartzbach efficient recursive subtyping mathematical structures computer science lin zhang automatic reverse engineering data structures binary execution network distributed system security symposium marlow yakushev peyton jones faster laziness using dynamic pointer tagging international conference functional programming pages mauborgne rival trace partitioning abstract interpretation based static analyzers programming languages systems pages noonan loginov cok polymorphic type inference machine code extended version url http palsberg keefe type system equivalent flow analysis acm transactions programming languages systems palsberg wand keefe type inference subtyping formal aspects computing pottier essence type inference pierce editor advanced topics types programming languages chapter mit press rehof flow analysis polymorphic subtyping principles programming languages pages richard regular canonical systems archive mathematical logic robbins howe king theory propagation principles practice declarative programming pages robertson brief history invsqrt university new brunswick phd thesis schwartz lee woo brumley native decompilation using structural analysis iterative structuring usenix security symposium pages siff chandra ball kunchithapadam coping type casts software pages slowinska stancescu bos howard dynamic excavator reverse engineering data structures network distributed system security symposium steensgaard analysis almost linear time principles programming languages pages aiken niehren priesnitz treinen theory subtyping constraints principles programming languages pages constraint generation type constraint generation performed parameterized abstract interpretation ype parameter abstract interpreter used transmit additional analysis information reaching definitions propagated constants available let denote set type variables set type constraints primitive tsl types ype given basetypet ypea basetypea map ypea map since type constraint generation syntactic flowinsensitive process regain flow sensitivity pairing abstract semantics carries summary information parameterizing type abstract interpretation allows factor particular way program variables abstracted types ssa form reaching definitions register loads stores basic reinterpretations proceed pairing abstract interpreter example regupdate reg let regupdate reg reg reg produces type variable register reg register map register loads handled similarly regaccess reg let reg regaccess reg example suppose represents concrete semantics reg yields type variable reg additional constraints expression mov ebx eax represented tsl expression regupdate ebx regaccess eax initial state sconc abstract interpretation become eax ebx changing parametric interpreter generated type constraints may made precise example continue example mov ebx eax suppose represents abstract semantics aware register reaching definitions define reg reg case reaching defs reg regp defs let fresh regp defs yields set definitions reg visible state ype program point update constraint set eaxp ebxq lone reaching definition eax multiple reaching definitions constraint set become eaxp ebxq addition subtraction useful track translations value additions subtraction constant end overload add sub operations cases constant values example concrete numeric value add let add case neither operand constant generate fresh type variable representing result constraint type variables add let fresh add add similar interpretations used sub memory loads stores memory accesses treated similarly register accesses except use dereference accesses handling sets abstract let denote set type variables representing address context furthermore define ptstoa set type variables representing values pointed context semantics load store functions memaccessn memupdaten given memaccess let load ptsto memaccess load memupdate let store ptsto memupdate store achieved acceptable results using bare minimum analysis tracks constant pointers local activation record data section use accessors allows track pointer information without need explicit data minimal approach tracks enough information resolve references local global variables procedure invocation earlier analysis phases responsible delineating procedures gathering data procedure variables including information parameters stored stack registers data transformed collection locators associated function locator bound type variable representing formal locator responsible finding appropriate set type variables representing actual callsite corresponding local within procedure example consider simple program invokes identity function push ebx writes local call begin procedure mov eax esp ret procedure two locators within procedure locator returns type variable returns eaxr resulting constraint set idi eaxr ido procedure may also associated set type constraints locator type variables called procedure summary type constraints may inserted function calls model known behavior function example invocation fopen result constraints fopeno support polymorphic function invocation instantiate fresh versions locator type variables tagged current callsite prevents type variables multiple invocations procedure linked example cont using callsite tagging callsite constraints generated locators would idqi idqo eaxq callsite tagging must also applied procedure summary example call malloc result constraints mallocpi mallocpo malloc used twice within single procedure see effect like use typed independently operations floating point types produced calls known library functions abstract interpretation reads writes floating point register bank track moves floating point registers though would straightforward add ability theory causes lose precision attempting distinguish typedefs floating point values practice typedefs appear extremely rare bit manipulation locator single parameter bound type assume operands results bitmanipulation operations integral special exceptions locator single return value bound type common idioms like xor reg reg reg variable idi variable ido procedure call site locator return type variable representing stack location tagged reaching definition likewise return type variable eaxq indicate held version eax defined point locator results combined locator type variables resulting constraint set idi ido eaxq used initialize registers certain constants instructions encoded immediates saving space relative equivalent versions mov mov reg assume results operations integral type discard constraints generated computing value used update flag status particular operation test implemented like discards result retaining effect flags specific operations act equivalent specific operations often used example requiring pointers aligned boundaries frees lower two bits pointer purposes marking garbage collection additive constraints special constraints used conditionally propagate information type variables represent pointers represent integers variables related addition subtraction deduction rules additive constraints summarized figure obtained good results inspecting unification graph used computing graph used quickly determine whether variable capabilities practice constraint set also updated new subtype constraints additive constraints applied fully applied constraint dropped omit details simplicity normal forms proofs finite constraint set recursive constraints infinite entailment closure order manipulate entailment closures efficiently need finite small representations infinite constraint sets first step towards achieving finite representation entailment find normal form every derivation appendix finite models entailment closure constructed manipulate representations normal forms lemma statement provable exists derivation use rules efl nherit proof redundancy efl immediate nherit use rule proof replaced followed eft ield followed ight following make simplifying assumption closed eft ight refix concretely requiring contains subtype constraint axiom also contains axioms var var contains term declaration var also contains term declarations prefixes lemma closed eft ight refix statement provable proven without use refix proof previous lemma may assume efl nherit used proof prove lemma transforming subproofs end use refix remove use end assume proof ending derivation var var using refix enumerate ways var may proven var var already entire proof tree leading var may replaced axiom way var could introduced eft ight simplicity let consider eft case ight similar point derivation tree looks like var var introduced axiom assumed closure property would imply already var cases introduced one ield axioms either case side one antecedents ield whole subderivation including refix axiom replaced single use eft ight introduced via ointer case var already antecedent ointer may elide subproof refix inductively reduce problem simpler proof tree finally must due rans case left antecedent form may elide subproof get simpler proof tree cases either removes instance refix results strictly smaller proof tree follows iterated application simplification rules results proof var remaining instances refix corollary var either var simplify statement theorem recall defined variadic rule rans remove degrees freedom regrouping repeated application rans rans figure inference rules lower case letters denote known integer pointer types upper case letters denote inferred types example first column says integral types constraint integral well theorem normal form proof trees let constraint set closed eft ight refix statement provable derivation also eliminate need subderivations except first last antecedents rans pulling term neighboring subtype relation schematically always replace uses rules refix nherit var efl every instance rans every pair adjacent antecedents least one result ield application proof previous lemmas already handled first point second suppose two adjacent antecedents result var var first note neighboring applications must use field label else rans would applicable observe may move ield application upwards combining two ield applications one var completeness also compute simplifying transformation adjacent uses ield derivation var var simplified var var transformation even works several ield applications row var var simplified var var var taken together demonstrates var antecedents ield automatically satisfied consequent eventually used rans application middle variable remains true even several ield applications sequence since ield var antecedents automatically satisfiable use simplified schematic depiction proof trees omits intermediate var subtrees simplified depiction previous proof tree fragment would written algebraic representation finally consider form simplified proof tree elides var subderivations leaf tree subtype constraint unique way fill necessary var antecedents ield applications glue constraints proof tree normal form words normal form proof tree completely determined sequence leaf constraints sequence ield applications applied final tree effect term algebra constants representing constraints associative binary operator combines two compatible constraints via rans inserting appropriate ield applications unary operator represents application ield proof tree manipulations provide additional relations term algebra normalization rules described section demonstrate every proof subtype constraint entailed represented term form pop pop pop push push push fragment automatically converted full proof var subderivations results input derivations simplify presentation sometimes write stackop weight domain section develop weight domain useful modeling constraint sets pushdown systems weight domain generated symbolic constants representing actions stack definition weight domain idempotent generated symbols pop push subject relation push pop definition monomial called reduced length shortened applications push pop rule every reduced monomial form pop pop push push elements understood denote functions operating stack symbols fail push cons pop case nil fail cons else fail interpretation semiring operations interpreted nondeterministic choice nondeterministic iteration note action pushdown rule stack configuration given application pop pop push push cons result either cons configuration obtained applying rule configuration fail rule applied lemma correspondence elements regular sets pushdown system rules definition variance operator extended defining hpop hpush hxi constructing transducer constraint set building pushdown system let set type variables set field labels equipped variance operator furthermore suppose fixed set constraints finally suppose partition set interesting uninteresting variables definition suppose interesting type variables proof call proof elementary normal form proof tree involves uninteresting variables internal leaves write elem elementary proof goal construct finite state transducer recognizes relation elem derived type variables defined elem elem proceed constructing unconstrained pushdown system whose derivations model proofs modification caucal saturation algorithm used build transducer representing derivpc definition define tagging rules lhs rhs lhs rhs definition unconstrained pushdown system associated constraint set given triple lhs rhs tart write element stack alphabet essentially extra tokens added represent interesting variables define transition rules first introduce helper functions hlhs hui hrhs hvi rule hlhs hrhs rules rule rule transition rules partitioned four parts rules rules tart tart hvl hvr note superscripts control states used track current variance stack state allowing distinguish uses axiom contravariant position tagging operations lhs rhs used prevent derivations making use variables preventing admitting derivations represent proofs note note although finite contains rules every derived type variable therefore infinite carefully adjust saturation rules rules considered lazily derivpc constructed lemma pair derivpc also derivpc proof immediate due symmetries construction lemma pair derivpc relation hui hvi must hold proof inductive consequence defined relation holds every rule due use rules function sign exponent propagated top stack application rule back stack application rule since every derivation begin rule proceed applying rules conclude rule completes proof definition let relation derived type variables induced derivpc follows hui hvi derivpc lemma well instead proof immediate symmetry considerations lemma derivpc partitioned derivpc hui hvi derivpc particular derivpc entirely reconstructed simpler set theorem elementary proof constraint proof prove theorem constructing bijection elementary proof trees derivations call configuration positive hui lemma positive positive well also call transition rule positive configurations appearing rule positive note construction every positive transition rule form axiom constraint positive transition rule let denote positive rule hphu hphu note construction redundant restriction one existing transition rules rule specific stack configuration particular adding set rules affect derivpc suppose positive state phwi apply transition rule hphui hvi must side rule exactly hphwi let rstart rend sequence rule cations used arbitrary derivation tart phui hvi let denote application rule followed rule side exactly matching side given initial stack state phui sequence rule applications unique rule rstart rend describes derivation exactly normalize using rule process clearly reversible producing bijection normalized expressions derivations complete proof obtain bijection normalized expressions normal forms elementary proof trees bijection straightforward represents introduction axiom represents application rule rans represents application rule ield results algebraic expression describing normalized proof tree completes proof constructing initial graph section construct finite state automaton accepts strings encode behavior next section describes saturation algorithm modifies initial automaton create finite state transducer automaton constructed follows add initial state tart accepting state side hpa rule add transitions tart pop pop pop side rule add transitions push push push rule hpa add transition definition following carayol hague call sequence transitions productive corresponding element sequence productive contains adjacent terms form pop push lemma correspondence productive transition sequences accepted elementary proofs use axiom ointer proof productive transition sequence must consist sequence pop edges followed sequence push edges possibly insertions unit edges intermediate sequences form push push pop pop following push pop edge corresponds observing forgetting part stack following edge corresponds applying pds rule algorithm converting transducer set pushdown system rules procedure paths tarjan path algorithm return finite state automaton recognizing label sequence paths graph return end procedure procedure ransducer set pushdown system rules hpa tart hpa tart pop push pop end push end end aturated paths tart return end procedure saturation label sequences appearing lemma tantalizingly close simple structure building pop sequence representing initial state pushdown automaton building push sequence representing final state intermediate terms form push pop unwieldy remove necessity sequences saturate adding additional transitions providing shortcuts subsequences modify standard saturation algorithm also lazily instantiate transitions correspond uses rules lemma reachable states automaton partitioned covariant contravariant states state variance defined variance sequence reaching state tart proof construction lemma involution defined tart tart proof immediate due use rule constructors rule forming section automaton saturated adding transitions create new automaton asat definition sequence called reduced productive contains factors form pop push reduced productive sequences form sequence pops followed sequence pushes goal saturation algorithm twofold ensure productive sequence accepted equivalent reduced sequence accepted asat ensure asat represent elementary proofs use ointer saturation algorithm proceeds maintaining state set reaching push set contain pair transition sequence weight push outgoing pop edge add new transition special propagation clause responsible propagating facts rules instantiated allowing saturation algorithm work even though corresponding unconstrained pushdown system infinitely many rules special clause justified considering standard saturation rule added axiom example appears figure saturation edge added due pointer saturation rule edge would also added states transitions corresponding depicted dotted edges nodes added saturation algorithm completes automaton asat property transition sequence weight equivalent pop pop push push path ignoring edges exactly label sequence pop pop push push shadowing automaton asat accepts sequences representing changes stack legal derivation pushdown system saturation guarantee every derivation represented path first pops sequence tokens pushes another sequence tokens unfortunately asat still accepts unproductive transition sequences push immediately pop token sequences complete construction form automaton intersecting asat automaton language words consisting pops followed pushes final transformation yields automaton property every transition sequence accepted asat accepts sequence stackop consists sequence pops followed sequence pushes ensures accepts productive transition sequences asat finally treat transducer treating transitions labeled pop reading symbol input tape push writing output tape manipulated determinization minimization produce compact transducer representing valid transition sequences taken whole shown interesting variables elementary derivation make use process two places type analysis first computing relative type variable function get transducer represents elementary derivations relationships function inputs outputs used convert transducer back pushdown system describes valid derivations rules interpreted subtype constraints resulting simplification constraint set relative formal type variables second computing relative set type constants obtain transducer efficiently queried determine derived type variables bound type constants used olve procedure populate lattice elements decorating inferred sketches algorithm converting transducer pushdown system procedure ype cheme new pds pop pdsrule else pdsrule end end return end procedure lattice sketches throughout section fix lattice atomic types assume anything except finite height infinite subtype chains eventually stabilize example purposes take lattice semantic classes depicted figure initial implementation sketches use auxiliary lattice found adding decorations sketches helped preserve types interest end user type inference allows recover highlevel windows typedefs file handle socket useful program understanding reverse engineering noted lin decorations also enable simple mechanism user extend retypd type system adding semantic purposes types known functions example extend add seeds tag attached int parameter signal approach also allows distinguish opaque typedefs underlying type handle void since semantics handle quite distinct void important mechanism preserve typedef name basic definitions definition sketch regular tree edges labeled elements nodes labeled elements set sketches implicitly fixed denoted may alternately think sketch regular language function algorithm saturation algorithm procedure aturated end push end repeat rold eold end pop end end end end end rold eold return end procedure set vertices partitioned set edges represented triples src tgt label initialize sets standard saturation rule lazily apply saturation rules corresponding ointer see figure example tart pop push store pop store pop load pop load push store push load push figure saturation using implicit application initial constraint set modeling simple program right dashed edge added saturation rule fires lazy handling dotted states edges show graph would look corresponding rule explicitly instantiated str str figure sketch representing linked list strings struct str struct url figure sketch representing url num str url figure sample lattice atomic types fiber regular convenient write value node reached following word collapsing equal subtrees represent sketches deterministic finite state automata state labeled element figure since regular language associated sketch states associated automaton accepting lemma set sketches forms lattice meet join operations defined according figure top element given sketch accepting language single node labeled finite also bottom element accepting language label function hwi hwi use denote partial order sketches compatible lattice operations modeling constraint solutions sketches sketches choice entity modeling solutions constraint sets definition solution constraint set type variables set bindings type constant var sketch corresponding subtree reached following path root main utility sketches almost free tree model constraint language constraint set satisfiable lattice sketches long prove impossible subtype relation theorem suppose constraint set variables exist sketches var proof languages computed algorithm similar spirit steensgaard method pointer analysis begin forming graph one node derived type variable appearing along prefixes add labeled edge derived type variable form graph quotient equivalence relation defined whenever edges either relation symmetrization first defining rule roughly corresponding nherit nherit second rule corresponding ield unusual condition due ointer rule construction exists path label sequence starting equivalence class var take definition language accepted working lattice elements label trickier problem basic idea use pushdown system construction appears constraint simplification answer queries type constants upper lower bounds given derived type variable computation upper lower lattice bounds derived type variable appears sketch narrowing function calls noted rule nherit leads system structural typing two types subtype relation must fields language sketches means two type variables subtype relation corresponding sketches accept exactly languages superficially seems problematic modeling typecasts narrow object motivated idioms missing piece allows effectively narrow objects instantiation callee type schemes callsite algorithm computing sketches constraint sets procedure nfer hapes cinitial ubstitute cinitial compute constraint graph modulo ind quiv ind quiv end end ind quiv ind quiv nify end repeat apply additive constraints update cold cold pplya nify end end cold infer initial sketches new sketch paths rom states hwi else end end end end procedure procedure nify make ake quiv nify end end end end procedure demonstrate polymorphism enables narrowing consider example type scheme figure function invoked providing actualin type particular capabilities pass constrainted capable type function expected less constrained input way recover aspects physical nonstructural subtyping utilized many programs via idioms described example suppose reverse dns lookup function type signature void num addr url result furthermore assume implementation works writing resulting url location pointed result yielding constraint form url inferred type scheme form num url void suppose structure figure representing linked list strings instance mylist safely invoke addr url mylist intuitively see possible since linked list payload first field value type also looks like value type str furthermore mylist const used store function output intuition borne retypd type system answer yes though takes bit work see let write instantiated sketch callsite question constraints require satisfy url type sketch seen figure copy generate constraint since already satisfied two constraints coming use freely add words freely set node labels almost value please subject constraint simple enough add every word word accepted must added define thus shape node labeling match one possible exception must check nested field node labels satisfy relation since contravariant indeed required field url str function invocation judged additional algorithms appendix holds algorithms referenced main text appendices algorithm type inference procedure nfert ypes callgraph map type variable sketch everse ostorder end olve efine parameters onstraints olve end end ketch ppx ype end return end procedure procedure olve nfer hapes ransducer end end end end procedure algorithm procedure specialization procedure efine parameters end end end end end procedure algorithm type scheme inference procedure nfer roc ypes callgraph map procedure type scheme ostorder end onstraints end nfer hapes ransducer ype cheme end end end procedure procedure onstraints bstract nterp typeinterp calls nstantiate end end return end procedure language node labels hwi hwi hwi hwi figure lattice operations set sketches type resolution policies example initial stage results types general possible often means types found general strictly helpful human observer policy called efine parameters used specialize type schemes specific scheme compatible uses example object may include getter function highly polymorphic type scheme since could operate equally well structure field right type right offset expect every calling context getter called specific object type perhaps derived types specializing function signature make use contextual clues exchange generality presenting final type user example suppose class class myfile public char filename const return private file char binary implementation myfile inlined roughly equivalent code typedef dword dword const void char char dword dword return accordingly would expect inferred type scheme myfile dword indicating accept pointer anything value type offset return value type function truly used polymorphically exactly kind precision wanted type system maintain common case called values type myfile perhaps subtype include inheritance every callsite passes pointer myfile may best specialize type monomorphic type const myfile char function efine parameters used specialize function type enough match function actually used program cost reduced generality example useful less sound heuristic represented reroll policy handling types look like unrolled recursive types reroll sketch sketch replace else policy apply practice often need add guards inspect shape determine application reroll appears appropriate example may require least one field help distinguish details figure example results constraint generation example program figure appears figure algorithm builds automaton figure recognize simplified entailment closure constraint set recognizes exactly pairs form finally two remaining transitions start end generate int successz generate simplified constraint set gather constraints applying lattice operations combine int filedescriptor inequalities differ lattice constant close introduced introducing quantifier result constraint set figure int successz generate simplified constraint set type variable synthesized single internal state path leading start state generates constraint loop transition generates two transitions generate filedescriptor int successz int int filedescriptor figure automaton constraint system figure proc near mov edx dword esp eax jmp times mov edx eax mov eax dword edx load test eax eax jnz mov eax dword edx load mov dword esp eax jmp close close close filedescriptor close int close eax int close eax endp figure constraints obtained abstract interpretation example code figure
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good things come threes three beads learn swim lattice boltzmann rigid body solver aug kristina pickla jan klaus iglbergerc jayant pandeb klaus meckeb smithb ulrich lehrstuhl systemsimulation erlangen germany cluster excellence engineering advanced materials erlangen germany zentralinstitut scientific computing erlangen germany institut theoretische physik erlangen germany abstract simulate devices fluid regime low reynolds numbers device consists three bodies spheres capsules connected two damped harmonic springs sinusoidal driving forces compress springs resolved within rigid body physics engine latter consistently coupled lattice boltzmann framework fluid dynamics simulations devices find propulsion velocity agrees well theoretical predictions simulations spheres replaced capsules find asymmetry design strongly affects propelling efficiency keywords stokes flow microorganism lattice boltzmann method numerical simulation introduction engineered developed way able move alone fluid simultaneously emit signal corresponding author email address kristina pickl preprint submitted journal computational science august cial use various fields science engineered devices developed last decade one approach device consisted magnetically actuated superparamagnetic particles linked together dna double strands another approach miniature semiconductor diodes floating water used powered external alternating electric field voltage induced diode electrodes changed resulted flow latter case devices could respond emit light thus providing signal could used carrier third example swimmer based creation traveling wave along piezoelectric layered beam divided several segments voltage frequency different phases amplitudes applied segment propulsion fluid induces flow surrounding fluid turn affects propulsion device generally fluid motion described equation velocity density pressure dynamic viscosity fluid applied force fluid first term left hand side describes acceleration fluid whereas second term accounts inertial effects may give rise effects turbulence right hand side first term driving pressure gradient second term takes account viscous dissipation case low reynolds numbers inertia terms left side equation dropped one left together incompressibility equation describes socalled stokes flow main characteristics stokes flow emerge domination viscous forces consequently flow always laminar turbulence vortex shedding characterized small momentum since flow proportional forces applied linear superposition solutions valid furthermore stokes flow instantaneous means given solely effects boundaries finally coasting flow important implications swimming strategies low reynolds numbers propulsion strategy regime must involve motion thus involving one degree freedom requirement limiting large number degrees freedom available however one tries design device swimmer degrees freedom balance simplicity functionality found number analytic numerical studies performed understand behavior single couple swimmers various conditions however types approaches limited simple geometries swimmers fail numerous swimmers involved leading collective behavior addition effective treatment many body interactions regimes inertia starts play role well problem motion confined geometry transport turbulent flow inaccessible circumstances extensive simulations become essential difficult perform efficiently direction linear movement figure swimming device two translational degrees freedom order meet challenges augmented already existing massively parallel lattice boltzmann simulation framework walberla widely applicable lattice boltzmann solver erlangen swimmers motion latter included walberla coupling rigid multibody physics engine framework simulation rigid fully resolved bodies arbitrary shape allows resolve driven motion swimmer within fluid induced motion fluid consistent manner swimming device choose simplest possible design consisting three rigid bodies connected two springs figure design studied extensively three spheres geometry analytical methods paper starts elucidation background numerical methods form basis simulation section lattice boltzmann method lbm walberla framework mechanics framework finally coupling special focus swimmers briefly explained section new implementation approach integration swimmers considered detail including cycling strategy moreover design variants presented afterwards section quantitative validation simulation results engine conducted section validation criteria swimmer defined asymmetric symmetric designs simulated analyzed detail demonstrate capability extension framework finally section conclusion achieved results given propositions future work made numerical methods lattice boltzmann method lbm used alternative classical solvers simulate fluid flows fluid modeled set imaginary discretized particles positioned equidistant grid lattice cells moreover particles allowed move fixed predefined directions given finite set velocity vectors resulting discretization velocity space unlike methods rely automatic generation body fitted mesh simulation walberla framework uses lbm structured mesh even objects moving eliminates need dynamic hence offers significant advantage respect performance study use common velocity phase discretization model originally developed qian lallemand particle distribution functions pdfs spatial time domain respectively figure illustrates directions corresponding dimensionless discrete velocity set denoted model shown stable efficient work presented paper adopt lattice boltzmann collision scheme proposed figure velocity phase discretization model bhatnagar gross krook called lbgk cell discretized simulation domain current time step next time step relaxation time units time step set represents equilibrium distribution time evolution distribution function given equation usually solved two steps known collision step streaming step respectively denotes state distribution function collision step local relaxation towards equilibrium streaming step advects pdfs except neighboring lattice site depending velocity thus time step information next neighbors needed boundary condition simple scheme used distribution functions pointing neighboring wall reflected normal tangential velocities vanish representing index opposite direction explicitly denoting fluid cell due locality cell updates lbm implemented extremely efficiently see instance reason parallelization lbm comparatively straightforward based subdomain partitioning realized walberla framework patch data structure described feichtinger rigid body physics engine simulating dynamics rigid bodies involves treatment movement discretization newton equations motion well handling collisions rigid objects forces involved may external gravity bodies springs furthermore number velocity constraints available hinges sliders ball joints like implemented collisions rigid bodies either treated directly application restitution laws form linear complementarity problem lcp applying hertzian contact mechanics instance dem approach work fully resolve geometry rigid bodies contrast approaches instance enables easily exchange geometries swimmers simulations see subsection rigid body framework used rigid multibody physics engine due highly flexible modular massively parallel implementation framework allows direct selection time discretization scheme collision treatment easily adjusted various simulation scenarios instance already successfully used simulate granular flow scenarios efficiently coupled simulations particulate flows see subsection springs modeled damped harmonic oscillators bodies subject harmonic damping force first one springs force constant deformation spring deformation defined difference rest length spring current length equals current distance connected bodies whose current positions denoted respectively order account dissipation spring damping force proportional velocity spring contracting implemented following springs damping coefficient velocities bodies damping force hence models internal friction resulting deformation spring consequently two bodies connected spring feels force respectively standard way classify damped harmonic oscillator damping ratio system overdamped upon excitation system returns exponentially equilibrium without oscillating system underdamped oscillate frequency decreasing amplitude zero equilibrium state reached choose parameters springs swimmer way latter regime achieved algorithm gives overview necessary steps within single time step framework time step collisions rigid bodies detected resolved appropriately order keep rigid bodies interpenetrating therefore initially contacts pairs colliding rigid bodies detected added set constraints already includes potentially existing velocity constraints subsequent constraint resolution step acting constraining forces including forces harmonic potentials determined moreover detected collisions resolved depending selected collision response algorithm important note interpenetration completely prevented due discrete time stepping small penetrations corrected collision response phase final step algorithm rigid body time step detection rigid body detect contacts add set constraints end resolution constraint determine acting constraint forces end integration rigid body update position velocity apply forces end rigid bodies moved forward time according newton laws acting motion depending current velocity total forces given instance time simulations swimmers size time step chosen rigid bodies interpenetrate another limitation size time step originates force constant harmonic oscillators frequency driving forces cases time step must small enough enable bodies respond sufficiently quickly changes acting forces coupling lattice boltzmann swimmers coupling lattice boltzmann flow solver rigid body dynamics simulation rigid bodies represented moving boundaries flow simulation whereas flow corresponds hydrodynamic forces acting rigid bodies figure use explicit coupling algorithm shown algorithm adopt swimmers additional steps highlighted driving force acts corresponding bodies according carefully defined protocol see subsection first step coupled algorithm mapping rigid bodies figure illustration coupling walberla fluid solver rigid body dynamics solver initial setup velocities object cells set velocity object example object translational velocity component fluid cells marked updated setup two fluid cells transformed object cells two object cells pdfs reconstructed figure mapping example algorithm coupled solver swimmers swimmer map lattice grid end lattice cell stream collide end surface cell fluid rigid objects add forces end body swimmers add external forces end step rigid body simulation see algorithm rigid body end onto lattice boltzmann grid see figure example objects thus represented flag fields flow solver implementation lattice node cell center inside object treated moving boundary cells apply boundary condition variation standard boundary conditions moving walls fluid density close wall weight lbm depending stencil direction velocity object cell corresponds velocity object cell position time way fluid surface object given velocity object surface depends rotational translational velocities object final part first step account flag changes due movement objects two cases occur see figure fluid cells turn object cells resulting conversion cell moving boundary reverse case boundary cell turns fluid cell missing pdfs reconstructed implementation missing pdfs set equilibrium distributions macroscopic velocity given velocity object cell density computed average surrounding fluid cells details procedure discussed previously iglberger formal ways initialization missing distribution functions one described paper mei subsequent stream collide step fluid flow acts hydrodynamic forces rigid objects fluid particles stream cells neighboring cells case enter cell occupied moving object reversed causing momentum exchange fluid object resulting particles total hydrodynamic force momentum exchange easily evaluated due kinetic origin lbm obstacle cells object neighbor least one fluid cell comparison different approaches momentum exchange given lorenz resulting cycling strategy third step driving force added previously calculated hydrodynamic force body together constraint force results total force object fourth final step coupling algorithm consists determining rigid body movements objects due influence rigid body framework results position change objects mapped lbm grid next time step validation method described binder iglberger integration swimming device modeling swimmer always involves similar physical setup one hand connections objects need modeled stiff rods springs commonly used hand cycling strategy responsible characteristic movement swimmer needs defined elementary design figure swimming device three spheres mass msph radius rsph similarly two damped harmonic oscillators identical force constant damping parameter rest length elementary design figure inspired analytical modeling golestanian ajdari felderhof consists three spheres identical radius rsph mass msph connections three rigid objects swimmer realized two damped harmonic springs equal force constant damping parameter forces applied spheres along main axis swimmer case resulting translation swimmer direction accordance equations different objects feel force osci respectively since bodies swimmer collide total force acting body swimmer given equation rewritten alternative designs since want investigate effect different geometries swimming behavior device replace spheres elementary swimmer design capsules example figure depicts characteristic parameters swimming device comparably smooth edges capsule thus resulting smooth behavior fluid substantiate choice particular geometry figure shows assembly variants figure swimming device three capsules mass mcap length lcap radius rcap similarly two damped harmonic oscillators identical force constant damping parameter rest length cycling strategy purcell scallop theorem necessitates motion swimmer low reynolds numbers additionally total applied force swimmer vanish one cycle means displacement center mass swimmer one cycle zero absence fluid contrast common approach imposing known velocities constituent bodies swimmer used earl najafi golestanian driving protocol imposes known forces objects forces applied along main axis swimmer center mass body given figure design options swimming device capsules designs rotated spheres design larger radii equal lengths capsules sin sin sin fdri fdri fdri amplitude driving frequency oscillation period phase shift kept constant value equation apply negative sum forces two outer bodies middle body order ensure net driving force acting system instant time zero figure illustrates forces force force body force body force body time step figure fragments driving forces three bodies applying force protocol swimmer end cycling strategy illustrated figure starting rest position delay force body fourth pulse length step depicts theoretical positions prescribed cycling strategy position plot simulation zpositions given lattice cells figure cycling strategy motion exemplified threesphere swimmer damped harmonic potentials lmin minimum length one arm rest length lmax maximal extended armlength distance covered swimmer one cycle initial position swimming device two arms connecting harmonic oscillators rest length following denote current lengths respectively step step first apply driving force body body shown figure also apparent equation starts negative direction reaches negative maximum point oscillator obtains maximum length lmax also influenced force therefore also gets extended length lmax step onwards increases start exert positive driving force fdri body reach intermediate step iii oscillators length lmax goes decreasing transition step iii step length lmax moreover fdri reaches positive maximum thus still increasing obtains maximum length lmax starting decrease attain another intermediate state step length lmin length lmax reaches positive maximum result takes step goes accordingly relaxes minimum length lmin furthermore also decrease length lmin start decrease passing temporary step vii oscillators length lmin pass step viii reaches negative maximum resulting obtaining minimum length lmin additionally goes therefore relaxes length lmin fdri continues grow negative direction decreases swimmer moves forward another transitional state conditions lmax lmin hold oscillators lengths finally reaches negative maximum becomes resulting length lmax oscillator reaching state equal state one swimming cycle completed subsequently swimmer begin another cycle motion validation simulation swimmers order quantitatively validate application springs physics engine perform simulations motion swimmer figure vacuum compare results analogous analytic model basis model lagrangian nondissipative assembly refers derivative positions three spheres respect time denotes rest length springs first term lagrangian consists kinetic energy three spheres second gives energy stored two springs due deformation remaining terms account driving forces equations discussed previously equation damping forces simulation proportional velocity spring deformation factor proportionality taking damping account equation motion sphere becomes sphere set three differential equations solved analytically assuming appropriate initial conditions simulation analytical calculation sphere sphere time step figure plot three spheres time weak damping one simulations values parameters used lattice cells discretization lattice cells lattice cells lattice cells simulation run time steps driving force left body switched time steps right hence also middle body time steps springs obtain starting configuration time steps figure shows simulation analytically calculated results typical variation sphere positions time vacuum case weak damping used parameter values stated caption particular setup damped harmonic oscillators driving forces equal setup fluid simulation section number time steps lattice boltzmann algorithm stable also end stokes regime expected sphere performs oscillatory motion steady state frequency driving frequency moreover figure indicates force neutrality total cycle switch driving force left body time steps right body hence also middle body time steps equal total time steps springs swimmer start revert rest lengths swimmer soon obtains starting configuration time steps causing flat ends position curves analytical calculation simulation figure amplitude oscillation scaled amplitude driving forces given lattice example sphere fixed value weak damping figure shows graph amplitude oscillation sphere scaled amplitude driving forces function natural spring frequency scaled constant fixed characteristic damping ratio equation graph mass sphere fixed stiffness spring varied values accordingly damping lies within bounds ranges yield damping ratio run order guarantee achievement steady state simulations perform time steps figure demonstrate excellent agreement simulation results analytically obtained results figure selected simulation scenarios match analytically obtained curve two characteristic maxima less digits simulation swimmers fluid following perform simulations swimmers highly viscous fluid apart exploring parameter space order achieve simulations regime also show shape swimmer considerable effect swimming velocity parameters simulations setup domain design parameter study cuboidal channel lattice cells cube side length corresponds axis movement device highly viscous fluid kinematic viscosity density typical values room temperature honey order make different designs comparable mass spheres capsules set msph mcap spheres designs radius rsph capsules used designs radius rcap length lcap order observe influence distances walls simulation box also consider swimmer larger radii rsphbig swimmer centered channel center middle body center channel respect three dimensions increase different bodies swimmer change also springs rest length lattice cells designs order make respective armlengths corresponding designs equal springs designs rest length lattice cells perpendicular capsules sphere perpendicular capsule rest length lattice cells springs designs rest length lattice cells springs characterized stiffness damping constant results damping ratio system weak damping regime simulations run time steps performing five swimming cycles oscillation period time steps sinusoidal driving forces left middle body applied whereas force right body implemented delay time steps amplitude driving force right left body force middle body instant negative sum forces two bodies swimmer net external driving force time step means amplitude driving force middle body time steps otherwise simulations terminated switching driving time steps left body time steps equals total simulation time bodies system allowed relax remaining time noted driving forces applied bodies simulation differences motion due different shapes stability show choice parameters results swimmers regime low reynolds numbers latter given kinematic viscosity fluid characteristic velocity characteristic order simulate swimmer stable regime define two different reynolds numbers first reynolds number whole rest length total width reswim design table parameters different swimmers rebi denotes reynolds number ith body swimmer reswim indicates reynolds number whole swimmer swimming device table fourth column defined reswim uswim lswim respect characteristic velocity swimmer uswim latter calculated movement central body swimmer one swimming cycle chosen last full cycle divided uswim characteristic lswim different swimming devices rest length direction motion different swimmer designs table second column additionally verify reynolds number whole swimmer also bodies table third fourth fifth column may move considerably higher velocities swimming device reynolds number body rebi ubi lbi ubi maximum velocity body lbi length direction motion table confirms reynolds numbers satisfy condition swimming velocities following investigate performance swimmer function design geometry also detailed pickl specifically study dependence amplitudes oscillations bodies swimmer shapes relate overall swimming velocity swimmers first look two swimmers consist solely spheres table shows velocity swimmer obtained simulations uswim find swimmer consisting smaller spheres moves significantly faster one larger spheres figure shows plot simulation trajectories body two swimmers previous work design analytically modeled golestanian ajdari predict velocity swimmer called uga table third column amplitude oscillation two arms uswim uga error efficiency design table comparison swimming velocities simulations uswim prediction golestanian ajdari uga case spheres quantities given terms values lattice known arm defined distance centers two bodies connected spring model golestanian ajdari assume deformation arm given cos average length instantaneous length arm amplitude frequency phase shift oscillation arm conditions formula average velocity swimmer uga sin geometrical factor given radii three spheres also assumed since model assumes velocity protocol contrast force protocol start known forces applied bodies whereas model golestanian ajdari assumes known deformations arms regardless forces cause first extract simulation trajectories table use known values find uga equation extract fit armlength trajectories last complete period motion equation define error uswim compared theoreticallyexpected uga value error uga uswim uga sphere large swimmer small swimmer sphere sphere time step figure trajectories sphere two swimmers dashed curves solid curves show trajectories spheres large small swimmer respectively simulation run time steps driving force left body switched time steps right hence also middle body time steps springs obtain starting configuration time steps swimmer large spheres error comes possibly explained effect boundaries simulation box spheres larger swimmer theoretical result assumes infinite fluid whereas simulations conducted finite box dimensions stated another possible reason disagreement two velocities swimmer condition satisfied well swimmer small spheres effect finiteness swimming domain smaller condition also satisfied greater extent consequently find case excellent agreement uga uswim error less table also observe swimmer small spheres amplitude sin design table parameters swimmers equals quantities given terms values lattice oscillation arm larger corresponding arm larger swimmer table third column swimmer amplitude oscillation leading arm greater trailing arm table also shows swimming efficiency two swimmers defined lighthill efficiency reff dri reff effective hydrodynamic radius swimmer approximated sum radii three spheres found simulation curves using last full swimming cycle integration also done period order work values table shows efficiency swimmer small spheres greater expected due significantly greater swimming velocity swimmers capsules rest swimmers contain capsules separated three distinct families first family consists swimmers labeled table contain capsules parallel swimming direction henceforth referred parallel capsules springs rest length lattice cells second family consists swimmers labeled table contain capsules perpendicular swimming direction henceforth referred perpendicular capsules springs rest length lattice cells third family consists swimmers labeled table also contain perpendicular capsules springs rest length making swimmers shorter swimmers second family uswim uga error efficiency design table comparison swimming velocities simulations uswim prediction golestanian ajdari uga case swimmers parallel capsules quantities given terms values lattice tables second column show measured uswim swimmers parallel capsules perpendicular capsules longer springs perpendicular capsules shorter springs respectively three tables swimmers arranged increasing order uswim observed corresponding designs family occupy position respective tables two designs said corresponding one needs replace spheres swimmer get designs albeit different kinds capsules possibly springs different lengths one also observes speed swimming decreases increase number capsules attributed fact small spheres greater amplitudes oscillation capsules means lengths arms contain spheres greater amplitudes oscillation greater amplitude oscillation arms results higher swimming velocity simulation trajectories swimmers capsules show irrespective number capsules swimmer body exhibits smooth sinusoidal movement steady state one may attempt use formula golestanian ajdari equation analysis swimming motion formula developed assuming bodies spherical one may hypothesize small deviations spherical shapes bodies mostly affect geometrical uswim uga error efficiency design table comparison swimming velocities simulations uswim prediction golestanian ajdari uga case second family swimmers capsules quantities given terms values lattice uswim uga error efficiency design table comparison swimming velocities simulations uswim prediction golestanian ajdari uga case third family swimmers capsules quantities given terms values lattice equation general expression test hypothesis extracting amplitudes body oscillations phase shifts fitting trajectories equation geometric factor obtained approximating capsule sphere diameter arithmetic mean dimensions capsule direction movement direction perpendicular mean armlengths original swimmers ease reference values parameters given appendix tables tables show case swimmers amplitude oscillation leading arm tables larger trailing arm tables swimmer second third columns tables show estimated uga error measured uswim relative uga three families interesting note uga approximation still good agreement results simulations especially first two families error less third family swimmers perpendicular capsules shorter springs errors increase probably approximation causes spring lengths become smaller equal sphere radii therefore conditions necessary validity golestanian ajdari formula compromised similar case threesphere swimmer large spheres calculated observed velocities swimmer also disagreed due condition satisfied well tables also show efficiencies defined equation swimmers family hydrodynamic radius calculated assuming capsule replaced sphere diameter results show family order efficiencies accords order uswim expected since driving forces swimmer simulations exactly efficient swimmer would expected move faster less efficient one comparison provided useful insights order fully understand simulation results theory golestanian ajdari expanded account capsules task hope undertake future conclusions future work work demonstrated successful integration selfpropelled coupled framework furthermore shown validity approach comparing results simulations analytical models taken advantage flexibility framework simulate symmetric asymmetric swimmers combining capsule sphere geometries possible far establishment framework especially important study systems inaccessible via experimental analytical methods future aim harness capabilities address issues problem swimmer regimes beyond dictated approximations felderhof golestanian ajdari another interesting application would study behavior swimmer narrow channel explore role boundary conditions wall would also interesting investigate hydrodynamic interactions amongst many swimmers swimming together figure show flow fields averaged five cycles case single swimmer also three swimmers swimming together simulation three swimmers took hours total single intel core exclude effects walls channel one needs set even larger flow fields around swimmers case case single swimmer moreover work indicates steady state reached significantly longer time case many swimmers swimming together compared case single swimmer factors suggest current setting accurate simulation many swimmers would take long reduce simulation time significantly large domains simulation processed efficient way parallelizing framework walberla already feature massively parallel simulation algorithms large scale particulate flow scenarios parallelization force generators springs velocity constraints yet implemented achieved massive simulations large number swimmers accessible allowing finally address problem swarming acknowledgements work supported kompetenznetzwerk und bayern konwihr figure flow field one three swimming devices averaged five swimming cycles used parameters simulation setup section except channel lattice cells der project walberlamc bundesministerium bildung und forschung skalb project moreover authors gratefully acknowledge support cluster excellence engineering advanced materials university funded german research foundation dfg within framework excellence initiative sin design table different parameters swimmers parallel capsules equals quantities given terms values lattice appendix values different parameters swimmers capsules present values different parameters swimmers capsules found simulation curves used equation sin design table different parameters swimmers perpendicular capsules long springs equals quantities given terms values lattice sin design table different parameters swimmers perpendicular capsules short springs equals quantities given terms values lattice references alouges desimone lefebvre optimal strokes axisymmetric microswimmers european physical journal soft matter biological physics anitescu potra formulating dynamic contact problems friction solvable linear complementarity problems nonlinear dynamics bhatnagar gross krook model collision processes gases small amplitude processes charged neutral onecomponent systems physical review binder feichtinger schmid peukert simulation hydrodynamic drag aggregated particles journal colloid interface science chang paunov petsev velev remotely powered particles micropumps based miniature diodes nature materials chen doolen lattice boltzmann method fluid flows annual review fluid mechanics cundall strack discrete numerical model granular assemblies geotechnique dreyfus baudry roper fermigier stone bibette microscopic artificial swimmers nature earl pooley ryder bredberg yeomans modeling microscopic swimmers low reynolds number journal chemical physics feichtinger donath walberla hpc software design computational engineering simulations journal computational science press corrected proof felderhof swimming animalcules physics fluids golestanian ajdari analytic results swimmer low reynolds number physical review golestanian yeomans uchida hydrodynamic synchronization low reynolds number soft matter iglberger feichtinger donath coupling multibody dynamics computational fluid dynamics processor cores parallel computing iglberger direct numerical simulation particulate flows processor cores international conference high performance computing networking storage analysis ieee iglberger software design massively parallel rigid body framework thesis iglberger massively parallel granular flow simulations particles computer science research development iglberger simulation moving particles lattice boltzmann method computers mathematics applications kanevsky shelley tornberg modeling simple locomotors stokes flow journal computational physics pohl zeiser parallel lattice boltzmann methods cfd applications bruaset tveito eds numerical solution partial differential equations parallel computers volume lecture notes computational science engineering springer verlag kosa shoham zaaroor propulsion method swimming microrobots robotics ieee transactions lauga powers hydrodynamics swimming microorganisms reports progress physics lighthill squirming motion nearly spherical deformable bodies liquids small reynolds numbers communications pure applied mathematics lorenz caiazzo hoekstra corrected momentum exchange method lattice boltzmann simulations suspension flow physical review mei luo lallemand consistent initial conditions lattice boltzmann simulations computers fluids proceedings first international conference mesoscopic methods engineering science mei shyy luo lattice boltzmann method flows curved boundary journal computational physics najafi golestanian simple swimmer low reynolds number three linked spheres physical review pickl rigid body dynamics links joints master thesis computer science department system simulation university pickl pande iglberger mecke smith effect shape efficiency swimmers submitted physical review letters pohl kowarschik wilke iglberger optimization profiling cache performance parallel lattice boltzmann codes parallel processing letters pooley yeomans lattice boltzmann simulation techniques simulating microscopic swimmers computer physics communications purcell life low reynolds number american journal physics putz yeomans hydrodynamic synchronisation model microswimmers journal statistical physics qian lallemand lattice bgk models equation europhysics letters epl succi lattice boltzmann equation fluid dynamics beyond clarendon press taylor analysis swimming microscopic organisms proceedings royal society london series mathematical physical sciences wellein zeiser donath hager single processor performance simple lattice boltzmann kernels computer fluids yang marceau gompper swarm behavior rods swimming flagella physical review mei luo shyy viscous flow computations method lattice boltzmann equation progress aerospace sciences
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identity testing high probability aug ilias usc diakonik themis csail mit tgoule john csail mit jpeebles eric price austin ecprice august abstract study problem testing identity given distribution focus high confidence regime precisely given samples unknown distribution elements explicitly given distribution parameters wish distinguish probability least whether distributions identical versus total variation statistical distance existing work focused constant confidence regime case sample complexity identity testing known typical applications distribution property testing require small values confidence parameter correspond small statistical hypothesis testing terminology prior work achieved arbitrarily small values via amplification multiplies required number samples log show upper bound suboptimal give new identity tester achieves optimal sample complexity new upper lower bounds show optimal sample complexity identity testing log log special case uniformity testing given distribution uniform distribution domain new tester surprisingly simple test whether versus dtv simply threshold dtv empirical probability distribution believe novel analysis techniques may useful distribution testing problems well supported nsf award career sloan research fellowship supported nsf grant supported nsf graduate research fellowship grant nsf grant introduction background distribution property testing studies following family problems given sample access one unknown distributions determine whether satisfy global property far satisfying property broad inference task originates field statistics extensively studied hypothesis testing somewhat different formalism work study following standard formalization statistical task given family distributions domain size parameters sample access unknown distribution domain want distinguish probability least following cases completeness soundness dtv call problem property denote dtv total variation distance statistical distance distributions def dtv similarly denote dtv minimum total variation distance goal characterize sample complexity problem number samples necessary sufficient correctly distinguish completeness soundness cases high probability past two decades general problem received significant attention within computer science community emphasis pinning sample complexity testing constant probability success regime setting see sample works two surveys constant confidence regime fairly well understood range fundamental properties testers matching lower bounds sharp contrast explain high confidence regime case poorly understood even basic properties since testing decision problem algorithm succeeds probability used along standard bpp amplification boost success probability desired accuracy specifically sample complexity upper bound method implies log sample complexity upper bound essentially distribution properties studied literature standard amplification method known way achieve high confidence probability discussion naturally leads following questions given distribution property amplification give testers specifically log multiplicative increase sample size best possible design testers whose sample complexity optimal function parameters believe fundamental question merits theoretical investigation right settings sample size known priori presented sample size able draw samples distribution cases interested characterizing optimal tradeoff curve proximity parameter error probability previous work distribution testing characterized special case tradeoff corresponding note analogous question context distribution learning intensely studied statistics probability theory see tight bounds known range settings moreover understanding regime small critical practical importance applications hypothesis testing biology corresponds bounding probabilities type type errors perhaps surprisingly one exception basic problem previously investigated finite sample regime conceptual contribution work raise problem fundamental goal distribution property testing results paper characterize aforementioned tradeoff problem identity testing goodness fit identity testing corresponds property explicitly given distribution domain size important special case uniform distribution domain referred uniformity testing identity testing arguably fundamental distribution testing problem problem show amplification suboptimal give new identity tester achieves optimal sample complexity specifically prove following theorem theorem main result exists computationally efficient tester distributions support size sample complexity log log moreover sample size optimal constant factor since sample complexity identity testing standard amplification gives sample upper bound log problem hard observe bound optimal values example extreme case gives sample complexity hand values learn underlying distribution therefore test identity case subtle priori clear improve upon naive amplification theorem provides smooth transition extremes constant thus provides quadratic improvement dependence naive bound shows best possible turns additive log term necessary outlined section learning distribution optimal main technical contribution obtain first uniformity tester high confidence regime identity tester follows uniformity tester applying recent result goldreich provides reduction identity uniformity also show matching lower bound sample complexity uniformity tester introduce remarkably simple distinguish cases uniform distribution elements versus dtv simply compute dtv empirical distribution tester accepts value statistic threshold rejects otherwise noted tester previously known work sample complexity even constant confidence regime surprisingly literature several different uniformity testers one previously proposed using empirical total variation distance fact would natural assume suggested tester possibly work likely reason following observation sample size smaller domain size empirical total variation distance far true distance uniformity suggests empirical distance statistic gives little information setting follows fact distribution elements empirical probability distribution pbm obtained log samples drawn total variation distance probability least show paper contrast intuition natural estimator relying empirical distance uniformity actually works following reason empirical distance uniformity noticeably smaller uniform distribution far uniform distributions even sample size moreover obtain stronger statement estimator uniformity tester parameters discussion prior work uniformity testing first one problems distribution testing already mentioned literature almost exclusively focused case constant error probability first uniformity tester introduced goldreich ron counts number collisions among samples shown work samples related tester proposed paninski relies number distinct elements set samples shown optimal sample complexity long recently tester shown achieve optimal sample complexity without restrictions finally original tester recently shown also achieve optimal sample complexity thus situation constant values well understood uniformity knowledge property regime previously considered itp shown tester achieves optimal sample complexity log long emphasize case many practically relevant settings see polish lottery example tester known fail completely even constant confidence regime also see next paragraph important note previously considered uniformity testers achieve optimal sample complexity function parameters including inherent failure previous analyses roughly speaking since collision statistic statistic lipschitz shown performance poor specifically completeness case many samples happen land bucket domain element test statistics become quite large leading suboptimal behavior formal justification reader referred section hand tester work well example sufficiently small necessitate log typically domain elements appear completeness soundness cases hence test statistic provides information interestingly empirical total variation distance statistic seen closely related tester regime show empirical total variation distance written linear function number elements never sampled contrast statistic number elements seen exactly however prove tester fact optimal parameter settings whereas distinct elements tester problem identity testing arbitrary explicitly given distribution studied gave sample complexity tight bound first given using type tester inspired subsequent work similar tester also achieves sample complexity bound given note testers sample complexity high confidence regime even case uniformity related work obtained reduction identity uniformity preserves sample complexity constant factor constant error probability regime recently goldreich building gave different reduction identity uniformity preserves error probability use latter reduction paper obtain optimal identity tester starting new optimal uniformity tester techniques upper bound uniformity testing would like show test statistic dtv high probability larger dtv start showing among possible alternative distributions dtv suffices consider simple family show test statistic highly concentrated around expectation expectations significantly different two cases simplify structure show section majorizes another distribution test statistic dtv stochastically dominates dtv fact statement holds test statistic convex symmetric function empirical histogram therefore average large small entries test statistic becomes harder distinguish uniform lets reduce considering either form coordinate remark aforementioned stochastic domination lemma may also useful rigorous empirical comparisons test statistics major difficulty empirical studies distribution testing space alternative hypotheses large therefore impossible experiment distributions structural lemma reduces space dramatically uniformity testing convex symmetric test statistic includes existing ones worst case distribution coordinates value rest value hence possible worstcase distributions notably reduction lose absolute constants could used identify optimal constants given set parameters returning uniformity tester need separate expectation test statistic two situations achieve providing explicit expression hessian expectation function note hessian diagonal two situations entry within constant factors value giving lower bound eigenvalues since expectation minimized use strong convexity show desired expectation gap specifically prove gap min finally need show test statistic concentrates expectation follows mcdiarmid inequality since test statistic samples probability lies within log expectation larger desired sample complexity less expectation gap concentration trickier since expectation gap smaller need establish tighter concentration get using bernstein variant mcdiarmid inequality stronger standard version mcdiarmid context upper bound identity testing shown reduce arbitrary distribution reduction preserves error probability applying gives identity tester sample complexity uniformity tester constant factors sample complexity lower bound match upper bound need two lower bounds lower bound log straightforward lower bound distinguishing fair coin bound challenging coin log intuition start log lower bound constant chance samples distinct least hence log would happen probability significantly larger hand uniform random subset coordinates samples also distinct probability two situations thus look probability tester could accuracy intuition easily extended include dependence getting desired dependence requires work first poissonize number samples independently see poi mpi samples coordinate exponentially high probability poissonization affects sample complexity constant factors alternative hypothesis set independently random unfortunate property longer sums rather actual distribution still exponentially likely sum using techniques sufficient purposes point considering situation number times see coordinate either poi poi poi every coordinate independent others two distributions hellinger distance least coordinate composition property hellinger distance independent coordinates implies log necessary success probability notation basic definitions write denote set consider discrete distributions functions use notation denote probability element distribution denote also sometimes think vector denote uniform distribution distribution identified corresponding vector kpkr distributions defined vector difference total variation distance distributions defined def dtv hellinger distance def denote poi poisson distribution parameter structure paper section formally describe analyze uniformity tester section give matching sample complexity lower bound finally section establishes stochastic domination result crucial analysis soundness section may useful rigorous empirical evaluation test statistics uniformity testing section describe analyze optimal uniformity tester given samples unknown distribution tester returns yes probability probability dtv test statistic define natural statistic yields uniformity tester optimal dependence domain size proximity parameter error probability statistic thresholded version empirical total variation distance unknown distribution uniform distribution tester niformity described following pseudocode algorithm niformity input sample access distribution output yes dtv draw log log iid samples let histogram samples number times domain element appears set samples define random variable set threshold universal constant derived analysis algorithm expected value statistic completeness case return otherwise return yes main part section devoted analysis niformity establishing following theorem theorem exists universal constant following holds given log log samples unknown distribution algorithm niformity uniformity tester point appendix value computed efficiently hence overall tester computationally efficient prove correctness tester need show expected value statistic completeness case sufficiently separated expected value soundness case also value statistic highly concentrated around expectation cases section bound difference expectation statistic completeness soundness cases section prove desired concentration completes proof theorem bounding expectation gap expectation statistic algorithm niformity viewed function def variables let aforementioned function samples drawn distribution analysis number complications following reason function linear combination sums indefinite closed form even distribution assigns two possible probabilities elements domain statement made precise appendix hope obtain approximation quantity natural approach try obtain approximation would produce separate closed form approximations combine quantities obtain approximation difference however one expect approach work context reason difference much smaller even arbitrarily small obtaining separate approximations fixed accuracy would contribute much error difference overcome difficulties introduce following technique novel context directly bound difference using strong convexity specifically show function strongly convex appropriate parameters use fact bound desired expectation gap main result section following lemma lemma let distribution dtv note bounds right hand side tight constant factors asymptotic improvement would yield uniformity tester sample complexity violates tight informationtheoretic lower bounds proof lemma requires number intermediate lemmas starting point follows intermediate value theorem quadratic expansion hessian matrix function point lies line segment expression simplified follows first show fact minimized probability distributions input thus gradient must orthogonal direction space probability distributions words must proportional vector formally since symmetric gradient symmetric function implies symmetric given symmetric input moreover direction within space probability distributions therefore sums making orthogonal vector thus obtain denotes minimum eigenvalue hessian line segment majority section devoted proving lower bound however must first address technical consideration considering function space probability distributions hessian gradient respect depend definition statistic also parameterization purposes subsection parameterize max analysis perform helpful sometimes think free parameter thus define max note also note compute hessian treating function following lemma derive exact expression entries hessian result perhaps surprising light likely nonexistence closed form expression expectation may closed form prove hessian fact closed form lemma hessian viewed function diagonal matrix whose ith diagonal entry given hii define follows let distance next largest integer words derive formula integral prove value nonintegral found linearly interpolating closest integral values proof note separable function separable function hence hessian diagonal matrix equation diagonal entry hessian written explicitly following expression dpi notice sum starting instead sum equals expectation bin minus notice observation fact summand switch values summing negate expression first prove case case view sequence respect fixed denote derive generating function observe derivatives respect formal variable commute taking generating functions generating function sequence avoid potential convergence issues view generating functions formal polynomials ring infinite formal polynomials formalism need deal convergence note coefficient claimed right hand side generating function thus expression gives entry hessian claimed consider case integer case dpi dpi dpi dpi last equality change bounds sum get partial derivative thus flip terms summing negate expression note expression subtracting alternatively written dpi thus desired completes proof lemma convenient simplify exact expressions lemma something manageable done following lemma lemma fix constant hessian viewed function diagonal matrix whose diagonal entry given hii assuming similarly bounds tight constant factors improvements would violate sample complexity lower bounds proof lemma exact expression ith entry hessian first consider case substituting gives consider case note case follows fact fractional linearly interpolates value nearest two integral values analyses cases thus left prove case since convex combination suffices bound quantities tasks accomplished simultaneously bounding quantity arbitrary follows let note stirling approximation tight constant factors long number taking factorial zero note thus apply stirling approximation factorials definition binomial coefficient substitute obtain following approximation tight constant factors npi completes proof lemma ready prove desired expectation gap proof lemma start reducing soundness case much simpler setting use following fact established section fact distribution exists distribution supported whose probability mass values set dtv one element mass fact proven section fact distribution satisfies conditions lemma total variation distance uniform distribution therefore suffices prove lower bound expectation gap completeness soundness cases distributions form note probability distributions line also form different larger values thus lemma gives lower bound diagonal entries hessian points line since hessian diagonal also bounds minimum eigenvalue hessian line therefore equation obtain first two cases lemma well third case final case lemma follows immediately folklore fact one takes least many samples empirical distribution approximates true distribution expected error completeness give proof var mpi equation follows fact sum symmetric concave function maximized setting equal concentration test statistic proof theorem let samples let number let empirical total variation test statistic prove theorem two parts one one require bernstein form standard bounded differences mcdiarmid inequality lemma bernstein version mcdiarmid inequality let independent random variables taking values set let function every exp addition varyj exp case first form mcdiarmid inequality since independent implies exp similarly applying exp let side equation lemma soundness case since threshold tester find completeness soundness cases success probability least exp hence need show log since regime two possible cases lemma need log need log log log theorem assumption implies conditions hold completes proof theorem case case establish theorem require bernstein form mcdiarmid inequality equation lemma apply form lemma suffices compute test statistic function note equal whenever particular implies hence value parameter test statistic since affect number nonzero particular function value varies kept fixed written sum deterministic quantity plus bernoulli random variable sample collides another sample otherwise thus variance varies kept fixed given var variance probability collides another thus variance varies kept fixed applying equation lemma find exp lemma soundness case expectation gap constant substituting concentration inequality yields tester correct probability long log appropriately chosen constant true assumption completes proof theorem matching lower bound section prove matching sample complexity lower bound namely prove theorem algorithm distinguishes probability least uniform distribution distribution uniform total variation distance requires least log log samples theorem immediately follow separate sample complexity lower bounds log log prove start simple sample complexity lower bound log lemma uniformity tester requires log samples proof odd set last probability subtract invoke following lower bound instance remaining elements even following consider distribution probability element element clearly dtv note probability sample comes first half domain probability comes second half domain therefore distinguishing equivalent distinguishing fair coin coin see chapter task requires log samples rest section devoted proof following lemma gives desired lower bound lemma uniformity tester requires least log samples prove lemma construct two indistinguishable families measure similar probability distribution except probabilities may sum something require always sum quantity within constant factor pair families said using samples tester exists every pair one two families distinguish product distributions poi mwi versus poi failure probability technique fairly standard used establish lower bounds distribution testing problems benefit method much easier show pseudodistributions indistinguishable opposed working ordinary distributions moreover lower bounds proven using imply lower bounds original distribution testing problem require following lemma whose proof implicit analyses lemma let property distributions extend unique property pseudodistributions agrees true distributions preserved rescaling suppose two families following properties property total variation distance property using samples every family within interval constants exist two families probability distributions following properties distributions property distribution property total variation distance tester distinguish error probability constant requires samples case property distributions simply uniform distribution families use lower bound family contains thepuniform distribution family form note constraint sum ensures first last conditions needed invoke lemma furthermore lemma required number samples satisfies log ignoring constant factors may assume log constant particular selecting appropriately guarantee constant last statement lemma thus error probability guaranteed lemma distinguishing true distribution families least thus remains show using samples order show families indistinguishable show impossible distinguish whether product distribution poi mwi uniform generated according following random process pick independently setting probability distribution generated process small probability specifically happens iff fails satisfy constraint sum within however application constraint first condition would satisfied proportional vector chernoff bound follows happens probability mots since bound lemma larger constant factors lower bound presently wish prove case assume case following lemma implies still invoke lemma probability absorbed overall indistinguishability probability get final indistinguishability probability least following lemma implicit lemma let property suppose two families pseudodistributions two distributions respectively following properties probability least distribution output distribution output generate according algorithm determining family came given access poi mwi worst case error probability least using samples thus simply need show hard distinguish let random variables equal number times element sampled completeness soundness cases respectively require technical lemma used bound hellinger distance pair corresponding coordinates completeness soundness cases poi poi denote uniform mixture corresponding distributions fact lemma poi poi poi constant ready prove lemma proof lemma follows preceding discussion suffices show hard distinguish use fact show hellinger distance overall distributions small implies total variation distance small hence distinguished probability better recall coordinates vectors output distributions distributed according poi uniform mixture poi single coordinate collection coordinates latter quantity least need log samples desired result follows lemma lemma completes proof stochastic domination statistics histogram section consider set statistics symmetric convex functions histogram number times domain element sampled arbitrary random variable start following definition definition let probability distributions denote vectors values respectively sorted order say majorizes denoted following theorem gives equivalent definition theorem let pair probability distributions exists doubly stochastic matrix remark shown multiplying distribution doubly stochastic matrix equivalent performing series called robin hood operations permutations elements robin hood operations operations probability mass transferred heavier lighter elements details reader referred note definition defines partial order set probability distributions see uniform distribution minimal element partial order directly follows special case following lemma lemma let probability distribution let distribution identical every denotes cardinality proof let aij doubly stochastic matrix aij entries aij otherwise observe therefore theorem implies rest section use following standard terminology say real random variable stochastically dominates real random variable holds state main result section see section proof lemma let symmetric convex function distribution suppose draw samples let denote number times sample element let random variable distribution stochastically dominates simple consequence obtain following fact let distribution let distribution identical probabilities averaged denotes expectation statistic defined section particular proof recall statistic applies symmetric convex function histogram sampled distribution since averaging probability masses subset lemma gives therefore lemma conclude stochastically dominates implies shown following lemma shows given arbitrary distribution uniform distribution average heaviest elements lightest elements get distribution uniform lemma let probability distribution distribution obtained averaging heaviest lightest elements separately following holds note averaging suggested lemma obtain distribution supported following set three values hence reduce computation expectation gap arbitrary distribution computing gap distribution form fact immediate corollary lemmas proof lemma establish lemma going use following intermediate lemmas lemma let symmetric convex function proof consider set convex functions defined show every possible choice holds since symmetric therefore points collinear since coordinates satisfy equation since applying jensen inequality get desired stochastic domination two statistics established following lemma lemma let symmetric convex function distribution also let distribution identical suppose take samples let denote number times sample element let random variable stochastically dominates proof prove stochastic domination going define coupling always true takes larger value initially define auxiliary coupling follows get sample first sample use sample unless output element case output probability suppose draw samples also convert samples using rule relation coupling define following random variables xlow number times element sampled xhigh number times element sampled swapped element xmid number times element sampled swapped element xlow xhigh xmid xlow xmid xhigh number times element sampled want show stochastically dominates want show xlow xhigh xmid xlow xmid xhigh condition events xlow xhigh xmid without loss generality let xlow xhigh xmid xlow xhigh xmid suffices show every xlow xhigh xmid xlow xmid xhigh point conditioned everything except xlow xhigh conditioning event xlow xmid xhigh xlow xhigh xmid note coupling fix value given fixed value defined convenience still show stochastic domination coupled random variables using second coupling simplify notation pick without loss generality since assumption lemma holds trivially equality remaining values equivalent xlow xhigh xmid xlow xmid xhigh hence xlow xhigh xlow xhigh also equivalent version less restricted conditioning xlow xhigh xlow xhigh neither event occurs added regime false rethink samples drawn find equivalent showing pya qbz pza qby holds since concluding proof proof lemma since theorem remark follows constructed repeated applications therefore lemma fact stochastic domination transitive imply stochastically dominates proof lemma recall denotes vector entries rearranged order suppose least elements least probability therefore uniform distribution thus even odd moreover since averaging since assumed majority elements mass least know total variation distance given dtv thus dtv dtv desired without loss generality since use essentially argument case conclusions future work paper gave first uniformity tester constant factors function confidence parameter tester remarkably simple novel analysis may useful related settings using known reduction identity uniformity also obtain first identity tester setting result step towards understanding behavior distribution testing problems highconfidence setting view direction one fundamental theoretical important practical interest number interesting open problems remain perhaps appealing one design general technique see yields testers high confidence regime wide range properties practical standpoint would interesting perform detailed experimental evaluation various algorithms see references acharya das jafarpour orlitsky pan competitive closeness testing journal machine learning research proceedings track acharya daskalakis kamath optimal testing properties distributions advances neural information processing systems nips pages barry arnold majorization lorenz order brief introduction volume springer science business media batu testing properties distributions phd thesis cornell university batu dasgupta kumar rubinfeld complexity approximating entropy acm symposium theory computing pages batu fischer fortnow kumar rubinfeld white testing random variables independence identity proc ieee symposium foundations computer science pages batu fortnow rubinfeld smith white testing distributions close ieee symposium foundations computer science pages batu fortnow rubinfeld smith white testing closeness discrete distributions acm batu kumar rubinfeld sublinear algorithms testing monotone unimodal distributions acm symposium theory computing pages balakrishnan wasserman hypothesis testing densities multinomials sharp local minimax rates corr complexity massive data set computations phd thesis berkeley usa canonne survey distribution testing data big blue electronic colloquium computational complexity eccc canonne diakonikolas gouleakis rubinfeld testing shape restrictions discrete distributions symposium theoretical aspects computer science stacs pages canonne diakonikolas kane stewart testing bayesian networks proceedings conference learning theory colt pages canonne diakonikolas stewart testing families distributions corr chan diakonikolas valiant valiant optimal algorithms testing closeness discrete distributions soda pages daskalakis diakonikolas servedio valiant valiant testing distributions optimal algorithms via reductions soda pages diakonikolas gouleakis peebles price testers optimal uniformity closeness electronic colloquium computational complexity eccc diakonikolas kane new approach testing properties discrete distributions focs pages full version available diakonikolas kane nikishkin optimal algorithms lower bounds testing closeness structured distributions ieee annual symposium foundations computer science focs pages diakonikolas kane nikishkin testing identity structured distributions proceedings annual symposium discrete algorithms soda pages diakonikolas kane nikishkin closeness testing discrete histogram distributions corr appear icalp devroye lugosi combinatorial methods density estimation springer series statistics springer goldreich uniform distribution complete respect testing identity fixed distribution eccc william gosper decision procedure indefinite hypergeometric summation proceedings national academy sciences goldreich ron testing expansion graphs technical report electronic colloquium computational complexity huang meyn generalized error exponents small sample universal hypothesis testing ieee trans inf december indyk levi rubinfeld approximating testing distributions time pods pages lehmann romano testing statistical hypotheses springer texts statistics springer levi ron rubinfeld testing properties collections distributions ics pages albert marshall ingram olkin barry arnold inequalities theory majorization applications volume springer neyman pearson problem efficient tests statistical hypotheses philosophical transactions royal society london series containing papers mathematical physical character paninski test uniformity given discrete data ieee transactions information theory peter paule markus schorn mathematica version zeilberger algorithm proving binomial coefficient identities journal symbolic computation petkovsek wilf zeilberger online edition peters series taylor francis rubinfeld taming big probability distributions xrds rubinfeld taming probability distributions big domains stoc workshop efficient distribution estimation http valiant testing symmetric properties distributions siam talk given available van der vaart wellner weak convergence empirical processes springer series statistics new york applications statistics valiant valiant automatic inequality prover instance optimal identity testing focs yang minimax rates entropy estimation large alphabets via best polynomial approximation ieee transactions information theory june ying mcdiarmid inequalities bernstein bennett forms city university hong kong appendix computation expectation completeness case statistic written max therefore linearity expectation get max max need compute max single value completeness case note bin expectation written max sum terms computed constant time giving runtime overall indefinite components expectation appendix formalize prove assertion section function linear combination sums indefinite closed form recall equation says expectation linear combination sums summands form various values values different variables closed form would need depend addition variables sum said indefinite closed form upper lower limits sum replaced new variables resulting sum closed form valid values variables closed form mean closed form defined definition far aware main formal sense phrase used combinatorics definition closed form says function written sum constant number rational functions numerator denominator linear combination constant number products exponentials factorials constant degree polynomials example function indefinite closed form prove sum summands limits sum variables closed form would need function one run gosper algorithm summand index summation observe returns indefinite closed form solution sense described theorem
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large system analysis power normalization techniques massive mimo may meysam sadeghi student member ieee luca sanguinetti senior member ieee romain couillet senior member ieee chau yuen senior member ieee precoding widely studied context massive mimo together two common power normalization techniques namely matrix normalization vector normalization despite effect performance massive mimo systems thoroughly studied yet aim paper fulfill gap using large system analysis considering system model accounts channel estimation pilot contamination arbitrary pathloss channel correlation compute tight approximations ratio rate user equipment system employing maximum ratio transmission mrt zero forcing regularized precoding techniques approximations used analytically reveal choice power normalization affects performance mrt uncorrelated fading channels turns resembles sum rate maximizer provides notion fairness numerical results used validate accuracy asymptotic analysis show massive mimo noncoherent interference noise rather pilot contamination often major limiting factors considered precoding schemes index mimo linear precoding power normalization techniques large system analysis pilot contamination ntroduction massive mimo multiuser mimo system employs large number antennas base stations bss serve relatively smaller number user equipments ues large number antennas enables focus radiated energy specific location space intercept power transmitted electromagnetic waves efficiently therefore massive mimo higher spectral efficiency energy efficiency compared classical multiuser mimo systems due nature channels copyright ieee personal use material permitted however permission use material purposes must obtained ieee sending request sadeghi meysam yuen yuenchau singapore university technology design sutd singapore sanguinetti university pisa dipartimento ingegneria dell informazione italy also large systems networks group laneas rue france couillet signals statistics group paris france work supported star serc project number sanguinetti couillet supported erc starting grant massive mimo linear precoding detection schemes perform channel reciprocity exploited overhead channel state information csi acquisition independent number antennas moreover recently shown capacity massive mimo increases without bound number antennas increases even pilot contamination remarkable features candidate massive mimo one promising technologies next generation cellular networks linear precoding central role massive mimo extensively studied past years spectral efficiency energy efficiency maximum ratio transmission mrt zero forcing precoding massive mimo systems investigated multicell linear precoding proposed mitigate effect pilot contamination multicell processing also considered performance mrt regularized rzf precoding singlecell mimo systems studied considering channel correlation model seminal treatment mrt rzf precoding schemes multicell massive mimo systems presented followed downlink training linear pilot contamination precoding also considered approximations achievable downlink rates mrt precoding schemes presented multicell massive mimo systems linear truncated polynomial expansion based precoding proposed reduces complexity rzf precoding effect phase noise sinr mrt rzf precoding schemes studied order utilize linear precoding power adjusted meet power constraint done either optimized power allocation among downlink data streams simply uniform power allocation among downlink data streams jointly precoder power normalization although latter approach may provide weaker performance compared former used massive mimo literature reason power allocation presents following major issues finding global solution challenging task certain level coordination cooperation among cells required iii performed frequently even static users scheduling may change rapidly practice two commonly used power normalization techniques massive mimo matrix normalization vector normalization precoding matrix adjusted multiplying scalar power constraint met hand precoding matrix normalized equal amount power allocated satisfying power constraint note two methods yield performance optimal power allocation practical suboptimal power allocation although linear precoding largely studied massive mimo detailed treatment impact power normalization exist literature first attempt direction carried extended wherein authors study impact mrt precoding schemes however grasp essence practical massive mimo system since network composed three radio units considered perfect csi assumed thus csi acquisition pilot contamination accounted iii attenuation neglected though fundamental impact power normalization detailed later goal paper study effect performance mrt rzf massive mimo simple practical case uniform power allocation particularly following contributions provided extend analysis multicell massive mimo system accounts channel estimation pilot contamination arbitrary pathloss model peruser channel correlation asymptotically tight approximations ratio sinr rate provided validated numerical results mrt rzf explicit asymptotic approximations sinr rate given rayleigh fading channel model results used elaborate two different normalization techniques affect signal noise interference powers well pilot contamination experienced system prove fading fundamental role performance provided two normalization techniques perform neglected iii show conveys notion sum rate maximization fairness asymptotic approximations sinrs used together numerical results study main limiting factors investigated schemes particularly reveal massive mimo interference noise rather pilot contamination often major limiting factors schemes remainder paper organized follows section introduces network model channel estimation scheme precoding power normalization methods well downlink achievable rates large system analysis provided section iii effect power normalization techniques elaborated section uncorrelated fading channels provided asymptotic approximations verified means numerical results section conclusions drawn section notations following notation used throughout paper scalars denoted lower case letters whereas boldface lower upper case letters used vectors matrices denote identity matrix size represent element ith row kth column symbol denotes circularly symmetric complex gaussian distribution trace transpose conjugate transpose real part expectation operators denoted respectively notation represent almost sure convergence ommunication cheme next introduce system model channel estimation method precoding power normalization techniques compute downlink achievable rates system model consider downlink massive mimo system composed cells set cells denoted cell antennas serves ues resource set ues belonging cell denoted assume transmissions channels employ double index notation refer user cell convention let hjlk channel cell within block assume hjlk zjlk zjlk accounts corresponding channel correlation matrix note enables assign unique correlation matrix pair includes many channel models literature special cases channel estimation assume bss ues perfectly synchronized operate according duplex tdd protocol channels found uplink training phase used downlink exploiting channel reciprocity using orthogonal pilots cell reusing cells correlating received training signal pilot sequence observed channel user cell yjk hjjk hjlk njk njk noise spectral density proportional training snr applying mmse estimation estimated channel computed follows qjk yjk also qjk given qjk qjk note due orthogonality principle mmse jjk hjjk independent estimation error hjjk notational simplicity denote matrix collecting estimated channels cell precoding power normalization techniques mentioned earlier consider mrt rzf denoting gjk precoding matrix gjk precoding vector cell fjk determines precoding scheme characterizes power allocation strategy therefore takes one following forms mrt rzf regularization parameter arbitrary hermitian nonnegative definite matrix used leverage system performance mentioned introduction finding optimal values elements challenging practice usually employed case diagonal entries chosen satisfy following average power constraint trgj used kth diagonal element computed djk fjk hand employed trfj downlink achievable rate received signal user cell written yjk jjk gjk sjk jjk gji sji ljk gli sli njk among many others assume downlink pilots ues knowledge current channels learn average channel gain jjk gjk total interference power note common approach massive mimo due channel hardening using technique ergodic achievable information rate cell obtained rjk given jjk gjk ljk gli jjk gjk expectation taken respect channel realizations result holds true precoding scheme obtained treating interference cells channel uncertainty gaussian noise using sinr takes respectively form given top next page precoding schemes depend statistical distribution hjlk makes hard compute overcome issue large system analysis provided next find tight asymptotic approximations hereafter called deterministic equivalents associated achievable rates iii arge ystem nalysis consider regime grow large ratio lim inf lim sup represent assumption provide asymptotic approximations also called deterministic equivalents des mrt rzf either represented applying continun ous mapping theorem almost sure convergence results illustrated implies rjk rjk rjk denotes one asymptotic approximations computed limiting cases considered following conditions widely used literature needed lim lim inf hll lim rank sli signal intended cell assumed independent across pairs zero mean unit variance njk proportional downlink signal power large system results vector normalization subsection derive des mrt rzf precoding schemes used djk var jjk fjk var jjk fjk djk jjk fjk dli dlk flk ljk fli jjk fjk ljk ljk dlk dli zli zli proof proof provided appendix theorem let hold true rzf used almost surely ljk uli uljk ulk ulk ulk znl also given theorems appendix proof proof provided appendix theorem let hold true employed almost surely ujk uli theorem let hold true mrt used almost surely ulk uli uljk ulk ulk uli uli uli computed entries given uli proof proof provided appendix notice computation des precoding either considered multicell massive mimo system involved mrt rzf precoding schemes mainly due fact straightforward start precoder compute des applying common techniques matrix inversion lemma therefore proving theorem also theorem start rzf use bounding limiting technique compute large system results matrix normalization next des given mrt rzf note des mrt rzf obtained theorem theorem let hold true mrt used almost surely zli zli given theorem theorem let hold true rzf almost used surely ffect ower ormalization echniques section use asymptotic approximations provided gain novel insights interplay different system parameters power normalization techniques massive mimo end consider special case general channel model djlk hjlk djlk zjlk zjlk djlk accounts arbitrary fading coefficient including pathloss shadowing note corresponds uncorrelated fading channel model quite popular model massive mimo allows capture essence technology circumstances corollary let channel modelled ljk trtl theorem let hold true almost surely used theorem uji uli uljk ulk uli uljk given dllk dljk dljk zdl noise interference dlnk proof see appendix corollary let channel modelled mrt used proof sketch proof follows procedure proof theorem presented appendix start triangle equality bound find letting asymptotic expressions provided theorems shown tight even systems finite dimensions means numerical results section allows use evaluating performance practical massive mimo systems without need monte carlo simulations moreover lay foundation analysis different configurations massive mimo systems distributed massive mimo systems next used get insights system investigation uncorrelated fading channels pilot contamination pilot contamination znl given theorem theorem also uli uljk defined theorem pilot contamination dllk dljk pilot contamination dljk zdl noise interference proof proof follows similar procedure corollary results corollaries instrumental obtaining following insights mrt either remark effect terms means normalization techniques exactly effect resulting noise interference terms experienced system hand affect differently signal pilot contamination powers expressions explicitly state relation sinr contributions signal interference noise pilot contamination propagation environment two normalization techniques mrt precoding schemes remark mutual effect ues employed signal power pilot contamination cell mrt precoding depends coefficients dlnk means affected gains ues network using pilot hand terms depend coefficients mrt thus influenced ues network even though make use different pilot sequences remark fading power normalization assume fading neglected every network dljk expressions mrt become equal means fading fundamental impact ignored consider simplicity setup dropping cell index reduces also assume ues operate high training snr regime conditions lemma precoding outperforms terms sum rate sum rate gap given log log proof corollary setting assuming obtain result follows applying jensen inequality convexity log notice lemma extends results system accounts csi acquisition arbitrary pathloss ues distribution also observe simplify follows provides higher sinr ues closer lower sinr far away resembles opportunistic resource allocation hand provides uniform quality experience ues proves evidence fact resembles sum rate maximizer hand provides notion fairness notice fairness means similar sinr quality experience confused equal power allocation results observations validated section means numerical results also des provided corollaries used investigate main limiting factors massive mimo umerical results simulations used validate asymptotic analysis different values consider multicell network composed cells one center six around cell radius meters mhz channel considered thermal noise power assumed ues randomly uniformly distributed within cell excluding circle radius meters channel modeled particular assume matrices given dljk given sin sin antenna spacing also dljk attenuation modeled dljk ljk xljk denotes distance cell exponent let corresponds practical setting results obtained different channel distributions realizations figs validate accuracy des provided theorems particular figures report ergodic achievable sum rate center cell versus respectively solid lines correspond asymptotic sum rate whereas markers achieved monte carlo simulation depicted asymptotic approximation match perfectly numerical results notice figs also table extend results sense account csi acquisition pilot contamination arbitrary pathloss ues distribution lemma shown conveys notion sum rate maximization resembles fairness provisioning precoder use table table sinr average sum rate center cell ant number antennas average sum rate center cell fig ergodic achievable sum rate center cell mrt rzf number antennas fig ergodic achievable sum rate center cell mrt rzf validate observation also verify accuracy computed des simplified channel model first column table reports number antennas second one index third fourth columns asymptotic given simulated sinrs corresponding results reported sixth seventh columns fifth eighth columns report percentage error estimating specific sinr computed des predicted lemma provides uniform experience ues provides high sinrs specific ues ues much lower sinrs others precisely sinr variance equal equal notice also percentage error always less proves high accuracy des therefore one simply use des achieve insight network performance instead using timeconsuming monte carlo simulations moreover des contain randomness purely based largescale statistics network hence used network optimization purposes des given corollaries theorems used investigate common belief massive mimo literature uncorrelated fading noise interference contributions vanish asymptotically pilot contamination becomes unique bottleneck system performance follows also results corollaries letting grow large kept fixed however shown desirable massive mimo systems work regime therefore interesting see major impairment massive mimo practical regime pilot contamination coherent interference noise interference exactly noncoherent interference iii answer related choice power normalization technique precoding scheme answer questions employ pilot ratio pcinr metric computed using des provided corollaries fig plots pcinr function number degrees freedom system although optimal operating regime maximal spectral efficiency consider cover wider range massive mimo configurations moreover interference increases ues system consider three different scenarios fig divided regions based significance pcinr term move away region towards region importance pilot contamination increases interference plus noise reduces region noise interference dominant limiting factors pilot contamination negligible noise interference depicted mrt operates within regime therefore pilot contamination never bottleneck scheme mainly limited noise interference notice adding ues system pcinr reduces pilot contamination becomes even less important hence mrt studied massive mimo effect pilot contamination safely neglected region represents regime noise based corollary based theorem based corollary based theorem based corollary based corollary based corollary based theorem based corollary based theorem based corollary based corollary region region average pcinr average pcinr region region region region degree freedom per degree freedom per average pcinr region based corollary based theorem based corollary based theorem based corollary based corollary region region degree freedom per fig pcinr versus degree freedom different values ference main limiting factors pilot contamination negligible interesting observe schemes massive mimo often operates within region shows although pilot contamination major challenge massive mimo interference noise still leading role limiting system performance finally region presents superiority pilot contamination effect fig shows superiority interference noise pilot contamination zfmn antennas system requires experience superiority pilot contamination interference noise increases fig see also given value value pcinr considered schemes ordered based discussion clear choice precoding scheme normalization technique change importance pilot contamination interference noise dramatically considered carefully designing massive mimo systems onclusions linear precoding schemes mrt fundamental role massive mimo although precoding schemes employed optimized power control policies usually implemented simple matrix vector power normalization techniques due complexity attaining optimal power control policies requires coordination cooperation among cells computationally demanding algorithms hand two precoding power normalization techniques simple efficient work made use large system analysis compute tight asymptotic approximations sinr experienced system using results used evaluate performance practical massive mimo systems without need monte carlo simulations uncorrellated fading channels analytically showed treat noise interference manner different effects pilot contamination received signal power also revealed key role played fading positions ues pilot assignment power normalization explained simple change power normalization resemble two totally different behaviors namely maximization fairness provisioning moreover showed numerically choice normalization technique change main bottleneck massive mimo systems asymptotic approximation lli computed follows lli lli lli tre lli applied lemma appendix used fact ppendix let define convenience follows lemma appendix obtained applying lemmas follows theorems znl applying continuous mapping theorem dominated convergence theorem yields uli main idea first compute rzf obtain letting using triangle inequality bounded follows plugging yields divide numerator denominator replace term notice des signal power component variance component interference components given therefore need compute coefficient therefore term made arbitrarily small applying theorem second term consider third term let define observe dli lli next show term right hand side made arbitrarily small smaller given let start notice different different form rzf lim flk lim flk ppendix begin plugging given obtain divide numerator denominator define dli applying continuous mapping theorem placing component obtained notice des signal power component variance component interference components given therefore need compute coefficient latter given lli ppendix dli lim lim lim equivalent ljk define ulk every based replacing ulk theorem ulk lim lim uli also defining uljk every uljk lim term dli obtain lim lim uli uli uli lim lim uli lli given theorem appendix notice lim lim ult ult replacing ult reduces uli therefore ulk thus conclude uli hand lim ulk ulk lim ult ult ult theorem lim given respectively therefore follows using results completes proof ppendix brevity consider steps used channel modelled dljk dllk dljk yields dlnk ulk call plugging therefore solving respect yields eventually ulk also uljk term dllk dljk therefore pilot contamination reduces ulk let compute defined using results yields uli similarly compact form may write btl applying lemma llk btl llk ulk plugging result produces lli uli thus left evaluating using results yields dljk dljk dllk dllk using obtain dljk uli defined llj ulj uli therefore dljk dljk uli plugging produces dljk dljk uli collecting results together completes proof ppendix seful esults theorem theorem let random independent column vectors hermitian nonnegative definite assume matrices uniformly bounded spectral norms respect define mbl trql grow large lim inf lim sup mbl given trql given uli initial values ulk uln lemma matrix inversion lemma let invertible matrix invertible xxh xxh lemma trace lemma let assume uniformly bounded spectral norm respect mutually independent independent tra lemma perturbation lemma let deterministic uniformly bounded spectral norm random hermitian eigenvalues probability exist large tran tran exist probability elements ulk defined uli uli ulk given theorem computed theorem let hermitian nonnegative definite uniformly bounded spectral norm respect conditions theorem trql trql eferences marzetta noncooperative cellular wireless unlimited numbers base station antennas ieee trans wireless vol november rusek persson lau larsson marzetta edfors tufvesson scaling mimo opportunities challenges large arrays ieee signal proc vol larsson edfors tufvesson marzetta massive mimo next generation wireless systems ieee commun vol liu yuen guan convergence analysis assurance gaussian message passing iterative detector massive systems ieee trans wireless vol hoydis ten brink debbah massive mimo cellular networks many antennas need ieee sel areas vol ngo larsson marzetta energy spectral efficiency large multiuser mimo systems ieee trans vol zuo zhang yuen jiang luo downlink transmission multicell massive das pilot contamination ieee trans veh vol andrews buzzi choi hanly lozano soong zhang ieee sel areas vol marzetta much training required multiuser mimo ieee asilomar conf signals sys pacific grove hoydis sanguinetti pilot contamination fundamental asymptotic limitation massive mimo proc ieee int conf commun icc paris france may online available https hoydis sanguinetti massive mimo unlimited capacity ieee trans wireless submitted online available https larsson edfors tufvesson marzetta massive mimo next generation wireless systems ieee commun vol boccardi heath lozano marzetta popovski five disruptive technology directions ieee commun vol yang marzetta performance conjugate zeroforcing beamforming antenna systems ieee sel areas vol jose ashikhmin marzetta vishwanath pilot contamination precoding tdd systems ieee trans wireless vol wagner couillet debbah slock large system analysis linear precoding correlated miso broadcast channels limited feedback ieee trans inf theory vol jul zuo zhang yuen jiang luo multiuser massive mimo transmission downlink training pilot contamination precoding ieee trans veh khansefid minn achievable downlink rates mrc precoders massive mimo uplink downlink pilot contamination ieee trans vol kammoun debbah linear precoding based polynomial expansion mimo systems ieee sel topics signal vol krishnan khanzadi krishnan amat eriksson schober linear massive mimo precoders presence phase noise analysis ieee trans veh vol may bjornson larsson zhou wang multicell mmse precoder massive mimo systems new large system analysis ieee int conf global commun globecom san diego dec sadeghi yuen massive mimo multicasting asymptotic analysis proc ieee global commun conf globecom san diego chien bjornson larsson joint power allocation user association optimization massive mimo systems ieee trans wireless vol zhang jin mckay zhu power allocation schemes multicell massive mimo systems ieee trans wireless vol sadeghi sanguinetti couillet yuen reducing computational complexity multicasting antenna systems ieee trans wireless appear online available https lee chae kim choi lee network massive mimo users precoding normalization perspective ieee globecom workshops wkshps anaheim lim chae caire performance analysis massive mimo users ieee trans wireless vol caire jindal kobayashi ravindran multiuser mimo achievable rates downlink training channel state feedback ieee trans inf theory vol jun bjornson larsson marzetta massive mimo ten myths one critical question ieee commun vol medard effect upon channel capacity wireless communications perfect imperfect knowledge channel ieee trans inf theory vol may billingsley probability measure john wiley sons sanguinetti moustakas debbah interference management reverse tdd hetnets wireless backhaul large system analysis ieee sel areas vol jun sanguinetti moustakas debbah large system analysis energy consumption distribution mimo systems mobility ieee trans wireless vol mar truong heath viability distributed antennas massive mimo systems asilomar conf signals syst pacific grove sadeghi yuen chew sum rate maximization uplink distributed massive mimo systems limited backhaul capacity ieee globecom workshops wkshps austin ngo marzetta larsson analysis pilot contamination effect large multicell multiuser mimo systems physical channel models ieee int conf acoustics speech signal process icassp prague czech may
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may arxiv text goes stochastic block coordinate algorithm biological network alignment yijie xiaoning department electrical computer engineering texas university college station texas usa dated may abstract increasingly big data available biomedical research deriving accurate reproducible biology knowledge big data imposes enormous computational challenges paper motivated recently developed stochastic block coordinate algorithms propose highly scalable randomized block coordinate algorithm convex optimization general compact convex constraints diverse applications analyzing biomedical data better understanding cellular disease mechanisms focus implementing derived stochastic block coordinate algorithm align interaction networks identifying conserved functional pathways based isorank framework derived stochastic block coordinate sbcfw algorithm convergence guarantee naturally leads decreased computational cost time space iteration experiments querying conserved functional protein complexes yeast networks confirm effectiveness technique analyzing biological networks introduction methods convex optimization attracted significant attention statistical learning recent years appealing many learning problems lasso regression matrix completion diverse applications analyzing biological systems biomedical measurement profiles optimization methods scale well current big data many biomedical applications due advantages low computation burden per iteration easy implemented parallel computational resources paper focus algorithm also known conditional gradient method one advantages iteration step decomposes complex constrained optimization problem easier solve additionally projection free algorithm avoids solving projection problem constrained optimization done many algorithms original algorithm developed smooth convex optimization polytope dates back frank wolfe dunn harshbarger generalized algorithm solve optimization general smooth convex objective functions bounded convex feasible regions recently researchers proposed stochastic optimization ideas scale original algorithm based previous seminal efforts main contribution paper generalize stochastic block coordinate algorithm proposed previously block separable constraints solve general optimization problems convex compact constraints including problems block inseparable constraints generalized algorithm broader range biomedical applications including biological network alignment prove convergence generalized stochastic block coordinate algorithm evaluate algorithm performance querying conserved functional protein complexes interaction ppi networks following sections first describe model formulation optimization problems generally interested specifically address potential difficulty general convex compact constraints derive new stochastic block coordinate algorithm provide convergence proof formulate isorank problem network alignment convex programming problem develop algorithm based new stochastic block coordinate frankwolfe algorithm last experiments show efficiency effectiveness algorithm solving ppi network query problem arxiv text goes stochastic block coordinate descent algorithm consider minimization problem min objective function convex differentiable domain compact convex subset vector space assume optimal solution problem bounded without loss generality assume decompose solution space subspaces rni rni denotes ith subspace along corresponding coordinates decomposition enables scalable stochastic optimization algorithms based decomposition introduce matrices sum identity matrix matrix rni diagonal entries equal zero typical stochastic optimization algorithms instead computing gradient iteration partial gradient randomly selected subspace rni used generalize previous stochastic block coordinate algorithm derived solve general optimization problems compact convex constraints new generalized stochastic block coordinate sbcfw algorithm illustrated algorithm pseudo code operation randomly selects one subspaces update partial gradient iteration probability addition denotes condition elements jth block equal elements jth block generalized sbcfw algorithm let stopping criteria satisfied randomly divide blocks rni choose find ski ski arg min determine step size arg min update endwhile note generalized sbcfw algorithm similar algorithm aims solve optimization problems block separable constraints convergence property however algorithm provides generalized framework manipulate convex compact constraints matter whether block separable setup algorithm general without specific structure difficult obtain theorectical convergence rate guarantees paper provide proof sbcfw converges global optimum convergence guarantee generalized sbcfw algorithm provided theorem based lemma arxiv text goes lemma iteration sbcfw algorithm following inequality holds ski ski expectation ski respect random selection ith cordinate block corresponding subspace proof assuming kth iteration solve following optimization problem min zki solution ski ski achieving minimum zki ski zki zki ski ski therefore taking expectation sides inequality respect random blocks obtain ski ski ski inequality third line derived based fact ski vector ith coordinate block values parts pzeros summation second line written inner product vectors ski analyze convergence new sbcfw algorithm based lemma two cases first case ski simply means stationary point original objective function convex conclude global minimum another case ski indicating ski decent direction based definition hence ski move along direction get closer global minimum expectation furthermore compute optimal step size iteration therefore objective function values guaranteed present theorem theorem sequence generated sbcfw algorithm nonincreasing arxiv text goes biological network alignment optimization model formulation section involved optimization problem network alignment isorank address potential computational challenges aligning multiple networks new formulation mathematical programming structure problem let two biological networks align two networks vertices respectively define cartesian product network denote one vector rna diag diag considered degree matrix diagonal entries equal zero contains transition probabilities underlying markov random walk isorank well known connected networks neither bipartite graph corresponding markov chain represented irreducible ergodic exists unique stationary distribution underlying state transition probability matrix goal isorank algorithm find right maximal eigenvector matrix corresponds best correspondence relationships vertices across two networks two networks reasonable size spectral methods well power methods implemented solve isorank problem however networks transition probability matrix extremely large quadratic spectral methods computationally prohibitive paper problem searching maximal right eigenvector constrained optimization problem min expanding objective function obtain therefore equivalent optimization problem min gradient easily computed furthermore find hessian matrix positive matrix proven lemma lemma positive proof written proves lemma lemma obvious objective function convex also constraint set unit simplex convex compact hence isorank problem problem structure generalized sbcfw algorithm used solve much better scalability efficiency due efficiency randomized partial gradient computation iteration similarly addition network topology incorporate information formulation biologically significant alignment results replacing normalized similarity vector size cancatenated doubly indexed similarity estimates based sequence function similarity vertices algorithm shown section convex constraint set convex compact set therefore apply generalized sbcwf algorithm proposed section solve corresponding optimization problem detailed algorithm illustrated algorithm want emphasize arxiv text goes algorithm time complexity space complexity achieved tracking vectors exk eski step iteration algorithm respectively stopping criterion kxk efficiently estimated ptk taken line algorithm algorithm algorithm input randomly divide parts choose initialize ith block endif compute exk solve ski arg min eski compute ptk kxk break endif compute step size ptk ptk min ptk qtk ski endfor output initialization order guarantee time space complexity iteration initialize algorithm randomly generated avoid multiplication matrix size vector size whose time space complexity would propose initialize following way first randomly divide parts equal sizes randomly pick ith part initialize every elements ith part makes feasible space defined constraint set using initialization strategy time space complexity computating easy verify algorithm solve shown algorithm iteration need solve fortunately solved straightforward manner optimization problem following iteration min arxiv text goes optimalp solution lej vector except jth element index coordinate smallest value ith block arg min optimal step size obtain optimal step size iteration need solve following optimization problem min ptk ptk ptk qtk solution without constraints optimal solution minimum value otherwise definition given lines algorithm classic quadratic form respect time space complexity iteration computationally expensive operations updates lines calculation partial gradient line calculation similar line algorithm know exk second equation derived replacing equation line algorithm keep tracking iteration need recompute therefore need compute takes operations vector block parts zeros additionally memory consumption also similar argument similarly compute esk lej lej lej vector ith block values therefore computation operations consumes memory also takes equation calculating follows operator get rows matrix corresponding ith coordinate block hence easy verify time complexity space complexity computing summary based time complexity space complexity analyses isorank iteration arxiv text goes collins krogan figure query subnetwork aligned result target network query subnetwork krogan yeast ppi network aligned result collins yeast ppi network experiments section apply algorithm two network query problems first set experiments take known protein complex archived yeast interaciton ppi network one database query search subnetwork another yeast ppi network different archived interactions call network query problem goal set experiments check correctness algorithm ground truth target subnetwork aim test convergence property algorithm different partitions relationship number iterations number partitions second experiment query yeast ppi network intact find similar subnetworks proteins similar cellular functionalities known protein complex human ppi network aim experiment show new algorithm help transfer biology knowledge model organisms study potential functionalities molecules different organisms network query problem test algorithm ppi network query problem solving optimization problem introduced previous section take subnetwork proteins fig krogan yeast ppi network query example search conserved functional complex target network collins network proteins interactions query subnetwork transcription factor tfiiic complex krogan network interested testing whether find subnetwork collins network dimension optimization problem run preliminary example compare stochastic optimization results results power method typically done original isorank algorithm theoretically time space complexity iteration based analysis section compared time space complexity power method isorank algorithm scale better properly selected query example target network contain interactions among proteins easily check correctness query result define accuracy number corrected aligned proteins divided total number proteins query subnetwork implement algorithm different numbers partitions use stopping criterion kxk table find stochastic optimization algorithm obtains objective function value arxiv text goes iterations figure change objective function values increasing number iterations different numbers partitions biologically meaningful results power method fig shows changes objective function values respect increasing number iterations illustrated fig algorithm converges different additionally find larger number partitions larger number iterations need algorithm converge global optimum stopping criterion clearly demonstrates tradeoff efficiency scalability stochastic optimization algorithms interestingly notice number iterations increase much indicates may achieve fast computation reasonably large algorithm efficient larger iteration table comparison different decompositions partitions computational time iterations accuracy investigate performance different run algorithm times show average computational time average number iterations average accuracy score table table observe different algorithm obtain accuracy demonstrates effectiveness convergence generalized sbcfw algorithm also notice increasing number iterations increase however computational time first reducing increasing example algorithm converges smallest number iterations computational time best iteration algorithm takes operations contrast though number iterations larger reaches global optimum arxiv text goes proteasome core complex human proteasome core complex yeast figure querying human protein complex yeast ppi network proteins annotated gene names solid lines protein interactions dash lines denote orthologous relationships based protein sequence similarity blast proteins different organisms human proteasome core complex aligned proteasome core complex yeast found least computation time indeed twice faster trend computational time implies may exist best number partitions empirically computational time decreases computational time increase however difficult provide theoretical proof observed phenomenon finally scalibility algorithm always prefer larger make memory requirement low possible network query problem study biological signficance network query results algorithm extract subnetwork query example human ppi network archived intact query subnetwork proteasome core complex induced interactions among corresponding proteins intact proteasome core complex human consists proteins total shown fig target network yeast ppi network also obtained intact proteins interactions goal find similar subnetwork human proteasome core complex target yeast ppi network based interaction topology protein sequence similarity computed blast first construct alignment network vertices sbcfwisorank algorithm instead operating matrix size power method need handle matrix size iteration computational time well memory requirement reduced times matlab implementation macpro notebook ram takes around seconds converge reaching stopping criteria identified subnetwork target yeast ppi network algorithm illustrated fig evaluate biological significance obtained subnetwork check based gene ontology enrichment analysis using goterm finder identified subnetwork significantly enriched term fact proteasome core complex experiment demonstrates algorithm find biologically consistent groups proteins cellular functionalities proteins query subnetwork hence capability transferring existing biology knowledge model organisms yeast example less studied organisms group proteins query subnetwork require better understanding cellular functionalities arxiv text goes conclusions paper generalize block coordinate algorithm solve general convex optimization problems convex compact constraint set generalized sbcfw algorithm convergence guarantee isorank problem convex programming problem solve biological network alignment problem algorithm scales better size networks study scalability efficiency effectiveness algorithm solving isorank demonstrated ppi network query problems future work consider derivation optimal partition number better tradeoff computational efficiency scalability acknowledgements authors would like thank simon pointing error original conference paper work partially supported awards national science foundation well award national institute diabetes digestive kidney diseases national institutes health references boyle elizabeth weng gollub jin botstein cherry sherlock source software accessing gene ontology information finding significantly enriched gene ontology terms associated list genes bioinformatics dunn convergence rates conditional gradient sequences generated implicit step length rules siam journal control optimization dunn harshbarger conditional gradient algorithms open loop step size rules journal mathematical analysis applications frank wolfe algorithm quadratic programming naval research logistics quarterly hasty mcmillen issacs collins computational studies gene regulatory networks numero molecular biology nat rev genet kerrien aranda breuza intact molecular interaction database nucleic acids research klau new method pairwise global network alignment bmc bioinformatics suppl krogan global landscape protein complexes yeast saccharomyces cerevisiae nature jaggi schmidt pletscher optimization structural svms international conference machine learning ortega rheinbold iterative solution nonlinear equations several variables society industrial applied mathematics singh berger global alignment multiple protein interaction networks application functional orthology detection proc natl acad sci uryasev pardalos eds stochastic optimization algorithm application springer arxiv text goes zaslavskiy bach vert global alignment interaction networks graph matching methods bioinformatics zhang schwartz wagner miller greedy algorithm aligning dna sequences comput biol
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lifting linear extension complexity bounds setting dec alfonso cevallos stefan rico december abstract mathematical programs among commonly used models wide set problems operations research related fields however still little known expressed small programs particular prior work open whether classical problems like minimum problem expressed compact mixedinteger program even constantly many integer variables stark contrast linear formulations recent breakthroughs field extended formulations shown many polytopes associated classical combinatorial optimization problems even admit approximate extended formulations size provide general framework lifting inapproximability results extended formulations setting extended formulations obtain almost tight lower bounds number integer variables needed describe variety classical combinatorial optimization problems among implications obtain show extended formulation size matching polytope cut polytope traveling salesman polytope dominant polytope needs log many integer variables number vertices underlying graph conversely polyhedra admit formulations log traveling salesman polytope many integer variables results build upon new decomposition technique convex set allows approximating description intersection union small number affine subspaces keywords extension complexity programs extended formulations introduction linear extended formulations milefs one common models mathematically describe wide variety problems operations research related fields due high expressive power made tool choice numerous optimization problems also led large ecosystem commercial solvers modeling languages supporting models despite prevalence relation expressed formulations number integer variables used remains badly understood particular open many integer variables needed obtain compact milef classical combinatorial objects including matchings traveling salesman tours cuts stable sets vertex covers odd cuts name department mathematics eth zurich zurich switzerland email department mathematics eth zurich zurich switzerland email department mathematics eth zurich zurich switzerland email ricoz supported swiss national science foundation grant moreover natural problem classes beyond classical combinatorial optimization problems efficient algorithms known yet prior work open whether could well solved efficiently via milef small number integer variables maybe even constantly many use lenstra algorithm bimodular integer programming example expand section whereas milefs mostly used describe hard problems hope find efficiently solvable milefs driven desire cast efficiently solvable problems specialized procedures known common framework moreover strong solvers available prior result lower bounds number integer variables milefs shows compact formulation matching polytope complete graph vertices needs integer variables unfortunately presented technique highly specialized matching polytope heavily exploiting facet structure moreover lower bound leaves large gap compared canonical description matchings using many integer variables one edge goal work address lack understanding expressive power milefs function number constraints integer variables used presenting general framework lift linear extension complexity results approximate extensions setting better put results context start brief summary basics linear extensions also allows introduce notations terminology used later formalize notion milefs typical settings discrete optimization linear models developed ask optimizing linear function set vectors often even set could example correspond characteristic vectors matchings graph case clearly problems restated optimizing linear function corresponding polytope simply convex hull points conv matchings graph would therefore correspond matching polytope hence discrete optimization problem gets described linear program algorithmically well understood ideally solve linear program would like inequality description unfortunately many polytopes arise combinatorial optimization descriptions require exponential number inequalities however polytopes admit much smaller descriptions allow use additional variables allow describing system preferably linear inequalities description called extended formulation allows stating original problem linear program solutions understanding polytopes admit small extended formulations scope field extended formulations refer chap many examples background material formally extension complexity polyhedron minimum number facets extended formulation clearly compact extended formulations desirable since allow rephrasing original problem small linear program whereas definition extended formulation requires projection one lift restriction allow affine image generalization use paper convenience easily seen impact notion extension complexity formally say polyhedron linear extended formulation lef exists affine map moreover size lef equal number facets study linear extensions received considerable attention recently due breakthrough results stating various prominent polytopes arise combinatorial optimization number inequalities every description type see situation changes dramatically allow imposing integrality constraints leads notion linear extended formulations setting describe conv words described polyhedron intersected integrality constraints projected subset coordinates convex hull resulting set finally considered complexity description captured parameters namely number facets called size number integer variables case linear extensions lift restriction projection axisparallel allow imposing integrality constraints affine forms without impact values achieved formally say polyhedron milef complexity number facets number integrality constraints affine maps conv conv need specific also say triple milef description terms milef corresponding maps allows reducing linear programming problem problem maximizing linear function set even difficult structures like stable sets sets efficiently optimize small extended formulation exists like matchings easily described small even integer formulations downside linear programs considerably harder problem class linear counterparts currently fastest algorithms terms dependence number integer variables running time dependence hence one needs log log expression become polynomial corresponds input size original problem thus hard problems like maximum stable sets expect milefs exist small size integer variables however mentioned achievements field extended formulations seem give rise general techniques obtaining lower bounds number integer variables setting work demonstrate recent generalizations results extended formulations namely inapproximability results extended formulations leveraged general way obtain lower bounds milef complexity problem question actually lower bounds obtain even hold milef close approximation sense define formally later problem consideration consequence show lower bounds compact approximate milef classical polytopes like matching polytope cut polytope dominant polytope main results consequences exemplify type results derive technique highlight breadth first state hardness results classical combinatorial problems efficiently solvable class integer programs bimodular integer programs formally define section later discuss general framework allowing derive results consequences end let denote complete undirected graph vertices polytope matching dominant cut tsp stable set knapsack matroid bimodular lower bound upper bound log figure bounds number integer variables milefs certain size first four rows refer respective polytopes complete undirected graph nodes lower bounds hold milefs size constant see theorem lower bounds latter four rows refer stable set polytopes graphs knapsack polytopes instances independence polytopes matroids cardinality integer hulls conic bimodular integer programs variables respectively bounds hold milefs size bounds interpreted guaranteeing existence polytopes respective family lower bound holds see theorem right column contains upper bounds number integer variables sufficient obtain milefs size polynomial respective family bounds valid members family see section case independence polytopes matroids milef known theorem constant following holds let let exists milef complexity either matching polytope dominant polytope cut polytope traveling salesman polytope see later cases lower bound number integer variables tight factor theorem constant let let exists milef complexity either stable set polytope graph knapsack polytope instance independence polytope matroid ground set cardinality convex hull feasible points conic bimodular integer program variables summary stated lower bounds together upper bounds number integer variables needed milefs found figure actually techniques imply even slightly stronger versions theorems rule milefs said problems closely approximate get back later previously lower bound number integer variables needed milef known matchings bound obtained results apply much broader class problems often also nearly tight terms revealing many integer variables needed key parameter milefs precisely milefs cuts minimal well known matchings textbook integer formulation uses one integer variable per edge thus leading compact milef integer variables show section also matchings admit compact milef integer variables hence theorem shows even allow milefs satisfying constant least nearly linearly many integer variables needed describing matching polytope cut polytope dominant polytope whereas may natural expect hard problems interesting polytopes corresponding efficiently solvable problems like maximum matchings minimum bimodular integer programming described milef much fewer linearly many integer variables particular rules possibility efficiently solve problem bimodular integer programming milef classical algorithm programs like lenstra algorithm whose running time dependence number integer variables would thus need log log lenstra algorithm run efficiently moreover since matching polytope complete graph vertices linear projection traveling salesman polytope tsp graph see results also extend tsp polytope note tsp variants heavily studied context milef formulations see references therein provide first lower bound number integer variables needed formulations whereas theorems give nice overview type results obtain main contribution work leads results general technique transform hardness results approximate lefs setting fashion first need formal notion approximate lefs milefs boils defining two convex sets think relaxation well approximates various notions used literature depending context particular viewpoint optimization natural consider notion related ratio optimal values optimizing linear objective respectively like integrality gap therefore use notion maximization gap minimization gap defined follows inf sup sup gap inf inf inf clearly maximization gap relevant maximization problems like maximum matchings maximum stable set minimization gap used minimization problems like minimum many approximation hardness results lefs stated terms linear programming gap notions however techniques geometric notion convenient particular one invariant basic operations like bijective affine transformations case therefore introduce notion relative distance interpreted normalized notion gap many helpful properties particular invariant respect affine bijections easily related gap notions two convex sets define relative distance rdist sup diam supremum taken linear maps hausdorff distance inf diam diameter function diam supa definition interpret fraction denominator well fraction one easily observe definition relative distance change supremum taken orthogonal projections onto line goes origin illustration notion relative distance see figure moreover extend definition empty figure relative distance light shaded dark shaded polygons sets setting rdist rdist even though one could define relative distance broader context without assuming restrict setting since one relevant derivations using notion relative distance define approximate lefs milefs natural way definition convex set pair polyhedron space affine map rdist analogously triple polyhedron affine maps conv satisfies rdist note classical lefs milefs respectively also remark approximate lefs milefs even convex sets ready state main reduction result shows existence approximate milef convex set implies existence approximate lef thus allows lifting results approximate lefs setting theorem let convex set admitting complexity every admits size derive theorem results stated theorem proceed follows first observe existing lef approximation hardness results imply constant size polyhedron consider result follows choosing theorem observing appropriately chosen constant theorem implies existence lef size strictly less thus leading contradiction particular proof approach implies even exist complexity constant due relation rdist gap establish result rephrased terms approximate milefs respect gap expand connections precise statements resulting later sections organization paper start summarizing key properties relative distance exploit later including relation distance done section section provide thorough overview techniques lead theorem main result reduce approximation hardness results lefs milefs key ingredient proof new decomposition technique convex set allows approximating description intersection union small number affine subspaces since result may independent interest present separately section section expands implications techniques different polytopes provides particular formal proof strengthened versions theorems section provide milefs polytopes mentioned theorems discuss quality bounds derived statements close main part paper general discussion bounds achieved techniques section appendix contains deferred proofs properties relative distance relative distance basic properties due extensive use relative distance throughout paper start stating key properties used main part paper remark notion relative distance closely related difference body metric convex bodies introduced shephard namely two convex bodies difference body metric defined via log rdist thus properties stated direct consequences results sake completeness since deal arbitrary convex sets provide proof lemma appendix lemma consider three convex sets rdist inf affine map rdist rdist equality case invertible iii rdist rdist rdist rdist rdist convex sets rdist conv conv max rdist next two lemmas highlight relation relative distance gap notions allows first translate lef approximation hardness results often stated terms gap gap terms relative distance lemmas allow translating hardness results stated respect relative distance back notion gap call convex set every moreover polytope whose vertices within say implies lemma two convex sets rdist lemma let convex set dim dim otherwise rdist rdist proofs lemmas postponed appendix convex body convex set bounded closed outline techniques section explain approach proving theorem reduce problem approximating hull convex set intersection set affine subspaces conv figure representation milef convex set conv milef integrality constraints picture lives space convex set lives space number integer constraints projection onto integer space highlighted left picture projection onto space highlighted bottom picture exemplify approach consider convex set milef complexity see figure means conv conv second equality follows fact convex hull commutes affine maps assume two constants set admit size smaller goal transform milef without blowing size much integer constraints milef cut polyhedron fibers fiber set consider first simple special case number fibers large end let points correspond fibers assume notice rewrite conv conv refer hull respect remark lef via affine map moreover bound extension complexity technique known disjunctive programming allows obtaining inequality description convex hull union family polyhedra given inequality description polyhedron family case polyhedra fibers polyhedra facets intersection whose facets bounded affine subspace disjunctive programming technique implies assuming implies moreover since assumed every size least must thus words milef fibers must large size ideally one could show milef complexity small number fibers would done particular number fibers would sufficiently small integer variables involved milef bounded range unfortunately hold general number fibers bounded key aspect approach overcome hurdle precisely given milef instead describing terms fibers show one approximate sets form comes family affine subspaces small cardinality whenever small particular affine subspace form case fibers typically contain many subspaces form moreover family may contain subspaces different dimensions price pay resulting description exact anymore yields approximation concretely resulting set conv conv property show good approximation error terms relative distance constant choose impacts constants statements consequently good approximation well find family subspaces recursively slice along different directions small width rely celebrated result convex geometry find good directions commonly known flatness theorem shows convex sets small width formally state flatness theorem start defining flatness constant lattice width definition flatness constant lattice width let flatness constant flt dimension smallest convex closed set exists vector sup inf moreover quantity width inf sup inf called lattice width hence flt smallest real upper bounds lattice width convex closed fulldimensional set notice term flatness constant may slightly misleading remainder paper denote general convex sets part keep notational convention original space extended space auxiliary space see figure since flt depend term historical comes fact lattice width often studied settings thus also flt constant finally term flatness theorem used theorems bound quantity flt terms many versions state one coming sec convenient follows theorem flatness theorem see flatness constant flt always finite moreover bounded polynomial following theorem key technical ingredient approach guarantees existence good family subspaces theorem let convex set affine map every exists family affine subspaces sets flt conv conv satisfy rdist since statement independent notion extended formulations might independent interest discuss proof next section let demonstrate theorem indeed implies theorem end show theorem implies following slightly stronger version theorem let convex set complexity every size flt first argue theorem indeed implies theorem proof theorem let convex set admitting complexity let applying theorem replaced obtain size flt notice implies hence remains prove since constants flt finally since must constant thus completes proof claim remains show theorem implies theorem proof theorem assumption exists polyhedron facets affine maps satisfies rdist applying theorem obtain set affine subspaces flt rdist let closure balas theorem thus exists polyhedron facets affine map let define show theorem proving already shows size bounded flt desired clearly hence remains show rdist end first observe rdist rdist rdist rdist first inequality follows lemma second equality fact rdist rdist always holds last inequality definition finally recall rdist hence using lemma iii obtain rdist rdist rdist rdist rdist claimed approximating hulls unions slices demonstrated previous section key technical ingredient proof theorem statement theorem section prove latter throughout section make extensive use sets defined theorem give idea proof let argue easily derived following statement proposition let convex set affine map every exists family affine subspaces satisfying flt iii rdist proof theorem let family affine subspaces described proposition sition monotonicity flt immediately obtain bound flt next property theorem follows proposition conv conv conv hence remains show property theorem end define leads conv conv second relation follows proposition finally obtain bound rdist max rdist first inequality follows lemma second one proposition iii proposition states small family affine subspaces cover fibers set property slice approximates well hull idea behind proof sketched follows set already good approximation hull need intersect proper affine subspaces simply choose otherwise next statement claims small family affine subspaces covering fibers hull slice described fewer integer constraints recurse intuitively idea exploit flatness theorem following good approximation point certifying rdist large however possible set fibers extremely dense respect every direction exploit flatness theorem find good direction respect slice polynomially many slices mentioned proposition obtained recursive slicing following lemma shows find family affine subspaces slicing thus reducing number integer constraints one recursive use lemma eliminate integer variables proof proposition becomes straightforward postponed end section lemma let convex set affine map rdist exists set affine subspaces affine map satisfying flt iii affine subspaces map needed lemma implicitly given next lemma shows fibers covered small number parallel lattice hyperplanes lemma let convex set affine map rdist width flt proof assume set otherwise lattice width statement holds trivially notice enough show value rdist lattice width bounded flt value follows lemma exists point turn means sets disjoint scaling setting obtain sets consider convex closed set notice obtained via scaling factor followed translation hence width width width thus prove lemma suffices show lattice free consequently width flt assume sake deriving contradiction point definition given exists consider point clear definition moreover convex combination points finally therefore obtain contradiction ready prove lemma proof lemma hypothesis lemma width exists vector set flt hence cardinality flt may assume gcd otherwise replacing gcd decrease cardinality definition set covered many hyperplanes take hyperplanes define family hence let clearly satisfies property lemma moreover property follows immediately fact hyperplanes remains show fulfills property iii lemma since gcd well known exists unimodular matrix det first row integral let matrix arises removing first row clearly defining via rephrase follows since thus obtain second equality follows hence setting fulfills property iii lemma desired existence unimodular matrix first row easily follows fact hermite normal form vector gcd since column hermite normal form obtained integer column operations operations described unimodular matrix hence unimodular matrix one choose finally provide proof proposition proof proposition proceed induction note claim trivial choosing let observe may assume rdist otherwise choose lemma exists family affine subspaces affine map flt induction hypothesis applied exists family affine subspaces flt rdist defining set clearly satisfy due furthermore shows finally iii direct consequence fact applications section demonstrate framework applied obtain strong lower bounds number integer variables milefs several relevant settings among results obtain statements mentioned theorems using theorem existing inapproximability results lefs fact prove stronger versions statements also rule existence approximate milefs end first derive following direct consequence theorem suited applications consider corollary let constants let convex set size admit size integer variables complexity proof let convex set hypothesis suppose admits may assume theorem admits size constant assumption must constant second inequality follows choosing sufficiently large third inequality follows last inequality due implies log yields claim note statement allows quickly translating inapproximability result lefs certain inapproximability result milefs besides proofs theorems main purpose section demonstrate several existing inapproximability results lefs literature usually stated using different notions approximations transferred inapproximability results lefs required statement corollary matching polytope start applying framework matching polytope complete graph simply refer matching polytope defined convex hull characteristic vectors matchings complete graph nodes denote polytope pmatch recall matching edge subset every vertex degree one result edmonds polytope exponential number facets even though linear function optimized strongly polynomial time question whether matching polytope admits extended formulation size polynomial open long time proved extension complexity exponential recently even proved polytope well approximated polytope low extension complexity theorem see also thm exist constants every polytope pmatch pmatch satisfies let translate result using notion relative distance corollary exist constants pmatch admits size proof let constants defined theorem may assume show every polyhedron pmatch rdist pmatch satisfies constant end first note since rdist pmatch finite pmatch bounded must also bounded defining polytope clearly rdist pmatch since pmatch lemma pmatch implies pmatch thus theorem obtain hence universal constant corollary directly obtain corollary exist constants pmatch size integer variables using lemma statement phrased similarly theorem corollary exist constants following holds let polytope pmatch pmatch milef size integer variables textbook milefs pmatch usually require integer variables section give simple milef pmatch uses integer variables thus lower bound number integer variables corollaries tight factor log cut polytope let complete undirected graph vertices define cut subset written convex hull pcut characteristic vectors cuts called cut polytope recall optimizing linear function pcut least hard solving maximum cut problem cut polytope first specific shown dimension extension complexity see specifically every lef pcut size least exponential see also follows lift bound milefs log integer variables constant end make use following inapproximability result refers correlation polytope conv affinely isomorphic pcut exists affine bijection aff pcut see theorem thm constant every polyhedron diag satisfies let translate result using notion relative distance size polyhedron corollary exist constants pcut admits proof define let constant theorem let pcut rdist pcut remains show holds end let aff affine map satisfies pcut clearly well rdist claim contained set defined statement theorem otherwise matrix diag satisfies hand one see sec well definition relative distance would imply rdist contradiction thus hence theorem obtain claim follows since highlight allowed equal empty set sometimes define cuts one requires discussion easily transferred case bit simpler also allowing trivial sets two matrices denote aij bij frobenius inner product corollary directly obtain corollary exist constants integer variables pcut size several known milefs pcut use integer variables however similar case matching polytope simple milefs pcut use integer variables see section bound number integer variables given corollary tight factor log traveling salesman polytope section use result matching polytope obtain lower bound number integer variables milefs traveling salesman polytope ptsp defined convex hull characteristic vectors hamiltonian cycles known constant every exists face ptsp affinely projected onto pmatch see proof thm following lemma implies whenever ptsp admits milef complexity also pmatch admits milef complexity lemma let polyhedra affine projection face admits milef complexity also admits milef complexity proof hypotheses face affine map additionally exists polyhedron facets affine maps conv remark face affine map define affine subspace notice face characterization face fact implies conv thus obtain hence admits milef complexity number facets since number facets number facets obtain yields claim corollary directly obtain corollary exists constant milef ptsp size integer variables textbook milefs ptsp require integer variables section give milef ptsp uses log integer variables thus bound number integer variables tight factor explicitly claim conv conv inclusion follows immediately fact conv convex set containing must thus contain conv smallest convex set containing opposite inclusion consider point conv must convex combination points point must however implies forces inequalities tight thus proves conv desired stable set polytope stable set polytope pstab undirected graph defined convex hull characteristic vectors stable sets graphs polytope pstab easily described arguably complicated polytope general example lem shown every exists graph vertices face pstab affinely projected onto pcut thus using lemma corollary conclude corollary exists constant holds every exists graph milef pstab size integer variables highlight result also reduced milef extension complexity result matchings corollary follows fact matching polytope graph stable set polytope corresponding line graph whose number vertices equal number edges hence matching polytope stable set polytope graph many vertices note pstab conv hence pstab admits milef integer variables every graph however aware milefs integer variables particular believe bound corollary significantly improved comment issue section knapsack polytope given item sizes capacity corresponding knapsack polytope defined pknap conv similar case stable set polytopes certain item sizes capacities corresponding knapsack polytopes simple structure general however knapsack polytopes turn complicated polytopes indeed shown every exist item sizes capacity pcut affine projection face pknap analogous previous section using lemma corollary conclude corollary exists constant following holds every exist item sizes capacity milef pknap size integer variables clearly definition pknap admits milef integer variables aware milef general knapsack polytopes uses integer variables clear whether bound corollary significantly improved dominant polytope section consider polytope particular dominant since polyhedra contain perfect matching polytope face surprising obtain lower bounds complexity milefs polyhedra however main purpose section obtain lower bounds approximate lefs essential establishing lower bounds dominant odd cut polytope next section let even complete graph vertices recall edge subset called every vertex odd degree polytope defined convex hull characteristic vectors denoted pvjoin dominant polytope defined pvjoin next statement derive lower bound approximate lefs exploiting following relation pmatch first note every cardinality least hence set face furthermore subset edges cardinality perfect matching matching cardinality since every matching consists edges ppmatch pmatch also face pmatch polytope ppmatch called perfect matching polytope furthermore easy see pmatch ppmatch holds follows use relations together theorem obtain similar statement theorem constants every even polyhedron satisfies proof brevity write pmatch ppmatch shorthands pmatch ppmatch respectively consider polyhedron fixed later consider hyperplanes better structure proof divide three claims claim vjoin since ppmatch suffices show vjoin attained point ppmatch end let vertex ppmatch observe next since nonnegative finite hence attained vertex claim must contained ppmatch indeed otherwise would satisfy hence contradicts previous inequality whenever next show approximates ppmatch well claim first let arbitrary since nonnegative since obtain min min min equality follows previous claim clearly implies min min min min min min holds every equivalently obtain max max holds every let since satisfies contained ppmatch clearly hence obtain since claimed inequality define observe thus enough prove holds universal constant since ppmatch pmatch ppmatch pmatch together following claim finally obtain pmatch implies desired setting constant theorem using theorem claim pmatch pmatch suffices show holds every end fix write previous claim obtain max max max max max max max max max first equality follows fact nonnegative last inequality implied ppmatch holds due following next demonstrate statement theorem implies particular inapproximability result terms relative distance end set define lemma let let every polyhedron satisfies admit size less proof show every polyhedron rdist satisfies end define observe rdist rdist note latter implies dim dim since lemma implies rdist equivalent note implies equality follows implying thus polyhedron also obtain since equal dominants implies equivalent assumption conclude holds recall defined via hence shows claimed following inapproximability result direct consequence theorem lemma corollary exist constants every even admit size using corollary immediately implies corollary exist constants every even size integer variables finally use following lemma deduce inapproximability result milefs lemma let let every size least integer variables furthermore let polyhedron every milef size least integer variables proof may assume dim dim otherwise intersect affine hull observe milef certain complexity milef complexity assumption first claim implies see let show min min holds note since min min let since obtain hence min min min shows hence established thus using facts dim dim invoke lemma together inequality implies rdist suppose milef size integer variables milef size integer variables means size integer variables assumption must yields claim finally able prove following lower bound complexity milefs approximating dominant polytope direct consequence corollary lemma corollary constants following holds let even polyhedron every milef size integer variables remark polytope hence also dominant exact milef integer variables see section dominant polytope using bounds obtained previous section ready provide lower bounds complexity milefs dominant polytope let even let complete undirected graph vertices odd cut defined subset written set odd cardinality polytope pocut defined convex hull characteristic vectors odd cuts dominant defined pocut easy check every odd cut intersects every however even stronger wellknown link polyhedra blockers convex set blocker defined see sec information blocking polyhedra using notation mentioned relation reads another important fact use follows observation every linear extended formulation polyhedron turned one adding additional inequalities precisely use following fact see prop ready transfer theorem dominant polytope corollary constants every even polyhedron satisfies proof let denote constants statement theorem brevity let use notation may assume let polyhedron note yields since using theorem obtain universal constant claim follows analogously case polytope obtain corollary lemma corollary exist constants every even admit size using corollary immediately implies corollary exist constants every even size integer variables finally corollary lemma yield corollary constants following holds let even polyhedron every milef size integer variables remark polytope hence also dominant exact milef integer variables see section conic bimodular integer programming given consider problem optimizing given linear function integer hull conv polyhedron without assumption describes general integer program hence solve special case problem becomes solvable totally unimodular largest absolute value determinant square submatrix equal open question integer programming community whether integer programs still solved efficiently described integer constraint matrix absolute value determinant square submatrix bounded constant recently answered question affirmative showing integer programs tractable constraint matrix bimodular integer matrix full column rank determinants submatrices lie within totally unimodular case solvability easily explained observing integer hull coincide hence problem reduces solving linear program contrast argumentation bimodular case much involved gives evidence whether simple polyhedral representation well compared section show bimodular integer programs integer programs bimodular constraint matrices lead polyhedra described small milef result follow showing dominant odd cut polytope captured bimodular integer program end let complete digraph vertices let consider polyhedron first note described system linear inequalities bimodular coefficient matrix integer side see observe first constraint matrix full column rank due constraints moreover notice described inequalities forming identity matrices incidence matrix totally unimodular plus additional row related containing entry value otherwise empty column variable thus developing last column see determinant submatrix bounded absolute value second note polyhedron conic vertex constraints tight point satisfies linear constraints equality third easy see conv affinely projected onto formal proof provided section optimizing integer points conic polyhedron described bimodular constraint matrix conic bimodular integer program discussion corollary thus obtain theorem exists constant following holds every conic bimodular integer program variables milef size convex hull feasible points requires log integrality constraints importance fact hardness result even holds conic bimodular integer programs motivated result veselov chirkov implies suffices find efficient algorithm conic bimodular integer programming solve bimodular integer program efficiently thus natural approach solve bimodular integer programs would try find compact lef milef integer variables describes feasible solutions conic bimodular integer programs thus avoiding partially involved combinatorial techniques used far method efficiently solve bimodular integer programs also one could hoped approach based extended formulations may amenable extensions beyond bimodular case theorem shows approach succeed still hope one may able design combinatorial approaches solve natural generalizations bimodular integer programs step direction done large families one first results establishing lower bounds size lefs shown every constant following holds family exists polytope also observed fact many ground set cardinality matroid ground set size whose corresponding matroid polytope exponential extension complexity section extend results setting end make use recent generalization result two compact sets recall hausdorff distance respect euclidean norm defined via max sup inf sup inf theorem thm let family polytopes dimensions least one let holds every two distinct polytopes matroid tuple finite ground set family subsets satisfying element matroid polytope corresponds convex hull characteristic vectors sets exists polytope log note family theorem restricted contain next show every large enough family polytopes even contains polytopes admit small approximate lefs end make use following lemma whose proof given appendix rdist lemma let convex sets proposition every constant exists constant following holds every family exists polytope admits size proof may assume contains polytopes dimensions least one suppose size thus every convex set every admits rdist clearly set satisfies rdist well lemma every hence every two distinct polytopes obtain first inequality follows fact two distinct lower bound hausdorff second inequality follows triangle inequality hausdorff distance hence implies applying theorem family obtain exists depending shows yields claim statement together corollary implies following result proposition every constant constant following holds let family exists polytope every integer variables size using fact many matroids see thus obtain theorem constant following holds let let exists complexity matroid polytope matroid ground set cardinality deduced observing hausdorff distance vertex hypercube convex hull vertices upper bounds section provide milefs polyhedra considered section complement bounds number integer variables obtained section end consider different polytopes convex hulls characteristic vectors certain edge subsets complete graph vertices denoted polytopes exist textbook milefs use integer variables usually consist binary variable every edge however follows present rather milefs use log even integer variables respectively matching polytope polytope start considering polytope pvjoin recall edge subset every vertex odd degree construct milef pvjoin integer variables let fix orientation edges denote sets edges enter leave according respectively furthermore let write finally edge set vector use notation proposition every even pvjoin conv particular pvjoin admits milef size integer variables proof let denote polytope side show pvjoin suffices show every vertex pvjoin contained end let vertex pvjoin since characteristic vector every odd thus every exists integer satisfies hence remains show pvjoin suffices show every vertex contained pvjoin end let vertex observe exists vector vertex polytope note defined totally unimodular matrix constraints described nodearc incidence matrix directed graph defined orientation thus since integral obtain furthermore every clearly odd shows characteristic vector hence pvjoin immediate corollary proposition obtain following corollary every even admits milef size integer variables shows lower bound provided corollary tight factor log since perfect matching polytope ppmatch face pvjoin since matching polytope pmatch equal ppmatch observation shows pmatch also admits milef size integer variables provide alternative even simpler milef complexity general graphs end let undirected graph fix orientation edges proposition matching polytope graph conv every particular admits milef size integer variables proof let denote polytope side clear contained show let satisfies every let support claim bipartite subgraph see first observe every suppose assume since implies furthermore hence thus edge incident node node showing bipartite since bipartite since holds every restriction contained matching polytope embedding matching polytope obtain face hence contained case complete graph shows lower bound obtained corollary tight factor log cut polytope polytope next let consider cut polytope pcut polytope pocut latter assume even recall cut edge subset written called odd cut odd remind reader allow let first start two simple milefs pcut pocut use integer variables proposition every pcut conv furthermore every even pocut conv satisfy particular pcut pocut admit milefs size integer variables proof let denote polytope side first claim definition cut clear pcut contained let satisfy straightforward check integrality forces integral well furthermore easy see characteristic vector cut defined thus contained pcut shows pcut second claim description pocut follows argumentation fact equivalent requiring odd proposition immediately implies following corollary every even dominant odd cut polytope admits milef size integer variables proposition corollary show bounds obtained corollaries respectively tight factor log recall reasoning section used another milef whose validity want prove next prove proposition shows polytope defined satisfies affine projection whose convex hull proposition every even let complete digraph vertices conv proof let denote polyhedron side straightforward check every characteristic vector odd cut contained clearly equal dominant shows see reverse inclusion let fix odd remains show projection onto polyhedron contained end let let smallest integer odd note exists since odd definition set odd cardinality thus greater equal characteristic vector odd cut induced hence traveling salesman polytope finally argue milef traveling salesman polytope ptsp uses log integer variables let let fix set cardinality furthermore pick bijective map finally fix hamiltonian cycle consider polytope conv use construct milef ptsp described proposition bound number constraints used milef later show small extension complexity proposition every ptsp conv proof let denote polytope side claim let characteristic vector hamiltonian cycle exists bijective map every choose every clearly well means thus every edge hence shows ptsp reverse inclusion let consider every edge since every vertex incident edge definition requires furthermore since every two vertices adjacent pairwise distinct consider set fix edge note conv thus exist points coefficients since implies analogously must also definition hence words means characteristic vector hamiltonian cycle edge set recall pairwise distinct thus obtain ptsp hence ptsp corollary every ptsp admits milef size log integer variables proof proposition suffices show polytope defined described extended formulation size see observe vertices since pset convex combinations vertices projection simplex linear map defined matrix whose columns vertices thus indeed extended formulation size shows lower bound obtained corollary tight factor aware milef ptsp uses log integer variables towards tight bounds work obtained lower bounds number integer variables required size milefs variety polyhedra relying lower bounds sizes approximate extended formulations polyhedra close paper highlighting gaps left techniques stable set polytopes case stable set polytopes show lower bound certain graphs aware milef size uses integer variables fact believe exist graphs integer variables needed large gap explained current approach simply uses lower bound either cut polytope matching polytope way considering stable set polytopes graphs vertices faces affinely projected onto pcut pmatch respectively promising family graphs study one considered recent work jain watson exhibit graphs whose stable set polytopes extension complexities log work however refers exact rather approximate extended formulations would require analysis lift results setting techniques despite fact believe stable set polytopes admit milefs integer variables another motivation improving lower bound following prop mentioned family polytopes vertices admits algorithm decide whether point belongs described milef whose size polynomial uses integer variables aware family polytopes shows bound number integer variables asymptotically tight believe stable set polytopes good candidates traveling salesman polytopes proved every milef size ptsp requires least integer variables exists milef log integer variables likely lower bound improved clear whether ptsp admits milef integer variables closing logarithmic gap formulations even though get nearly tight lower bounds number integer variables required size milefs matching polytope cut polytope dominant polytope still gap remaining precisely graph vertices show lower bound polytopes whereas descriptions using many integer variables leaves logarithmic gap believe lower bounds tight integer variables needed whereas know get rid log general show stronger lower bound different technique restricted class milefs matching polytope namely milefs live original space space matching polytope words milefs original space allowed use additional variables formally say milef polyhedron integer constraints original space identity following see conv milefs matching polytope show lower bound however highlight derive linear lower bound matching polytope original space open whether technique may extend general milefs beyond matching polytope notice lower bound number integer constraints small milefs matching polytope original space tight constant factor milef given proposition original space theorem exists constant milef pmatch size original space integer constraints proof recall exists constant extension complexity pmatch least every suffices prove milef pmatch original space complexity inequality must hold definition inequality clearly holds whenever assume proceed induction since milef pmatch complexity must satisfy inequality clearly satisfied whenever hence holds let assume pmatch admits milef complexity denoting complete undirected graph vertices exist matrices vectors pmatch conv start simplifications done integrality constraints without loss generality vector contained pmatch vector must integral thus assume zero integral integral next characteristic vector single edge contained pmatch learn integral matrix finally remark add row integer multiple another row operation change integrality vector fix edge corresponding column zero performing integral row operations described assume single entry column let row corresponding entry let collection rows hence obtain pmatch conv let contain edges adjacent let let column matrix corresponding edge let matching polytope claim identify face pmatch defined setting following identity conv min min operator min taken note proof complete show linearly isomorphic pmatch hence pmatch admits milef complexity induction hypothesis implies yields inequality show inclusion holds consider vertex notice characteristic vector matching augments matching add edge therefore must pmatch using deduce inequalities hold min holds well conditions side clearly satisfied opposite inclusion let satisfying min clearly must satisfies furthermore column corresponds column moreover inequality min thus vector satisfies constraints formulation contained pmatch since pmatch also contained pmatch finally recall satisfies hence references artmann weismantel zenklusen strongly polynomial algorithm bimodular integer linear programming proceedings annual acm symposium theory computing stoc pages averkov kaibel weltge maximum semidefinite linear extension complexity families polytopes mathematical programming avis tiwary extension complexity combinatorial polytopes proceedings international colloquium automata languages programming icalp pages balas disjunctive programming hammer johnson korte editors discrete optimization volume annals discrete mathematics pages elsevier barvinok course convexity volume american mathematical society providence braun fiorini pokutta steurer approximation limits linear programs beyond hierarchies mathematics operations research braun pokutta matching polytope admit size relaxation schemes proceedings annual symposium discrete algorithms soda pages conforti zambelli extended formulations combinatorial optimization annals operations research conforti zambelli integer programming graduate texts mathematics springer conforti kaibel walter weltge subgraph polytopes independence polytopes count matroids operations research letters simone cut polytope boolean quadric polytope discrete mathematics dukes bounds number generalized partitions applications australasian journal combinatorics edmonds maximum matching polyhedron journal research national bureau standards fiorini massar pokutta tiwary wolf exponential lower bounds polytopes combinatorial optimization journal acm jain watson extension complexity independent set polytopes proceedings ieee annual symposium foundations computer science focs pages gouveia classification formulations traveling salesman problem european journal operational research hildebrand weismantel zenklusen extension complexity lower bounds mixedinteger extended formulations proceedings annual symposium discrete algorithms soda pages kaibel extended formulations combinatorial optimization optima kaibel weltge short proof extension complexity correlation polytope grows exponentially discrete computational geometry kaibel weltge lower bounds sizes integer programs without additional variables mathematical programming series lenstra integer programming fixed number variables mathematics operations research sudakov zenklusen submodular minimization congruency constraints proceedings symposium discrete algorithms soda laporte comparative analysis several asymmetric traveling salesman problem formulations computers operations research orman williams survey different integer programming formulations travelling salesman problem volume advances computational management science chapter optimisation econometric financial analysis pages springer berlin heidelberg berlin heidelberg padberg sung analytical comparison different formulations travelling salesman problem mathematical programming pokutta van vyve note extension complexity knapsack polytope operations research letters rockafellar convex analysis princeton university press polytopes need exponential size extended formulations mathematical programming matching polytope exponential extension complexity proceedings annual acm symposium theory computing stoc pages schrijver theory linear integer programming john wiley sons shephard inequalities mixed volumes convex sets mathematika veselov chirkov integer program bimodular matrix discrete optimization yannakakis expressing combinatorial optimization problems linear programs journal computer system sciences relative distance proofs part provide proofs lemmas proof lemma order prove let define inf straightforward check rdist cases value value thus follows may assume sets since rdist suffices show implies rdist implies rdist suppose first holds clearly implies hence linear map obtain thus point must points equivalently recalling treat fraction obtain inequality inf diam inequality holds every linear map every point obtain inf sup rdist diam diam sup conversely suppose clearly implies let moreover notice convex follows convexity invoke classic convex separation theorem see thm properly separate precisely using convex sets whose relative interiors holds trivially relative interior empty one find hyperplane properly separates means contained one two closed halfspaces defined contained closed halfspace defined iii fully contained shifting one assume fully contained implies linear map sup inf inf hence diam furthermore inf implies obtain rdist otherwise inf sup sup inf sup diam diam follows assumption inf finite quantities diam finally rdist diam diam claim follows directly fact every affine map satisfies order show iii let denote side claimed inequality since rdist suffices show rdist implies definition rdist note rdist implies exists linear map diam diam due particular proper interval unbounded must rdist also unbounded rdist bounded thus unbounded inequality holds remains consider case proper intervals notice intervals need closed case exist numbers describing closures intervals using notation setting max max rdist max diam well rdist max diam thus obtain max diam claimed last inequality follows prove first notice claimed inequality holds trivially sets empty also ignore pair empty sets removal modify terms inequality thus assume follows sets suffices show rdist implies rdist conv conv suppose rdist holds implies let conv write every since exist obtain conv conv shows conv conv implies relative distance conv conv completes proof proof lemma shall use alternative definition relative distance provided lemma since rdist suffices show implies rdist rdist implies suppose first contained implies clear therefore inclusion inequality rdist conversely rdist inclusion set definition means thus since contained implies proof lemma proper line segments whose endpoints since line segment contains third since obtain thus assume holds prove two inequalities first argue may assume see let affine hull exists set case let denote projection onto coordinates straightforward verify rdist rdist hold assuming inequalities hold fulldimensional sets directly obtain claimed inequalities since dim dim thus may assume dim dim holds show since rdist suffices show implies rdist assume gap holds implies exists direction min inf first observe implies second argue may assume kck holds end let denote vertex set clearly every replace smallest nonnegative number value change modification clearly still valid furthermore every must exist point hence implies claimed third since contains vector hence denoting linear projection thus obtain inequality also finally establish inf rdist show since suffices show rdist implies assume rdist holds lemma implies denoting vector particular means exists equivalently obtain since since polyhedron obtain exists vector min min holds equivalent furthermore note since must contain support size hence clearly implies hence using previous inequality obtain inf since inf must hence inf shows proof lemma since suffices show rdist rdist note lemma rdist implies every implies thus exists inequality follows fact thus every exists point yields claim
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creating modular reusable dsl textual syntax definitions feb andrey breslav petersburg state university information technology mechanics optics abreslav abstract paper present grammatic tool textual syntax definition grammatic serves parser generators tools brings modularity reuse development artifacts adapts techniques separation concerns apsectoriented programming grammars uses templates grammar reuse illustrate usage grammatic describing case study bringing separation concerns antlr parser generator achieved without common technique building ast separate semantic actions grammar definition introduction adopting concept languages dsls see developing textual syntax using tools extensively thus need different tools cases use grammars define language syntax call grammarware engineering tools paper tools use grammar definitions according paper strong need applying software engineering practices area present paper address problem modularity reuse grammar definitions grammarware engineering tools support reuse input artifacts requires tools authors quite effort implement examined popular parser generators three strong reuse capabilities though even could improved sense grammarware engineering limited parser generators may natural working new tool addressing problem implementing new parsing algorithm new concept probably one last things developer todo list grammar definition reuse since complicated feature mostly irrelevant working anyway likely appear first version unix world problem solved following principle make program one thing well probably would ideal grammarware tools could use common grammar definition language common solution reuse problems would easy support modular reusable syntax definitions addition tools would common data format using could interoperate make step towards solution propose common grammar definition language named grammatic provides strong modularity reuse capabilities box one main problems making suitable wide range tools tool requires different information attached grammar almost tool takes mere ebnf definition input one extends extra data cope grammatic allows extend grammar definition arbitrary metadata represented common format attached grammar externally sake separation concerns author new tool may use grammatic follows use grammatic grammar definition language define grammars define extensions grammar definitions terms metadata write custom back end processing definition using grammatic api allows developer implement modularity reuse features easily concentrate tool specific functionality many great tools already rarely strong terms reuse even rarely interoperate well benefit tools grammatic use latter front end namely identify extensions tool adds pure grammar definition language decide express extensions metadata attached grammar elements write generator converts properly annotated grammatic grammar definition tool input language use modular grammar definitions throughout development process necessarily modularized input tool question generated needed never modified hand paper present case study latter case demonstrate using grammatic front end antlr parser generator antlr popular due flexibility clearness many target languages supported hand lacks modularity supports reuse rather weakly paper organized follows next section give short overview grammatic main features section gives overview case study subsection describes simple way attaching grammatic antlr mentioned subsection describes creating usable though somewhat less general parser generator grammatic antlr concluding remarks given section grammatic features overview give overview four languages constituting grammatic core languages used define modular grammars attach metadata grammar definitions grammar definition language allows define grammar set symbols associated set productions concrete syntax separate productions example grammar definition use throughout paper describes simple language constants typed variables assigned values form arithmetic expressions const sum vardecl type sum type sum mult mult mult factor factor factor num sum alpha alpha alpha num example characters single quotes represent embedded lexical definitions separate notion lexical rule since necessarily required see use syntax ebnf regular expressions grammar symbol sum virtually nonterminal also virtually terminal since regular expressions right side imports templates told grammatic grammar definition language provides strong reuse techniques ideas behind techniques generalized ones implemented rats sdf lisa first focus reusing grammar definitions popular way reuse importing grammar definition might imported grammar definition means rules inserted rules may refer symbols way two grammar definitions connected frequently customize imported rules add productions symbols replace existing productions paper referred rule overriding grammatic decided use general form concept namely templates language grammar templates allows creating grammar definitions placeholders replaced actual objects upon template instantiation placeholders might defined roles identifier expression production symbol template instantiation might result grammar object type specified template declaration example template usage symbol binaryoperation name expression sign expression argument name argument sign argument import binaryoperation product factor import binaryoperation sum product factor number sum example define template named binaryoperation makes infix binary operation symbol name sign argument expression instantiate twice import instantiation results current grammar definition use new symbol product create sum sum define factor use templates overriding things put customizable set rules template provide placeholder production subexpression replaced put right thing upon instantiation symbol attributevalue production morevaluetypes attributevalue string int annotation valuesequence morevaluetypes import attributevalue expression defines template attributevalue symbol instantiates adding new production use expressions attribute values metadata told grammatic allows attach arbitrary metadata grammar definition order express various extensions used specific tools metadata annotations might attached grammar symbol individual production subexpression annotation may contain several attributes pairs attribute values may different types several predefined value types string integer tuple number pairs sequence values punctuation symbols somename str string int class name myclass super object astproduction left string integer tuple name super sequence right users may add types attribute fixed semantics metadata passive tools like analyzers transformers generators etc may use according needs even without adding custom types many things might expressed annotations powerful type sequence allows define small embedded dsls inside grammatic use dsls describe complicated custom properties see section queries attach metadata grammar many cases done directly embedding annotations grammar definition therefore different concerns mixed together results problem system modular hard understand extend employ ideas programming paradigm aop see solve problem grammatic grammar definition knows nothing metadata metadata attached outside aop done defining join points described point cuts language point cuts kind addressing notation way find object found object may attach metadata perform actions see grammatic language analogous aop point cuts call query language example query matches rules defining binary operations arg sign arg names represent variables query matches following rules sum mult mult mult factor factor default variable matches symbol may match subexpression whole production symbol production alt variables match symbol references alt matches subexpression use wildcards queries following query matches immediately rules rec rec two dots represent wildcard matches arbitrary subexpression consider metadata queries restrict particular attribute certain type value require attribute presence absence type nonterminal operation associativity commutative query matches symbol type attribute value nonterminal operation attribute present associativity attribute value type commutative attribute present aspects query selects objects grammar definition attach metadata rec rec rec leftrecursive rule adds leftrecursive attribute value symbols matched rec variable query set rules constitutes aspect many aspects independent might assigned single grammar even many grammars since queries tied concrete objects grammar structure aspects might generally reusable told query language require hard linking grammar objects objects located structural context properties rule example constitutes reusable aspect use grammar case study antlr one popular parser generators antlr mature tool based recursive descent parsing algorithm empowered syntactical predicates backtracking many projects including sun netbeans use antlr generate parsers hand antlr weaknesses sense modularity reuse main issue uses embedded semantic actions means syntactical structure language physically mixed java code describing semantic actions thus antlr grammars look bloated grammar structure clear also issues grammar reuse capabilities though resolved newer versions see want use antlr powerful features working modular grammar definitions java code clearly separated grammar sections describe could done grammatic straightforward solution general way achieving grammatic described section identify antlr extensions ebnf express grammatic metadata write generator convert annotated grammatic definitions antlr input language let look antlr extensions ebnf sake brevity focus valuable specifying rule parameters return types embedding semantic actions written java specifying syntactic predicates express extensions metadata define following attributes used grammar elements rules returns return type params sequence tuple type name parameters productions predicate string syntactic predicate production string semantic action performed production string semantic action performed production expressions string semantic action performed expression rule calls arguments sequence arguments rule call define semantic actions simple strings close antlr actually treats example let define aspect assigns antlr metadata arithmetic expressions grammar see want parser computes value expression parsed thus semantic actions perform arithmetic operations return values type int sample metadata assignment rule sum sum returns int result result mult assigns returns attribute symbol sum semantic action production semantic action occurrences mult right side action bodies result mult correspond result variable rule variable denotes value returned mult semantics defined generator convert grammatic definition antlr language depends generator treat metadata assigned grammar elements handle syntactic predicates way get following antlr definition newline define grammar rule newline metadata assignment rule newline predicate metadata also properly treated generator specific features antlr like grammar names java imports etc expressed way method general enough applied cases imagine adds original tool separation concerns reuse techniques available grammatic however generator may add value original tool show example sophisticated solution nowadays programming language usually supported strong ide makes common activities easier example features brought eclipse ide java include syntax highlighting code completion semantic highlighting templates refactorings many things developer glad enter java code outside specialized editor say antlr editor grammatic editor although editors may provide basic features like highlighting folding unlikely provide refactorings complicated features therefore want separate java code grammar definitions way could edited separately java editor using power tools solve problem generating parser builds ast processed external code approach following disadvantages consumes memory storing ast time walking also another drawback many parsers dsls simply build models close asts slightly different specific additional attributes etc case work done ast transformer program converts ast model simply waist resources since additional information might assigned parsing process thus want avoid building asts instead making parser always build ast propose use builder design pattern parser call methods interfaces builders implemented outside builder interfaces abstract semantic actions parser generated along parser code illustrate bit let look sum rule antlr sum returns int result result result left result right semantic actions abstracted like sum returns int result init isumbuilder builder left right result verbose immediately embedded actions generated one write hand approach requires less memory time since need build ast objects requires memory consumption proportional input length traverse across need create builder objects requires build one object call simultaneously present call stack requires memory consumption proportional stack depth grammatic help going define metadata give generator enough information generate builder interfaces antlr grammar definition embedded builder calls metadata need able generate builders along antlr grammar following information sufficient return values parameters rule arguments rule call give illustrative example create flexible system also allow many rules grammar symbol useful since one signature specification parameters return value syntactical rule rule might different semantics called different contexts example although bit strained may distinguish constant expressions ones containing variables since constant ones may calculated compile time writing compiler want constant expressions evaluated place rule must return value variable expressions stored objects expression trees hence one grammar rule sum mult mult get two rules different return types parameters varsum scope scope returns expression result varmult scope varmult scope constsum context context returns int result constmult context constmult context assume scope maps variable names objects denoting variables context maps constant names values want duplicate rules grammar sake matters express metadata single rule example sum builders expression varsum scope scope int constsum context context mult varsum varmult scope constsum constmult context see small dsl inside grammatic metadata actually two dsls one specifying return types parameters another one specifying called rules arguments builders attribute symbol defines signatures names return types parameters antlr rules generated symbol two rules generated example call attribute symbol reference specifies antlr rule arguments called case brevity metadata definition given achieved grammatic ability define internal dsls done parsing attribute values type sequence externally supplied parser lexical structure dsls fixed sequence elements identifiers strings numbers tuples sequences punctuation values serve tokens generator produce two rules given form example omitted builder calls rule definitions clear full rule looks like varsum scope scope returns expression result init ivarsumbuilder builder scope scope scope conclusion future work paper addressed solving problems modularity reuse grammar definitions defining general front end grammatic front end adopted newly developed tools api attached existing tool creating converter universal format tool specific input format grammatic provides language defining modular grammars supporting templates imports extensible attaching arbitrary metadata also supports separation concerns defining reusable aspects showed two ways using grammatic bring reuse separation concerns popular parser generator antlr straightforward way based expressing extensions added tool general grammar definition language terms metadata creating generator transforms annotated grammatic definition tool input also presented way using grammatic separate custom code written target language java grammar definition metadata done using builder design pattern generating set interfaces implemented manually allows developer use power ide working java code case study shows grammatic helps adding reuse modularity capabilities existing tools plan apply practices tools find things supported grammatic plan support metadata templates grammar testing facilities text generation error tracking facilities helping convert errors reported back end grammatic errors long term goal create common grammar definition platform usable wide range grammarware engineering tools references martin ward programming software concepts tools paul klint ralf chris verhoef toward engineering discipline grammarware acm trans softw eng steven johnson yacc yet another compiler compiler unix programmer manual volume pages holt rinehart winston new york usa terence parr definitive antlr reference building languages pragmatic bookshelf raleigh hanspeter mssenbck johannes kepler compiler generator user manual etienne gagnon laurie hendren compiler framework proceedings tools pages open architecture ware xtext http heering hendriks klint rekers syntax definition formalism sdf reference manual sigplan marjan mernik mitja leni enis generator system lisa ieee proceedings hawaii international conference system sciences pages robert grimm better extensibility modular syntax pldi proceedings acm sigplan conference programming language design implementation pages new york usa acm mcilroy pinson tague unix time sharing system forward bell system technical jounal part scheerder vinju visser disambiguation filters scannerless generalized parsers compiler construction pages springerverlag bryan ford packrat parsing simple powerful lazy linear time functional pearl icfp proceedings seventh acm sigplan international conference functional programming pages new york usa acm terence parr reuse grammars embedded semantic actions international conference program comprehension gregor kiczales john lamping anurag mendhekar chris maeda cristina videira lopes jean marc loingtier john irwin gregor kiczales john lamping anurag mendhekar chris maeda cristina lopes jean marc loingtier john irwin programming ecoop springerverlag gregor kiczales erik hilsdale jim hugunin mik kersten jeffrey palm william griswold overview aspectj ecoop proceedings european conference programming pages london eclipse foundation eclipse ide http erich gamma richard helm ralph johnson john vlissides design patterns elements reusable software
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aug duality equivalence results majid rahro zargar abstract let relative local ring respect ideal set paper investigate properties matlis dual hca hca show modules treat like canonical modules local rings also provide duality equivalence results respect module hca results lead achieve generalizations known results local duality theorem provided local ring admits canonical module contents introduction notation prerequisites generalized local duality generalized dualizing modules maximal relative modules references introduction throughout paper commutative noetherian ring proper ideal case local maximal ideal denotes completion denotes injective hull residue field denotes matlis dual functor hom theory canonical modules local rings developed bruns herzog chapter general setting arbitrary local ring canonical module finitely generated special case condition turns equivalent bruns herzog definition namely canonical module precisely maximal module type one finite injective dimension mathematics subject classification key words phrases local cohomology canonical module complex relative module zargar literature called dualizing module remarkable result foxby reiten sharp proved local ring admits canonical module homomorphic image gorenstein local ring particular complete hnm canonical module hand local duality theorem provides fundamental tool study local cohomology modules respect maximal ideal local ring although applies local rings expressed homomorphic images gorenstein local rings great restriction class local rings includes local rings points affine varieties mentioned complete local rings theorem provides functorial isomorphism ext modules canonical module local cohomology modules respect maximal ideal local ring local ring canonical module exists plays important role studying algebraic homological properties ideals modules thus finding modules preserve beneficiaries canonical modules aim many commutative algebraists direction principal aim paper study properties hca case relative local ring respect precisely one local cohomology module respect provide connection module hca local cohomology module respect ideal indeed show modules treat like canonical modules local rings recently modules studied authors hellus hellus hellus schenzel khashyarmanesh schenzel led interesting results organization paper follows section collect notations definitions used present paper section first generalization local duality theorem provide following result theorem let ideal injective let arbitrary complex hia integer integer complexes following isomorphisms tor ext hom hom also among things application theorem proposition provide connection module local cohomology modules respect ideal next theorem state one main results shows module basic properties ordinary canonical modules local rings indeed prove following result duality local cohomology modules theorem let local ring following statements hold true ideals hib iii one ext hia ext hand lemma shown hcb ideals also proposition provide result leads determine endomorphism ring local cohomology module support next section first introduce concept maximal relative cohenmacaulay modules respect ideal proposition generalize result remark raise quite known conjecture related maximal modules addition theorem generalize another known result related ordinary canonical modules local rings indeed show relative local ring respect exists natural isomorphism ext ext particular maximal hom hom also theorem relative local rings establish characterization maximal relative modules suppose moment local ring dualizing module finitely generated kawasaki theorem showed finite projective dimension next theorem khatami yassemi generalized result indeed showed result hold true whenever finite gorenstein dimension theorem establish another main result using instead provides generalization mentioned result khatami yassemi finally proposition application theorem could find another generalization results theorem theorem notation prerequisites hyperhomology sequence together maps zargar defined supp right derived functor functor exists complex defined injective resolution complex integer local cohomology module respect defined hia also functorial isomorphism denotes derived category complex respect hence implies hia details refer let tensor product complex denoted defined flat resolutions respectively also let right derived homomorphism complex denoted rhom defined rhom hom hom hom projective resolution injective resolution respectively two complex set ext rhom tor integer shift complex complex given also next contravariant additive exact functor denotes category definition say finitely generated relative respect precisely one local cohomology module respect clearly case grade largest integer hia convenance use notation relative respect furthermore finitely generated said maximal observe definition provides generalization concept cohenmacaulay maximal modules also notice notion relative modules connected notion cohomologically complete intersection ideals studied led interesting results furthermore recently modules studied definition proper ideal set grade inf ext notice grade least integer duality local cohomology modules grade hia local ring define hom case convenience set generalized local duality starting point section following result plays essential role present paper result provides generalization local duality theorem indeed generalization done ideals given ring complexes local cohomology modules respect precisely one generalization local duality becomes special case maximal ideal given local ring theorem let ideal injective let arbitrary complex hia integer integer complexes following isomorphisms tor ext hom hom proof let elements let denotes complex respect notice hia thus assumption hence one deduce exists isomorphism derived category hca therefore use following isomorphisms tor third isomorphism follows lemma compte proof note since injective rhom hca hom hca therefore follows following isomorphisms ext hom rhom hom rhom rhom rhom hom hom tor hom hia required notice third isomorphism follows theorem zargar following result special case theorem category proposition let ideal injective let arbitrary hia integer integers tor following isomorphisms tor ext hom hom proof notice since tor isomorphism derived category hence immediately follows theorem consequence result following result corollary let local ring dualizing finite gorenstein flat dimension see definition following isomorphisms tor ext proof first notice maximal also view proposition tor hence one use known fact hom proposition complete proof following result generalization local duality theorem consequence proposition recovers theorem proposition let local ring rmodules flat hca faithfully flat rmodules following isomorphisms tor ext hia proof first notice theorem directed index set family finitely generated free limfi notice relative respect therefore hja limhja hence one use proposition complete proof application theorem prove known local duality theorem already prove theorem duality local cohomology modules corollary let local ring dualizing integers one ext proof first notice artinian also therefore using flat base change theorem matlis duality theorem theorem fact implies following isomorphisms hom hom hom ext hom ext hom ext hom ext hom hom ext ext notice third isomorphism follows proposition last isomorphism follows notice local ring dualizing finitely generated sup ext depth generalization result provide following result immediate consequence proposition corollary let local ring one sup ext grade grade generalized dualizing modules notice local complete ring respect topology coincides ordinary dualizing module main aim section determine properties module show much modules behavior similar dualizing modules direction need following two lemmas play essential role proof results lemma let integer following conditions equivalent hia ext iii ext zargar proof let straightforward see hom hom hence view theorem obtain grothendieck third quadrant spectral sequence ext ext note therefore consider following filtration ext see ext implication iii trivial implication iii follows similar argument following lemma needed prove next results proved propositon using spectral sequence tools convenience provide new proof using derived category tools lemma let ideal hia integer integer one tor tor proof let let elements let denotes complex respect notice hia assumption hence one deduce exists isomorphism derived category also similar argument one therefore use following isomorphisms tor tor complete proof first main result section provide following result determines basic properties module duality local cohomology modules theorem let local ring following statements hold true ideals hib iii one ext hia ext proof first notice since one use theorem see hca let ideal part hib show hib end lemma enough show ext proposition ext hence assertion done module iii since vdim ext easily follows proposition follows proposition assumption first notice supp ext supp exists rected index set family finitely generated submodules ext since supp supp ext lim theorem therefore hia ext next show hia ext end using lemma enough show ext ext since view lemma tor tor therefore one use proposition get following isomorphisms ext ext tor tor complete proof following corollary known results dualizing module consequence previous theorem corollary let local ring dualizing following statements hold true zargar iii dim one ext ext proof straightforward see one may assume complete local ring therefore one using theorem nothing prove following lemma needed proof next proposition recovers corollary lemma let situation theorem ideals exists isomorphism proof first one use theorem theorem see hcb injective view proposition theorem get following isomorphisms hom hcb ext last isomorphism follows proposition therefore melkerson lemma theorem hcb artinian hence exists integer hand corollary hcb therefore follows form theorem iii immediate consequence previous lemma next result corollary let local ring depth let system parameters next provide example justify theorem lemma example let ring power series field set notice local ring maximal ideal consider following exact sequence duality local cohomology modules applying functor exact sequence one gets exact sequence notice hom injective hence show end enough show injective assume contrary injective since exists element element hence exists element hand exists element since divisible exists element contradiction hence also next applying functor obtain exact sequence therefore since hia let hia since ext also one use together fact natural map one one see ext hom since hom hom one hom hand one hom hom must prove end lemma enough show ext hom since ext hom tor need show tor end consider exact sequence induce exact sequence tor use fact complete proof proposition let local ring let ideal set inf hib consider following statements hib iii ext ext implications iii hold true moreover condition hom satisfied ext zargar proof first notice one use lemma proposition deduce hence finite number view proposition assumption one isomorphism therefore one use theorem lemma hia htb complete proof iii let since lemma plies ext htb hand since injective one use proposition see ext htb ext hom hence proof complete iii first view lemma iii assumption deduce hia htb notice view proposition ext since hib hib one use lemma see ext htb hence hia htb therefore one use proposition assumption deduce hom ext hence hom artinian therefore theot rem artinian thus one use corollary see hence injective module therefore use corollary see finial assertion suppose statement hold true use proposition proposition get following isomorphisms ext htb htb ext one use complete proof following corollary immediate consequence proposition theorem notice first part next corollary proved theorem corollary let local ring following statements hold true hom hca hca hom ext ext hca hca duality local cohomology modules maximal relative modules given finite free resolution define ker starting point section following proposition recovers well known fact finitely generated module local ring zero maximal dim proposition let finitely generated let finite free resolution following hold true grade min grade zero maximal proof proceed induction nothing prove let suppose result proved let finite free resolution following exact sequences hence one use proposition inductive hypothesis see grade min grade grade min grade min grade min grade let grade next since supp spec theorem fore one use complete proof immediate application previous proposition next corollary corollary let local ring let finitely generated whenever remark important conjecture indicates local ring admits maximal module raise conjecture maximal relative modules indeed following conjecture conjecture let proper ideal local ring exists maximal relative module respect zargar let local ring dualizing exists known isomorphism ext ext whenever cohenr macaulay dimension following result use instead provide generalization result theorem let local local ring grade nonzero exists natural isomorphism ext ext particular maximal hom hom proof let view theorem one ext therefore one use following isomorphisms ext rhom rhom rhom ext deduce hom ext ext notice second isomorphism follows lemma hence obtain isomorphism hom rhom derived category hand view corollary rhom hom hence one use theorem obtain following isomorphism rhom rhom rhom therefore use following isomorphisms ext ext rhom ext rhom hom rhom rhom rhom rhom rhom rhom rhom rhom complete proof forth isomorphism follows lemma fifth isomorphism follows lemma following theorem provides characterization maximal relative modules relative ring duality local cohomology modules theorem let local ring let finitely generated following statements equivalent maximal ext iii following conditions hold true hom hom ext hom proof implications follows corollary fact also implication iii follows theorem iii view theorem hia hom therefore one use proposition complete proof iii view proposition one ext hom hom therefore assumption iii one gets hja hom hand argument proof theorem one see hom hence hja hom therefore one use proposition assumption iii lemma obtain following isomorphisms ext ext hca hom tor hom tor hom hence ext lemma hia therefore using flat base change theorem independence theorem implies hia hence grade maximal described introduction following theorem one main results provides generalization result theorem khatami yassemi next theorem shall use notion grade defined grade inf ext theorem let local ring dimension let finitely generated tor following statements hold true ext ext zargar suppose sup ext depth grade depth dim hia grade proof let projective theorem exists directed index set family finitely generated free limfj therefore view theorem hence one lim use proposition see ext hence using theorem corollary third quadrant spectral sequence ext tor ext since tor therefore spectral sequence collapses column hence one gets isomorphism ext ext flat theorem next notice since ext ext therefore required suppose ext grade hence one use proposition complete proof converse follows proposition next recall concept gorenstein dimension introduced auslander definition finite said gorenstein dimension zero write ext ext hom iii canonical map hom hom isomorphism integer said gorenstein dimension exists exact sequence sequence exist write let local finitely generated finite corollary proposition implies sup ext duality local cohomology modules depth depth tor dualizing rmodule next provide remark shows converse fact longer true sup ext depth depth tor necessary finite remark let local ring admits dualizing module let finitely generated sup ext depth depth next suppose contrary converse mentioned fact true case prove finite end may assume let dim let syzygy finite free resolution view proposition maximal also since ext ext one ext therefore hom maximal macaulay next view following isomorphisms rhom rhom rhom derived category since ext deduce rhom hom hand view corollary ext hom tor therefore thus using contrary assumption lemma get hence indeed corollary dim therefore could prove every finitely generated satisfying condition sup ext depth depth finite contradiction exists local artinian ring admits finitely generated satisfying condition ext see considering mentioned remark following proposition generalize result theorem indicates local ring dualizing module finitely generated finite proposition let local ring dualizing suppose finitely generated sup ext depth depth tor proof first notice may assume complete also since one gets grade depth dim therefore assertion follows theorem zargar acknowledgements grateful professor hossein zakeri professor kamran professor olgur celikbas kind comments assistance preparation manuscript references auslanser anneau gorenstein torsion commutative commutative notes mangeney peskine szpiro normale jeunes filles paris brodmann sharp local cohomology algebraic introduction geometric applications cambridge university press cambridge bruns herzog rings cambridge university press cambridge christensen gorenstein dimensions lecture notes mathematices berlin christensen foxby hyperhomological algebra applications commutative rings preparation christensen foxby holm beyond totally reflexive modules back noetherian perspectives edited fontana kabbaj olberding swanson springer media llc new york naghipour tousi cohomological dimension certain algebraic varieties proc amer math soc enochs jenda relative homological algebra gruyter berlin foxby gorenstein modules related modules math scand hellus finiteness properties duals local cohomology modules commun algebra hellus schenzel cohomologically complete intersections algebra hellus schenzel notes local cohomology duality algebra hellus endomorphism rings local cohomology modules proc amer math soc herzog kunz der kanonische modul eines lect notes jorgensen sega independence total reflexivity conditions modules algebras representation theory kawasaki modules local rings math khashyarmanesh matlis duals local cohomology modules arch math khatami yassemi tensor products rocky mountain math lipman lectures local cohomology duality local cohomology applications guanajuato lecture notes pure appl dekker new york rahro zargar zakeri injective gorenstein injective dimensions local cohomology modules appear algebra colloquium rahro zargar zakeri flat gorenstein flat dimensions local cohomology modules duality local cohomology modules reiten converse theorem sharp gorenstein modules proc amer math soc rotman introduction homological algebra second springer new york schenzel matlis duals local cohomology modules endomorphism rings arch math sharp gorenstein modules complete local ring quart math oxford ser faculty mathematical sciences department mathematics university mohaghegh ardabili ardabil iran address address
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information observer external internal information processes information cooperation origin observer intellect vladimir lerner usa lernervs observing interactive processes conversion observed uncertainty observer certainty natural phenomenon creating actions information bit information observer information observer emerges interacting random field kolmogorov probabilities link kolmogorov law probabilities bayesian probabilities observing markov diffusion process probabilistic impulses action cuts maximum minimal entropy impulse correlation following action transfers maxim impulse performing dual principle converting process entropy information merging actions generate microprocess within bordered impulse producing information bit free information microprocess probability approaches free information follows cutting correlation connecting markov process impulses impulse free information attracts interacting bits borderer impulse attracting interaction captures energy interactive action memorizes bit multiple bits connected free information move macroprocess bits triplet macrounits memorized information binds reversible microprocess within impulse irreversible information macroprocess along process observation automatically converts cutting entropy information consecutively automatically converts cutting entropy information conveying process information causality certain logic complexity macrounits logically information networks encoding units information geometrical structures enclosing triplet code selects objective subjective information observer depending encoded units geometry observer dynamical geometrical hierarchical structures limited boundary timespace distributed structure cooperates information decreasing complexity triplet units built attraction resonance limited stability leading finite triplet structure ending triplet bit encloses observing process multiple finite triplet number bit free information potentially loses stability evolving chaos possesses ability cooperating triplet multiple binds ending triplets encloses subjective observer information cognition intelligence observer cognition assembles common units multiple attraction resonances forming hierarchy accept units concentrates recognizes node ended triplet observer hierarchical informational networks measures level observer intelligence maximal number accepted triplet levels multiple measures observer maximum information intelligence comparative intelligent observers intelligent observer recognizes encodes digital images message transmission enables understanding message meaning variation problem integral measures observing process entropy functional bits information path integral formalizes minimax law describes regularities processes solving problem mathematically defines processes selective objective subjective information observers invariant conditions observer trajectory observation process carries wave function probabilistic certain selforganizes hierarchical structures functional regularities create united information mechanism whose integral logic mechanism transforming multiple interacting uncertainties physical human information cognitive logic information structure encoding intelligence coding structure minimax information law creates invariant information physical regularities applications practical implementations confirm formalism theoretical concepts results keywords impulse probabilistic observation cutting correlation minimax information law wave function processes integral information measure causal logic cooperative information dynamics hierarchical network objective subjective observers cognition intellect designing observer applications implementations introduction revealing information nature various interactive processes including multiple physical interactions human observations communications biological social economic interactive systems integrated information observer becomes important scientific task physical approach observer developed copenhagen interpretation quantum mechanics requires act observation physical carrier observer knowledge observer role describes formalism quantum mechanics according bohm ontological interpretation quantum physics physical processes determined information difference form makes difference content unfolds intention intention arises previous perception meaning significance certain total observer entails mental processes eccles quantum approach find way control wheeler physical theory origin observer introduces doctrine bit wheeler hypothesized bit participates creating origin physical processes however wheeler theory remained unproven theory include bit existence wheeler theory many physical scientists including einstein penrose define observer physical origin weinberg focusing probability quantum mechanics gets trouble probability natural origin follows physical quantum field vacuum quantum fluctuations natural probability fluctuation originates physical particles weller included observer wave function according standard paradigm quantum mechanics natural nonetheless quantum bayesianism combines quantum theory probability theory states wave function exists world rather merely reflects individual mental since information initially originates quantum process conjugated probabilities study focus physics observing process interacting particles essence kolmogorov establishes probability theory foundation information theory logics shannon information measures relative entropy applies random states information process divergence probabilities distributions measures relative information connections states observed process jaynes applies bayesian probabilities propose plausible reasoning mechanism whose rules deductive logic connect maximum bayes information entropy human mental activities subjective observer references along many others studying information mechanisms intelligence explain various physical phenomena whose specifics still mostly unknown science knows interactions built structure universe fundamental phenomena many studies interactions specifics however one approach unified study common information origins regularities differentiation first approach unifying studies published extends results review approach focuses observations interactions producing observer unified approach shows information observer emerges observing random interactive process observation uncertainty random interactive process converts certainty certainty information single certain action known bit elementary unit information multiple observations generate bit dynamics informational dynamics bits organize triplets logically assemble informational network process network assembling triplets merge interact interaction gets memorized becomes node informational network nodes also logically organize sequence logically organized nodes defines code network code encloses information network code integrates carries prior observations immerged information observer informational observer emerges probabilistic observation without preexisting physical law even unknown particles planets could revealed probable real interactions occur identifies interactions primary indicator potential probabilistic object observation introduced approach based informational origin observer explains observer emerges random observations shannon approach defines entropy probability measures uncertainty observation entropy erased uncertainty disappears instead appearing equal certainty uncovering certainty uncertainty scientific path determine facts reality entropy erased physical energy exerted converted enclosing certainty certainty information turn physical entity contains physical energy equal energy spent erase entropy process elementary unit information bit created summarize physical bit evolved removing uncertainty observation bit evolves abstract probability observation elementary observer since information initially originates quantum process conjugated probabilities study focus physics observing process interacting particles essence approach substantiates every step origin unified formalism mathematics logic formalism allows understand describe regularity law informational processes preexisting physical law irrelevant emerging regularities approach approach initial points interaction objects particles primary indicator origin field probability source information physics interactions abstract actions impulse probabilistic real multiple interactions create random process whose interactive impulses model markov diffusion process process observes objective probabilities linking kolmogorov law axiomatic probabilities bayesian probabilities sequence probabilistic probabilistic impulses initiates bayes probabilities within markov process evolving markov process generating correlation observing process impulses particular objective probability observes specific set events entropy correlation holds removing entropy correlation uncertainty produces certainty originating information emerges particular set observing probabilistic events create specific information observer points describe details observing objective probabilities measure idealized virtual process impulses virtual observer observer processing random interactions generates virtual probability measurement random process uncertainty observable process virtual observer probabilistic model potential observer hidden random bits models markov observable process located surrounded random field therefore affected field random growing probabilities virtual impulses observing process correlations increase real impulses emerge observable markov process correlation virtually cuts probabilistic impulses hidden impulses release observing multiple interactions integrate growing correlation process entropy impulses cutting removing entropy correlations create real observing information moves information process integral measure objective probabilities connecting interactive observations starts virtual probabilistic observation virtual observer evolves objective interaction subjective real integrates observing impulses enable unities observing information information observer merging actions markov probabilistic impulse generate microprocess within bordered impulse microprocesss produces information bit free information probability approaches one free information follows cutting correlation connecting markov process impulses impulse free information attracts interacting bits borderer impulse attracting interaction approaching certainty captures energy interactive action memorizes bit multiple bits connected free information move macroprocess transitive gap separating edge reality overcomes injection needed energy macroprocess bits continue attracting creates resonance resonance process links bits duplets free information one bit pair gets spent assembling duplet free information duplet bits attracts third forming bit assembling memorizing three basic elementary structure macrounits triplet third bit free information attracting another duplet bits creates two bound triplets enclose another triplet continuing process creates levels bound triplets informational network levels hierarchical structure triplet information network collects encloses entire network information bit memorized information binds reversible microprocess within impulse irreversible information macroprocess along observing process triplets emerging process cooperates bayes sequential probabilities generate probabilistic logic observing process information process transforms certain logic observation consecutively automatically converts cutting entropy information conveying process information causality certain logic bit logical complexity bound cooperative complexity triplet three segments information macroprocess generates three symbols one free information composing minimal logical code encodes macroprocess physical information process triplet macrounits logically information networks encoding units information geometrical structures enclosing triplet code select objective subjective information observer depending encoding units geometry observer dynamical geometrical hierarchical structures limited boundary distributed structure cooperates information decreasing complexity attraction triplets ended triplet contains maximum amount free information enables ins attraction creating multiple networks domain informational networks connect free information third bit triplet sufficient attract another bound duplet triplet becomes ended triplet network completes finite network process network stops loses stability creating chaos bits dna ended triplet code forms telomerase controls dna life cycle however chaos could pair bits bind creating duplet enough free information attract another bit attracting bits create triplet building network may continue hence finite chaotic process could bound triplets creating next generation assembling observer hierarchy observing probabilistic certain logic assembling triplets networks domains free information logic build observer cognitive logic intelligence code every observer different amount observed logic bits needed building specific individual cognitive logic intelligence code process analogous human brain cognition neuron actions modeling bit moreover described finite informational networks multiple networks well performs human brain information machine process probabilistic observation attracting free logical bit forming triplet selects bits equal speed creating resonance bits cohere resonance assembles common logical loop loop bits involved resonance recognizes cognition ability recognizing binding bits resonance process attracting logical bits sufficient performing cognitive actions cognition emerge build duplets triplets enough observed information cognition arises levels informational networks along distributes units hierarchy node accepts units node concentrates recognizes hierarchy distributed logical loops builds chain multiple logical hierarchical units resonance frequencies cognitive logic units provide interactive actions attracting impulses external energy attracting actions carry free logic assembling logical unit opens external impulse carrying landauer energy starts erasing entropy memorizing information bit bit free information encoding memorized bit hierarchy generates multiple triplet codes observer logical code integrates observation process carries wave probabilistic real spinning space trajectory wave function frequencies observer hierarchical structures movement along trajectory observing process generating triplets logic composes double spiral space dss integral triple logic code memorizes dss information helix structure dss helix structure rotates spinning physical wave function frequencies multiple local bits coding units encodes observer triplet coding structure call observer intelligence logical switching free information hierarchical level performs intelligence functions generate local code functions distributed hierarchically along assembling logical units cognitive chain distributed intelligence coding actions hierarchical level control entrance needed external physical processes dss encodes triplet units information macrodynamic process related physical irreversible thermodynamic process implements observer encoding logic thermodynamic process forces determine power physically implement encoding actions following observation performance actions provides feedback observer performance approach results describe emerging observer information regularities intelligence satisfying simple natural law conversion uncertainty certainty observer interactive probabilistic observation environment natural real interactions converts entropy information interactions phenomena approach formalism comes feynman concepts variation principle process integral problem solution mathematically formulates physical law regularities process variation problem integral measures observing process entropy functional bits information path integral formalizes minimax law describes regularities observing processes solving problem mathematically defines processes selective objective subjective information observers invariant conditions observer functional regularities create united information mechanism whose integral logic mechanism transforming multiple interacting uncertainties physical human information cognition selforiginating observer information intellect logic holds invariance information physical regularities following minimax information approach focuses formal information mechanisms observer without reference specific physical processes could originate mechanisms formally described information regularities contribute basic theory brain function information mechanisms interactions allow finding information structure artificial designed observer toward artificial brain information formalism describes self information machine creates humans nature essence approach main stages approach works forewords interactions natural fundamental phenomena multiple events common environment universe interactions built nature elementary natural interaction consists action reaction represents abstract symbols actions impulse modeling bit physical examples sequence opposite interactive actions models rubber ball hitting ground reversible microfluctuations produced within irreversible macroprocess physical biological processes physical actions connected naturally bit logical sequence originate information structure logic basis dna brain information mechanism forms many physical micro show probabilistic interactions instead interacting particles create information physical processes observer information probability measures multiple events therefore probability measures interacting probability interactive actions predict real interaction particles interaction objects particles primary indicator origin measure probabilities field probability source information physics approach aim formal principles methodology explaining procedure emerging interacting observer information aim derives unifying different interactions independent origin focusing observation interactive observer objective probabilities measure idealized virtual impulses observation virtual observation observation correlates random interactive action observing interactive random process multiple interactions evaluates probabilities measured equivalent entropy correlation cutting correlation impulse high probable observation removes entropy uncertainty producing certainty information integrating cutting correlations finally produces information observer natural real interactions innately convert entropy information information observer information observers may reproduce brain memorized image starting points multiple interactive actions random events surrounding random field interacting random events formally describe probabilities kolmogorov theory probabilities probability field defines mathematical triple sets possible events subsets sets probability borel function sets defined condition triple formally connects sets possible events sets actual events probability function abstract axiomatic kolmogorov probability predicts probability measurement experiment whose probability distributions tested relative frequencies occurrences events satisfy condition symmetry equal probable events form multiple infinite sequence independent events satisfying kolmogorov law random field sequence random events collected independent series forms random process including markov diffusion process modeling multiple interactions markov diffusion process describes probability distributions random field events satisfying kolmogorov law probability affect markov diffusion process probabilities distributed field via transitional probabilities randomly switch drifts speeds markov process switching markov speeds sequentially change process current posteriori bayes probabilities whose ratio determines probability density random impulses part markov process links kolmogorov probabilities markov process bayes probabilities markov impulses common markov diffusion process within markov process bayes probabilities densities randomly observe process composing observing process virtual observation kolmogorov law probabilities observe initiate discrete probabilities actions bayes probabilities observe thus observing markov probability impulses different former observing process holds random impulse actions probability accordingly multiple random actions describe probability distributions observing sequence specific set events formally define observing triple probability field natural fluctuations elementary events random probabilistic impulse virtual observation represents immanent randomness moving stochastic process time surrounding random field observing random probabilistic interactions become logical sequence encoding bits observation virtual observer uncertainty information certainty objective probabilities quantify idealized virtual impulses whose actions represent act virtual observation observation measures probability possible events potential observer multiple virtual observations transit probabilities along interactive random process generated virtual probability measurement models observable process potential virtual observer impulse virtual reversible actions second recursion affects predecessor yes connecting weak correlation arising correlation connection memorizes action indicating start observation following impulse correlation encloses mean time interval begins time observation observing bayes posteriori probabilities determine arrow time course observation process continuous correlations uncertainty observation measures conditional entropy bayesian posteriori probabilities maximal uncertainty measures posteriori probabilities connection approaches zero theoretical uncertainty infinite entropy measures whose conditional entropy time exist finite uncertainty measure nonzero correlating finite priori posteriori probability interactive events finite time interval following finite conditional entropy example finite uncertainty process allows measuring observable process uncertainty relative white noise common formal model natural random process markov diffusion process includes considered interactive observable process elementary dirac increases bayes posteriori probability concurrently increases probability virtual impulse real impulse probability decreases related uncertainty information notion uncertainty measures reduction uncertainty maximal posteriori probability assume evaluates observing probabilistic fact actually shown action cuts maximum impulse minimal entropy following action transfers maxmin impulses decreasing following entropy process thus impulse interactive actions express impulse minimax principle impulse observation imposes minimax principle increasing posteriori probability simple example rubber ball hits ground energy interaction partially dissipates increases entropy interaction ball following reverse movement holds less entropy part dissipated leading entropy bouncing ball adding periodically small energy compensating interactive dissipation supports continuing bouncing observation kronicker analog dirac formally imposes minimax principle automatically impulse minimax principle imposed sequential bayesian probabilities leads growing posteriori probability correlations reducing process entropy along observing process correlation freezes entropy bayes probability hidden within correlation connecting hidden process correlations conveys probabilistic causality along process particular probability observes specific set events entropy correlation holds correlation connects bayesian posteriori probabilities temporal memory store virtual connection renews virtual events actions observed observing process automatic renewal virtual calls virtual observer acts actions resume virtual observer belongs process whose virtually starts next impulse action sending probing impulses process observer temporal ending stopping observation starting virtual limits identified threshold new virtually observing temporary memorizes whole events starting observation including summarized integrated entropy virtually cutting correlations memory last current correlation connection automatically holds integrated entropy correlations memory temporary holds difference probabilities actions virtual measure distance impulses actions probabilistic accuracy measuring correlation measuring beginning starting observation identifies interval start also virtual disappearing new connection identifies next interval memorized connection random process impulses hold virtually observing random time intervals hidden entropy events collecting measuring uncertainty along random process integrate entropy functional minimax process interacting impulses carries minimax variation principle imposed brings invariant measure running time intervals correlation indicates appearance invariant time interval difference probabilities actions temporary holds memory correlation identifies virtual measure adjacent distance impulse actions originates space shift quantified curved time space displacement shifts virtual observation source random field space process initiating probabilistic emergence coordinate system gradient entropy force depending entropy density space coordinates displaced process actions continues requesting virtual observation probing impulses intends preserve probabilities invariant impulses observations impulses enclose reducing entropy movement forming volume initiates observer entropy correlation connections starting virtual observation shape correlation virtual observer volume entropy force rotates curved coordinate system within volume developing rotating coriolis force moment moving space trajectory along coordinate system depends gradient velocity running movement gradient entropy along rotating interval trajectory could engage next impulse rotating action increases correlation temporally memorizing observation memory temporary holds difference starting correlation accuracy closeness determines observer location shape evolving shape gradually confines running rotating movement formation shape observer virtual observer displaced initial virtual process sends discrete impulses virtual probes test preservation kolmogorov probability measure observer process probes frequencies test checks abstract probability via symmetry condition indicating probability correctness observing structural location observer increasing frequencies observer probes check growing probabilities symmetry virtual observer virtual geometrical structure virtual observation gains real form transforming integrated entropy correlated events equivalent specific information observer real physics affecting virtual observation virtual probing impulses replicate information impulses start probabilistic path maximal entropy uncertainty maximal real certainty cutting entropy functions within markov process related impulse cutting satisfying microprocess within impulse information observer since discrete actions forming virtual real controls cut within markov process preserve additive multiplicative properties requirement limits admissible controls class two real two complex functions applying control functions identify cutting invariant impulse extreme imposed process impulse cutting proves three invariant virtual impulses process time intervals enable generating single invariant information impulse instead three emerges last information delivered impulse control transferred nearest impulse keeps information connection impulses providing persistence impulse sequence condenses two previous impulses intervals entropies following information impulse interval also proves action starting information impulse captures markov multiplicative entropy increments impulse includes three parts multiplicative action capturing entropy random process impulse cut process entropy delivered impulse control transferred nearest impulse keeps information connection impulses provides persistence continuation impulse sequence process time three parts holds invariant portion within impulse measure since cutting impulse preserves invariant information measure third sequential cutting impulses triples condensed information density implies final ipf impulse condenses previous ipf impulses final time interval limited depending process time course final time interval evaluates density impulse information entropy also limited well total ipf conclusively ipf finite maximal information density limits finite minimal physical time interval accessible time course finite three times intervals within invariant impulse parts allow identifying related discreet correlation functions cutting increments results verify estimated entropy contributions parts impulse following information increments increment measures memorized correlation impulse probability events impulse time interval invariant measure growing probability correlations intensity entropy per interval entropy density increases following interval indicating shift virtual actions measured time interval unit growing density opposite actions merge jumping impulse whose cutting action curves emerging time units starting impulse time interval following rotating curved action initiates displacement within impulse opposite rotating actions rotating displacement space shift quantifies curved time displacement shift two space units replaces curved time units transitional impulse within impulses rises transitional impulse preserves probability measure initial time impulse holding impulse probability two space units counterpart curved time illustration origin impulse space coordinate measure curving time coordinate measure transitional movement right initial impulse actions jump locality obvious relation ratio leads ratio measuring units elementary space curvature equals inverse radius thus transitional probability equal related time interval impulse vice versa allows appearance transitional time interval simultaneously random observation merging impulse border cutting curving time spots curve transition cutting time finite space unit thus growing bayes posteriori probability along observations merges neighbor impulses generating interactive jump impulse border jump brings extreme discrete displacement rotates jump opposite actions anti symmetric entropy increments starting microprocess within bordering impulse merging entropy increments relate impulse actions superposition jump initiates inner transitional impulse within microprocess displacement preserves probabilities transitional impulse emerging movement conserves probability measures discrete form impulse probabilities time interval interactive jump impulse estimates impulse curvature entropy measure invariant measure equal satisfaction symmetry condition impulses axiomatic probability begins transforming microprocess probability pairs conjugated entropies correlated movements rotating conjugate movement starts discrete forming transitional impulse rotating angle makes symmetrical mirror copies observation holding within transitional impulse maximal correlation adjoins conjugated symmetric entropy fractions uniting entropies running pair entanglement confines entropy volume pair superposition transitional impulse angle rotation entangled pair entropy fractions appears simultaneously starting space interval correlation binging couple maximal probability extremely tangible pair correlated conjugated entropies virtual impulse separable real action entangled increments captured rotation forming volumes adjoin entropy volumes stable entanglement conjugated entropies reach equalization anti correlations cohere arising correlated entanglement opposite rotating conjugated entropy increments condenses correlating entropies entropy volumes microprocess stable entanglement minimizes quantum uncertainty entangled virtual impulses increases bayes probability maximal priori probability approaches entropy volume rotating moment grow still maximal priori probability virtual process posteriori probability real process small microprocess gap associated probabilistic transitive movement separating entropy appearance information gap implies distinction statistical possibilities entropies uncertain reality informationcertainty reality bayes probabilities measure may overcome transitive gap gap holds hidden real locality within hidden correlation rotating momentum growing increased volume intensifies time volume transition gap acquiring physical property near gap end rising probability last posterior probability approaching overcomes last prior virtual probability curving momentum may physically cut transferred entropy volume growing posteriori probability virtual impulse successively brings reality posteriori action injects energy capturing real interaction like bouncing ball cuts erases hidden correlations conjugated pair correlated entropies cut action transforms adjoin entropy increments real information binds real bit entangled couple changing one acts control cut kills total entropy volume finite rotation memorizes dynamically cutting entropy equivalent information asymmetric geometry ability observer overcome gap depends amount entropy volume enclosing observed events collected virtual probes probes entropy force momentum spin rotating momentum transition gap real control jump adds energy covering transition cutting curved volume places real needle pleat time interacting impulse cutting action killing process entropy near produces interactive impact impulse yes actions requires impulse access energy overcome gap impact emerges virtual ending preceding imaginary microprocess follows real delivers equivalent information compensating impact virtual cuts avoid thus information impulse appears energy within transition maximal probability observation gap killing resulting entropy runs physical microprocess local nonlocal entangled information units real preempt memorizing information posteriori probability closed reality impulse positive curvature action interacting external impulse negative curvatures action transits real interactive energy opposite asymmetrical curvatures actions cover curved interaction asymmetrical curvature action compensates asymmetrical curvature real impulse real asymmetry memorized erasure supplied external landauer energy fig fig virtual impulse starts action probability potential cutting part impulse middle part transitional impulse transitive logical action changes holding end interacting part cut transforms impulse entropy information bit fig impulse fig starting instance probability transits instance interaction interacting impulse negative curvature impulse action opposite curvature ending action analogous beginning impulse opposite curved interaction provides difference barrier actions necessary creating bit interactive process provides landauer energy maximal probability certainty interactive impulse ending state memorizes bit certain interaction injects energy overcoming transitive gap including barrier toward creation bit action external natural process curvature equivalent potential entropy nat carries entropy impulse total entropy nat interacting part internal process impulse invariant entropy nat potential entropy interacting curvature enclosing entropy density lowers initial energy related temperatures ratio follow conditions creating bit interacting curved impulse opposite curving impulses interactive transition require keeping entropy ratio interacting process possess landauer energy moment ending interaction interacting impulse hold invariant measure entropy nat whose topological metric preserves impulse curvatures last follows impulse law actions generate invariant measure topological metric preserving opposite curvatures theoretically pure probability predictability challenges kolmogorov probability measure quantum mechanics entanglement additive symmetry probability mutual exchangeable events vanish observing markov probabilities additive multiplicative properties changing within microprocess specifically merging jump violates regular properties markov process leading process starting following jump launching space interval transitional impulse microprocess holds additive properties probability related conjugated entropy transitional impulse confining entanglement ends jump initiating multiplicative property entangled entropy probability microprocess within transit ional impulse reversible microprocess within whole impulse reversible impulse ending action cuts entropy concurs property quantum wave function interactive measurement random impulse proceeding temporary fixed correlated random actions microprocess multiple whose manifold decreases growing probability measure maximal probability pair additive entropy flows symmetric probabilities hich contains states advance superposition multiple impulses initiate manifold virtual observers random space shape collective probabilistic movement maximal observing probabilities mani fold virtual observers also decreases microprocess different quantum mechanics since rises virtual inside probing impulse growing probability jumping action evolves real final physical superposing rotating anti entropy flows additive time complex amplitudes correlated entanglement carry bind energy connects entropy joint correlation whose cuts models ementary interaction physics probabilistic particles carry analogous conjugated probability amplitudes correlated entanglement virtual microprocess dissipate integral entropy decreases along reversible probes observation real microprocess builds information within cutting impulse real time becoming irreversible cut erasure operations creating bit impulse reveal structure weller bit memorizes logic virtual actions bit free information participates getting multiple bits information primary information observer formed without priori physical law whereas observing probable triple field specifies information observer information observer starts real impulse cutting observing process extracting hidden information bit identifies information observer extractor holde information emerging observation killing physical action converts entropy virtual observer equivalent information real information observer killing distinct volumes densities converts bits distinguished information density curvature curving impulse bit accumulates impulse actions carrying free information initiates bits attraction curved interaction primary virtual asymmetry measured equivalent entropy compensates asymmetrical curvature real external impulse real asymmetry memorized information entropy erasure supplied external landauer energy entropy cutting interaction curving asymmetry producing information performs function maxwell demon emerges curving asymmetry different bits rises information gradient attraction minimizing free information finally binds bits connects observer collected information bit units information process finally builds observer information structure principle rising impulse observation leads following attributes emerging information bit information delivered capturing cutting entropy virtual observing correlated impulses free information transferring nearest impulse keeps persistence continuation impulse sequence via attracting bits persistent bits sequentially automatically convert entropy information holding cutoff information random process correlation connects bits sequences cutoff bit holds geometry following geometrical form discrete entro impulse information memorized bit cuts symmetry virtual process free information rising bits cutting random process spending binding attracted bits three bits free information allows nding triple bits new bit different primary bit cutoff random process primary attracting bits persistence continuation integrate real information process composing elementary information units information structure information observer bit preserves origin information growing information condensed integrated bit finite impulse size limited speed light increases bit inf ormation density increasing density conserves growing energy equivalent interacting physical particles information microprocess formalism allows contribute explanation known paradox problems optimal estimation lower limit increment probability events evaluates probabilities limitations process observations evaluation information constraints limitation microprocess conditions observer start predicts probability path virtual probes probability approaching real cutoff path observing uncertainty certainty emerge causality information complexity information bit geometry encloses impulse time curvature space coordinates edge reality within gap evaluates plank fine structural constant region uncertainty minimal displacement within region estimates number probing impulses reach region concurs virtual probes approaching real cutoff macroprocess rotating movement information bits binds information macroprocess describes extremals variati problem solved macroprocess free information integrates multiple bits information path functional ipf encloses bits time geometry process information structure estimation extremal process shows information collected diffusion process ipf approaches entropy functional ipf formalizes extreme integral cutting interactive impulse whose information approaches bit number ipf integrates flows bits units finite distances sizes ipf maximum integrating unlimited number bits limits total information carries process bits intensifies bit information density running infinite process dimension infinitive dimension macroprocess describes ipf extremals limited number process units free information assembles leads limited free information transforming impulse croprocess macroprocess restricted dimension randomly applied deterministic real impulses cutting process correlations transforms initial random process limited sequence independent states forming space spiral trajectory current radius bsin sin conic surface points spatial discrete interval corresponds angle radius vector projection cone base vertex angle ratio primary posteriori probabilities beginning probabilistic observation identifies initial conditions extremals initial conditions determine entropy function starting virtual observations moment depends minimal entropy uncertainty arising observations entropy allows finding unknown posteriori entropy starting virtual observer initial conditions bring complex real imaginary entropies starting conjugated processes virtual observer microprocess process minimal interactive entropy becomes threshold starting information microprocess beginning real observation information observer starting extreme hamiltonian processes evaluate two pairs real states conjugated rotating dynamic process allows finding equations unifying description virtual real observation microprocess dynamic macroprocess applying macrodynamic equations traditional form dynamic model unknown control function solves initial determines optimal control traditional model controls formed feedback function macrostates bring actions sequentially start terminate constraint imposed model allows extracting cutoff hidden information localities joining extremal segments observing trajectory extracted information feeds observer macrodynamics recent inf ormation current observations feedback process concurrently renovates observing markov correlations connecting macrostates controls correlations identify markov drift function transferred equations information macrodynamics within extremal segment information dynamics reversible irreversibility rises termination constraint segments feedback identifies cutting correlations automatically transforms ipf reveals integrated information hidden observing process randomness sum cutting correlation functions identifies ipf integrant lagrangian integrates impulses constraint inform ation intervals identified constraint leads invariant relations impulse information interval intervals information impulses holding three invariant entropies virtual impulses well invariant conjugate vector extremal invariants estimate segment information macrotrajectory locality segments predicting potential feeding information transfer feedback measuring invariants allow encoding observing process using shannon formula average optimal code word length code alphabet letters information analog plank constant evaluates maximal information speed observing process estimates time interval entropy equivalent gap separating macroprocess shifting time real time course automatically converts entropy information working maxwell demon enables compensating transitive gap equations observation finalize math description micro validate numerically information process basic structure cutoff attracting bits start collecting three primary basic iplet unit equal information speeds resonates coheres joining triplet units upo information process specifically three bits processing joint attractive movement cooperate created new attracting bit composes primary basic triple unit composes secondary triple unit cooperates nested sequence enclosed triplet unit emerging cooperative rotation following triplet unit enfolds information cooperating units composite final bit upo size limits unit starting maximal ending minimal information speeds attracting forming new triplet unit free information along macroprocess process movement selects automatically upo attracting minimax movement joining two cutoff bits third bit delivers free information next cutting bit forming next cooperative triple unit upi forming multiple triplet trajectory follows procedure fig opposite conjugated trajectories spo spo form spirals located conic surfaces analogous trajectory connected bridges spi binds contributions process information unit impulse joint actions model line switching controls two opposite space helixes middle curve shown right triplet joins two segments positive eigenvalues reversing unstable eigenvalues attracting third segment negative eigenvalues third segment rotating trajectory moves two opposite rotating eigenvectors cooperates three information segments triplet knot fig triplet unit generates three symbols three segments information dynamics one control composing minimal logical code encodes elementary physical information process illustrative dynamics assembling units trajectory adjoining knot along sections max trajectory changing information speeds max max accordingly dynamic information invariant impulse shows minimum three self bits assemble optimal upo triplet whose free information requests binds new triplet joins binds three basic triplets ending knot accumulating memorizing information trees current unit upk composes bits triplet code encodes connects units fig simulation forming triplet space structure knot node composed spirals according indicated time intervals measure real times simulation shown cones diameters pair opposite directional rotating units equalizes eigenvalues rotating third one attracts binds triple starting attracting force assembling information network joint triple unit free information transfers triple code next forming triplet assembles network building forming triplet code multiple triples sequentially adjoin time hierarchical network whose free information requests observation attaches new triplet unit higher level knot concurrently encoding triple logic information geometrical structure hierarchy spiral dynamics triplet nodes space iot ranged string initial eigenvalues cooperating locations iok iok number nearest triplets triplet unique position hierarchy defines exact location code logical structure node hierarchical level classifies quality assembled information currently ending node integrates information enfolding levels node knot automatically feeding novel information concurrently requests deficit needed information deficit creates equal requesting new information initiates probing impulses frequency observing entropy impulses information probing impulses interact observing cutting entropy provides information verifies nodes requesting information new information delivers requested node interactive impulses whose impact probing impulses observing frequency cutoff memorizes entropy observing appearing new quality information triplet currently builds temporary hierarchy whose high level enfolds information triplet logic requests new information running observer attaches extends logic code emergence observer current level indicates observer information surprise measured attach ing new information logic encodes double spiral space dss triple code rotates encircli conic structures multiple ins logic extends simulation forming double spiral cone structure dss cells arising along switching control line hyperbola shown uncertainty zone geometry surrounding curved macro trajectory models line whose rotation around forms enfolding geometry volume space surface space structure encloses spiral models fig ending knots higher level assemble three units next level triplet starts assembling three formed units higher triplet level connecting units equal speeds attracting process assembling knots forms rotating loop shown forming following cooperative triplet scales curves figs distinct interacting knots curves fig since units reaching equal speeds resonates increases size curves fig rotating process loop harmonized frequencies different levels analogous efimoff scenario loop could temporal new formed triplet memorized loop includes borromean knot ring early proposed borromean universal relation multiple information geometry shapes observer asymmetrical structure cellular geometry fig dss triple code fig structure cellular geometry formed cells dss triplet code portion surface cells illustrating space formation structure geometry integrates information contributions modelling macroprocess integrates imaginary entropy merging impulses micro processes cutoff information real impulses sequentially convert collected tropy information physical process macro observation process entropy processes observer information one observer distinct information observers however invariant information minimax law leads invariant information regularities different observers observing process different probability field triple observer gets specific information information requests current optimal time information dynamics optimal information process determines extremal trajectories entropy functional solving minimax variation problem observing process information macrodynamic equations describe information macroprocess whi averages observing microprocesses holds regularity observations maxmin impulses equations predict optimal information path starting virtual observation information process physical macrodynamics forming structure information observer regularities rotating coordinate system forms observer information structure confining multiple ins determine observer time inner communication self requesting accumulating information observer owns inner time information processing time scale required information micro macrolevels depending density nodes information current information cooperative force initiated free information evaluates observer selective actions attracting new information quality delivers high related observing information selective mechanism requested information actions engage acceleration observer information processing coordinated new selection quick memorizing encoding node information logic structure minimizes spending information determines observer feedback loop optimal criterion growing quality observing information collected selects needed information observer acquires observer optimal multiple choices limit implement minimax strategy evaluates amount information emanated integrated node identifies attracting cooperative force nested structure holds cooperative complexity measuring origin complexity interactive dynamic process cooperating free information anticipates new information requests automatically builds hierarchical dss decreases complexity cooperating yet information units information structure self feedback drives self evolution macrodynamics ability observer cognition emerges evolution process evolving intentional ability requesting integrating predicting observer needed information builds observer growing networks evolving free information builds observer specific time information logical structure conserves logical structure possesses virtual probabilistic real information causality complexity whose information measures cognitive intentional actions rotating cognitive movement connects impulse microprocess bits macroprocess composing elementary growing levels quality information integrates multiple nested information logic information domains observer cognition assembles common units multiple resonances forming hierarchy accept units node concentrates recognizes cognitive movement forming nodes level processes temporary loop might disappear new formed triplet memorized along hierarchy runs distributed resonance frequencies spreading chain loops chain rotates thermodynamic process cognitive thermodynamics minimal landauer energy performs natural memorizing bit evolution level cognitive actions model correlated levels controls highest domain level cognitive process cognitive actions emerge evolving observations maintain cognitive functions emerging properties encodes cognitive logic information language emerging information intelligence observer intelligence emerges path staring observation virtual observer creation microprocess bits information macroprocesses nested networks growing quality information nested logic observer selective actions ability prediction concurrent renovation extension complex domains enclosing maximal quality condense information logic structure drives evolution macrodynamics ability within evolving processes integrating coherent structure emerges observer cognition starts creation elementary units virtual observer holding memory microlevel coordinated selection involving verification synchronization concentration observed information necessary build logical structure growing maximum accumulated information unites observer selforganized cognitive actions performing functional organization functional organization intelligent actions spent action evaluate memorized amount quality information ending hierarchical level functional organization integrates interacting observers levels domains observer highest ending hierarchical level maximal level measures maximal cooperative complexity enfolding maximal number nested ins structures memorize ending node highest dss helix structure multiple local bits coding units encodes observer triplet coding structure fig call observer intelligence intelligence ability observer build informational networks domains includes cognitions ended triplet observer hierarchical informational networks domains measures level observer intelligence observers different levels intelligence classify observer levels quality information memorized ending node observer highest levels measures observer information intelligence cognitive process triplet level preempts memorizing observer builds maximum number hierarchical informational networks domains maximum intelligence observes imbedded ability control observers observer intelligence holds ability uncover causal relationships enclosed evolving observer networks selfextends growing quality information cognitive logic building collective observer intellect intelligence multiple interactive observers integrates joint ending node observer cognition emerges two forms virtual rotating movement processing temporal memory following real information mechanisms rotating double helix geometrical structure memorizing dss concurrently places organizes observing information bits nodes whose sequential knots memorize information causality logic strategy develops multiple logical operations computation enhances collective logic knowledge organization diverse intelligent observers measure allows evaluating information necessary build minimal intelligent observer increasing ins hierarchy enfolds rising information density accelerates grow intelligence concurrently memorizes transmits time course observing time scale intelligence growing time interval increases observer life span evolving distributed information structure models artificial intellect detail specifics cognition intelligence described observer individuality determines probability field triple observing specific set probabilistic events arise emerging particular information observer observation measuring quantity density information delivered bits observing information limited number nodes limited number observer depends individual observer selective actions selective actions define cooperative information forces depends number nodes minimal cooperative force forming first triplet defines minimal selective observer individual ability selection classifies information observers levels hierarchy time space geometrical structure inner time scale whose feedback holds admissible information spectrum observation individual observer ins determine explicit ability specifics classify observers also level cognition intelligence information mechanism building observers invariant describes invariant equation information dynamics following minimax variation principle information structure artificial designed observer basic stages toward artificial brain observers everywhere including people animals species multiple particles objects communicating interacting accepting transforming exchanging information physics elementary particles interact starting four known fundamental interactions nature form interacting atoms molecules different intermolecular forces chemical interactive reactions biological interaction building genetics organisms biological interactions involve multilevel interspecies interactions ecology various interactive communications creates social interactions mutual technological economic financial interactions interactive communication different languages computer interactive telecommunications internet connect society technology business science education media prospective artificial interactions based brain would generate future technology societies interactions simplify example rubber ball hitting ground bouncing consists actions term information observers understands differently whereas single certain impulse known bit elementary unit information many study observer specific interactions however one approach ever unified study common information origins regularities conditions differentiation first approach unify studies aiming understand common notion information since multiple observers interact manifold actions interactive actions random multiplicity generates random process retrieving information units random process requires observation searching information web potential observer information sends probing impulses evaluates interacting result probability occurrences material particles observation probable occurrence allows selecting needed result estimating certainty observed information simple example observing uncertain planet moving around star probability observation increases making brings copy probable observation copy removes observer uncertainty spending energy shot copying film consumes film exposes certain observation enclosing spending energy brings current information observing planet therefore probabilities observation process generates sources random uncertainty removed brings equivalent certainty information observation process removing uncertainty natural interaction creates information natural phenomenon interaction observation searching information interacts current observing probabilities sending multiple probabilistic impulses decreases uncertainty observing probabilities getting certain impulse elementary bit information active observation builds observation process decreasing random uncertainty probabilistic impulses observe probabilities random uncertainty active observation models observer whose increasing probabilistic impulses generate multiple bits certain information observer aim information observer emerging observation integrating information codes selfcreated structure models brain information logic cognition intellect probabilistic observation measured discrete probabilistic impulses observing process since probabilistic observation runs multiple interactive impulses probability impulses probing random environment formally describes random field multiple interactions field axiomatic probabilities defines mathematical triad sets possible events sets actual events probability function triad specifies particular observer whose probability belonging field observes current events multiple triads extend impulses probabilistic observation toward artificial brain emerging observations according formal theory probabilities axiomatic probabilities objective definition formalizing multiple experimental occurrences random event artificial observation probabilistic impulses produce random generator probabilities chosen triad formally source discrete axiomatic probabilistic impulses kolmogorov law probabilities generates sequence mutually independent variables events field triad probabilistic impulses interact observable random process field reflects multiple impulses probabilities priory probability posteriori probability bayes probabilities connecting sequence impulses observing random process classical process connecting multiple interacting impulses markov diffusion process widely used many physical chemical biological many observations applications allows assigning markov process observable random process sequence probabilistic impulses random field initiates bayes probabilities within markov process evolving markov process inner probabilities generating observing process sequential bayes probabilities provide virtual observation observable process virtual potential observer also axiomatic probabilities analogy standard bayes probabilities evaluate probability hypothesis priori probability updated posteriori probability evidence since impulse probabilities start specific triad within random field initiated markov process probabilities begin observation process triad multiple observing markov processes start multiple triads probabilistic model potential observer hidden random bits models markov observing process located surrounded random field therefore affected field random since universe build multiple interacting processes running multiplicity actual source random field formal probabilistic model axiomatic artificial probabilities objectively measure observation process build observing process observing process starts observation specific field triad spots current observing environment enclosing observing events artificial observer designing observation probabilities identify multiple interacting events observable spot artificial designed observer generates random impulses observable process current environment observing process hidden bits process probabilities identify multiple interacting events observable spot finally measures probability revealing bits uncovered bits provide future real code generates information observer reduction observable process entropy probing impulse rising probabilistic logic uncertainty random events observable process evaluates logarithmic measure event relative probabilities defines relative entropy events connect correlation along observable process uncertainty random events impulse probabilities identifies relative probabilities logarithmic measure relative entropy probabilistic impulse entropy immanent probabilistic probing impulses interactive actions observing process observable process correlation evaluates time interval correlation connection interactive actions equivalent time interval actions impulse unit measure probing impulse virtually cuts entropy correlation observable process within interacting impulse action maximizes cutting entropy yes reaction spending entropy action creation minimizes cutting entropy provides principle impulse along multiple observing impulses impulse actions maximal cutting action minimizes absolute entropy conveys yes action rising probability leads relational entropy transferring probabilities simple example bouncing rubber ball ball hits ground energy interaction partially dissipates increases interaction total entropy ball following reverse movement holds less entropy part dissipated leading entropy bouncing ball adding periodically small energy compensating interactive dissipation supports continuing bouncing soon initial impulse actions involve minimax principle imposed along impulse observation leads reduction observable process entropy multiple probing impulses increase posteriori probability observing process sequential posteriori probabilities determine probabilistic causality along observing process carries probabilistic logic hidden bit covering probabilistic causality hides probabilistic logic logarithmic ratio probabilities defines relational entropy measures logic impulse sequence observing process logic integrates entropy functional along trajectories markov process measures process integral entropy minimax principle formalizes variation problem applying allows analytically describing observing process minimax trajectory extreme solution solution brings invariant entropy increment discrete impulse preserving probability measure impulse time interval measure entropy equivalent nat connects entropy process total time synchronizes adjoins local impulse time measure along process absolute time scale impulse logical measure includes logical bit measures nat difference carries wide impulse yes action transferring next impulse evaluates measure nat impulse free logic logic originates observing impulse relational bayes probabilities reducing entropy along observation process growing correlation dependencies impulses observing process reduction observable process entropy observing probabilistic impulses determines growing logical dependency impulses observation involves virtual impulse correlational attraction free logic logically connecting impulses measures virtual attraction growing correlation increasing posteriori probability logic observing process probability less one probabilistic logic even growing probability still uncertain virtual closeness uncertainty certainty probability one measures particular relational probability entropy finalize reduction uncertainty emergence geometry impulse observing process growing bayes posteriori probability along observations neighbor impulses may merge generating interactive jump impulses border merge converges action reaction superimposing cause effect probabilities merge squeezes time interval initiates microprocess bordered impulses jump increases markov process drift speed infinity generates high entropy density curving correlation actions virtually cuts time measure correlation emerging units bordered impulse dissolve correlation measures orthogonal correlations jump curving time related orthogonal correlation rotates impulse reaction unit measure orthogonal displacement time measured action units measure orthogonal displaced second unit within impulse becoming orthogonal space impulse measure originates curved space shifts quantified impulse discrete probability measure measures two impulse parts units including counterpart primary curved time thus displacement within impulse changes impulse second time discrete space shift preserves measure emerging coordinate system measure conserved following correlated movement transferring minimax beginning space precisely starts part impulse invariant measure curved impulse measured curvature virtual observer displaced initial virtual process possesses discrete impulse sent virtual probe observing probabilities define probe frequencies preservation axiomatic probability measure example bernoulli formulas minimax observer probes increasing frequencies checking probability grows test checks probability also via symmetry condition imposed axiomatic probability indicating probability correctness closeness axiomatic probability temporary memorized current correlation encloses difference starting current correlation identifying current location virtual observer sequence bayes probabilities transfers events causal relationship along observation process probabilistic logic impulses cutting entropy temporary memorize correlation logic code virtual observer rising correlations memorize shape evolving observer evolving shape gradually confines running rotating movement developing curved shape observer time space geometrical structure virtual observer virtual geometrical structure virtual observation gains real form sequential transforming integrated entropy equivalent information impulse transformation observing process entropy information transformations emerge interactive impulse observation posteriori probability last probing action impulse cutting final minimal entropy observation process follows yes action impulse brings posterior probability one certainty growing correlation observing interactive impulses interaction connects impulse ending yes action following opposite action next interacting impulse virtual attraction interaction curving impulses brings opposite topological curvatures positive yes pushing action negative opposite action creates topological transitive transformation interaction impulse opposite curvatures creates topological transitive transformation transformation brings logical asymmetry evaluates part logical bit entropy transferring probabilistic logic certain logic covers transitive barrier entropy information opposite curved interaction observing impulse external impulse brings asymmetrical entropy could overcome barrier opposite curving impulses interactive transition require entropy ratio transfer entropy equivalent forming logical bit opposite curved interaction decreases difference entropies ratio relative amount counts entropies impulse potential bits transformation deduct entropy measure nat transforming impulse process impulse interaction part logical impulse ratio identifies moment following appearance potential previously hidden logical bit since entropy measure evaluates free logic curved interaction moment relative impulse invariant measure interval nat measures part impulse delivering free logic creates asymmetrical logic information bit interval follows interval nat appearance logical bit indicates logical certainty observing hidden bit logical bit certain logic appears certain free logic carries certain logical attraction logical entropy bit becomes information bit memorizing antisymmetrical logic information certain free logic evaluates appears time interval measured relatively impulse time interval bit thus begging impulse time interval appears evaluated relatively impulse interval measure interval deducts external impulse measure relative time interval ten estimates interval needed memorizing logical bit observation process reducing uncertainty random interactive process concurrently integrates maximal growing correlation holding entropy equivalent interval ten temporary memorizes logical bit certain logical bit becomes physical information bit erasure entropy logic allows replacing logic memorizing bit logical bit conceals hidden entropy carrying energy initial real interactive process covered entropy multiple random injecting energy equivalent integrated removes entropy interval creation asymmetrical logical bit memorizing asymmetrical logical bit erases logical bit entropy memory physical bit must stored placed physical entity performs encoding memorized bit extracting initial position physical reality revealing bit brings also energy potential erasing entropy acquiring entropy satisfying second law irreversible process requires cost energy equivalent demon maxwell according landauer principle logically irreversible manipulation information encoding leads erasure information dissipative irreversible process erasure information bit requires spending minimal energy boltzmann constant absolute temperature delivered outside environment therefore transformation observing impulse entropy information includes getting logic memorizing erasure encoding memorized bit storing position environmental process impulse process needs energy getting asymmetrical logical bit memorizing ending encoding bit observing virtual process ending minimal entropy reversible symmetrical logical bit physical bit transitive opposite curving interaction curries asymmetrical logic correlation actual cost since interaction reduces entropy getting information logic requires less energy erasure memorizing bit estimate components process information logic covers time encoding ien nat transferred next interacting impulse equivalent entropy nat free information needed encoding carry information logic call encoding logic since landaurer energy allows memorizing bit free information currying attraction information cost encoding becoming physical external impulse brings additional landaurer energy adds cost landaurer energy whereas encoding costs total additional cost temperature energy delivered time interval whose start identifies impulse interacting time transferring entropy moment ends interval curved opposite interaction adds bringing total interval hence moment delivering total energy identifies potential encoding time interval ending internal impulse interacts external impulse opposite curvatures since external process movement within impulse ends impulse stopping states thermodynamic process delivering energy stop state hence erased impulse entropy memorizes equivalent physical information impulse ending state encoding stores information impulse starts producing physical bit moment ending transitive logic ends producing moment ending delivering landaurer energy time interval estimates time intervals memorizing free information encoding estimate accordingly ten free information attract next logical bit first memorized bit encoding attracting encoding automatically produces curving interaction next logical impulse next real environmental process since time interval encoding equals time interval transitive curving interaction starts producing next physical bit moment ending transitive logic ends producing moment ending delivering landaurer energy time interval also estimates time intervals memorizing bit free information encoding estimate ten follows necessity proper concurrence time curse impulse inner external impulses coordination sequence moment appearance bit memory encoding allow consecutive integration process entropy transformation process information transformation integrated process entropy process integral information growing information density process impulses functional integrates sequential bayes probabilities observing impulses virtually observe cut observable process measures total process entropy process integral information minimax impulses process observing process time trajectories emerging trajectories analytically describes minimax extreme trajectory observing process impulses integrated process reach certainty impulses cut maximal probabilities transforming impulse entropy invariant measure equivalent information invariant measure multiple information impulses integrate information path functional ipf path discrete impulses information measures observing process probabilities approaching certainty measure automatically transform virtual observing probabilistic ipf information trajectories curved impulse invariant measure encloses information measure includes bit free information information needed encoding free information nearest impulse bits attracts bit per impulse ipf integrates bits free information connecting bits sequence minimax principle applied along trajectory ipf information process maximizes information enclosed current impulse squeezes time interval growing attracting free information minimizes time interval nearest impulses proportionally third impulse interval information distance becomes proportional bit information invariant impulse measure hence invariant impulse along ipf extreme trajectory squeezes proceeding impulses distance ipf extreme trajectory sequentially condenses increasing information third following impulse grows information density invariant impulse free information impulse increases intensifies information attraction invariant impulses proportion end ipf integration integrated information concentrated last impulse whose information density approach maximal limit since free information encloses information logic multiple bits triple growing density raise process information logic ipf integral information logic condensed last integrated impulse time interval volume multiple information impulses third curved impulse invariant measure appears information process time frequency information density dii nat vis concentrating nat third impulse increases growing volume vis holds invariant measure invariant impulse time coordinate flat surface space coordinate measure orthogonal space coordinate space coordinate measure determine impulse volume vis determines dii nat current impulse geometry encloses information density frequency concentrating information logic information previous impulses along path efextreme trajectories starting observing process transformation converts orthogonal processes whose curved impulses hold information measures extremal trajectories rotates forming spirals located conic surfaces starts virtual entropy process continues information process since bit trajectory creates cutting entropy impulse observation trajectory consists segments information process dynamics segments intervals delivering following bit segment segment starts cone ends point connects vertex following cone observing bit delivering cone vertex segment includes impulse process logical bit intervals free logic correlation connecting nearest segment temporary memorizes segment logic logical information dynamics describes process sequential logical interaction multiple impulses rotating information speed determined impulse density dynamics cone vertexes reversible symmetrical analogous hamiltonian dynamics logical brings logical bit prior interaction external impulse starts delivering external energy bit supplying cone vertex external energy generates physical multiple bits physical information process starts starting process follows time interval logical interaction since impulse curved measure relative time interval appears information process frequency frequency appearance interval equals frequency encoding time interval ten frequency spectrum identifies time opening supply external energy equal spectrum frequency encoding time interval time interval memorizing bit identifies bit information measure equivalent invariant impulse part frequency appearance interval within impulse determines frequency spectrum necessary delivering energy memorizing encoding bit frequencies present spectrum along information process spectrum frequency currying proportional frequency supplying external energy time intervals whose sum equals invariant impulse time interval whole impulse becomes segment physical information process therefore physical dynamics describe ipf extremal trajectory rotating sequential cones cone vertex encodes bit memorized previous frequency current segment repeats frequency transfers next cone vertex frequency encoding current impulse bit hence physical information impulse carries spectrum sequential pair trajectory carries spectrum impulses resonance frequency two impulses whose distance shortening allows closely connecting impulses resonance along trajectory pairs appears growing frequency impulses appearance since fixed time intervals ten relative invariant impulse measure repeating invariant impulse increasing information density growing frequency thus along extreme trajectory third impulse time intervals deliver triple time intervals sequentially proportional impulse distance measuring bit proportion invariant time measure impulse shortening accordingly sequentially shortens distance impulses extreme trajectory assembling three impulses triple resonance frequencies triplet logical structures information network hierarchical logic observing process minimum three logical bits free logics appear attracting would cooperate logical triple multiple probabilities interacting impulses process produce numerous frequencies minimum three generate attractive resonance cooperating triple triple logic starts temporary memorizing two sequential pair time correlation memorizes asymmetrical logic locally asymmetric time process process ending triple correlations temporary memorize triple logical bits according minimal entropy cross correlation memorized cost equivalent minimal energy logical bit information cost memorizing triple logical bit includes additional free logic attracting free logic emerging three logical bits starts bits following sequence free logic emerging logical bit holding frequency attracts next logical bits toward resonance equal frequency next bit free logic assembling two joint resonance resonance process links bits duplets free logic one bit pair gets spent binding duplet free logic duplet bit attracts third bit binds three knot bit creating triplet logical structure knot bit still free information used attract different bound pair emerging bits creating two bound triplets process continues creating nested layers bound triplets three triplets figs hence triplet logical structure creates resonance frequencies attracting logic joining triple bits free logic attraction toward triple resonance equal frequencies core information mechanism structuring elementary triplet trajectory forming triplet describes rotating segments cones whose vertexes join knot starting base following cone knot frequency joins cone vertexes resonance along cone base next spiral segment starts connects next triplet resonance creating nested layers logical information network layers knots hierarchy identifies nested nodes hierarchy triplets basic elements form nested informational network hierarchical structure triplet unit generates three symbols three segments information dynamics one segment attracting triple logic binding logical triplet knot symbols produce triplet code knot logic symbol binds triple code potential encoding triple knot encoding release free information logic transfers triple code next triplet node thus nodes logically organize code attracting free information resonance three bits frequencies creates triplet information logical structure carries unbound free information logic including encoding logic related frequencies frequency free information logic determines moment time external energy starts memorizing bit frequency triple encoding logic determines moment time external energy starts encoding knot triple code emerging logical structure carries triple code node space last triplet network collects encloses entire network information network built resonance limited stability therefore encloses finite structure observing process multiple limited ins free information ending nods final triplet every network contains maximum amount free information networks attraction ended triplets even potentially loses stability evolving chaos possesses ability multiple ins hierarchical domain starting cooperation tree ended triplets free information knot joining ins triples resonance ending knot free information resonates three ins ending free information forming triplet structure analogously core triplet high level triplet joins three ins structuring next domain hierarchy hierarchical logical trajectory describes spiral structure also presents trajectory information process evolving observations hierarchy enables generating sequential triple code locating rotating trajectory cone vertexes distributed different hierarchical levels multiple domain hierarchy code integrates observing process information geometry observer hierarchical distributed logical structure cognition multiple moving ins sequentially equalizing nodes attracting information logic resonance assembles total observer logic logic consists mutual attracting free information sequentially interacting cooperative logical rotating spiral loops enclosing observing information call observer cognitive logic encloses probabilistic information causalities distributed along observer hierarchy logical functions free information resonance perform cognitive functions distributed along hierarchy assembling units triplets nested nodes ending nodes local functions observer cognition assembling runs distributed resonance frequencies spreading along hierarchy since unit ending high level structure encloses levels information logic unit impulse invariant interval containing information concentrates information density unit lower level hierarchy attracting resonance frequencies attracting free information hold cognitive logic loop includes local loops structuring local information units unit hierarchy thus frequency delivering spectrum increasing growing triple sequentially shortening intervals trajectory segment specify details following propositions proved following conditions spiral trajectory extremal describes sequence curving rotating segments representing interacting impulses observing process integrates observing process logic segment impulse invariant entropy measure moving along trajectory rotating curved impulse invariant measure includes time coordinate measure flat surface space coordinate measure orthogonal space coordinate space coordinate measure measure includes impulse logical bit free logic nat holds asymmetric logic segment logic density per third segment increases according dii nat vis vis asymmetric logic divides sequential segments barriers transfer segments interaction logic interval following interval memorizing bit interval ten free information ending segment along trajectory sequential segment repeats triple invariant frequency spectrum ratio alternating sequences along segments trajectory identify frequencies spectrum propositions segment rotates three dimensional space wave functions spinning like top rotating speed around spiral square radian radian orthogonal rotation space speed volume radian iho radian related frequencies orthogonal rotations iso accordingly segment rotation spreads space rotation space interval segments invariant measure wave function distributes space rotation along segments trajectory invariant speeds delivering invariant spectrum spectrum frequencies identifies alternating ratio along sequential segment let consider dimensional segments among rotating segments extreme trajectory segment delivers invariant spectrum rotation speeding space rotation distributes spectrum along threedimensional space dimensional segments along extreme trajectory segment equal measure increasing density proportional segments shortening intervals locations along trajectory wave consecutive space movements sequentially picks segments specific locations dimensions simultaneously starts rotating interval placed segments accordingly densities increase proportionally squeezing time interval measures along dimensions trajectory first wave rotation moves segment rotating interval equivalent space interval density proportional second wave rotation moves segment interval equivalent space interval density proportional third wave threedimensional rotation moves segment time interval equivalent space interval density proportional movements repeat shortening intervals triple segment increasing frequency growing information density along trajectory since segments deliver equivalent spectrums equal frequencies sequential segment spectrum synchronized time intervals sequence according proposition condition invariant spectrum frequency repeats time interval logical antisymmetrical interaction bridge separating segments trajectory interval end indicates beginning time interval middle segment repeating frequency time segment bit memorized end end indicates beginning time interval ten free information logic identifies beginning bridge separated segments free information attracts separated segments sequentially squeezing segments time intervals allows performing first double synchronization interval next double synchronizes interval sum equals first interval two doublets forming three segments finally deliver three memorized bits three free information intervals sequentially attract synchronizing doublets rotation movement information attraction time intervals adjoins synchronized intervals information triple dimensional interval forming triplet completes free information delivers segment triple frequency holding invariant spectrum delivering three ending free information join tree memorized bits triple knot additional free information interval bits encode triple frequencies shortening time intervals distribute orthogonal space rotations along segments multiple dimensional observing trajectory moving three dimensional space wave function trajectory dimension three dimensions shortening time intervals rotation moves bring triplet knot joins one sequentially forming triple knots squeezing initial observing process first rotation single dimensional information process encoding bits multiple knots finally periodical wave function includes sequence repeating arguments along orthogonal rotations ush iws iws performs multiple movement three dimensional space wave functions like top shape multiple wave functions describes extreme trajectory formalizing minimax observation process models rotating segments cones illustrative schematic spinning top google top trajectory schematic bellow illustrates rotating segment bridge trajectory distributing wave space movement wave function frequencies properties wave function speeds frequencies emerges observation process space interval appears within impulse microprocess reversible time interval impulse invariant measure equivalent observing trajectory described probabilistic time function whose probability indicates appearance probabilistic wave probabilistic time observation entropy bayes probabilities measures probabilistic symmetric logic sequence probabilities thus wave function starts emerging probabilistic observation probability wave probability field beginning microprocess probabilistic wave measures time propagation asymmetrical logic emerges appearance free logic interval repeating equal wave frequency indicates beginning interactive rotating asymmetry segment bridge observation logic trajectory becomes asymmetric part total free logic nat asymmetric logical wave emerges asymmetric logic probability approaching exp appears observing previously hidden asymmetrical bit logic temporary memorizes correlation probability carries logical bit certain logic certain logical bit may carry energy real interactive process covered entropy multiple random interactions markov process path creation bit includes increment probability starting injection energy interacting impulse markov process thus coming certain free impulse logic carries certain logical attraction therefore wave function microprocess probabilistic certain logical information bit appears certain asymmetrical logical bit become physical bit erasure entropy logic allows replacing logic memorizing bit wave function starts observation process minimax extreme trajectory prognoses carrying probabilistic wave transforming observation process certainty real observation spinning movement space trajectory describes invariant speed around rotating spreads invariant rotation space speed along segments trajectory segments invariant spectrum repeats triple frequencies three time intervals shortens distance equal spectrum frequencies assembles resonance creating joint logical units hierarchies domains frequency absolute maximum indicates finite end creation minimal energy resonance supports forming logical loop distribution hierarchy hierarchy triplet units distributes space rotation emerging along segments extreme trajectory third impulse progressively increases information density measure bit triple hierarchy units starts emerging observation symmetrical logic appearance space interval microprocess logic hierarchy logical unit structures impulse mutual attracting free logic sequentially attracting moving unit speeds equalizes frequencies resonance assembles observer logic along units hierarchy hierarchy logical cooperating units becomes asymmetrical appearance certain logical bit extremal trajectory repeating free logic interval indicates wave frequency rotating trajectory three segments equalizes information speeds joining resonance frequency space rotation cooperates third logical bit segment trajectory logically composes triplet structure unit space hierarchy appearance asymmetrical logical bit extreme trajectory indicates entrance ipf information measure path forming logical path starts relative time interval logical asymmetry identifies segment bridge triple impulses third time interval indicates end triple cooperative logic builds triplet knot forming cooperative knot triplet needs time interval triple free logic binds triplet bit time interval creating bit approaches difference evaluates time binding triplet thus wave space interval delivers logical bit wave spectrum frequency triplet knot repeats spectrum frequency delivering external energy memorizing logical bit identifies relative moment ending interval asymmetry resonance frequencies asymmetrical logic moment already created along ipf path trajectory moment follows interval creation logical bit ending emergence knot binds free logic interval memorizing physical bit requires interval entropy logical bit erases needed external impulse erasing asymmetric logical bit starts interval ends interval encoding bit ten external energy supplied time interval includes erasure logical bit encoding whereas interval information free logic left attracting new bit adding opposite asymmetrical interaction brings total interval external bit bit free information starts encoding interval ten interval already spent reidentifies interval encoding teno external impulse interval asymmetry therefore frequency spectrum initiating encoding equals sequence triple sequence identifies segments alternating trajectory repeating ratio next bridge ratio holds bridge relative interval ten thus sequence segments extreme trajectory carries wave function frequencies selfstructure unit logical bits hierarchy observer logic logic controls memorizing encoding physical bits hierarchical structure units information geometry impulses spiral trajectory sequentially interact frequencies repeating bridges locations connect segments trajectory segments sequence extreme trajectory carries wave function frequencies unit logical bit hierarchy selfassembles total observer logic logic controls memorizing encoding physical bits units hierarchical structure observer logical structure observer logical structure attracting free information self hierarchy logical triplet units assembling resonance frequencies triplet logical structure models borromean ring consisting three topological circles linking brunnian spinning top space trajectory figa well trajectory includes borromean rings chain modeling distributed hierarchical logic observer logical structure carries wave along trajectory segments third segment delivers triple logic information spectrum sequentially shortening intervals kio kio increasing segment information density two sequential segments synchronize resonance frequencies triplet synchronizes resonance frequency triple logic holds one bit observer triplet logical structure unit sequential triplets attracting free logic conveys resonance spectrum progressively shortening time intervals growing frequencies cooperate logical units nested hierarchy needed spectrum increasing frequencies automatically carries consecutive segment along trajectory emanating wave function delivers frequencies cooperating growing hierarchy logical units hierarchy logical structures observed logic structure encloses hierarchy distributed logical loops logical chain logical chain wide determines invariant impulse relative interval enclosing assembled logical code growing density consecutive impulses along trajectory sequentially squeezes absolute value interval whose ratio preserves invariant impulse absolute sizes logical chain squeezing distributed hierarchy logical chain rotation carrying frequencies synchronized spectrum requires minimal energy support chain energy equivalent logical bits code integrated chain logic holds code observer logic chain synchronizes triple rhythms along trajectory melody hierarchical chain harmonizes melody rhythms therefore wave function frequencies initiating observer cognition emerges along extreme trajectory form probabilistic time wave probability field probabilistic impulse observation starts microprocess entangled space rotation code develops rotating probability wave emerging opposite asymmetrical topological interaction shapes wave function becoming certain well observer cognitive logic results conclusively numerically determine structure functions cognition hierarchical cognitive logic intelligence code enclosing observer information geometrical structure hierarchy distributed logical loops along chain multiple logical hierarchical units resonance frequencies cognitive logic units provide interactive actions attracting impulses external energy attracting actions carry free logic assembling logical unit opens external impulse carrying landauer energy starts erasing entropy memorizing information bit bit free information encoding memorized bit memorized bits encodes triplet information structures cooperating units ins hierarchy multiple units bits hierarchy integrating local units codes encoded triplet structures highest level observer information geometry structures double spiral triplet code dss dss integrates observing process minimax impulses triplet codes optimal structure units enclose structured units bits energy dss physical rotating helix structure spinning physical wave function frequencies multiple local bits coding units encodes observer triplet coding structure call observer intelligence logical switching free information hierarchical level performs intelligence functions generate local code functions distributed hierarchically along assembling logical units cognitive chain dss encodes triplet dynamics information macrodynamic process implements observer encoding logic cognitive movement beginning virtual observation holds imaginary form composing entropy microprocess memorized bit transfers information macromovement brings two forms cognitive helix process imaginary reversible temporal memory moving irreversible thermodynamics memorizing incoming information basics detail memorizing encoding physical information structure unit hierarchy memorize encode unit logical hierarchy physical information structure observer logic provides sequence frequencies along chain according growing triple density current trajectory impulse identifies sequence moments entrance external information energy memorizing units hierarchy physical encoding sequence frequencies hold observer logic delivers observer logic minimal energy attracting free information memorized bit encodes physical bits connecting first triplets second ins nested nodes ending triplet code therefore along observing trajectory emerges triplet code sequence segments whose frequency brings logical triple following physical encoding cognitive process triplet level preempts memorizing spinning space wave functions frequencies synchronizing segments synchronize triplet code rotating spiral structure code switching time clock synchronizes rhythmical sequence time intervals windows observer logical structural unit gets needed external energy clock time course assigns frequency repeating time intervals determine local resonance frequency assembling structural unit rhythms identify moments ending interval free information unit level interacting cognitive intelligence local actions code free information holds frequency identifying moments physical encoding code multiple bits frequency following integral cognitive logic chain connects local cognitive loops identifying moments physical encoding unit bits code final frequency encoding high level physical bit brings free information integrates energy free information units local code bits high level coding bit moment encoding gets physical energy necessary encoding thus following unit frequency open widow integral encoding previous units bits therefore cognitive logic logically encodes intelligence logic stable observer conserves widows according variation law regularities bit memorized conjugated interactive bridge left divides trajectory reversible process section excluding bit bridge irreversible bridge reversible sections triplet knot located cone vertex information logical dynamics memorize information physical dynamics double spiral structure dss observing process impulse trajectory realized information dynamics compose triplet dynamics multiple triplet build dss memorizes information dynamics optimal trajectory predicts information dynamics observer optimal dss encloses predicting code hierarchy observer logical structure local codes observer triplet dss code encodes structures information structure information observer observer triplet code memorizes observer cooperative information structure enhances multiple rhythms local structural units dss coding structure memorizes total collected observer information quantity quality determines observer cooperative complexity coding structure assembled information integrates function cognition intelligences observing process information dynamics artificially design dss multiple observations build numerous dss structures integrate information geometrical structure information observer quality information memorized ended triplet observer hierarchical informational networks domains measures level observer intelligence maximal level emerging intelligence measures maximal cooperative complexity enfolds maximal number nested structures memorized ending node highest number level limits wave function minimal space speed imposing information limitation information observers different levels intelligence classify observer levels multiple levels observer interacting logical intelligent functions develop computation enhance collective logic knowledge organization diverse intelligent observers intelligent actions intelligence different observers connect level knowledge build organizes observers information logical structure highest level ins enfolds growing information density expands intelligence concurrently memorizes transmits time course observing time scale intelligence growing region increases observer life span limits memory multiple final ending node extended region multiple information limited well total time existence arises enhances collective intelligence extends develops expanding intellect growth trajectory wave functions evolving distributed information structure models artificial intellect invariance information minimax law information observer preserves common regularities accepting proceeding information building information structure guarantees objectivity identity basic observer individual actions common information mechanisms common mechanism enables creation specific information structures particular observed information individual goal preferences energy material carriers various implementations communications numerous observers sending quality messenger qmess enfolding sender cooperative force requires access observers allowing observer increase personal intelligence level generate collective logic multiple observers enhances collective intelligence also extends develops expanding intellect growth artificial designed dss information measures total observer particular observer dss encodes difference iqs measures distinctness intelligence maximal information obtained observation allows designating maximal achievable measures optimal observer dss code designed information structure encoding integral observed information analytically designs information observer observing information particular observer limited considered constrains observation constrains limit conversion observing process information process thresholds evolving stages observation limit stages evolution integral cognitive information following intellective actions limit amount free information reducing ability making intelligent connections believe observer current mind integrated information causal logic distributed along observer hierarchical levels integral cognitive logic memorized mind integrates physical dss codes integral according enables prognosis new observation process creates new logic extended code intelligence renews multiple observations developing evolving observer regularities individuality finally intelligent observer two main attributes cognition intelligence interacting intelligence observers understand meaning communication intelligent observer sends message containing information emanates intelligent observer node another intelligent observer receiving information enables recognize meaning information equivalent observer nodes information quality satisfying coherence cognitive logic since dss code invariant information observers observer encodes message coding language whose logic length depend sending information possibly collected different ins nodes observer request growing quality needed information measures specific qualities free information emanating distinctive nodes need compensation observer request initiates recognition needed information includes understanding meaning process comprises following steps node requesting needed quality bayes high posteriori probability correlation closed certainty memorizes message logical information making temporary copy copying logical information builds temporary logical number nodes triplets enable adjoining cohere resonance automatically constraining forming temporary resonance temporal cognition initiates requested nodes high level probabilistic free logic allows involving incoming copy observer cognitive logic copies mirror transitive impulses providing asymmetrical free logic intervals logic interval allows access external impulse interval erasing copy temporal logic reveals information bit starting process memorizing bit decoding decoding memorized bit holds interval ten coherence observer cognitive logic actually allows starting decoding low level observer hierarchical structure structure needs updating information using part frequency spectrum message information delivers wave function frequencies along observer space hierarchy decoding finalizes requested nodes whose acceptance message comparative qualities indicates ability cohere cooperating message quality quality node enclosed observer structure since acceptance message quality changes existing observer logic encoded ins hierarchy understanding meaning message logic requires high level observer intelligent logic observer logic coherence message logic permits memorizing encoding decoded message information delivering message logic related observer logic needs high frequency wave function spectrum generates cognitive loop recognizing message logic message recognition allows memorizing encoding hierarchy observer coding structure accepting message quality intelligent observer recognizes logic encodes copying digital images space codes understanding message meaning therefore intelligent observer uncovers meaning communicating message process using common message information language temporary memorized logic cognitive acceptance logic memorized decoding whose coherence intelligence observer cognitive logic permits memorizing encoding accepted message understanding meaning observing process includes coherence information observer current coding structure integrated observer created evolved previous observations interactions communications multiple experimental studies conclusively demonstrate large monopolar cell lmc retinal neuron performs cognitive model main actions brain neurons communicate presynaptic dopamine terminals demand neuronal activity neurotransmission response depolarization dopamine vesicles utilize cascade vesicular transporters dynamically increase vesicular gradient thereby increasing dopamine vesicle content confirms communication interacting bits modeling neurons intelligence code observer physical irreversible processes distributed intelligence coding actions hierarchical level control entrance needed external physical processes dss encodes triplet dynamics information macrodynamic process implements observer encoding logic assume observer requests needed energy implement actions levels hierarchical structure enclose requested code request follows steps perform communication except encoding levels information macrodynamic related physical irreversible thermodynamic request approval observer cognition request interactive action attracts impulses needed external energy bringing entropy gradient interactive actions states gradient provides equivalent information force impulse correlation determines diffusion force acting diffusion initiates thermodynamic flow speed needed external thermodynamic process thermodynamic process forces determine power process hamiltonian physically implement requested actions following observation performance actions provides feedback observer performance observer regularities observer regularity rises impulse observation virtual observer real observers impulse action transferred following action variation principle imposes information form law encloses following regularities process extreme trajectory implementing law mathematical form releases regularities general information form physical process trajectory information macrodynamics form irreversible thermodynamics observer evolution develops without preexisting laws following observer trajectory includes levels stages domains potential thresholds observer specific regularities prolonging observation law extending regularities abilities initiate chain virtual logical information causalities extreme trajectory includes information units observer computation using code serves common external internal communications allowing encoding different interactions universal information language conduct cooperative operations within outside domains observer unites observers emergence observer time space information multiple hierarchical levels follows emerging evolution information dynamics creating multiple evolving observers information mechanisms cognition intelligent observers interacting communication enable message recognition involves cognitive coherence reading information selection acceptance selective requirements limitations acceptance intelligent observer uncover meaning observing process message based sequential memorized observing information moving rotating cognitive mechanism gives start succeeding level meaning accumulates formal analysis shows observer cognition intelligence self observer evolution numerical analysis evaluates highest level observer intelligence maximal quantity potential accumulated information estimates intelligence intelligent observer humans may overcome threshold requiring highest information information universe observer conquers threshold possess supper intellect control intellect control intelligent observers information observer converting mechanism coordinates connection observer inner external time allowing transform observing wave function observer inner processes considered stages artificial designed information observer open path toward artificial brain brain information physical structure models dss coding rotating structure materialized advanced technological computation observer brain main information function cognition encoding integrates distributed logical intelligence actions multiple requests needed information extend number nodes mixes organized hierarchy growing natural net logical intelligence distributed functions carry multiple frequencies analogy neural nervous systems materializes advance electrical conducting system material satisfy propagation optimal physical irreversible process trajectories integrating reversible segments implement described observer functions observation includes selecting getting information cognitive logic memorizing encoding intelligence code distributing cognition intelligence implementing others observer movements power forces momentum flows information physical dynamics thermodynamics mathematical basic observer formalism includes conditional entropies random events integral measure observing process trajectories formalizes entropy functional expressed regular stochastic components markov diffusion process impulse determines increments information impulse path functional ipf unites information cutoff contributions taking along dimensional markov process impulses total time interval feynman path integral quantum analog action principle physics expresses probabilistic causally action principle cutoff memorizes certain information casualty integrated ipf equation microprocess inverse actions interactive function starting impulse opposite time measured space rotating angle determine entropies entangling rotation process conversion entangled entropy equivalent qubit bit information macrodynamic equations whose information force gradient information path functional macroprocess trajectories information flow speed macroprocess following markov drift averaged along microprocesses well averaged diffusion macroprocess information hamiltonian equations information form equation irreversible thermodynamics information macrodynamic process generalizes extends observer relativity connecting information curvature differential hamiltonian per volume density information mass cooperative complexity approach formalism comes feynman concepts physical law regularities mathematically formulate variation principle process integral variation problem integral measures observing process entropy functional bits information path integral formalizes minimax law describes regularities processes theoretical concepts scientifically proves mathematical logical formalism allows uncovering regularities results simulate mathematical models various experimental studies applications confirm information observer regularities arises without physical law significance main finding composite structure observer generated information process including reduction process entropy probing impulse observing probabilities link increases posterior correlation impulse cutoff correlation sequentially converts cutting entropy information memorizes probes logic bit participating next finding curved interactive creation bit creation wave function emerging probabilistic observation whose frequencies observer cognition wave space distributes multiple bits hierarchy becoming certain along observer cognitive logic identifying process stages information macrolevels govern minimax information law revealing functional space structures cognition intelligence mutual influence finding information triplet macrounit information cooperative distributed network enables adaptive cognitive intelligent actions finding information structure artificial designed observer toward artificial brain results analytical computer simulations validate illustrate experimental applications appendix selected examples reviews scientific investigations different area natural sciences illustrating information regularities supporting theoretical information results general physics physicists demonstrate first direct observation vacuum fluctuations using short light pulses employing highly precise optical measurement techniques proving absolute nothingness positive red negative blue regions randomly distributed space change constantly high speed vacuum filled finite fluctuations electromagnetic field representing quantum ground state light radio waves quantum light field found access elementary time scales shorter investigated oscillation period light waves confirms approach initial assumption initial random probability field observer field probabistic observation gluons standard model particle physics exists virtually mediating strong forces interactions carries combination colors charges whopping colors holding two colors pair interactions increasing interaction forces conserve shape like string higgs particle also matched probabilistic observation illustration looks similar virtual processing process quantum sates confined phase space volume characterized classical action develop structure scale state positions orthogonal distinguishable unshifted original orthogonality factor moves classical plank uncertainty random direction reduces limit sensitivity perturbations relates origin structure virtual observer sec measurement probability distributions mapping quantum paths quantum states rich interplay measurement dynamics typically associated wave function collapse unitary evolution quantum state described wave function collapse final measurement measurement starts time distributed ensemble trajectories whose rotation waveguide cavity produces space coordinate ensemble study statistical mechanics hamiltonian systems topological constraints form adiabatic casimir invariants affecting canonical phase space reveals correct measure entropy built distorted invariant measure consistent second law thermodynamics decreasing entropy negative entropy production arises arbitrary priori variables nature differential entropy associated time evolution uncertainty applying jaynes entropy functional invariant entropy measure requires euler rotation angular momentum identifying appearance cartesian coordinate satisfies topological invariant results agree applied functional invariant measures impulse entropy appearance space coordinate rotation preserving impulse measure follows japan sea wave statistics entropies pronounced tail negative entropy values indicating higher probability rogue experiments confirms results tracing quantum particle observing wave function probabilities paper experimentally confirms results regarding wave function propagation frequencies inside range frequencies found experimentally neural dynamics integrating observing information neurodynamics analyzed selected multiple examples reviews different neurodynamic processes add recent results studying following publications according discussions cognitive science philosophy mind defended theory according live virtual world generated model perceived external perceptionindependent even though neither two view mind brain world entailed theory peculiar virtual brain thoughts experimental results show bayesian probabilistic inference governs special attentional belief updating though trials directional influence explains changes cortical coupling connectivity frontal eye fields modulate optimal frequency oscillations strongly modulated attention causes shifts attention locations neurons increasingly discriminate stimuli learning modify sensory representations adjust processing preferring rewarded stimulus causal reflecting anticipation choices predicts features observer formalism brain learns distinction important discriminating images optimizing stimulus processing anticipation reward depending importance relevance existence dss triple code confirms uncovered neuron communicates trinary code utilizing zeros ones binary code also minus ones experiment provides analog magnitude code triple model arabic digits represents knowledge numerical quantities results demonstrate decreasing entropy brain neurodynamic measured frequency spectral densities growing neurodynamic organization influence rhythms visual selection report results importance decisional uncertainty learning focuses results impulse information greater information comparing correct choice evidence experiments show region brain appears essential resolving uncertainty build progress everyday sequence tasks key node network preventing errors keeping track study shows learning enhances sensory multiple representations primary visual cortex neuron paper describes build performing cogitation produces signals forms neural connections synapses making readily producible test beds neuroscience author proposes cells comprised series highly sophisticated nanomachines carrying life vital functions nanomachines incorporated single complex cell descendant combination earlier cells built complex signaling networks quorum sensing allowed one microbe live inside communicate host forming binary organism third entity bacterium could photosynthesize gained ability synchronize mechanism binary organism became photosynthetic ancestor every plant earth driven life since origin resulting complex nanomachine forms photosynthetic triplet selected multiple examples reviews different neurodynamic processes substantiates approach functional regularities create united information mechanism whose integral logic mechanism transforming multiple interacting uncertainties physical human information cognition originate observer information intellect information mechanism enables specific predictions individual collective functional neuron information activities time space neurons microprocesses retrieve external information including spike actions related impulses generate inner macrodynamics identified cooperative communications among neurons assemble integrate logical information structures hierarchy information network revealing dynamics creation geometrical information structure triplet code limitations found information forces hold neuron communication whose information generated automatically neuronal interactions multiple cooperative networks assemble integrate logical hierarchical structures model information brain processing information mechanism integral logic reveals information quantities required attention portioned extraction speed including needed internal information dynamics time intervals information quality observer accumulated information specific location within hierarchical logic information needed verification digital code generated observer neurons cooperative space logic internal cooperative dynamics build information network hierarchical logic information units integrates observer required information temporary build high level logic requests new information enclosing running observer nodes enfold memorize logic cooperative information dynamical geometrical structures limited boundary shaped geometry hierarchical locations nodes provide measuring quality information node memorizes whole information operations sequentially enclosed memorized information units perform logical computing using code cooperative force hierarchical levels selects requested information observer dynamic efforts multiple choices needed implement minimax optimal strategy information quantity quality required sequential creation hierarchical ins values brain cognitive intelligence action leading multicooperative brain processing extension intelligence dynamic motion elementary macrosystems experiments computer simulations collective motion exhibit systems ranging flocks animals selfpropelled microorganisms cell migration established similarities systems illustrates following specific results emergent correlations attribute spontaneous dynamic selfassembling persistence coherent angular motion collective rotation within circular areas persistence coherent angular motion increases cell number exhibits geometric rearrangement cells configuration containing central cell cell density kept constant increasing cell number emerging collective rotational motion consists two eight cells confined circular micropatterns experimentally observed gradual transition increasing system size predominantly erratic motion small cell groups directionally persistent migration larger assemblies underlining role internal cell polarity emergence collective behavior nucleus angular position evaluated respectively circle center angular velocity normalized averaged individual angular velocities system circle size increases way average area per cell constant approximately probability distribution mean angular velocity systems containing two eight cells fitted single gaussian mixture two gaussians cells probability distribution displays symmetry breaking clockwise counterclockwise rotations directionalities almost equally represented small bias towards clockwise rotation average mean squared displacement indicates ballistic angular motion cell numbers averaged displaced intervals nucleus exhibited diffusive behavior experiments simulations showed consistently persistence coherent state increases number confined cells small cell numbers drops abruptly system containing five cells attributed geometric rearrangement cells configuration central weakly polarized cell reveals decisive role interplay local arrangement neighboring cells internal cell polarization collective migration similarities suggest universal principles underlying pattern formation interactions rules systems generic symmetries confinement stabilizes bacterial suspension spiral vortex ordering interacting filaments molecular motors demonstrate emergence collective motion concentrated filaments propelled immobilized molecular motors planar critical density filaments form coherently moving structures persistent density modulations experiment allows backtracking assembly disassembly pathways underlying local interactions identified weak local alignment interactions essential observed formation patterns dynamics presented minimal system provide new insight emerging order broad class bacteria colonies particles confining surfaces play crucial roles dynamics transport order many physical systems studying flow orientation order within small droplets dense bacterial suspension reveals influence global confinement surface curvature collective motion observing competition radial confinement selfpropulsion interactions induces steady state cells align inward spiraling patterns accompanied thin counter rotating boundary layer cited experiments validate dynamic persistence coherent angular motion collective rotation within circular displacement angular motion diffusive behavior displaced intervals emergence collective order confined curved surface others according recent discovery protein stable shapes adopted proteins contained parts trapped act changing shape changes relate proteins convert one observable shape process rna translation dna triplets enzymes aminoacids proteins start linear chains building blocks quickly fold proper shape going many transitions proteins multiple biological functions experimental results encoding practical implementations last theoretical results many previous confirm following applications natural increase correlations demonstrates experimental results coding genetic information reveals multiple experiments experimental coding spiking neurons demonstrates evolutions genetic code randomness reviews supports natural encoding cutting correlations physically verifies reliability natural encoding information process impulse method practically applied different solidification processes impulse controls automatic system method reveals unidentified compulsive appearance centers crystallization indicators generation information code integrated ipf impulse metal extraction withdrawing metallic alloys diffusion creating density gradients often frequency impulse withdrawing computes regulates designed automatic system reach maximum ipf information indicators detailed experimental data industrial implemented system automatic control regulator impulse frequency cutting movement implemented different superimposing processes interacting naturally comparative experimental results confirm advanced thermodynamic description casting process coincides information description imd moreover imd solutions leads optimal casing process automatic computer system controlling horizontal casting process implemented casting factory examples method applications communications biological cognitive systems others retinal ganglion cells eyes discrete impulse receptors interacting observations generating information transmission integrates encoding natural chemical reactions connecting chemical molecules experiments confirm encoding coherent qubits spinning electron locked attractive examples quantum solar dots semiconducting particles using information coding retrieving images encoding quantum information natural encoding information interacting impulses published applications biology medicine economics along related theoretical results computer restoration simulation information model structure computer procedure diagrams implementing procedures model restoration simulation performance shown statistical data microlevel process used identify matrix macrolevel equation computation correlation function derivative discrete interval compose computed invariant fig diagram computation optimal model process calculating correlation function using microlevel random process derivative object macrooperator invariant discrete interval allow simulating optimal macroprocess inner output optimal controls inner output optimal controls invar illustrates scheme computation optimal model process information per model invariants invar time model differential operator simulates inner space using given space distributed discrete intervals eigenvalues output optimal controls methodology based connection model macrodynamics corresponding information geometry case microlevel stochastics used macromodel restoration instead restoration requires computation model basic parameters dimension uncertainty curvature indicator allow finding model optimal macroprocess synthesized optimal control well model hierarchy computation uses primary parameters basic model parameters object geometry known invar grad center coordinates invar diagram presents functional schema imd software operations computing invariants invar discrete moments space coordinates increment volume function speeds difference current space parameters polar coordinates gradients grad given space distribution calculating radius coordinates center square used compute object space model minimal optimal parameter output variables optimal dynamic process optimal controls eigenvalues model differential equation distributed process space current speed intervals moving stopping computed averaging speed estimated time computation diagrams approximately minutes conventional computation performed movement object cross section input calculated object current statistics solving considered complex problem traditional computation methods requires developing mathematical methodology software able overcome method high computational complexity solving even part problem existing techniques require many hours computation modern main frames structure imd software package software package transfers imd analytical methodology numerical procedures computer algorithms programs packet consisting programs includes following modules computation identification procedure restoration object equations parameters transformations movement opmc parameters processes controls structure function macrosystemic complexity transformation informational macromodel characteristics appropriate physical technological variables using particular applied programs main software modules compute basic optimal macromodel parameters spectrum model eigenvalues macromodel informational invariants intervals distribution optimal eigenvalues optimal controls geometrical macromodel coordinates space distributed macroprocesses procedure macrocoordinates cooperation aggregation hierarchical macromodel structure macrocomplexity formulas algorithms complete software numerical computation equations given program package included description imd software programs used practical solutions different applied problems including references bohr atomic physics human knowledge wiley new york dirac principles quantum mechanics oxford university press clarendon von neumann mathematical foundations quantum theory princeton university press wigner review quantum mechanical measurement problem quantum optics experimentalgravity measurement theory nato series physics series eds meystre scully wigner unreasonable effectivenss mathematics natural sciences communications pure applied mathematics bohm suggested interpretation quantum theory terms hidden variables phys rev bohm new theory relationship mind matter journal soc psychic res bohm hiley undivided universe ontological interpretation quantum theory routledge london eccles mental events cause neural events analogously probability fields quantum mechanics proceedings royal society wheeler recognizing without wheeler zurek information physics quantum search links complexity entropy physics information redwood california wesley wheeler computer universe international journal theoretical physics wheeler ford bit geons black holes quantum foam life physics new york norton wheeler quantum mechanics half century later include observer wave function episteme einstein podolsky rosen description physical reality considered complete phys rev penrose fashion faith fantasy new physics universe princeton university weinberg trouble quantum mechanics bricmont maudlin steven weinberg puzzle quantum mechanics new york review books april tong quantum fields real building blocks universe royal institution cambridge von baeyer quantum weirdness mind new version quantum theory sweeps away bizarre paradoxes microscopic world quantum information exists imagination scientific american kolmogorov logical basis information theory probability theory ieee trans inform theory shannon weaver mathematical theory communication univ illinois press kullback information theory statistics wiley new york jaynes information theory statistical mechanics statistical physics benjamin new york jaynes brain plausible reasoning stanford university feynman character physical law cox wyman ltd london lerner dynamic model origin order controlled macrosystem regulation biological processes walter gruyter new york lerner macro systemic approach solution control problems condition indeterminacy trans scripta technical automation kiev lerner mathematical foundation information macrodynamics systems lerner information systems theory informational macrodynamics review main results ieee transactions systems man applications reviews lerner information path functional informational macrodynamics nova science new york lerner observer information dynamics acquisition information origin cognitive dynamics information sciences lerner boundary value problem jensen inequality entropy functional markov diffusion process journal math anal solution variation problem information path functional controlled random process functional journal mathematical analysis applications lerner impulse interactive cuts entropy functional measure trajectories markov diffusion process integrating information path functional encoding application british journal mathematics computer science lerner impulse observations random process generate information binding reversible micro irreversible macro processes observer regularities limitations conditions self arxiv lerner arising information regularities observer arxiv kolmogorov foundations theory probability chelsea new york levy stochasic processes brownian movement deuxieme edition paris bennett logical reversibility computation ibm res develop landauer irreversibility heat generation computing process ibm journal research development jan yves markov paths loops lecture notes mathematics vol springer heidelberg lectures probability summer school held efimov states three interacting particles soviet journal nuclear physics lerner macrodynamic cooperative complexity information dynamics open systems information dynamics sidiropoulou pissadaki poirazi inside brain neuron review european molecular biology organization reports laughlin ruyter steveninck anderson metabolic cost neural information nature neuroscience alavash lim thiel sehm deserno obleser dopaminergic modulation brain signal variability functional connectome cognitive performance biorxiv aguilar neuronal depolarization drives increased dopamine synaptic vesicle loading via vglut doi http riek seletskiy moskalenko schmidt krause eckard eggert burkard leitenstorfer direct sampling vacuum fluctuations science doi discovery gluon european physical journal zurek spots schrodinger cats quantum decoherence arxiv chantasri dressel jordan murch siddiqi mapping optimal route two quantum states yoshida diffusion creating density proper entropy hadjihoseini lind mori hoffmann peinke rogue waves entropy consumption evaluation counterfactuality counterfactual communication protocols physical review zhang theta alpha oscillations traveling waves human neocortex biorxiv december vossel mathys stephan friston cortical coupling reflects bayesian belief updating attention networks neuroscience westerhoff means live virtual world generated brain erkenntnis liu local structural balance functional interaction excitatory inhibitory synapses hippocampal dendrites nature neuroscience magazine posner brain mechanisms quantity similar children adults proc natl acad ronald trysha modeling neurodynamic organizations interactions teams social neuroscience capotosto spadone tosoni sestieri romani penna corbetta dynamics eeg rhythms support distinct visual selection mechanisms parietal cortex simultaneous transcranial magnetic stimulation eeg study neuroscience seger braunlich wehe liu generalization category learning roles representational decisional uncertainty neuroscience desrochers chatham badre necessity rostrolateral prefrontal cortex sequential behavior neuron doi poort khan pachitariu nemri orsolic krupic bauza sahani keller hofer learning enhances sensory multiple nonsensory representations primary visual cortex neuron dingle boutin chirila livi labriola jakubek morgan darling kauer neural spheroid culture vitro model cortical studies tissue engineering part methods doi falkowski life engines microbes made earth habitable microbes made earth habitable princeton university press segerer alberola frey emergence persistence collective cell migration small circular micropatterns phys rev lett dombrowski cisneros chatkaew goldstein kessler coherence bacterial dynamics phys rev lett vicsek cziro cohen shochet novel type phase transition system particles phys rev lett weber hanke deseigne dauchot frey ordering vibrated polar disks phys rev lett ridley schwartz burridge firtel ginsberg borisy parsons horwitz cell migration integrating signals front back science huang brangwynne parker ingber mammalian cell cohort migration tissue pattern formation role persistence cell motil cytoskeleton doxzen vedula leong hirata gov kabla ladoux lim guidance collective cell migration substrate geometry integr biol marel zorn klingner frey flow diffusion channelguided cell migration biophys cisneros dombrowski wolgemuth kessler goldstein bacterial swimming oxygen transport near contact lines proc natl acad sokolov aranson physical properties collective motion suspensions bacteria phys rev lett sokolov aranson kessler goldstein concentration dependence collective dynamics swimming bacteria phys rev lett lushi peskin modeling simulation active suspensions containing large numbers interacting comput tam tang amplified effect brownian motion bacterial swimming proc natl acad sci cohen vicsekt formation complex bacterial colonies via vortices phys rev volfson cookson hasty tsimring biomechanical ordering dense cell populations proc natl acad sci ginelli peruani chate collective properties rods phys rev lett lushi peskin modeling simulation active suspensions containing large numbers interacting microswimmers comput giomi marchetti liverpool complex spontaneous flows concentration banding active polar crystalline films phys rev weber semmrich frey bausch polar patterns driven filaments nature wioland woodhouse dunkel kessler goldstein confinement stabilizes bacterial suspension spiral vortex phys rev lett grossman aranson ben jacob emergence agent swarm migration vortex formation inelastic collisions new phys lowen aggregation colloidal rods near confining walls phys rev woodhouse goldstein spontaneous circulation confined active suspensions phys rev porenta ravnikab geometrical frustration chiral ordering cholesteric droplets soft matter brereton karplus native proteins trap transit conformations science advances lerner emergence time curvature space casualty complexity encoding discrete impulse information process lerner information process composite stages cooperative structure cybernetics systems doi lerner entropy functional measure trajectories markov diffusion process estimation process cut applying impulse control gilson savin zenke emergent neural computation interaction different forms plasticity front comput neuroscience november cutts eglen bayesian framework comparing structure spontaneous correlated activity recorded different conditions biorxiv nirenbero jones leder clark sly pestka coding genetic information cold spring harb symposium quantum biology rodin origin genetic code trna translation biology direct brea urbanczik senn prospective coding spiking neurons plos computational biology june koonin novozhilov origin evolution genetic code universal enigma iubmb life lerner lerner solidification modeling continuous casting process material engineering performance lerner griffin development continuous gray ductil iron modern casting lerner rudnitski berger osipov chebanyuk noncontact power regulator electric furnace electrothermics moscow lerner sobolev comparison mathematical models contiguous horizontal casting gray iron ingots material resources foundry nto machprom press chelyabinsk lerner sobolev trefilov optimal condition iron solidity contiguous horizontal casting university news metallurgy lerner zhelnis dobrovolskis tzarev system automatization contiguous horizontal casting process foundry lerner treyger information virtual network optimal data proceedings international conference electronics communications computers mexico lerner information modeling processes transformation biological information biological systems lerner dennis herl novak niemi computerized methodology evaluation level knowledge cybernetics systems int journal koch mclean berry sterling balasubramanian freed efficiency information transmission retinal ganglion cells current biology chirgwin google tests quantum qubits prechtel kuhlmann houel ludwig valentin wieck warburton decoupling hole spin qubit nuclear spins nature materials doi chang zhou grover information coding retrieving using fluorescent semiconductor nanocrystals object identification optics express acin ant quantum information theory black boxes mao sun seeman assembly borromean rings dna nature lerner natural encoding information interacting impulses arxiv ieee xplore http lerner informational macrodynamics cognitive information modeling computation intelligent systems lerner information macrodynamics approach modeling biology medicine biological systems lerner cooperative information space distributed macromodels int control lerner systemic mechanism organizing assembling information kybernetes lerner portugal economic mathematic model optimal consolidation productions structures izvestia academy science ussr nauka moscow lerner elementary information macrodynamic model market economic system information sciences lerner information systems analysis modeling informational macrodynamics approach kluwer academic publisher lerner variation principle information macrodynamic academic publisher lerner information hidden markov diffusion lambert academic publisher information path randomness uncertainty information thermodynamics intelligence observer lerner riesel roykhel zaverchnev kogan applied software package niisl kovetz shiraishi antisymmetric galaxy cosmological probe
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calculus modeling floating authorizations jovanka ivan hugo torres faculty technical sciences university novi sad serbia imt school advanced studies lucca lucca italy feb abstract controlling resource usage distributed systems challenging task given dynamics involved access granting consider instance setting floating licenses access granted request originates licensed domain number active users within license limits licenses interchanged access granting scenarios given terms floating authorizations addressed paper first class entities process calculus model encompassing notions domain accounting delegation present operational semantics model two equivalent alternative ways informing specific nature authorizations also introduce typing discipline single systems never get stuck due lacking authorizations addressing configurations authorization assignment statically prescribed system specification introduction despite continuous increase computational resources usage always nevertheless subject availability mention accessibility regardless whether resources hardware software nature might finite virtually infinite capabilities physical examples mapped finite capabilities directly include actual devices printers cell phones components computing system memory processors virtual examples shared memory cell web service easily seen infinite potential often availability also finitely constrained general ensuring proper resource usage crucial yet non trivial task given highly dynamic nature access requests flexibility necessary handle requests ensuring secure efficient system operation particular security purposes crucial control access resources guarantee one hand access granted authorized users hand granting access subject availability concrete examples include wireless access point determined policy grant access limited capacity number connected devices software application licensed used internally institution bounded way examples include limited amount capabilities accessible shared way number authorized users involving notion floating authorizations borrowing terminology licensing essentially authorization use resource associated capacity access granted authorized user point capacity reached floating captures flexible nature access granting also identify notion implicit granting since users may granted access certain domain licensed institution even aware fact capacity bound present least point access denied pursuing licensing setting find examples hint dynamic dimension authorizations consider user deploys software application cloud licensing may provided user notion sometimes dubbed bring license scenario involves notion authorization delegation user may actually lose access use application locally given capacity constraints identify notion explicit authorization granting intention yield obtain authorization somehow signaled therefore distill following notions floating authorizations model domain specify access may implicitly granted accounting capture capacity delegation model explicit authorization granting paper present model encompasses notions developed focusing process calculus tailored communication centered systems model resources considered communication channels exploration authorization control carried considering authorizations refer channels usage communication controlled development builds provides basis model communicating systems including name passing model authorizations adopt language constructs authorization domain scope delegation adapt interpretation authorization domains encompass accounting thus allows model floating authorizations best knowledge first process calculus model addresses floating resources case authorizations first class entities presenting language also show typing discipline ensures systems never incur authorization errors systems blocked due lacking authorizations type analysis addresses systems authorization assignment statically prescribed system specification given particular combination name passing floating authorizations particular authorizations received names already held receiving parties notion call contextual authorizations start informally introducing model means examples examples section present language showing examples resorting licensing setting sake intuitive reading first model authorization domains consider scoping operator may write license university represent university domain holds one license means one use license within university authorized particular university comprises two students simultaneously active dub alice bob may write license alice bob case either alice bob use license floating idea support notion accounting one students uses license scope confined accordingly example bob uses license evolves licensedbob system evolves alice license lincensedbob change scope denotes license available alice anymore evolution system authorization directly confined bob using model implicit granting hence point alice implicitly grab authorization gets stuck tries use license hence name license may shared alice bob change scoping mean name privately held licensedbob authorization consider generous university license license alice bob carol specifies two authorizations resource available alice bob however none specified carol mentioned carol use authorization implicitly explicitly ask model explicit authorization granting introduce two communication primitives allow authorization delegated write represent explicit delegation one authorization license via communication channel auth activating configuration unlicensedbob instead auth license represent dual primitive allows receive one authorization license via channel auth activating configuration licensedcarol license auth auth auth license represent system authorization license transferred delegating user receiving user leading auth unlicensedbob auth license licensedcarol scope authorization license changes accordingly underlying communication carried synchronization channel auth remark respective authorizations auth present fact model resources considered communication channels immediate usage subject implicit authorization granting mechanism model supports form fairness allow greedy usage resources example license alice license lincensedbob user lincensedbob considered granted closest license confined floating one except case needs licenses possibly delegation remark name passing involved authorization delegation mechanism name license known ends first place instead name passing supported dedicated primitives namely comm comm comm comm represents system name license passed left hand side right hand side via synchronization channel comm leading activation alice dylan placeholder instantiated license notice synchronization take place since authorizations use channel comm given one endpoint name passing allows model systems access channels changes dynamically since communicated names refer channels hinted previous examples knowing name mean authorized use instance comm comm specifies reception comm received name used output reply leading inactive state represented receiving license initial duly authorized use channel comm leads comm license reply authorization comm present authorization license acquired result communication hence output specified using license authorized take place remark authorization reply required hence communicating name mean using purpose authorizations separating name passing authorization delegation able model systems unauthorized intermediaries brokers may involved forwarding names authorized parties without ever authorized use example comm comm forward forward receiving name typical pattern comm comm auth auth authorized reception comm authorization reception via authorized auth specified upon authorization use received name acquired another possibility enabling authorizations received names use authorization scoping construct comm comm licenseddylan authorization instantiated received name example hints fact authorization scoping powerful mechanism may therefore reserved trusted computing base resorting controllable authorization delegation mechanism general introduce last constructs language consider system license license license license license fresh licensing server specified left hand side used specification given right hand side domain represent creation new name private domain specification right hand side read first create name send via authorized channel license receive authorization use fresh name via channel license terminate received authorization actually used simple example left hand side find replicated repeatably available reception authorized channel license authorization received name specified may delegated via license two communication steps authorization newly created name therefore transferred returning university setting consider example exam minitest alice alice two authorizations available one exam another minitest alice waiting receive channel assessment made assuming take exam minitest authorizations specified sufficient carry reception task assuming extension language considering values use subject authorizations authorization actually used depends received name authorization implicitly taken directly using received channel naturally name viva sent student prescribed authorizations suffice inaction parallel restriction output input authorization send authorization receive authorization replicated input table syntax processes order capture fact configuration safe provided inserted context matches assumptions described previously types identify names safely communicated instance may say names exam minitest communicated channel alice also consider alice name subject instantiation exam minitest receive values subject authorization control denote alice exam minitest type channel alice circumstances subject replacement return point used communicate exam minitest turn used communication typed reading left right using information ensure specification given alice safe since names possibly used communications authorized analyse use input variable take account instantiated either exam minitest used channel communication type exam minitest hence need talk possible replacements name allowing uniformly address names bound inputs types channels built two parts one addressing possible replacements channel identity informing type names may exchanged channel denoted typing assumption alice alice exam minitest informs possible contexts system safely used instance safe compose system alice alice minitest minitest sent alice since name sent belongs names expected alice instead consider configuration exam minitest alice alice bob bob also safe addressed typing analysis considering typing assumptions alice alice exam bob bob minitest notice authorization needed student statically specified system safe exam minitest sent given authorization scopes confined accordingly clearly typing specification already informs association symmetric association also admissible typing analysis shown section addresses configurations authorizations received names may provided context section present operational semantics language considering two equivalent alternatives inform specific nature authorizations model model floating authorizations section present process model extension specialized constructs regarding authorizations adopted model authorizations syntax language given table assumes countable set names ranged briefly present syntactic constructs adopted inactive process represented represents two processes simultaneously active may interact via synchronization channels name restriction construct specifying creation channel name known process output prefixed process sends name channel proceeds input prefixed process receives channel name substitutes placeholder received name comment remaining constructs introduced model authorizations detail term another scoping mechanism names representing process one authorization use channel contrast name restriction name private term represents process delegates one authorization name along name proceeds table structural congruence term represents process receives one authorization name along name proceeds term allows specify infinite behavior process receives name along authorized name substitutes placeholder received name activating parallel copy original process name binding scope occurrences name binding scope binding occurrence said bound occurrence name bound term said free use denote sets free bound names respectively regarding language constructs authorization manipulation occurrence name free occurrences names processes also free remark model authorization scope extrusion applicable since free name specified unlike name restriction see table constructs send receive authorizations affect possible changes scope authorization involve name passing reduction semantics essence behavior processes seen communication specific model two processes ready synchronize channel must authorized use channel example evolve since sending receiving actions authorized lacks proper authorization receiving end hence synchronization occur another specific aspect language authorization delegation example consider actions along name authorized delegating process respective authorization hence authorization delegation take place leading notice authorization changed scope process received authorization actions along name authorized process delegating authorization authorized like synchronization possible formally define behavior processes means reduction semantics afterwards means labeled transition system reduction defined binary relation processes denoted specifies process evolves process one computational step order identify processes differ syntactically behavior introduce structural congruence relation least congruence relation processes satisfying rules given table rules standard considering structural congruence addition adopt rules introduced previously manipulate authorization scoping namely regarding authorization scoping remark rule relates authorization scoping parallel composition like name restriction due interpretation authorization scoping adopting rule sort would represent one authorization thus interfering authorization accounting hence distinguish authorization shared two authorizations specified one process another approach could rule sort also may affect computational power process example two processes considered equal since first one authorized perform output second one contrast notice output channel authorized action carried authorization table contexts one two holes drift drift drift drift drift drift drift table definition drift contexts one hole confined longer available since one authorization effectively used single thread structural congruence therefore expressive enough isolate two authorized processes willing communicate channel example process rewritten using structural congruence rules however processes able reduce process since actions scope proper authorizations define reduction relation thus introduce auxiliary notion static contexts one two holes operation allow single configurations communication occur intuitively two processes active prefixes ready synchronization scope appropriate authorizations reduction step possible static contexts defined table following standard lines use notation avoid ambiguity note table case name restriction construct allows identify specific names avoid unintended name capture remaining cases specify holes occur parallel composition underneath authorization scope contexts underneath processes deemed active omit symmetric cases parallel composition since contexts considered structural congruence operation drift plays double role one hand defined scope appropriate number authorizations context hand defined yields context obtained original one removing specific authorizations capture confinement model specific authorizations removed sake confinement ones nearest occurrence hole defined inductively derivation rules shown table operator drift present rules reading conclusion premise operator takes arguments name appear context one hole lists names first list names represents names authorizations removed context second represents names authorizations already removed operation briefly comment rules shown table rule authorization removed context specifies name passed first list second list hence removed removed rule authorization preserved context check name specified authorization second list removed hence authorizations already removed proceeding towards hole preserved ensures removed authorizations ones nearest hole rule parallel composition straightforward base rule defined first list empty implies operator defined authorizations first list actually removed context point hole reached noted second list names internal use operation top level defining operator context list names removed context authorizations removed respective list empty drift drift drift drift drift drift drift drift drift drift drift table definition drift contexts two holes drift drift table reduction rules example drift drift drift undefined drift drift undefined drift undefined sake defining reduction pair interacting processes must identified quire generalization operation contexts two holes operator drift defined inductively rules shown table takes arguments context two holes two lists names representing names authorizations removed two list names representing names authorizations already removed lists refer hole refer hole briefly describe rules reading conclusion premise rule describes case two contexts one hole case respective operation contexts used obtain resulting context considering name lists context left hand side context right hand side remaining rules follow exactly lines ones shown table duplicating authorization removal address two pairs lists dedicated way example drift drift drift drift undefined notice operation carried contexts two holes relies point operators contexts one hole fact derivation possible axioms empty contexts true thus operator undefined proper authorizations lacking lists used internally operator rest presentation abbreviate drift drift may present reduction rules shown table rule states two processes synchronize name passing name emitter receiver processes scope least one per process authorizations name yielded process considers context two authorizations removed drift operation specifies confined authorizations scope continuations communication prefixes analogously rule states two process exchange authorization name first process scope least one authorization processes authorized perform action name yielded process considers context authorizations removed drift operation notice authorization removed delegating process confined receiving process model exchange finally rule closes reduction structural congruence rule closes reduction restriction construct note rules close reduction parallel composition authorization scoping constructs already addressed static contexts also rule dedicated replicated input since thanks structural congruence rule single copy replicated process may distinguished take part synchronization captured illustrate rules drift operation consider process may say applying drift context remove two authorizations name drift thus observe authorizations confined continuations synchronizations model tightly coupled notion authorization sense absence proper authorizations synchronizations take place characterize undesired configurations referred error processes identifying redexes reduction semantics stuck due lack necessary authorizations roughly case premise reduction rules valid hence drift operation defined introduce abbreviations useful remaining presentation prefix stands communication prefix along name ahbi stands definition error process error drift undefined ahbi drift undefined notice definition errors aligned reduction structural congruence used identify configuration possibly scope number restrictions directly matches one redexes given reduction respective application drift undefined type analysis presented afterwards singles processes never incur errors first show alternative characterization operational semantics action semantics section introduce labeled transition system lts provides equivalent shown later alternative representation operational semantics model usual lts less compact albeit informative respect reduction semantics basic notion observable actions ranged identified follows ahbi form may recognize communication action prefixes together annotations capture authorizations bound names intuitively communication action tagged represents action carrying authorization represents action carrying authorization notice case authorization delegation two annotations present one name involved usual used denote name object communication bound bound output case internal steps annotation identifies authorizations lacking synchronization take place use abbreviate authorizations lacking given label represents set label names free names bound names respectively defined expected lines ahbi ahbi table transition rules transition relation least relation included set processes set actions satisfies rules table briefly describe rules capture actions correspond communication prefixes notice rule continuation activated scope authorization required action provided case authorization reception capture confinement notice labels decorated corresponding authorizations represents actions carrying authorizations omitting annotations contrast replicated input authorized construction label decorated corresponding authorization rule adopted lifting actions one branches symmetric rule omitted avoiding unintended name capture rules restriction follow lines ones given rule says actions also actions provided restricted name specified action captures bound output case opening scope restricted name thus allowing scope extrusion rule shows case synchronization lacks authorization level authorization scope action exhibited longer lacks respective authorization leads state longer specifies authorization scope use abbreviate case obtained respective exponent decrement remark contrast extrusion restricted name via bound output scope floats point synchronization rule explained authorization scopes actually float level communication prefixes rules capture confinement rule follows similar lines also refers lacking authorizations specifically case external action carrying necessary authorization use denote action specifies communication subject annotated including bound output form bhai includes ahai second authorization lacking also use denote respective annotation exponent increase rule captures case action lacking authorization case action crosses seamlessly authorization scope synchronization parallel processes expressed last three rules omitting symmetric cases rule one process able send receive name along name synchronization may take place notice sending receiving actions carrying appropriate authorizations transition label specifies lacking authorizations needed two minus existing ones rule scope bound name closed one process able send bound name receive along name synchronization may occur leading configuration restriction scope specified avoiding unintended name capture finalize scope extrusion authorization delegation expressed rule extra annotation considered given required authorization delegated authorization carried authorization annotations considered permutation thus identify compositional way requirements synchronization occur illustrate transition system let consider process ahbi ahbi using rule obtain using get parallel rule using rule obtain since action pending two authorizations apply twice applying get sake showing equivalence semantics induced reduction labelled transition system must focus transitions authorizations lacking theorem harmony proof get one implication induction derivation induction derivation see appendix presentations semantics inform particular nature authorizations model usual labeled transition system directly explicit compact reduction semantics allows global view authorization manipulation follows present type system allows statically identify processes never incur authorization errors type system section present typing discipline allows statically identify safe processes never incur error definition hence exhibit actions lacking proper authorizations mentioned introduction typing analysis addresses configurations authorizations granted contextually presenting typing language talks names safely communicated channels introduce auxiliary notions cope name generation namely symbol annotations since process model includes name restrictions types contain name identities require symbolic handling bound names included type specifications without loss generality refine process model purpose type analysis adding explicit symbolic representation name restrictions way avoid involved treatment bound names typing environments formally introduce countable set symbols ranged disjoint set names symbol also order introduce unique association table syntax types restricted names symbols refine syntax name creation construct two possible ways decorated symbol symbol respectively use sym denote set symbols process names associated symbols may subject contextual authorizations names associated symbol subject contextual authorizations latter communicated flexible way since receiver side expectation relying contextual authorizations purpose section adopt reduction semantics adapted considering refined definitions structural congruence reduction particular structural congruence omit rule decorate name restriction accordingly rules keeping side condition remark omission axiom new process models name restriction decorated typing information annotations symbols allow yield unique identification respective restricted names processes interested unique occurrences symbols contain occurrences symbols replicated input say process contain two occurrences symbol subprocess prefixed replicated input sym shown later typable process may show preserved structural congruence reduction may introduce type language syntax given table allows identify safe instantiations channel names subject contextual authorizations represent set names symbols use stands set symbol characterize either names may instantiated name subject contextual authorizations type carried type characterizes names communicated channel type represents ground names used communication usual define typing environments denoted set typing assumptions associating type name represent names set names occur type names set names occur entries may present type system defined inductively structure processes rules given table typing judgment states uses channels prescribed typing environment placed contexts provide authorizations given multiset names including multiplicities use multiset motivated considering process typed necessarily contains specifies process placed contexts offer two authorizations name one required per communicating prefix comment salient points typing rules inactive process typable using processes typed environment typed also consider enough authorizations placed contexts providing authorizations respectively process enough authorizations placed context providing sum authorizations represent addition operation multisets sums frequencies elements side condition sym sym necessary ensure unique association symbols names processes typed environment due fact scoping operator process owns enough authorizations placed context provides authorizations owns enough authorizations placed context provides authorizations process typable environment contains entry process restricted name typed environment removing entry occurrence name substituted symbol represent environment obtained replacing every occurrence every typing sym sym sym names names names names sym table typing rules assumption hence every type exclude case entry side condition sym necessary uniqueness symbol name pairings names says context provide authorization private name ensuring consistency typing assumption symbolic representation bound name typing environment enables avoid case restricted unforgeable name could sent process expects provide contextual authorization received name example consider process contextual authorization specified configuration excluded type analysis since assumption type channel carries symbol contextual authorizations provided notice typing process scope restriction uniformly handles name leaves open possibility considering contextual authorizations name within scope restriction difference respect rule substitution performed since environment must already refer symbol whatever pertains restricted name notice side condition example type process type must identifies names communicated never subject contextual authorizations process typed environment types names possible replacements name given safe communicated along name formalized carried type also carried type matches specification given type case process typed environment two possibilities ensure name authorized namely context may provide directly authorization name may provide authorizations replacements name formalized notice latter option crucial address contextual authorizations case contain symbols since definition rule readable light principles explained previous rule continuation typed considering assumption input bound name specifies expected context provides authorizations name case replicated input typable considering environment obtained removing entry must match carried type provided names since bound process process contain symbols necessary ensure unique association symbols names copies replicated process activated see discussion example shown end section case process placed context conforms rules typing environment premises conclusion handling subject communication similar rule way authorization addressed rule follows lines rule rule authorization added ones expected context notice way contextual authorizations provided delegation generalizing rule direct following rules prefixes say process contains assumptions form top level typing assumptions address free names process subject instantiation free names either characterized says replaced says subject contextual authorizations example process typable assumption name type typable assumption fact authorizations provided context means process terms authorizations may present results starting mentioning fundamental properties may show typing derivations enjoy standard properties weakening strengthening typing preserved structural congruence subject congruence usual prove typing preserved reduction need auxiliary result talks name substitution lemma substitution let names proof proof induction depth derivation see appendix remark name must contained set possible instantiations name replacing two names must carried type even though subtyping present inclusion principle used lemma already hints substitutability notion first main result says typing preserved reduction theorem subject reduction proof proof case analysis last reduction step see appendix surprisingly since errors involve redexes proof theorem intertwined proof error absence property included second main result proposition captures soundness typing analysis processes stuck due lack proper authorizations hence errors definition proposition typing soundness error proof immediate auxiliary result see appendix usual combination theorem proposition yields type safety corollary type safety error type safety ensures processes never lack necessary authorizations carry communications illustrate typing rules recall example introduction exam minitest alice alice task type alice exam minitest assigned channel name alice following typing rule see name typed exam minitest carried type alice knowing name instantiated exam minitest order apply rule considering contextual authorizations must check whether authorizations use names provided contained furthermore consider process shown composed parallel another process willing send name along alice specifically alice alice exam minitest alice alice task considering typing assumption alice type alice exam minitest assuming name exam type exam rule conclude safe send name exam along alice since exam contained exam minitest instantiation exam contained carried type alice consider latter process placed context name exam restricted alice alice exam minitest alice alice task order type process assumption alice considers type alice minitest representing alice restricted name communicated hence process shown composed others rely contextual authorizations names exchanged alice follows fact multiset provided authorizations definition contain names symbols use names received along alice one specify authorization possible receptions alice alice task rely authorization delegation let also consider process license license exam task models server receives name afterwards capable receiving lhs sending fresh name rhs along received name typing analysis excludes process since contains symbol replicated input see rule show names bound inside replicated input subject contextual authorizations even inside scope restriction like consider process shown may evolve receiving name alice twice alice alice alice alice task alice alice alice alice task two copies replicated process active parallel two different restricted names sent alice hence leading error contextual authorization match received name alice task order address name generation within replicated input use distinguished symbol captures fact names never subject contextual authorizations means names typed within scope outside granted contextual authorizations hence process obtained replacing annotation concretely license license exam exam task also typable since contextual authorization expected name exam notice configuration leads error like one shown however process license license alice alice typable hence may safely composed contexts comply assumption alice alice rely contextual authorizations names received alice remark approach generalized address contextual authorizations name generation context certain forms infinite behavior namely considering recursion together linearity constraints ensure race freedom like setting behavioral types concluding remarks literature find plethora techniques address resource usage control ranging locks guarantee mutual exclusion critical code blocks communication protocols token ring several typing disciplines developed ensure proper resource usage capabilities specified type language first class entity model therefore approaches possible separate resource capability like unauthorized intermediaries brokers mentioned introduction distinguish approach considers accounting sense types specify number messages may exchanged therefore relates notion accounting presented also find proposals models include capabilities first class entities addressing usage channels resources communication objects specifically constructs restricting hiding filtering behaviors allowed channels usage specification binding name scope construct authorization scopes resources based given access policies distinguish approach since proposed constructs static able capture notion floating resource capability usage specification directly embedded model resembles type given binding scoping operator contrasts authorization scoping also detailed usage policies provided associated authorization scopes resources models seem less adequate capture notion floating authorizations access granted explicitly controlled via specification instance notion confinement seem directly representable believe approach extended considering form usage specifications like ones mentioned associating authorization scoping precise capabilities form behavioral types would also interesting resort refinement types carry typing analysis given types seen extent refinements domain names investigation leave future work perhaps even relevant would convey principles licensing domain identified related patents purpose certifying license usage level would important extend work considering also authorizations sense authorizations placed back original scope used presented model addresses floating authorizations notion believe unexplored existing literature based development previous work extending model minimal way carry investigation even though required technical changes revealed far straightforward left form choice since focus interplay parallel composition authorization scope believe adding choice development carried expected lines remark certain form accounting inconsistency handling authorization delegation already identified work believe accounting handled consistently throughout model intend study behavioral theory model also sake illuminating notion floating authorizations accounting instance axiomatization behavioral semantics would surely informative authorization scoping construct also presented typing analysis addresses contextual authorizations also believe unexplored literature form present typing rules induce decidable procedure since rules syntax directed provided usual carried type annotation added name restrictions however already started working typechecking procedure nevertheless based typing rules focus efficiency namely level distributing authorizations provided environment parallel subsystems allows information authorizations actually required processes hopefully lead identifying principles may used sake type inference notion substitutability naturally arises typing analysis leave future work detailed investigation subtyping relation captures notion may mention preliminary assessment actually hinted non standard features respect variance covariance carried types references lucia acciai michele boreale spatial behavioral types inf william joseph armstrong naresh nayar kevin patrick stamschror management concurrent use license computer october patent paolo baratti paolo squartini license management system june patent chiara bodei viet dung dinh gian luigi ferrari checking global usage resources handled local policies sci comput ankush das jan hoffmann frank pfenning work analysis session types corr timothy freeman frank pfenning refinement types david wise editor proceedings acm sigplan conference programming language design implementation pldi toronto ontario canada june pages acm silvia ghilezan svetlana jovanka jorge hugo torres vieira dynamic role authorization multiparty conversations formal asp marco giunti catuscia palamidessi frank valencia hide new proceedings combined international workshop expressiveness concurrency workshop structured operational semantics volume eptcs pages daniele gorla rosario pugliese dynamic management capabilities network aware coordination language log algebr hans ivan lanese vasco vasconcelos caires marco carbone dimitris mostrous luca padovani ravara emilio tuosto hugo torres vieira gianluigi zavattaro foundations session types behavioural contracts acm comput naoki kobayashi kohei suenaga lucian wischik resource usage analysis pcalculus logical methods computer science davide sangiorgi david walker theory mobile processes cambridge university press nikhil swamy juan chen ravi chugh enforcing stateful authorization information flow policies fine programming languages systems european symposium programming esop proceedings volume lncs pages springer vivas nobuko yoshida dynamic channel screening higher order electr notes theor comput appendix proofs section lemma inversion labelling let ahbi proof proof induction inference comment first second assertions base case rule comment case last applied rule derived induction hypothesis implies base case derived first part proof thus comment base case derived rule first assertion lemma get directly follows lemma lts closure structural congruence exists proof proof induction length derivation detail case last applied rule two possible transitions assume derived possible transitions since derived get since get derived lemma get thus get two cases derived lemma get since assuming bound names different get thus derived bhci lemma get since get thus assume derived since lemma get could derived using get lemma inversion lts let processes drift undefined drift drift undefined drift drift undefined drift ahbi drift drift undefined ahbi drift drift undefined ahbi drift drift undefined drift either drift drift either drift drift drift undefined drift drift drift undefined drift drift undefined proof proof induction inference comment first two assertions suppose let show drift undefined get base case rule operator drift undefined since second parameter operator empty list induction steps next cases last applied rule last applied rule immediately get result induction hypothesis last applied rule get induction hypothesis drift undefined considering free bound names different get get drift undefined since case get similar reasoning consider get base case rule derived first part proof get drift undefined lemma get thus get get drift implies inductions steps possibilities last applied rule three cases similar first part proof proposition inversion drift drift defined case drift drift undefined case drift drift drift undefined case drift drift drift undefined case drift drift drift drift undefined drift defined case drift drift drift undefined case drift drift drift undefined case drift drift drift drift undefined case drift drift drift drift undefined rest cases analog lemma harmony lts proof proof induction derivation obtain two base cases drift get proposition distinguish four cases structure context comment case drift drift undefined thus contexts holes scope authorizations induction contexts using rules get get proceeding induction contexts using rules get induction structure context using rules get need note drift apply similar reasoning induction step two possible last applied rules assume derived induction hypothesis get contextually get assume derived induction hypothesis lemma get since structural congruence equivalence relation get lemma harmony reduction proof proof induction derivation base cases obtained derived lemma one possibility structural congruence drift drift drift get proof analogous drift drift get similar reasoning second apply instead rule induction step last applied rule one following derived induction hypothesis derived lemma three cases detail case drift drift drift undefined get drift thus get proof derived derived induction hypothesis drift drift thus get former case get proof latter case get proof similar reasoning theorem harmony proof proof follows directly lemma lemma proofs section proposition preservation also sym sym proof proof induction last applied structural congruence reduction rule detail case last applied reduction rule let derived since sym induction hypothesis get sym sym thus sym sym sym lemma inversion typing sym names names names sym names sym sym following two results weakening strengthening properties fundamental prove subject congruence turn crucial prove subject reduction write depending whether name bound process process context name bound respectively lemma weakening let sym proof proof induction depth derivation detail two cases last applied rule let derived two cases sym induction hypothesis since since follows follows thus get induction hypothesis since change get result directly let derived without loss generality assume two cases sym induction hypothesis since get induction hypothesis get result directly detail case last applied rule let derived names induction hypothesis without loss generality assume new since conclude get lemma strengthening sym proof proof induction depth derivation comment case last applied rule two cases sym without loss generality conclude lemma get induction hypothesis since follows thus using get using arguments assume lemma get induction hypothesis thus using get lemma subject congruence proof proof induction depth derivation comment three cases last applied rule lemma get lemma get get conclude show implication right left two cases lemma get sym sym applying lemma get sym sym sym get sym without loss generality assume lemma get lemma get applying lemma get lemma get show implication left right two cases lemma get sym lemma sym sym since sym get sym lemma using get lemma get lemma since lemma using get get show one implication suppose lemma get lemma get sym names get lemma substitution let names proof proof induction depth derivation detail two cases lemma induction hypothesis get implies get lemma induction hypothesis thus get lemma authorization safety drift defined proof induction structure notation use abbreviate ranges symbols lemma interaction safety drift defined drift drift defined drift proof proof induction structure context detail first statement lemma get consecutive application lemma get dom consecutive application lemma get multisets names lemma thus furthermore names hence lemma get get since lemma conclude drift defined proposition distinguish four cases structure context comment case drift drift undefined thus contexts holes scope authorizations consecutive application get also since consecutive application get consecutive application get lemma get theorem subject reduction proof proof case analysis last reduction step two base cases rules applied get directly lemma induction steps two cases last applied rule two cases derived let proposition get thus sym lemma get sym induction hypothesis get get derived let lemma get induction hypothesis get get last applied rule derived let lemma get induction hypothesis lemma get proposition typing soundness error proof directly typing rules immediate lemma corollary type safety error proof proof follows directly theorem proposition
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growth periodic grigorchuk groups feb anna erschler tianyi zheng bstract torsion grigorchuk groups construct random walks finite entropy tail decay poisson boundary random walks provide near optimal volume lower estimates groups particular first grigorchuk group show volume growth function pnq satisfies log log log pnq log log tive root polynomial ntroduction let finitely generated group let finite generating set word length element smallest integer exists growth function pnq counts number elements word length pnq finitely generated group polynomial growth exists constant pnq cnd exponential growth exists pnq pnq polynomial growth say group intermediate growth first examples groups intermediate growth constructed grigorchuk grigorchuk groups indexed infinite strings whose definitions recalled subsection following associate automorphisms rooted binary tree consider group generated string corresponding group called first grigorchuk group introduced grigorchuk eventually constant virtually abelian hence polynomial growth otherwise intermediate growth group periodic torsion contains three letters infinitely often open question since construction grigorchuk groups intermediate growth find asymptotics growth functions among first grigorchuk group investigated grigorchuk shown pnq exp upper bound shows growth based norm contracting property property considered follows integer injective homomorphism qdi key words phrases grigorchuk groups growth function poisson boundary work first named author supported erc grant groisran anna erschler tianyi zheng finite index subgroup respect word length constant contraction coefficient later bartholdi shown exist choices norm achieve better contraction coefficient word length therefore better upper bound growth upper bound states pnq exp log log positive root polynomial lower bound proved grigorchuk uses calls erties remained open whether grows strictly faster leonov introduced algorithm led better coefficient showed pnq exp bartholdi obtained pnq exp thesis brieussel corollary states pnq exp let probability measure group function pxq ypg among several definitions poisson boundary also called boundary one formulated terms measure theoretic boundary represents bounded functions see particular poisson boundary trivial bounded functions constant shannon entropy measure defined log finitely generated entropy criterion derriennic provides quantitative relation growth function tail decay measure finite entropy poisson boundary see details reviewed subsection paper construct probability measures poisson boundary torsion grigorchuk groups measures chosen finite entropy explicit control tail decay upper bound ptg pgq nuq main contribution paper collection grigorchuk groups including indexed periodic strings containing three letters construct measures good tail decay provide near optimal volume lower estimates main result theorem specialized first grigorchuk group following log theorem let log positive root polynomial exists constant symmetric probability measure finite entropy nontrivial poisson boundary tail decay satisfies ptg pgq ruq consequence exists constant pnq exp construct measures poisson boundary needed proof theorem introducing studying asymptotic invariants group related diameter shape call sets uniformised sets see subsection sketch proof main problem address growth periodic grigorchuk groups use geometric algebraic structure groups construct measures good control tail decay careful analysis induced random walks schreier graph performed show poisson boundary volume lower bound theorem matches exponent bartholdi upper bound particular combined upper bound conclude volume exponent first grigorchuk group exists equal lim log log pnq log open whether limit exists argument applies large class satisfying certain assumptions see theorem particular assumption theorem verified periodic strings containing letters strings products general volume upper bounds based norm contracting property first shown grigorchuk suitable choices norms given work bartholdi first named author combined upper bounds derive following two corollaries theorem periodic string contains three letters show volume exponent exists let matrices defined corollary let periodic string period contains three letters growth exponent grigorchuk group exists log log pnq lim log log log spectral radius matrix theorem family provides continuum mutually growth functions periodic grigorchuk showed choices growth exhibits oscillating behavior log log pnq log exist together upper bounds lower bounds show given numbers exists realizes prescribed numbers lower upper volume exponents corollary volume exponent exists grigorchuk group satisfies log log pnq log log log pnq lim sup log lim inf mention groups intermediate growth way obtain near optimal lower bound growth function using tail decay measures nontrivial poisson boundary extensions first grigorchuk group intermediate growth growth function arbitrarily close exponential random walk finite trivial poisson boundary see remark end paper anna erschler tianyi zheng igure orbital schreier graph right ray action first grigorchuk group sketch arguments first grigorchuk group nilpotent group probability measure known poisson boundary trivial measure finitely generated nilpotent group result due dynkin maljutov general case see azencott proposition assume growth function corollary entropy criterion finitely supported measure generally measure finite first moment trivial poisson boundary growth bounded pnq exppcn measures finite trivial poisson boundary see corollary since volume growth first grigorchuk group satisfies pnq look measures infinite order poisson boundary first grigorchuk group generated four automorphisms rooted binary tree subsection recall definition automorphisms see also figure group acts boundary tree homeomorphisms orbital schreier graph half line shown figure right ray tree random walks poisson boundary used show eventually constant contains finite number example volume growth satisfies pnq exp measures poisson boundary constructed come transient symmetric random walks easy see best tail condition measures situation remained completely unclear grigorchuk groups containing three letters infinitely often since group torsion argument relying infinite order element applied need introduce new construction meausures first grigorchuk group consider groupoid germs action isotropy group groupoid germs quotient stabilizer subgroup elements act trivially neighborhood first grigorchuk group orbit isomorphic elements listed tid see details section given group element denote germ growth periodic grigorchuk groups pxq germ neighborhood finitary automorphism tree depend choice see precise definition formula goal construct probability measure along walk trajectory pwn coset poq xby stabilizes probability almost sure stabilization provides events tail random walk implies poisson boundary first construction measure controlled near optimal tail decay poisson boundary form pus uniform measure symmetric measure supported subgroup subgroup consists germs pxq xby measure random walk orbit transient poisson boundary see remark situation containing infinitely often subgroup contains infinite order element thus case one apply well known results random walks xaby first grigorchuk group subgroup locally finite orbit full orbit rather rays cofinal locations digits see lemma since subgroup locally finite unlike simple random walk one take convex combinations convolution powers corresponding measure produce measure transient induced random walk orbit rather build measures sequence group elements satisfying injectivity property action orbit say sequence satisfies cube independence property map hypercube orbit injective call set set uniform measure ufn measure given sequence pgn take convex combination measures inverses cube independence property guarantees measures induced random walks orbit satisfy good isoperimetric inequalities prove sufficient criterion induced random walks transient proposition igure portrtrait pacq first grigorchuk group generally words paxq represents element stabilizer level labels sections vertices level anna erschler tianyi zheng choice elements satisfying cube independence property closely related words known called asymptotically maximal inverted orbit growth see let substitution tab defined abadac abab acac achieves asymptotic maximal inverted orbit growth see proof proposition considerations germs mind take sequence pacq mod padq mod xbygerms mod mod construction important sequence pgn satisfies cube independence property illustration proposition subsection find optimal moment condition measures induce transient random walks orbit near optimal measure taken form ufn set formed uniform measure section explain use subsequence related sequence adapted produce finite entropy symmetric measures supported induced random walks orbit transient particular already provides volume lower bound exp better previously known bounds see corollary however taking measure supported restrictive see remark end section section carry main construction removes restriction produces measures poisson boundary near optimal tail decay give informal sketch construction return full cube independent sequence pgn measure built sequence near optimal tail decay among measures induce transient random walks schreier graph however sequence germs types mod mod one check poq stabilize random walk driven pus goal achieve almost sure stabilization poq along random walk trajectory want perform modifications elements used construction prevent stabilization namely turns considerations transience induced random walks schreier graph sufficient observe almost sure stabilization poq walk need show weighted sum green function schreier graph finite see criterion proposition ptg xby thus behavior green function induced random walk essential informally call ray carries germ bad location assume green function decays distance control effect xcygerm elements advantage push bad locations away ray note perform modifications measure terms summation affected green function bad locations growth periodic grigorchuk groups recall divisible pacq stabilizer section vertex level tac cau therefore bad locations schreier graph distance locations bounded move locations without causing disturbances globally rely certain branching properties group let integer divisible introduce conjugation generator swaps level section section sibling explicitly take element hkn rigid stabilizer acts baba subtree rooted denote ckn conjugate ckn chkn portrait ckn illustrated figure divisible take packn still locations schreier graph distance locations comparable remark parameters pkn need chosen carefully word length element approximately therefore large tail decay near hand sufficiently large effect introduces strong enough achieve goal stabilization end take log constant large enough igure portrtrait mod levels tree shown picture portrait portrait level level left right branches permuted take parameter sequence pkn cube independent sequence pgn keep mod replace mod modified element modified sequence satisfies cube independence property form sets define measure form using sequence possible choice like close log one observe almost sure stabilization resulting measure pus however measure manage obtain good enough estimates induced green function verify summability condition issue modifications induced transition kernel schreier graph becomes singular compared one induced anna erschler tianyi zheng igure conjugated element usual notation first grigorchuk group levels form traced dotted lines two possiblities portraits picture shown indexing vertex overcome difficulty randomize choice conjugates ckn produce less singular induced transition kernels take collection conjugates indexed vertices form loooooooomoooooooon digit digits see figure given conjugate apply substitution obtain pacv call set formed random vertex chosen uniformly corresponding indexing set uniformised set final measure take similar form discussed previous paragraph involves call uniformised measures detailed description measures see example although construction appears complicated resulting induced transition kernel homogeneous transition kernels techniques based called meyer construction davies method heat kernel estimates jumping processes applied way obtain good enough decay estimate induced green function terms schreier graph distance orbit together explicit estimates bad locations show weighted sum green function schreier graph finite appropriate choice parameters construction provides finite entropy measures near optimal tail decay poisson boundary organization paper section reviews background poisson boundary connection growth groups acting rooted trees basic properties grigorchuk groups section formulate criterion stabilization cosets germ growth periodic grigorchuk groups configurations section provides quick examples measures without explicit tail control poisson boundary first grigorchuk group section define cube independence property show build explicit measures transient induced random walks sequences satisfying property section discusses measures restricted germs first grigorchuk group induce transient random walks orbit particular volume lower bounds measures provide limitations section contains main result paper construct finite entropy measures near optimal tail decay poisson boundary satisfies call assumption pfrpdqq section derive volume lower estimates measures constructed section close paper remarks questions conventions used throughout paper consider right group actions write action group element given probability measure say support generates semigroup let random variables distribution refer pwn walk regular rooted tree given vertex denotes concatenation strings ray prefix followed given denotes longest common prefix given finite set denotes uniform distribution let two functions constant pxq pcxq domain let equivalence relation note finitely generated infinite groups equivalence class growth function depend choice generating set reliminaries boundary behavior random walks connection growth recall notion invariant tail random walk let probability measure countable group let law random walk trajectory starting endow space infinite trajectories usual borel shift acts invariant defined paq also referred stationary correspondence space bounded functions particular invariant trivial event satisfies paq bounded functions constant see intersection called tail exit random walk aperiodic coincide therefore aperiodic bounded functions constant tail hence shannon entropy probability measure log anna erschler tianyi zheng case measure finite entropy avez asymptotic entropy defined lim entropy criterion derriennic states finite entropy bounded functions constant avez asymptotic entropy given probability measure finite entropy poisson boundary finitely generated group entropy criterion shannon theorem one derive lower estimate volume growth tail decay measure following lemma extends lemma lemma let finitely generated group word distance suppose admits probability measure finite entropy poisson boundary nontrivial let inf ptg ruq exists constant prn exp pcnq proof measure finite first moment words function bounded exponential growth case claim true follows assume let truncation measure radius pgq triangle inequality pgq pgq follows markov inequality ptg ruq pnq total mass convolution power satisfies pgq uqq take denote asymptotic entropy entropy criterion poisson boundary equivalent shannon theorem exists constant finite set pvn pbpe pbpe pxq consider intersection pgq gpvn growth periodic grigorchuk groups choose constant less large obtain sufficiently prn record following two corollaries lemma corollary suppose admits probability measure finite entropy nontrivial poisson boundary suppose satisfies ptg ruq prq slowly varying function pxq pxb exists constant pnq pnqq proof let asymptotic inverse prq pxq bruijn conjugate see proposition let pxq pxq see proposition follows pnqq assumption pxq pxb pxq pxq bound simplifies cpn lemma exists constants depend cpn exppcnq since asymptotic inverse pxq case conclude exists constant pnq pnqq corollary let finitely generated group prq probability measure finite gpg finite entropy trivial poisson boundary proof first show finite entropy write gpg let random variable distribution hpzq anna erschler tianyi zheng log gpg inequality log hpzq markov inequality ppz ppz log ppz ppz conclude finite entropy suppose contrary poisson boundary since finite inftr ruq therefore lemma prn contradicts upper bound prq permutational wreath products let group acting set right let group permutational defined semidirect product acts permuting coordinates support suppf consists points pxq ida elements viewed finitely supported functions action pxq follows psuppf elements recorded pairs multiplication given finitely generated finitely generated well information cayley graphs permutational wreath products see section groups acting rooted trees review basics groups acting rooted trees background details refer part chapter viii chapter let pdj qjpn sequence integers spherically symmetric rooted tree tree vertices root denoted empty sequence edge set tree index called depth level denoted denote tnd finite subtree vertices depth vertices level boundary btd tree set infinite rays group automorphisms autptd rooted tree group tree automorphisms fix root group autptd admits following canonical isomorphism autptd autptsd growth periodic grigorchuk groups permutation group set shift sequences refer isomorphism wreath recursion isomorphism canonical omit identify automorphism image autptd write pgq sections autptsd pgq section represents action subtree rooted vertex permutation pgq describes subtrees rooted permuted refer pgq root permutation apply wreath recursion times autptd autptsn oln autptnd pgv qvpln pgq finitary permutation pgq projection automorphism group finite tree tnd called section vertex since isomorphism canonical write pgv qvpln pgq let subgroup autptd given vertex vertex stabilizer stg puq consists elements fix stg puq level stabilizer stg pln consists automorphisms fixes vertices level stg pln xvpln stg pvq rigid vertex stabilizer denoted ristg puq consists automorphisms fix vertices prefix element ristg puq acts subtree rooted wreath recursion finitary permuation pgq equal identity section equal identity since automorphisms different rigid vertex stabilizers level act disjoint subtrees commute rigid level stabilizer defined ristg pln ristg puq upln recall definitions branch groups let rooted tree constant valency subgroup autptq acting level transitively rigid level stabilizers ristg pln finite index say branch group say regular branch group acts regular tree constant valency exists finite index subgroup contained finite index subgroup finite index subgroup loooomoooon copies case say branching group automorphisms subtree rooted embedded autptd rigid vertex stabilizer autptusn autptsn oln autptnd pgv follows definition branching vertex ristg puq contains subgroup particular anna erschler tianyi zheng grigorchuk groups groups introduced grigorchuk first examples groups intermediate growth recall definition groups let three homomorphisms tid tid ordered way vanish respectively example maps let space infinite sequences letters space endowed shift map given grigorchuk group acting rooted binary tree generated pid transposes automorphisms defined recursively according follows wreath recursion sends gsn gsn pbsn pbq bsn pcsn pcq csn pdsn pdq dsn string determines portrait automorphisms recursive definition first grigorchuk group corresponds periodic sequence often denoted see figure portraits tree automorphisms igure portraits generators first grigorchuk group eventually constant polynomial growth otherwise intermediate growth contains infinitely many three symbols infinite torsion group following substitutions used obtain words asymptotically maximal inverted orbit growth section define growth periodic grigorchuk groups tab adu abab acac abadac abab adacab adad acabad acac adad understood sends tabsn acsn adsn tabsn acsn adsn rules specified example pabsn absn absn one step wreath recursion gives tab adu pawa pwq contains even number pwqq pwa pwq contains odd number formula used section misprint see also lecture notes bartholdi section follows gsn represented word tabsn acsn adsn substitution applied resulting image gsn depend choice representing word composition maps gsn usually recursion first grigorchuk group recorded pid pid compared general notation following identification made note identification applying substitutions along string substitution mentioned introduction abadac abab acac example mod pabsn pacsn padsn general notation identification explained written substitution mention another useful substitution substitution aca gives homomorphism substitution used proof property leads lower bound grigorchuk also appears lysionok recursive presentation group later sections first grigorchuk group discussed adopt usual notation one keep mind identification discussed avoid possible confusions general notation sections along ray tree automorphism described portrait see example section purposes useful describe automorphism sections along ray finite infinite fixes let rooted binary tree let autptq suppose vertex write since fixed completely described sections denotes opposite anna erschler tianyi zheng igure portrait bscra along formally wreath recursion root permutation fixed record continue perform wreath recursion record section continue procedure stops reach level record sections sibling automorphism fixes infinite ray described collection sections unpn sections following terminology definition say automorphism directed along infinite ray example grigorchuk group generators directed along ray sections represented short words explicit collection draw picture sections along describe element although picture portrait strict sense effectively describes acts tree example take first grigorchuk group element bscra vertex fixed portrait along given cac see figure tabilization germ configurations main purpose section formulate prove proposition provides sufficient condition stabilization cosets germ configurations recall notion germs homeomorphisms let topological space germ homeomorphism equivalence class pairs homeomorphism neighborhood neighborhood gpxq two germs equal coincide neighborhood composition defined inverse germ groupoid germs homeomorphisms set germs homeomorphisms closed composition inverse contains identity germs pidx let group acting homeomorphisms right groupoid germs denoted set germs tpg isotropy group denoted set germs tpg stg pxqu words quotient stabilizer stg pxq subgroup consists elements acting trivially neighborhood growth periodic grigorchuk groups suppose group acting homeomorphisms point orbits isotropy group groupoid trivial refer auxiliary group trivial isotropy write conjugation germ easy see since trivial isotropy groups auxiliary group chosen let groupoid germs group isotropy group called group germs notation let homeomorphisms auxiliary group trivial isotropy suppose isotropy group point let proper subgroup point fix choice let proper subgroup let hpho following tpg discuss examples groupoid germs groups acting rooted trees let autptd acts boundary tree btd homeomorphisms let subgroup autptd consists finitary automorphisms words element autptd exists finite level sections trivial group locally finite autptd use group btu auxiliary group trivial isotropy groups particular acts level transitively used auxiliary group isotropy groups groupoid germs easy recognize directed groups called spinal groups chapter example let grigorchuk group acts level transitively use finitary automorphisms auxiliary group definition group cofinal isotropy group trivial eventually constant say eventually constant pid case cofinal eventually constant tid cofinal case group element finite level section eventually constant subgroup xby tid proper subgroup refer corresponding hpxbyq groupoid groupoid resp defined way subgroup tid tid resp proposition provides sufficient condition stabilization germs based green function induced random walk orbit criterion applied verify poisson boundary measures good control anna erschler tianyi zheng tail decay constructed section grigorchuk groups given probability measure induced transition kernel orbit given gpg green function walk refer book chapter general background random walks graphs many situations estimates transition probabilities induced walk available walk group hard understand proposition let countable group acting homeomorphisms right auxiliary group trivial isotropy assume isotropy group nontrivial point let probability measure let induced transition kernel orbit green function walk suppose exists proper subgroup ptg huq associated defined poisson boundary remark suppose following additional property exists finite subset element support condition proposition equivalent words induced walk transient orbit special case claim given proposition example applied measures form finite support element support case say supported subgroup restricted germs natural consider embedded subgroup unrestricted permutational wreath product follows denote product record elements pairs assigns point element isotropy group denote set points equal identity action given satisfies one readily checks depend choice left action multiplication given product isomorphic permutational wreath product describe embedding fact let defined pxq monomorphism growth periodic grigorchuk groups proof let pxq pxq pxq follows homomorphism clearly injective proof proposition let random variables let embedding described fact along random walk trajectory consider germ denote poqq right coset poq rule multiplication poq poq therefore event poqq poq definition germs tpxn poqq poq ppxn xqrhu gpg ptg huq summation finite lemma poqq stabilizes along random walk trajectory pwn coset consider tail event lim poqq given let group element poq note pid paho pid paho qpid pid pgqpg pgqpid paho since exists finite pgq pid paho would imply pid hand since poqq stabilizes probability pid pgo follows pid paho since pid pgqpid paho follows pid conclude aho tail event poisson boundary anna erschler tianyi zheng boundary behavior random walk proposition viewed lamplighter boundary embedding refer germ configuration let pwn random walk step distribution holds every point coset pxq stabilizes probability one along infinite trajectory random walk pwn thus view space germ configurations mod stabilization occurs limit configuration space cosets endowed hitting distribution viewed measures nontrivial oisson boundaries first rigorchuk group section give quick examples symmetric probability measures first grigorchuk group poisson boundary choices measures better quantitative control tail decay discussed section section recall first grigorchuk group acts rooted binary tree generated explained example generators nontrivial germs isotropy group list elements tid stands germ similarly stand corresponding germs take following proper subgroup tid group finitary automorphisms rooted binary tree plays role auxiliary group recall definition hpho refer consider subgroup consists group elements trivial given element one recognize whether sections deep enough level fact let group element exists section set tid proof direction follows definition show direction since contracting exists finite level sections nucleus tid perform wreath recursion levels mod suppose contrary exists vertex section contradiction thus level satisfies condition statement key property use orbit action infinite recall notation defined denotes automorphism rigid stabilizer acts subtree rooted note explicitly pababq homomorphism given substitution cac generally growth periodic grigorchuk groups igure portrtrait pababq first grigorchuk group usual notation divisible element belongs group germs equal germ trivial xra bsy vertex regularly branching subgroup see proposition portrait drawn figure subset contained since follows therefore orbit infinite theorem take measure poisson boundary form symmetric measure supported transient induced random walk orbit existence measure deduced fact orbit infinite following lemma lemma assumption finitely generated dropped since subgroup finitely generated include proof lemma let countable group subgroup given probability measure recall induced transition kernel left cosets pbx byq byu gpa cosets say measure transient respect lemma let countable group subgroup infinite index exists symmetric probability measure transient respect apply lemma group subgroup stg since orbit infinite symmetric measure induced random walk transient let uniform generating set take pus since obviously dirichlet forms satisfies comparison principle see example corollary induced random walk transient well since supported subgroup finite support explained remark transience anna erschler tianyi zheng verifies condition proposition conclude poisson boundary proof lemma first take symmetric measure support generate group induced random walk cosets bza irreducible since generates since bza infinite admits invariant vector pbzaq therefore strict convexity hilbert space prove convex linear combinations convolution powers measures form induce transient random walks bza following call discrete subordination let smooth strictly increasing suppose expansion coefficients one take class functions suitable operation bernstein functions see book induced random walk transition operator operator acting denote spectral resolution follows since vector pbzaq psq particular psq since write psq psq make green function finite suffices take psq near example one take measure bernstein function representing measure note coefficients expansion positive conclude choice walk bza transient growth periodic grigorchuk groups explained proof lemma measure transient induced random walk orbit obtained discrete subordination measure wish point general measure necessarily finite entropy section develop direct method construct measures transient induced random walks resulting measures finite entropy explicit tail decay bounds unlike general lemma construction heavily relies structure groups consideration crucial applications growth estimates urther construction measures transient induced random walk measures constructed cube independent elements subsection describe construction measures transient induced random walks particularly useful groups acting homeomorphisms possessing rich collection rigid stabilizers transience induced random walk deduced inequalities let transition kernel countable graph assume reversible respect dirichlet form defined pxq ypx denote respect measure given finite subset denote smallest dirichlet eigenvalue inf suppf inequality form referred inequality also known inequality following isoperimetric test transience theorem consequence connection inequalities heat kernel upper bound see coulhon proposition grigor yan proposition isoperimetric test transience suppose inf xpx assume holds finite subset continuous positive decreasing function markov chain transient best possible choice function defined profile pvq inf useful way obtain lower bounds cheeger inequality see theorem anna erschler tianyi zheng size boundary respect follows consider induced symmetric probability measure case symmetric reversible respect lemma let countable group acting point orbit infinite suppose sequence finite subsets constant inf let sequence positive numbers let ufn denotes uniform measure set induced walk transient proof denote transition kernel induced measure ufn assumption inf set indeed size boundary given gpf xpu ypu xpu assumption inf implies gpfn follows gpfn mention argument step proof isoperimetric inequality cheeger inequality profile satisfies pvq random walk induced convex linear combination definition clear epn therefore piecewise lower bound pvq growth periodic grigorchuk groups plug estimate isoperimetric test transience psq summation right hand side inequality finite proposition induced random walk transient example suppose sequence satisfying assumption lemma volume one choose sequence resulting convex linear combination statement lemma induces transient random walk orbit condition lemma forces stg pxq small orbit sets common observe especially situation totally actions applications grigorchuk groups subsets built sequence elements satisfy injective property refer cube independence defined mentioned definition cube independence property introduction formulate general definition sequence elements say sequence satisfies cube independence property parameters pkn map injective parameters constant omit reference say sequence cube independence property ordering elements definition would serve purpose choose formulate way examples consider easier verify inductively definition clear cube independence property inherited subsequences proposition suppose sequence elements satisfying cube independence property orbit pkn let let ufn normalizing constant total mass symmetric probability measure finite entropy induced walk transient proof cube independence property choose anna erschler tianyi zheng therefore lemma convex combination ufn induces transient random walk orbit entropy bounded log log series summable remark mentioned introduction refer set set measure ufn measure definition bounds length elements explicit estimates truncated moments tail decay precisely measure corollary pbpe one way produce cube independent sequence select suitable elements level stabilizers lemma let autptd suppose satisfies stg pln root permutation section pgn fixed point sequence pgn cube independence property proof given need show injectivity map since stabilizer level first two digits combined assumption root permutation second digit different thus gnn implies argument recursively shows critical constant recurrence subsection consider explicit sequence cube independent elements first grigorchuk group obtained applying certain substitutions application proposition evaluate critical constant recurrence stg notion critical constant recurrence subgroup introduced group equipped length function subgroup infinite index critical constant recurrence crt respect defined sup sup exists symmetric probability measure transient induced random walk hzg finite respect gpg growth periodic grigorchuk groups finitely generated word distance cayley graph omit reference rest subsection denotes first grigorchuk group use usual notation theorem stabilizer stg critical constant crt stg show indeed crt stg equal growth exponent permutational wreath product theorem let first grigorchuk group log log postivie root polynomial crt stg proof theorem consists two parts show lower bound construct explicit measure transient induced random walks orbit applying proposition recall substitutions tab used produce words asymptotic maximal inverted orbit growth proposition abadac abab acac direct calculation pca acq pda adq fact let word alphabet tab adu even number pwq level stabilizer section vertex given mod pwqqv awa mod proof claim verified induction even number pawa note even number applied pwq pwq pwqa pwqq induction hypothesis pwq implies statement pwq fact lemma pacqqn form cube independent sequence considerations germs mind although relevant subsection prefer take following sequence example mentioned introduction take following sequence elements pgn pacq mod padq mod fact lemma pgn cube independence property sequence cube independent elements take sets convex combination anna erschler tianyi zheng normalization constant probability measure proposition random walk orbit transient definition length estimated eigenvalues matrix associated substitution record number occurrences word column vector lpwq definition substitution matrix follows therefore spectral radius positive root characteristic polynomial proof lower bound theorem consider measure defined induces transient random walk orbit corollary since follows log tail estimate implies log finite since induced random walk orbit transient implies log crt stg log upper bound crt stg consequence volume growth estimate proof upper bound theorem show slightly stronger statement nondegenerate probability measure finite induced random walk orbit recurrent suppose claim true let probability measure finite transient induced random walk orbit let uniform measure lamp group uniform tid since random walk induced orbit transient measure poisson boundary proposition hand volume growth functions satisfy prq prq growth periodic grigorchuk groups lemma since finite lemma implies trivial poisson boundary contradiction note definition critical constant crt restrict symmetric random walks priori critical constant might become strictly larger one includes random walks example crt biased random walk finite range example transient however upper bound proved shows critical constant stg remains take sup moment measures transient induced random walk orbit remark measure transient induced random walk defined used show volume lower estimates certain extensions first grigorchuk group example consider double grigorchuk groups defined follows take shift eventually constant take directed automorphism defined subsection generator group let group generated since generators two strings called double grigorchuk group example group gpmq corollary corresponding string using measure one show improvement corollary isotropy group groupoid germs strictly larger take apply proposition measure pus measure defined poisson boundary follows corollary exists pnq exp remark gap proof proposition order guaranteethat measure considered finite entropy one needs strengthen assumption statement pnq exp assuming infinitely many sufficient ore examples measures restricted germs first rigorchuk group section exhibit examples symmetric measures first grigorchuk group poisson boundary form pus supported subgroup consists elements trivial compared section improvement measures finite entropy explicit tail estimates consequence derive volume lower bounds measures already better previously known estimates see corollary throughout section denote first grigorchuk group standard generating set recall subgroup consists elements trivial defined following example sequence elements satisfies cube independence property example take sequence elements pgn satisfying cube independence property defined definition mod mod filter elements keep subsequence convenience notation relabel elements anna erschler tianyi zheng sequence example take corresponding sets take proposition induced random walk transient let uniform generating set pus comparison principle corollary random walk transient since supported subgroup finite support proposition poisson boundary since finite entropy finite support also finite entropy estimate tail decay length estimate elements spectral radius positive root characteristic polynomial therefore explained remark corollary words corollary conclude prq exp exponent bound estimate sense exponent strictly larger however worse lower bound proved bartholdi clear construction skipping every third element cube independent sequence pgn leads significant loss improve construction choosing another cube independent sequence adapted note following property lemma subgroup defined locally finite orbit cofinal proof first show locally finite let finite collection elements fact exists level every sections level set tid follows subgroup finite rooted binary levels since finite conclude finite recall rigid stabilizer mod element consider collection since baba babababa follows set cofinal contained orbit growth periodic grigorchuk groups consider projection abelian group note projection subgroup xby indeed wreath recursion level pgv projections satisfy thus xby implies least one sections satisfies xby therefore claim follows induction length next show satisfies seen induction word length claim true suppose statement true length take word length representing perform wreath recursion level claim sections even number consequence claim digit verify claim odd number definitions generators projection section sibling contained xby contradicts assumption apply induction hypothesis conclude satisfies observe since subgroup acts trivially levels corresponding levels use substitution instead substitution given aca applied word tab always doubles length typically substitution results multiplicative factor larger define sequence words phn padq note abab padq abababab thus lemma sequence elements phn belong subgroup satisfy cube independence property orbit proof note pid ababq definition word isdin since element finite dihedral group satisfies follows pid word since pid combined fact level stabilizer level moreover level vertex section belongs tac cau otherwise section trivial since level stabilizer level vertex section belongs tad dau otherwise section trivial description sections conclude lemma form cube independent sequence cube independent sequence perform procedure take sets form convex combination uhn anna erschler tianyi zheng let pus proposition proposition poisson boundary length elements phn estimated using substitution matrices obtain following corollary exists symmetric probability measure form pus finite entropy nontrivial poisson boundary supported subgroup tail decay satisfies pbpe rqc number largest real root polynomial consequence pnq exp proof word tab recall records number occurrences column vector matrix recall matrix let largest real eigenvalue matrix words largest real root characteristic polynomial numerically follows therefore definition tail pus satisfies statement follows one measure following property supported subgroup smaller germ group xby finite support poisson boundary finite close critical constant crt stg main drawback action element fixes digits explained lemma orbit much smaller orbit indeed norm contracting calculations one bound growth respect induced word distance show measure transient induced random walk orbit tail decay close growth periodic grigorchuk groups construction measures non trivial oisson boundary rigorchuk groups section prove main result existence measures poisson boundary near optimal tail decay satisfies following assumption notation say string satisfies assumption pfrpdqq every integer string contains least one following two substring notation stands frequency given string satisfies assumption pfrpdqq define pmk follows let smallest number let set tkd note definition value important long finite let permutation letters string definition grigorchuk groups clear isomorphic call renaming letters may also take finite shift resulting string convenient example following examples satisfies assumption pfrpdqq finite depending sequence renaming letters every periodic sequence contains three letters sequence form uniformly bounded word obtained concatenating powers distance bounds schreier graph subsection review elementary facts distances orbital schreier graph action let denote schreier graph action vertex set two vertices connected edge labelled schreier graph constructed applying global substitution rules see example section let denote graph distance schreier graph unlabelled schreier graph depend sequence follows graph distance also depend reason omit reference write graph distance convenient read distance gray code enumeration orbit explicitly flip digits ray grey code mod note cofinal gray code finitely many regard element represented binary string schreier graph distance constant particularly easy read gray code point anna erschler tianyi zheng fact let point cofinal let npxq maxtk proof since maximum index digit npxq grey code form prefix length npxq general points similar upper bound holds fact let two points cofinal denote shift strings psn proof write gray codes grey code usn prefix length similarly vsn prefix length follows usn vsn psn consequence following estimate displacement recall denotes word length group equipped generating set lemma let pgv qvpln wreath recursion level proof write prefix therefore construction measures throughout rest section assume satisfies assumption pfrpdqq sketch first grigorchuk group introduction explicit descriptions example may help understand definitions avoid possible confusion indexing keep mind string starts tree vertices recorded first take sequence words obtained substitutions pabsn pacsn sequence pgn cube independence property lemma definition goal construct symmetric measure along corresponding random walk trajectory pwn germ stabilizes end introduce modifications modifications performed taking conjugations generators csj recall notation denotes group element rigid stabilizer acts subtree rooted following fact rigid stabilizers used repeatedly growth periodic grigorchuk groups fact let letter vertex rigid stabilizer pvq proof claim rigid stabilizer pvq equivalent asy pvq letter note determined although omit subscript lighten notation claim shown induction let letter pid claim true suppose true let letter pyq either case pid easy check similarly idq verified induction step claim follows see length estimate let reduced word representing apply substitution aya word letter pyq resulting word satisfies image satisfies pid written product elements form gsn substitution induces homomorphism gsn image idq therefore starting vertex apply substitution procedure recursively levels obtain word length image divisible define set wkn vertices depth except wkn recall set defined notation cardinality set wkn given integer denote residue mod define set vkj index set collection conjugates element csj gsj let integer divisible write define collection vertices vkj wkj defined notation words vkj obtained set wkj appending prefix match adding end important vkj ends digit cardinality set vkj cardinality wkj given vertex vkj denotes length take following sequence elements gsj prbsj definition vkj string either since digit case must fact rbsj rigid anna erschler tianyi zheng igure conjugated element label corresponding ray red picture shows turn left turn right general notation stabilizer vertex letter following different therefore rbsj acts bsj right subtree since hvi rigid stabilizer easy verify recursively definition elements hvi vkj take conjugation csj cvj csj following description element cvj portrait element along ray segment explained subsection lemma let vkj write length vertex fixed cvj along ray segment nontrivial sections cvj among section bsj absj otherwise section level section csj proof indices conjugation hvi nontrivial subtree rooted section conjugated rbsj effect conjugation rbsj drawn explicitly figure subscripts omitted portrait cvj obtained applying conjugations one one every nontrivial conjugation corresponding results turn illustrated pictures remark reason need assumption pfrpdqq perform conjugations following digit needed rigid stabilizer corresponding level second digit different implies acts next level last digit implies sections gets swapped consequence cvj fixes ray vertices form possible arguments work weaker assumption one find every simplifies calculations assume third digit growth periodic grigorchuk groups next apply substitutions acvj element obtain element given vertex vkj define acvj lemma element well defined substitutions applied level stabilizer sections vertices acvj cvj proof explained proof fact hvi defined prbsj represented reduced word tab adu understood bsj follows applying conjugations element cvj represented either reduced word form tab tab first case acvj represented word tba second case represented word tab either case substitutions applied acvj second claim follows let pkn sequence increasing positive integers divisible determined later divisible proceed construct measure replace measure first take following sets index define following set elements acvj denotes longest common prefix two rays set indexed subset consists vertices prefix cardinality indices specified keep single element set tgj defined follows always integer divisible take direct product product hypercube define map take measure uniform measure words random element distribution obtained follows take independent random variables uniform pgq anna erschler tianyi zheng uniform set group element distribution refer uniformised measure purpose symmetrization set measure pgq finally take convex combination measures npn uniform measure generating set normalization constant probability measure note although suppressed notations measure depends sequence pkn example explain definitions first grigorchuk group usual notation correspondence two systems notations usual general explained defining string shift two digits satisfies assumption frpdq notation shifted string sequence pgn defined formula pacq mod padq mod usual notation sequence appeared subsection section among mod perform conjugations follows divisible mod comes shift two digits set wkn defined wkn words wkn consists vertices length concatenations segments mod set vkj defined vkj wkj vertices form digit looooooomooooooon digits vkj cardinality set note set vkj contained vertex subgroup consists elements xbygerms see figure figure draws portrait conjugated element cvj indexed vkj defined note since periodicity mod cvj element thus omit reference mod take parameter divisible sequence pgj qnj mod let tgj mod growth periodic grigorchuk groups igure grey black vertices orbit rightmost ray vertices form subsets grey vertices levels mod show vertices indexing set conjugaction definition grey verticies level prefixes allowed vertices replace set defined prefix pac looooooomooooooon digits uniformised measure described follows independently choose uniformly set choose independent random variable uniformly random group element distribution finally measure defined parameter sequence pkn convex combination uniformised measures inverses mod poisson boundary goal subsection show appropriate choices measure defined poisson boundary recall explained example isotropy group groupoid germs isomorphic xby xcy element level sections tid bsn let xby set germs either trivial apply proposition need verify summation finite transition kernel induced end show following upper estimate green function anna erschler tianyi zheng corollary let exists constant green function satisfies corollary proved subsection summation following proposition let defined let function exists constants pxq exists constant pds xqq inequality holds replaced proposition proved subsection two estimates prove appropriate choice parameters along random walk trajectory germ stabilizes theorem let string satisfying assumption pfrpdqq integer divisible measure defined poisson boundary proof show summation finite poisson boundary follows proposition let small constant corollary constant depending satisfies assumption propolet psq sition proposition pds xqq growth periodic grigorchuk groups constant depending summing estimate pds last step plugged since follows summation finite rest section devoted proofs corollary proposition estimates induced transition kernel subsection give bounds transition kernel used derive green function upper bounds use definitions notations subsection definition set following fact let index finite level action element element proof definition ray segment required satisfy first level sees difference portraits previous fact following uniqueness property inherited pgn lemma let suppose derive displacement estimate lemma description elements lemma lemma lemma let index let proof show second bound first since definitions prefix implies case action therefore anna erschler tianyi zheng suppose definition lemma depending parity need consider psj acvj psj cvj description cvj lemma distance bound lemma ray cvj since max psj acvj psj cvj claim follows lemma implies max triangle inequality max purpose introducing conjugations cvj randomizing construction measure average tail decays measures lighter maximum indicated lemma following upper bounds let level ufj ufn proof lemma ufj ufj recall indexed strings ber strings bounded first digits prescribed agree definition size therefore uniform measure ufj since tgi triangle inequality growth periodic grigorchuk groups indices ufn sup ufj combining two parts obtain ufn modifications induced transition kernel schreier graph comparable transition kernel expressed terms distance function indeed jump kernel depends location rather distance following upper bounds somewhat consider worst scenario proposition consider defined integer divisible exists constant cpd cds yqq yqq proof given write let min either case contribute lemma unique exists latter case ufn yuq moreover second claim lemma ufn since induced transition probability satisfies bound terms anna erschler tianyi zheng split two parts yqq yqq coefficients first part upper bound implies since yqq yqq second part bound deduce combine two parts obtain statement bound tail note ufn ruq ufn follows lemma sup ufn growth periodic grigorchuk groups let min log log last step assumption used definition finally truncated second moment tail bound obtained obtain log upper bounds green function throughout subsection measure defined following upper bound depend choice pkn construction finite level transition kernel induced coincide one induced ufn argument lemma implies random walk orbit admits following upper bound proposition let transition kernel induced exists constant sup proof let transition kernel orbit induced proof lemma lemma derive set volume cheeger inequality fact convex combination profile satisfies pvq lower bound profile implies stated upper bound see proposition theorem anna erschler tianyi zheng need upper bound deduced meyer construction davies method originally davies method developed derive gaussian upper bounds extended general markov semigroups successfully applied jump processes see example references therein let jpx symmetric transition kernel countable set technical reasons convenient consider continuous time random walk let associated heat semigroup ppt transition density following proposition follows proof heat kernel upper bound section see also section states upper bound upper bounds tail jpx growth truncated second moment jpx davies method provides offdiagonal upper bound ppt note uniform volume condition rqq prq required upper bound proposition let jpx symmetric transition kernel countable set let metric suppose exists sup ppt xpx exists increasing function jpx ypx yqjpx ypx exists constant yqq ppt yqq sup jpu since bound crucial argument poisson boundary formulation proposition explain proof reader convenience end subsection plugging bounds propositions obtain upper bound transition probability upper bound green function stated corollary remark bounds sharp however sufficient purposes proof corollary let induced jumping kernel schreier graph ppt heat kernel corresponding jumping process continuous time proposition lower bound profile implies sup ppt ypx let growth periodic grigorchuk groups proposition exists constant jpx ypx yqjpx ypx plugging bounds proposition yqq ppt pds yqq sup jpu note pds yqq proposition exists sup jpu combining two parts obtain yqq ppt constant depending compared bound ppt note interval pds yqq yqq rqp prq better use bound conclude heat kernel satisfies ppt min elementary jpx continuous time transition probability discrete time transition probabilities comparable former poissonization later see example subsection theorem discrete time transition probability cppt constant depends pidq therefore admits upper bound ppt larger constant obtain estimate green function sum transition probability upper bound yqq pds yqq anna erschler tianyi zheng first part yqq yqq pds second part pds yqq pds yqq pds combining two parts obtain stated upper bound green function proof proposition following first split jumping kernel two parts jpx jpx let transition density jumping process jump kernel consequence meyer construction see lemma ppt dirichlet forms satisfy sup xpx ypx jpx follows sup sup ppt bound turns nash inequality theorem function exp constant quantity defined max gqpxq pyq pxqqpgpyq ypx growth periodic grigorchuk groups take parameter two points take function note respect metric inequality pez assumption yqjpx qpxq ypx calculation applies therefore prq inequality evaluated points states exp exp restrict case choose log conclude yqq statement obtained combining bound proof proposition addition green function estimates need bound probability point carries germ estimate contributions statement proposition proof proposition lighten notation write xbygerms let random variable uniform distribution set defined write definition measure pdpo xqq ptg huq xqq xqrhu expectation taken respect uniform distribution recall multiplication rule groupoid germs since follows indicator bounded xqrhu qrhu anna erschler tianyi zheng therefore xqq xqrhu xqq qrhu xqrhu xqrhu similarly pdpo xqq ptg xqq xqrhu xqq xqrhu recall definition levels element germs therefore summation split sums definition set element form acvj follows lemma collection points carry germs exactly bpj odd particular cardinality set schreier graph distance point set comparable lemma let level suppose function let random variable uniform distribution set defined let point growth periodic grigorchuk groups let point proof lemma let point set bpj described let position first occurrence note substring first claim positions remains see denote length prefix consider section note suffix recall fact action finite level definitions elements lemma dpy follows therefore follows particular since digit remains description sections product every factor belongs set tid bsn tcusn cusn section cusn cusn pcq cusn wkn action element set possible sections fixes vertex claim follows claim shown previous paragraph digit position ensures point denote pxq event pxq dpo since assumed pxqc pxq paj pxqq anna erschler tianyi zheng since dpo paj pxqq last step applied lemma proved proof similar replaced omit repetition return proof proposition pgj odd since case write xqrhu pgj xqrh note choose uniform uniform distribution first digits uniform therefore pgj xqrh euln zqqq use lemma combining two parts obtain xqrhu last line applied assumption pxq growth periodic grigorchuk groups calculation similar use fact first uniform xqrhu eulj zqqq combine lemma conclude xqrhu last line applied assumption pxq pplications volume lower estimates rigorchuk groups section derive volume lower estimate theorem satisfies assumption pfrpdqq defined notation throughout section use notations reviewed subsection let string satisfies assumption pfrpdqq measure nontrivial poisson boundary theorem construction subsection recall built measures bound length elements support let matrix associated substitution column vector records occurrence explicitly define number lemma let defined proof element support form indices tgj anna erschler tianyi zheng otherwise definition cvj gsj hvj defined element prbsj obtained applying substitutions gsj therefore gsj since obtained applying substitutions acvj triangle inequality obtain tail estimate immediate corollary lemma corollary let defined parameters pkn finite entropy exists constant depending entropy proof since size support bounded bounded lemma thus combine theorem tail estimate corollary deduce following theorem let string satisfying assumption pfrpdqq exists constant cpd symmetric probability measure finite entropy nontrivial poisson boundary tail decay pbpid number defined consequence exists constant cpd exp growth periodic grigorchuk groups proof given take theorem exists measure poisson boundary note corollary tail estimate stated follows volume lower bound follows lemma applied periodic strings contain three letters obtain corollary stated log introduction proposition set quantities log spectral radius coming periodic containing three letters dense therefore volume exponents collection periodic string three letters form dense subset proof corollary renaming letters periodic string containing three letters satisfies assumption pfrpdqq tail estimate theorem note length satisfies spectral radius therefore theorem exists constant pnq log qlog proposition period string permutation wreath extension growth log pnq exp log statement follows consider strings prove corollary strategy choose string approximate prescribed volume function follows main result states function grows uniformly faster exp realized growth function permutational wreath extension roughly speaking procedure relies fact string exhibits sum contracting coefficient string coefficient careful choice length pik product two types strings allows approximate prescribed volume function section proof corollary let volume estimate corollary lemma corollary exists constants exppcnq exppcnq permutation wreath product let function real root function proof section produces string form volume growth equivalent result one use string form instead properties lemma satisfied anna erschler tianyi zheng following notations similar lemma exists constants using property instead instead proof section carries verbatim shows following prescribed function function exists string volume growth equivalent theorem since satisfies assumptionpdq exists constant exp combined estimates pnq volume equivalent exists exp log pnq pnq pnq statement follows choosing satisfying condition lower exponent upper exponent inal remarks questions without assumption pfrpdqq string satisfy assumption pfrpdqq notation construction section apply example consider strings form known sequence pin grows rapidly growth close exponential along subsequence however different ideas needed obtain good volume lower estimates eventually constant one apply construction takes measure restricted germs section obtain measures poisson boundary argument proves corollary extends general omit detail periodic string theorem explicit calculation shows volume exponent log log log pnq satisfies number kind dependence frequencies letters observed general strings given let pxq count number occurrences symbol first digits let min exists absolute constants log pnq growth periodic grigorchuk groups critical constant liouville property kaimanovich vershik conjectured group exponential growth admits probability measure poisson boundary recent preprint frisch hartman tamuz ferdowsi answered conjecture positively authors completely characterize countable groups admit random walks poisson boundary countable group admits probability measure poisson boundary admits quotient infinite conjugacy class property particular finitely generated group exists symmetric probability measure finite entropy poisson boundary virtually nilpotent finitely generated group virtually nilpotent one ask whether tail decay measures poisson boundary provides useful information growth group similar flavor critical constant recurrence see subsection given countable group equipped length function one define critical constant liouville property respect sup sup exists symmetric probability measure finite entropy poisson boundary finite finitely generated word length omit reference example lamplighter group polycyclic groups virtually nilpotent critical constant liouville property examples exponential growth intermediate growth groups critical constant liouville property bounded upper growth exponent results paper imply periodic containing letters volume exponent grigorchuk group critical constant recurrence critical constant liouville property coincide three exponents equal contains exactly two letters infinitely often recall group zfc consists elements finite conjugacy classes theorem measure bounded functions constant zfc consequence critical constant liouville property respect quotient zfc respect quotient length function hand volume growth much faster zfc example known extensions first grigorchuk group growth arbitrarily close exponential arbitrarily large function see remarks therefore one expect random walks poisson boundary provide near optimal lower bound general intermediate growth groups gap conjecture strong version grigorchuk asks growth finitely generated group strictly smaller implies polynomial weaker formulation conjecture called gap conjecture parameter asks growth strictly smaller implies polynomial growth related gap conjecture type questions regarding various asymptotic characteristics groups see survey grigorchuk information gap conjectures grigorchuk strong weak hold true one ask mind results paper whether stronger statements hold true group growth let finitely generated group growth true exists finite entropy measure tail decay bound ptg ruq anna erschler tianyi zheng poisson boundary one always choose remark following construction provides extensions let group equipped finite generating set typically free group free product suppose sequence quotients quotient group marked generating set let largest radius cayley graph ball radius around ball radius around cayley graph take universal group sequence ker suppose sequence marked finite groups respect group extension see first note quotient suppose kerpf kerpf every follows kerpf need show element ker finite conjugacy class let word represents since lim finite ball radius around agrees ball radius particular image identity therefore conjugacy class contained product since assumed finite conclude conjugacy class finite remark shown intermediate growth groups arbitrarily large functions groups obtained direct products piecewise automatic groups returns factor group represented piecewise automatic group return finite quotient free product let generating set free product xay tid action first grigorchuk group tree contracting sequence satisfies property respect words converges chabautygrigorchuk topology also called cayley topology universal group sequence referred direct product piecewise automatic groups ker extension intermediate growth function arbitrarily large suitable choices parameters particular growth arbitrarily close exponential since group torsion theorem measure bounded functions factor quotient zfc since growth bounded exp follows measure finite trivial poisson boundary eferences robert azencott espaces poisson des groupes localement compacts lecture notes mathematics vol york martin barlow richard bass chen moritz kassmann dirichlet forms symmetric jump processes trans amer math soc martin barlow alexander grigor yan takashi kumagai heat kernel upper bounds jump processes first exit time reine angew math laurent bartholdi growth grigorchuk torsion group internat math res notices lower bounds growth group acting binary rooted tree internat algebra comput growth periodic grigorchuk groups growth groups wreath products groups graphs random walks laurent bartholdi anna erschler growth permutational extensions invent math groups given intermediate word growth ann inst fourier grenoble boundary growth groups probab theory related fields laurent bartholdi rostislav grigorchuk zoran branch groups handbook algebra vol bendikov random walks groups discrete subordination math nachr bingham goldie teugels regular variation encyclopedia mathematics applications vol cambridge university press cambridge brieussel croissance certains groupes automorphismes arbre doctorat diderot paris carlen kusuoka stroock upper bounds symmetric markov transition functions ann inst probab statist chen takashi kumagai heat kernel estimates jump processes mixed types metric measure spaces probab theory related fields thierry coulhon ultracontractivity nash type inequalities funct anal thierry coulhon laurent pour les groupes les rev mat iberoamericana davies explicit constants gaussian upper bounds heat kernels amer math heat kernels spectral theory cambridge tracts mathematics vol cambridge university press cambridge pierre harpe topics geometric group theory chicago lectures mathematics university chicago press chicago thierry delmotte parabolic harnack inequality estimates markov chains graphs rev mat iberoamericana yves derriennic quelques applications ergodique conference random walks kleebach french dynkin maljutov random walk groups finite number generators dokl akad nauk sssr anna erschler boundary behavior groups subexponential growth ann math critical constants recurrence random walks ann inst fourier grenoble piecewise automatic groups duke math joshua frisch yair hartman omer tamuz pooya vahidi ferdowsi groups infinite conjugacy class property arxiv preprint rostislav grigorchuk burnside problem periodic groups funktsional anal prilozhen degrees growth finitely generated groups theory invariant means izv akad nauk sssr ser mat problems dynamics group actions rooted trees mat inst steklova sovremennye problemy matematiki milnor problem growth groups consequences frontiers complex dynamics alexander grigor yan analysis graphs lecture notes available https heat kernel upper bounds complete manifold rev mat iberoamericana vershik random walks discrete groups boundary entropy ann probab anna erschler tianyi zheng gregory lawler alan sokal bounds spectrum markov chains markov processes generalization cheeger inequality trans amer math soc leonov lower bound growth function periods grigorchuk groups mat stud lysionok system defining relations grigorchuk group mat zametki volodymyr nekrashevych groups mathematical surveys monographs vol american mathematical society providence schilling renming song zoran cek bernstein functions second gruyter studies mathematics vol walter gruyter berlin theory applications wolfgang woess random walks infinite graphs groups cambridge tracts mathematics vol cambridge university press cambridge nna rschler applications cole normale cnrs psl esearch niversity rue paris ianyi heng epartment athematics iego iman olla
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manuscript manuscript provided authors final version please see http robots adapt like animals may antoine jeff danesh robots leave controlled environments factories autonomously function complex natural respond inevitable fact become however animals quickly adapt wide variety injuries current robots think outside box find compensatory behavior damaged limited selfsensing abilities diagnose anticipated failure require contingency plan every type potential damage impracticality complex introduce intelligent trial error algorithm allows robots adapt damage less two minutes without requiring contingency plans deployment robot exploits novel algorithm create detailed map space behaviors map represents robot intuitions behaviors perform value robot damaged uses intuitions guide learning algorithm conducts intelligent experiments rapidly discover compensatory behavior works spite damage experiments reveal successful adaptations legged robot injured five different ways including damaged broken missing legs robotic arm joints broken different ways new technique enable robust effective autonomous robots suggests principles animals may use adapt injury robots transformed economics many industries notably power deliver tremendous benefits society search disaster health also invaluable tools scientific exploration whether distant deep major obstacle widespread adoption complex environments outside factories robots presently pale comparison natural animals ability invent compensatory behaviors injury fig current damage recovery robots typically involves two phases selecting best contingency robots expensive sensors expensive difficult design robot engineers foresee every possible situation approach often fails either diagnosis appropriate contingency plan injured animals respond differently learn trial error compensate damage learning limp minimizes pain similarly learning algorithms could allow robots creatively discover compensatory behaviors without limited designers assumptions damage may occur compensate damage type however learning algorithms impractical curse dimensionality fastest sorbonne upmc univ paris umr isir paris cnrs umr isir paris france inria france cnrs loria umr france lorraine loria umr france university wyoming laramie usa corresponding author cully clune tarapore mouret algorithms constrain search behaviors tuning parameters requiring minutes require human algorithms without limitations take several damage recovery would much practical effective robots adapted creatively quickly animals minutes without expensive sensors show rapid adaptation achieved guiding intelligent learning algorithm automatically generated map predicts performance thousands different behaviors supplementary video key insight whereas current learning algorithms either start knowledge search minimal knowledge human animals better understand space possible behaviors value previous enabling injured animals intelligently select tests validate invalidate whole families promising compensatory behaviors robots store knowledge previous experience form map space guided map damaged robot tries different types behaviors predicted perform well tests conducted updates estimates performance types behaviors process ends robot predicts effective behavior already discovered result robot quickly discovers way compensate damage fig without detailed mechanistic understanding cause occurs animals call approach intelligent trial error fig map created novel algorithm simulation robot either standard physics simulator automatically robot designers describe dimensions space possible behaviors performance measure instance walking gaits could described much leg involved gait behavioral measure speed performance measure grasping performance could amount surface contact demonstrated effective poses human hand captured behavioral fill map optimization algorithm simultaneously searches solution point behavioral space fig extended data fig step requires simulating millions behaviors needs performed per robot design deployment methods low confidence assigned predicted performance behaviors stored map tried reality fig extended data fig robot mission senses performance drop selects promising behavior behaviorperformance map tests measures performance robot subsequently updates prediction behavior nearby behaviors assigns high confidence predictions fig extended data fig continues process finds satisfactory compensatory behavior fig extended data fig ideas technically captured via gaussian process approximates performance function arxiv preprint image michael brashier image michael lloyd trial goal fast straight walking trial trial learning guided compensatory behavior figure intelligent trial error robots like animals quickly adapt recover damage animals find compensatory behavior injury without relying predefined compensatory behaviors learn avoid behaviors painful longer effective undamaged hexapod robot one type damage hexapod may cope broken leg damage occurs case making robot unable walk straight damage recovery via intelligent trial error begins robot tests different types behaviors automatically generated map space test robot updates predictions behaviors perform well despite damage way robot rapidly discovers effective compensatory behavior already acquired data bayesian optimization exploits model search maximum performance function methods robot selects behaviors test maximizing information acquisition function balances exploration selecting points whose performance uncertain exploitation selecting points whose performance expected high methods selected behavior tested physical robot actual performance recorded algorithm updates expected performance tested behavior lowers uncertainty updates propagated neighboring solutions behavioral space updating gaussian process methods updated performance confidence distributions affect behavior tested next loop repeats robot finds behavior whose measured performance greater best performance predicted behavior map methods first test algorithm hexapod robot needs walk fast possible fig robot motors onboard computer depth camera allows robot estimate walking speed supplementary methods gait parametrized parameters describe amplitude oscillation phase shift duty cycle joint supplementary methods behavior space dimension proportion time ith leg spends contact ground duty factor supplementary methods created map contains approximately different gaits supplementary video shows examples tested robot six different conditions undamaged fig four different structural failures fig temporary leg repair fig compare walking speed resultant gaits classic tripod supplementary methods damage conditions ran adaptation step times independently generated maps default duty factor behavioral description leading experiments total also ran adaptation step times independently generated behaviorperformance maps defined alternate behavioral cully clune tarapore mouret tion body orientation see supplementary methods two damage conditions fig leading additional experiments robot undamaged fig approach yields dynamic gaits faster classic reference gait fig median percentiles suggesting intelligent trial error good search algorithm automatically producing successful robot behaviors putting aside damage recovery damage scenarios reference gait longer effective four damage conditions fig intelligent trial error compensatory gaits achieve reasonably fast speed times efficient reference gait damage condition experiments demonstrate intelligent trial error allows robot initially learn fast gaits reliably recover physical damage additional experiments reveal capabilities substantially faster algorithms extended data fig intelligent trial error help another major challenge robotics adapting new environments extended data fig undamaged repaired robot fig intelligent trial error learns walking gait less seconds fig undamaged seconds physical trials repaired seconds trials four damage scenarios robot adapts approximately one minute seconds trials results qualitatively unchanged using different behavioral characterizations including randomly choosing descriptors among possibilities fig extended data fig additional experiments show reducing parameter space behavior space via map key component intelligent trial error standard bayesian optimization original parameter space find working controllers extended data fig investigated map updated robot loses leg fig initially map arxiv preprint map creation confidence level performance simulation undamaged adaptation step walking robot figure creating map user reduces search space behavior space defining dimensions along behaviors vary simulation space automatically searched find behavior point behavior space creating map performance potential location space hexapod robot experiments behavior space portion time leg contact ground confidence regarding accuracy predicted performance behavior map initially low tests physical robot conducted adaptation step damage robot selects promising behavior tests updates predicted performance behavior map sets high confidence performance prediction predicted performances nearby confidence likely similar tested behavior thus updated accordingly loop repeated tested behavior physical robot performs better best predicted performance map value decrease test extended data fig algorithm selects behavior test next balances choosing behavior highest predicted performance behaviors different tested far overall intelligent trial error approach presented rapidly locates types behaviors least affected damage find effective compensatory behavior predicts large areas high performance adaptation areas disappear behaviors work well damaged robot intelligent trial error quickly identifies one remaining behaviors fig extended data fig damage recovery approach applied robot robotic arm tested different damage conditions planar robotic arm fig extended data fig map behavioral dimensions position performance measure minimizing variance specified motor angles supplementary methods adaptation performance measured distance target like hexapod robot approach discovers compensatory behavior less minutes usually less seconds fewer trials fig extended data fig natural animals use specific algorithm present parallels intelligent trial error animal learning like animals robot predefined strategy cope every possible damage condition face new injury exploits intuitions body works experiment different behaviors find works best also like intelligent trial error allows quick identification working behaviors diverse tests instead trying behaviors random trying small modifications best behavior found far additionally bayesian optimization procedure followed robot appears similar technique employed humans optimize unknown strong evidence animal brains learn probability distributions combine prior knowledge act bayesian additional parallel intelligent trial error primes robot creativity motionless period cully clune tarapore mouret generated ideas tested process reminiscent finding animals start day new ideas may quickly disregard experimenting generally sleep improves creativity cognitive final parallel simulator gaussian process components intelligent trial error two forms predictive models known exist told shown combining pieces nature algorithm even differently assembled moves robots towards animals endowing ability rapidly adapt unforeseen circumstances supplementary information methods end document appended acknowledgments thanks luigi tedesco doncieux nicolas bredeche shimon whiteson roberto calandra jacques droulez pierre florian lesaint charles thurat serena ivaldi jingyu joost huizinga roby velez henok mengistu tim clune anh nguyen helpful feedback discussions thanks michael brashier photo dog work funded anr creadapt project european research council erc european union horizon research innovation programme grant agreement number direction armement dga scholarship author contributions designed study performed experiments discussed additional experiments analyzed results wrote paper arxiv preprint walking speed adaptation time number trials trials dim initial map map dim dim dim dim dim dim dim dim dim dim figure conditions tested physical hexapod robot undamaged robot one leg shortened half one leg unpowered one leg missing two legs missing temporary makeshift repair tip one leg performance adaptation box plots represent intelligent trial error central mark median edges box percentiles whiskers extend extreme data points considered outliers outliers plotted individually yellow stars represent performance handmade reference tripod gait supplementary methods conditions tested times independently created maps duty factor behavior description experiments per damage condition supplementary methods damage conditions also tested times independently created maps body orientation behavior description supplementary methods time number trials required adapt box plots represent intelligent trial error robotic arm experiment planar robot drop ball bin example conditions tested physical robotic arm one joint stuck degrees one joint permanent offset one broken one offset joint total conditions tested extended data fig time number trials required reach within bin center condition tested independently created maps dim proportion maximum expected performance figure example map map stores behaviors point behavior space dimension portion time leg contact ground behavioral space discretized five values dimension colored pixel represents behavior discovered map creation point behavior space matrices visualize behavioral space two dimensions according legend map created simulated robot bottom left open dynamics engine physics simulator http left matrix map produced map creation algorithm adaptation map updated tests conducted case damage condition robot missing one leg fig right matrix shows state map compensatory behavior discovered arrows white circles represent order behaviors tested physical robot red circle final discovered compensatory behavior amongst areas behaviors found damaged robot first two columns third dimension columns represent behaviors least use leg leg missing cully clune tarapore mouret arxiv preprint author information correspondence requests materials addressed email bellingham rajan robotics remote hostile science issn doi yoerger underwater robotics springer handbook robotics springer broadbent stafford macdonald acceptance healthcare robots older population review future directions international journal social robotics sanderson mars rover spirit nature carlson murphy ugvs physically fail field ieee transactions robotics blanke diagnosis control springer siciliano khatib springer handbook robotics springer murphy trial fire rescue robots robotics automation magazine ieee nagatani kiribayashi okada otake yoshida tadokoro nishimura yoshida koyanagi fukushima kawatsuma emergency response nuclear accident fukushima daiichi nuclear power plants using mobile rescue robots journal field robotics thrun montemerlo dahlkamp stavens aron diebel fong gale halpenny hoffmann stanley robot darpa grand challenge journal field robotics verma gordon simmons thrun fault diagnosis robotics automation magazine bongard zykov lipson resilient machines continuous science fenton mcginnity maguire fault diagnosis electronic systems using intelligent techniques review ieee transactions systems man cybernetics part applications reviews kluger lovell apollo mariner books isbn jarvis worley hogy hill haussler reiser kinematic kinetic analysis dogs trotting amputation thoracic limb american journal veterinary research fuchs goldner nolte schilling ground reaction force adaptations tripedal locomotion veterinary journal issn doi kober bagnell peters reinforcement learning robotics survey international journal robotics research doi argall chernova veloso browning survey robot learning demonstration robotics autonomous systems thelen motor development new synthesis american psychologist santello postural hand synergies tool use journal neuroscience rasmussen williams gaussian processes machine learning mit press isbn mockus bayesian approach global optimization theory applications kluwer academic borji itti bayesian optimization explains human active search advances neural information processing systems nips holekamp innovative problem solving wild spotted hyenas proceedings royal society biological sciences pouget beck latham probabilistic brains knowns unknowns nature neuroscience wolpert bayesian integration sensorimotor learning nature mitra pytte tchernichovski sleep affects developmental learning bird song nature wagner gais haider verleger born sleep inspires insight nature ito control mental activities internal models cerebellum nature reviews neuroscience methods notations parameters controller vector location discrete behavioral space type behavior vector location discrete behavioral space tested physical robot vector map stores performance associative table map stores controllers associative table max performance yet encountered scalar controller currently stored vector previously tested behavioral descriptors time vector vectors performance reality candidate solutions tested robot time vector performance map candidate solutions tested robot time vector performance function unknown algorithm function observation noise parameter scalar kernel function see section kernel function function kernel matrix matrix kernel vector vector predicted performance mean gaussian process function standard deviation gaussian process function intelligent trial error algorithm intelligent trial error algorithm consists two major steps extended data fig map creation step adaptation step focus damage recovery intelligent trial error search type required adaptation learning initial gait undamaged robot adapting new environments map creation step accomplished via new algorithm introduced paper called archive phenotypic elites explained next section adaptation step accomplished via second new algorithm introduced paper called mapbased bayesian optimization algorithm explained adaptation step section map creation via algorithm map created new algorithm introduce paper called archive phenotypic elites algorithm searches solution point userdefined space user chooses dimensions space interested seeing variation example designing robots user may interested seeing highestperforming solution point space one axis weight robot axis height robot alternatively user may wish see weight cost see solutions throughout space weight cost height dimension vary could chosen user limit number dimensions chosen although becomes computationally expensive cully clune tarapore mouret fill map store number dimensions increases also becomes difficult visualize results refer space behavior space usually dimensions variation measure behavioral characteristics note behavioral space refer aspects solution example dimensions variation physical properties robot height weight behavior descriptors parameters controller one possible location behavioral space creating behaviorperformance map straightforward one simply needs simulate solution location behavior space record performance however known priori produce end specific location behavior space parameter space higher dimension behavioral space example many different robot designs specific weight height cost unknown make description produce robot specific weight height cost beneficial efficiently search solution point behavioral space efficient random sampling search space solutions often similar many ways randomly altering solution one type produce highperforming solution different type see extended data fig supplementary experiment reason searching solutions types simultaneously much quicker separately searching type example generate lightweight robot design tends effective efficient modify existing design light robot rather randomly generate new designs scratch launch separate search process new type design begins generating set random candidate solutions evaluates performance solution records solution located behavior space dimensions behavior space height weight records height weight robot addition performance solution performance better current solution location map added map replacing solution location words kept best type solution type defined location behavior space thus one solution kept location behavior space keeping could beneficial computational reasons keep one solution present map location newly generated candidate solution added location initialization step finished enters loop similar stochastic optimization algorithms evolutionary solutions map form population improved random variation selection generation algorithm picks solution random via uniform distribution meaning solution equal chance chosen copy selected solution randomly mutated change way performance evaluated location behavioral space determined kept outperforms current occupant point behavior space note mutated solutions may end different behavior space locations parents process repeated stopping criterion met fixed amount time expired experiments stopped run million iterations stochastic arxiv preprint online adaptation robot evaluation simulation random replace selection best far repertoire behavior type expected performance uncertainty performance threshold uncertainty actual performance unknown behavioral descriptor performance lowered performance threshold updated expectations behavioral descriptor stop solution performance threshold behavioral descriptor evaluation damaged robot evaluation damaged robot current best solution behavior type previously encountered solutions stored performance random parameter variation performance performance map generation behavioral descriptor map extended data figure overview intelligent trial error algorithm map creation initialized random controllers behavioral map stores controller found far behavior type improved repeating process depicted newly generated controllers rarely good enough added map million evaluations step occurs simulation computationally expensive needs performed per robot robot design prior deployment experiments creating one map involved million iterations lasted roughly two weeks one computer supplementary methods section running time adaptation behavior behaviorperformance map expected performance based performance simulation dark green line estimate uncertainty regarding predicted performance light green band actual performance robot black dashed line unknown algorithm behavior selected try damaged robot selection made balancing behaviors expected perform behaviors whose performance uncertain methods section acquisition function points initially equal maximal uncertainty first point chosen highest expected performance behavior tested physical robot performance predicted behavior set actual performance uncertainty regarding prediction lowered predictions uncertainties nearby controllers also updated according gaussian process model see methods section kernel function results seen process repeated performance damaged robot greater maximum expected performance behavior performance threshold orange dashed line lowers maximum expected performance highest point dark green line lowered occurs physical tests robot underperform expectations occurred search process resultant map different terms number locations behavioral space candidate found terms performance candidate location algorithm available supplementary figure details experiments available mouret clune adaptation step via bayesian optimization algorithm adaptation step accomplished via bayesian optimization algorithm seeded map call approach bayesian optimization algorithm mboa bayesian optimization optimization algorithm tailored expensive objective functions cost functions optimization algorithm bayesian optimization searches maximum unknown objective function samples obtained measuring performance robot like optimization algorithms bayesian optimization creates model objective function regression method uses model select next point acquire updates model etc called bayesian general algorithm chooses next point computing posterior distribution objective function using likelihood data already acquired prior type function use gaussian process regression find common choice bayesian gaussian processes particularly interesting regression model cost function also cully clune tarapore mouret tainty associated prediction cost function usually unknown gaussian process defines probability distribution possible values point probability distributions gaussian therefore defined mean standard deviation however different therefore define probability distribution functions denotes standard normal distribution estimate need fit gaussian process data assume observation sample normal distribution data set made several observations vector sample multivariate normal distribution defined mean vector covariance matrix gaussian process therefore generalization normal distribution number observations covariance matrix relates one observation another two observations correspond nearby values likely correlated prior assumption based fact functions tend smooth injected algorithm via prior likelihood functions two observations correspond distant values influence distributions correlated put differently covariance matrix represents distant samples almost uncorrelated nearby samples strongly correlated covariance matrix defined via kernel function called usually based euclidean distance see kernel function given set observations sampling noise parameter gaussian arxiv preprint variants using performance variants using trials performance intelligent trial error variant variant variant variant variant variants int lli int lli variant variant variant variant variant map creation none none intelligent trial error variant performance trials variants priors performance yes none none none none none search algorithm bayesian optimization random search bayesian optimization policy gradient bayesian optimization policy gradient equivalent approach lizotte kohl extended data figure contribution subcomponent intelligent trial error algorithm adaptation progress versus number robot trials walking speed achieved intelligent trial error several knockout variants missing one algorithm key components variants correspond learning algorithms policy gradient kohl bayesian optimization lizotte tesch calandra bold lines represent medians colored areas extend percentiles adaptation performance trials shown speed compensatory behavior discovered algorithm evaluations robot respectively panels data pooled across six damage conditions removal legs turn see supplementary experiment methods analysis cully clune tarapore mouret arxiv preprint damaged robot slope angle degree adaptation time iterations performance undamaged robot adaptation time iterations performance slope angle degree extended data figure intelligent trial error algorithm robust environmental changes plot shows performance required adaptation time intelligent trial error robot must adapt walk terrains different slopes adaptation performance undamaged robot slope angles physical trials intelligent trial error algorithm pink shaded region finds fast gaits outperform reference gait black dotted line adaptation performance damaged robot robot damaged six legs removed six different damage scenarios data pooled six damage conditions median compensatory behavior found via intelligent trial error outperforms median reference controller slope angles middle black lines represent medians colored areas extend percentiles black dashed line performance classic tripod gait reference reference gait tried six damage conditions median black line percentiles black colored area shown see supplementary experiment methods analysis process computed update equation mean function equation performance according simulation performance previous observations also according simulation replacing means gaussian process models difference actual performance performance map term prediction map therefore starts prediction behaviorperformance map corrects gaussian process algorithm available supplementary figure implementation bayesian optimization uses gaussian process model search maximum objective function unknown selects next test selecting maximum acquisition function balances exploration improving model less explored parts search space exploitation favoring parts models predicts promising use upper confidence bound acquisition function see information acquisition function section observation made algorithm updates gaussian process take new data account classic bayesian optimization gaussian process initialized constant mean assumed points search space equally likely good model progressively refined observation key concept bayesian optimization algorithm use output prior bayesian optimization algorithm thanks simulations expect behaviors perform better others robot incorporate idea bayesian optimization models difference prediction map actual performance real robot instead directly modeling objective function idea incorporated gaussian process modifying kernel function kernel function covariance function gaussian process defines influence controller performance physical robot performance confidence estimations controllers behaviorperformance map nearby behavior space tested controller extended data fig squared exponential covariance function kernel common kernels gaussian kernels variants bell curve chose kernel general includes squared exponential function special case allows control distance effects become nearly zero function parameter extended data fig also rate distance effects decrease function parameter kernel function computed exp euclidean distance behavior space cully clune tarapore mouret arxiv preprint trials performance duty factor erleg erleg erleg oll tch tiv tio via ela tiv lac ien fac tio reference gait trials performance duty factor erle erleg oll tch tiv tio via ela tiv lac tio fac reference gait extended data figure intelligent trial error algorithm largely robust alternate choices behavior descriptors speed compensatory behavior discovered intelligent trial error various choices behavior descriptors performance plotted evaluations panels respectively experiments performed simulated damaged hexapod damaged robot six legs removed six different damage scenarios data pooled across six damage conditions described supplementary experiment evaluated behavior descriptors characterize following time leg contact ground duty factor orientation robot frame orientation iii instantaneous velocity robot displacement energy expended robot walking energy total energy relative deviation straight line deviation ground reaction force leg grf total grf relative vii angle leg touches ground angle pitch angle roll angle yaw viii random selection without replacement subcomponents available behavior descriptors random reference gait yellow compensatory gaits found default duty factor behavior descriptor green bold lines represent medians colored areas extend percentiles data treatments including duty factor treatment black circles represent median colored area extends percentiles data colored circles outliers see supplementary experiment methods analysis cully clune tarapore mouret arxiv preprint map dim dim next tested behavior currently tested behavior tested behavior dim proportion maximum expected performance initial map dim dim dim dim dim dim dim dim dim extended data figure behavior performance map explored discover compensatory behavior normalized iteration highlight range remaining performance predictions colors represent performance prediction point map relative highest performing prediction map step process black circle indicates next behavior tested physical robot red circle indicates behavior tested note performance predictions surrounding changed versus previous panel arrows reveal order points explored red circle last map final selected compensatory behavior scenario robot loses leg number six dimensional space visualized according inset legend cully clune tarapore mouret arxiv preprint map dim dim next tested behavior currently tested behavior tested behavior dim proportion maximum expected performance initial map dim dim dim dim dim dim dim dim dim extended data figure behavior performance map explored discover compensatory behavior highlight performance predictions decrease discovered predictions simulated undamaged robot work well damaged robot colors represent performance prediction point map relative highest performing prediction first map black circle indicates next behavior tested physical robot red circle indicates behavior tested note performance predictions surrounding changed versus previous panel arrows reveal order points explored red circle last map sequence final selected compensatory behavior scenario robot loses leg number six dimensional space visualized according inset legend data visualized figure identical previous figure difference simply whether data renormalized new map sequence cully clune tarapore mouret arxiv preprint extended data figure intelligent trial error works completely different type robot supplementary data robotic arm experiment robotic arm experimental setup tested damage conditions example behavior performance maps colormaps behaviors overlaid arm configurations obtained left typical map produced example behaviors behavior described angle joints color point function performance defined low variance joint angles zigzag arm lower performing straighter arm reaches point right neighboring points map tend similar behaviors thanks performance function would penalize jagged ways reaching points neighbors similar behaviors justifies updating predictions performance nearby behaviors testing single behavior real damaged robot performance trial number intelligent trial error traditional bayesian optimization experiment conducted physical robot independent replications damage conditions performance pooled experiments success damage condition shown success rate replications damage condition defined percentage replicates robot reaches within bin center trials required adapt shown number iterations required reach within basket center accuracy physical trials performance physical trials damage condition stopping criterion disabled see supplementary experiment methods analysis cully clune tarapore mouret arxiv preprint procedure ntelligent rial rror lgorithm mission reate behavior performance map via lites algorithm simulation mission significant performance fall daptation tep via algorithm procedure lites lgorithm iter iter else simu simu return map creation empty map empty grid repeat iterations choose million iterations first controllers generated randomly next controllers generated using map randomly select controller map create randomly modified copy simulate controller record behavioral descriptor record performance cell empty better current stored performance store performance map according behavioral descriptor associate controller behavioral descriptor procedure based ayesian ptimization lgorithm map initialisation definition gaussian process initialize mean prior map initialize variance prior common case max max iteration loop arg maxx select next test argmax acquisition function performance evaluation physical robot update gaussian process update mean update variance compute observations correlation matrix compute observation correlation vector extended data figure intelligent trial error algorithm algorithm bayesian optimization algorithm notations described beginning methods section cully clune tarapore mouret arxiv preprint model update step directly depends one critical parameters intelligent trial error algorithm selected value extensive experiments simulation extended data fig section information acquisition function information acquisition function selects next solution evaluated physical robot selection made finding solution maximizes acquisition function step another optimization problem require testing controller simulation reality general optimization problem derive exact equation find solution specific behavior space example problem paper though discretized search space map small enough exhaustively compute acquisition value solution map choose maximum value several different acquisition functions exist probability improvement expected improvement upper confidence bound ucb chose ucb provided best results several previous equation ucb arg max parameter tunes tradeoff exploration exploitation acquisition function handles adaptation step ucb function emphasis exploitation exploration explicit easy adjust ucb function seen maximum value argmax across solutions weighted sum expected performance mean gaussian uncertainty standard deviation gaussian solution sum weighted factor low algorithm choose solutions expected highperforming conversely high algorithm focus search unexplored areas search space may solutions factor enables fine adjustments algorithm adaptation step describe chose value supplementary methods section code availability source code experiments paper available following url http implementation bayesian optimization algorithm freely available http hexapod experiment physical robot robot robot degrees freedom dofs per leg dof actuated positioncontrolled servos dynamixel actuators manufactured robotis first servo controls horizontal orientation leg two others control elevation camera xtion asus fixed top robot data used estimate forward displacement robot via slam robot operating system ros simulator simulator dynamic physics simulation undamaged robot flat ground fig weighted segment leg body real robot http http cully clune tarapore mouret used masses simulations simulator based open dynamics engine ode http parametrized controller angular position dof governed periodic function parametrized amplitude phase duty cycle duty cycle proportion one period joint higher position function defined square signal frequency amplitude duty cycle signal smoothed via gaussian filter order remove sharp transitions shifted according phase angular positions sent servos every order keep tibia leg vertical control signal third servo opposite second one consequently angles sent ith leg dof dof dof controller makes robot equivalent dof system even though motors controlled parameters leg therefore controller fully described parameters parameter one possible values different values parameters produce numerous different gaits purely quadruped gaits classic tripod gaits controller designed simple enough show performance algorithm intuitive setup nevertheless algorithm work type controller including central pattern evolved neural reference controller reference controller classic tripod involves two tripods legs legs fig controller designed always keep robot balanced least one tripods walking gait achieved lifting one tripod tripod pushes robot forward shifting backward lifted tripod placed forward order repeat cycle tripod gait static fast similar insect table shows parameters reference controller amplitude orientation parameters set produce fastest possible gait amplitude elevation parameters set small value keep gait stable phase elevation parameters define two tripods legs legs achieve cyclic motion leg phase orientation values chosen subtracting phase elevation values plus shift legs left side robot duty cycle parameters set motors spend proportion time two limit angles actual speed reference controller important comparisons made paper simply intended reference show performance classic gaits tend fail damage occurs random variation controller parameters parameter controller chance changed value set possible values new value chosen randomly uniform distribution possible values main behavioral descriptor duty factor default behavioral descriptor vector corresponds proportion time leg contact ground also called duty factor controller simulated algorithm records time step every whether leg arxiv preprint leg number first joint two last joints extended data table parameters reference controller contact ground contact contact result boolean time series ith leg behavioral descriptor computed average time series numtimesteps numtimesteps generation map behaviors stored maps cells discretizing dimension behavioral descriptor space five values adaptation phase behavioral descriptors used actual values thus discretized alternative behavioral descriptor orientation alternative behavioral descriptor tested physical robot investigated many descriptors simulation supplementary experiment characterizes changes angular position robot walking measured proportion intervals pitch roll yaw angles robot frame positive three dimensions negative three additional dimensions denote pitch roll yaw angles respectively robot torso hence end interval denotes number intervals seconds simulated movement unit step function returns argument exceeds returns otherwise discount insignificant motion around rad orientation angles defined positive exceed rad similarly orientation angles defined negative less rad performance function experiments mission robot forward fast possible performance controller set parameters section parametrized controller defined far robot moves direction seconds map creation step performance obtained thanks simulation robot odometry results reported physical robot adaptation step measured embedded simultaneous location mapping slam accuracy algorithm evaluated comparing measurements ones made hand different walking gaits experiments revealed median measurement produced odometry algorithm reasonably accurate lower handmade measurement extended data fig damage robot may make flip cases visual odometry algorithm returns pathological cully clune tarapore mouret traveled measurements either several meters backward forward remove errors set measurements less zero greater two meters zero result adjustment algorithm appropriately considers behaviors additionally slam algorithm sometimes reports substantially inaccurate low values outliers supplementary fig cases adaptation step algorithm assume behavior lowperforming select another working behavior thus overall algorithm substantially impacted infrequent performance stopping criterion addition guiding learning process promising area search space estimated performance solution map also informs algorithm maximum performance expected physical robot example controller map expected perform faster real robot unlikely faster solution exists information used algorithm decide worth continuing search better controller algorithm already discovered controller performs nearly well highest value predicted model stop search formally stopping criterion max max location discrete behavioral space type behavior predicted performance type behavior thus one tested solutions performance higher maximum expected performance behavior map algorithm terminates point solution found far compensatory behavior algorithm selects alternative way algorithm halt tests physical robot occur without triggering stopping criterion described equation event occurred experiments performed physical robot described main text case selected solution encountered search stopping criterion strictly necessary algorithm guaranteed stop worst case every behavior map tested allows practical limit number trials performed physical robot initiating adaptation step adaptation step triggered performance drops certain amount simplest way choose threshold let user specify automating selection value impact triggering algorithm prematurely interesting question future research area main parameters parameters controller parameter values controller increments size behavioral space possible behavioral descriptors arxiv preprint iterations million main parameters robotic arm experiment physical robot physical robot planar robotic arm degrees freedom extended data fig gripper robot release ball bin variant classic pick place task industrial robotics assess position gripper red cap placed top gripper tracked video camera visual tracking achieved cmvision ros package tracks colored blobs http eight joints robot actuated servos manufactured dynamixel maximize reliability arm type servo joints servos used near base robot lighter ones used end arm first joint fixed base moved two servos mounted parallel second joint moved servo subsequent servos single remaining servos robot joints limited motion range simulator generation map made simulated robot way hexapod experiment consistency simulated hexapod experiments used dynamic opposed kinematic version simulator based ode library joint configuration resulted arm colliding added map parametrized controller controller defines target position joint controller thus parametrized eight continuous values describing angle joint mapped total motion range joint joints activated simultaneously driven target position internal pid controllers chose simple control strategy make experiments easy reproduce highlight contribution intelligent trial error damage recovery advanced control strategies instance visual would realistic industrial environment would made hard analyze experimental results intelligent trial error controller would compensate damage time randomly varying controller parameters parameter controller section parametrized controller chance changed value new value chosen polynomial distribution described deb behavioral descriptor important aspect robot behavior task final position gripper use behavioral descriptor simu denotes position gripper joint reached target position cully clune tarapore mouret size working area robot rectangle measuring map rectangle discretized grid composed square cells robot long performance function contrary hexapod experiment robotic arm experiment performance function creation step adaptation step different demonstrate two different create map would work arbitrary locations target bin map generation step accomplished via algorithm performance function captures idea joints contribute equally movement specifically defined minimizing variance joint angles performance simu angular position joint radians mean joint angles performance tion depend target map therefore generic contains controller point robot working space adaptation step accomplished via algorithm map generic many tasks used particular task adaption step different performance measure step creates map problem predicted performance measure euclidean distance target closer better specifically behavior descriptor map performance position target bin note variance joint angles used create behaviorperformance map ignored adaptation step performance controller physical robot minimizing euclidean distance gripper measured external camera target bin performance position physical gripper joints reached final position position bin controller evaluated position simulation controller reached gripper reaches position outside working area camera see marker rare cases set performance corresponding controller low value control experiments traditional bayesian optimization physical robot see supplementary experiment frequent adaptation especially given initialize process purely random controllers random joint angles single unlikely break robot hundreds wear gearboxes servo continues apply force period time determines move minimize costs ran independent runs algorithm scenarios replicates first tested behavior simulation taking damage account check detected performance behavior set low value much less likely intelligent trial error map contains controllers undamaged simulated arxiv preprint robot consequence intelligent trial error experiments simulate controllers testing physical robot stopping criterion robot task release ball bin adaptation step stopped gripper bin bin circular diameter stopped adaptation step red cap within center bin main parameters robotic arm experiment parameters controller controller parameter values continuous dimensions behavioral space simulated evaluations create map million main parameters robotic arm experiment selection parameters data reported section comes experiments simulated hexapod robot unless otherwise stated selecting value counted number behaviors map would influenced single test real hexapod robot considered behavior influenced predicted performance affected magnitude update tested behavior update process affect neighbor map affects behaviors affects additional values shown extended data fig previous paragraph describes tests conducted determine number behaviors map affected different values experiments tell different values affect performance algorithm overall assess repeated experiments main paper set possible values simulation simulated damaged robot including testing separate damage scenarios robot loses different leg independently generated replicates default map algorithm stopped adaptation iterations passed without success according stopping criteria described main text section stopping criterion results reveal median performance decreases modestly significantly value increases changing decreases median value via matlab wilcoxon ranksum test extended data fig variance performance especially extreme low end distribution performance values constant range explored values around minimum performance extended data fig dotted red line higher minimum performance extreme values larger effect changing amount time required find compensatory behavior decreases value increases extended data fig value lower algorithm rarely converges less allotted iterations occurs many tests required cover promising areas search space know behavior exists hand high value algorithm updates cully clune tarapore mouret predictions entire search space observations fast strategy risks missing promising areas search space light data chose default value hexapod experiments represents good tradeoff high minimum performance low number physical tests robot value robotic arm experiment chosen method selection value hexapod robot experiments chose relatively low value emphasizes exploitation exploration chose value exploration search space already largely performed map creation step map suggests areas space thus tested areas space likely unprofitable thus avoided robotic arm experiments chose emphasizes exploration experimentally leads better results running time computing hardware computation physical robots simulation conducted hyperthreaded computer intel xeon ram computational power mainly required behaviorperformance map creation step creating one map hexapod experiment took weeks taking advantage fact map creation easily parallelized across multiple cores map creation needs performed per robot robot design happen robot deployed robot onboard computer need powerful enough create map hexapod robot experiment expensive part adaptation simultaneous localization mapping slam measures distance traveled physical hexapod robot slow processes millions points per second run less powerful computers lowers accuracy fewer frames per second processed computers become faster possible run slam algorithms onboard computers robots rest adaptation step needs much less computational power easily run onboard computer smartphone takes approximately arithmetic operations two evaluations physical robot requires less second two current smartphones measuring long adaptation takes hexapod robot reported time adapt includes time required computer select test time conduct test physical robot overall evaluating controller physical hexapod robot takes seconds median seconds percentiles seconds second initialize robot seconds robot walk second allow robot stabilize taking final measurement seconds run slam algorithm identifying first controller test takes seconds time select next controller test increases depending number previous experiments size kernel matrix matrix see methods extended data fig involved many arithmetic operations grows one row one column per test conducted example selecting second test takes seconds selection takes seconds arxiv preprint covariance kernel output distance measured performance proportion affected solutions percent parameter performance adaptation time iterations parameter parameter real performance extended data figure effect changing algorithm parameters shape kernel function different values parameter performance required adaptation time obtained different values value algorithm executed simulation independently generated maps different damage conditions case one leg missing number controllers map affected new observation according different values parameter precision odometry value distances traveled physical robot measured manually real performance compared measurements automatically provided simultaneous location mapping slam algorithm measured performance dashed black line indicates hypothetical case slam measurements thus manual measurements middle black lines represent medians borders shaded areas show percentiles dotted lines minimum maximum values gray bars show value chosen hexapod experiments main text cully clune tarapore mouret arxiv preprint supplementary experiments additional conditions robotic arm methods investigated damage conditions physical robot addition described main text fig used setup described main text see main text section extended data fig shows scenarios damage scenarios replicated experiments physical robot independently generated maps runs total also replicated control experiments consist traditional bayesian optimization directly original parameter space without maps times damage conditions runs total experimental control treatments experiment involved evaluations physical robot first trial counted many cases evaluations required reach target report number trials required reach goal results running algorithm million evaluations generated maps contain behaviors behaviors extended data fig generated maps regions different performance values behaviors arranged concentric shapes resembling cardioids inverted curves cover places robot reach extended data fig black line drawn shown map corresponds positions degrees freedom set angle theoretically highest achievable performance lowest possible variance servo angles performance behaviors tends decrease optimal line adaptation results extended data fig show intelligent trial error algorithm manages reach goal less center bin runs tested scenarios save two scenarios two scenarios algorithm still reaches target time respectively damage conditions intelligent trial error algorithm reaches target significantly often bayesian optimization algorithm specifically median number iterations reach target extended data fig iterations seconds scenarios except iterations required respectively robot able reach target recorded number iterations set explains median number iterations bayesian optimization algorithm equal damage conditions damage conditions except one scenario intelligent trial error algorithm used fewer trials reach target traditional bayesian optimization algorithm robot allowed continue experiment reaching radius tolerance total iterations extended data fig reaches accuracy around damage conditions except two difficult ones scenarios level accuracy never achieved classic bayesian optimization algorithm whose lowest median accuracy scenarios appear challenge intelligent trial error algorithm cases success rate improved though substantially case median accuracy actually lower results stem fact difference successful behaviors large solutions scenarios lie outside map illustrates limit proposed approach map contain behavior able cope cully clune tarapore mouret damage robot able adapt limit mainly comes behavioral descriptor choice chose simplicity capture important dimensions variation robot sophisticated descriptors likely allow algorithm cope situations hand experiment shows simple behavioral descriptor using final position approach able deal large variety different target positions significantly faster traditional bayesian optimization approach extended data fig maximum time step current state art technique direct policy search supplementary experiments contribution subcomponent intelligent trial error algorithm methods intelligent trial error algorithm relies three main concepts creation map simulation via algorithm searching map bayesian optimization algorithm find behaviors perform well physical robot initializing bayesian optimization search performance predictions obtained via algorithm note second step could performed without third step searching map bayesian optimization initial priors uniformly set value investigated contribution subcomponents testing five variants algorithm deactivated one three subcomponents replaced alternative algorithm literature tested variants hexapod robot variants follows variant dimensions random search evaluates benefit searching map via bayesian optimization searching map random search instead iteration behavior randomly selected map tested robot best one kept variant dimensions bayesian optimization use priors evaluates contribution initializing gaussian process performance predictions map variant gaussian process initialized constant mean average performance map location behavior space constant variance average variance map performance customary first trials bayesian optimization process selected randomly instead letting algorithm choose points known improve variant dimensions policy gradient evaluates benefit bayesian optimization compared classic local search obvious way use priors policy gradient algorithms variant bayesian optimization original parameter space dimensions evaluates contribution using map behavioral space variant searches directly original parameter space instead reducing space lowerdimensional behavior space thus variant map behaviors produced ahead time algorithm searches directly original space variant corresponds one best algorithms known learn locomotion variant gaussian process initialized constant mean set arxiv preprint zero constant variance described five first trials selected pure random search prime bayesian optimization variant policy gradient original parameter space dimensions stochastic gradient descent original parameter approach classic reinforcement learning algorithm baseline many necessary compare variants simulation physical robot would required months experiments would repeatedly worn broken robot modified simulator main experiments section simulator emulate different possible damage conditions involved removing different leg variants creates map variants used maps main experiments eight independently generated maps generated simulation undamaged robot cases launched ten replicates variant eight maps six damage conditions therefore replicates variants variants replicated experiment times six damage conditions also led replicates per variant simulated experiments roughly simulate distribution noisy odometry measurements real robot simulated performance values randomly perturbed multiplicative gaussian noise centered standard deviation analyze fastest walking speed achieved variant two different numbers trials first case trials maximum number iterations used intelligent trial error algorithm second case trials approximately number trials used previous results trials robot intelligent trial error significantly outperforms variants extended data fig intelligent trial error performance demonstrating three main components algorithm needed quickly find behaviors among investigated variants random search map performs best variant followed bayesian optimization map variant policy gradient map variant variants search directly parameter space find working behavior variant bayesian optimization variant policy gradient two reasons random search performs better one might expect first map contains solutions result intense search mapelites algorithm million evaluations simulation map thus already contains gaits nearly every possible type therefore variant testing random controllers randomly selecting solutions second bayesian optimization policy gradient designed low number trials without priors performance predictions introduced intelligent trial error algorithm bayesian optimization process needs learn overall shape search space model gaussian process trials low number effectively sample six dimensions uniform sampling two possible values dimension trials needed five possible values samples needed consequence low number trials gaussian process models performance function informed enough effectively guide search policy gradient algorithm gradient estimated empirically measuring partial derivative cully clune tarapore mouret performance function dimension policy gradient algorithm performs trials iteration consequently trials allowed iterates addition policy gradient local optimization algorithm highly depends starting point chosen randomly illustrated high variability performance achieved variant extended data fig issues faced bayesian optimization policy gradient exacerbated algorithms search directly original parameter space instead lowerdimensional behavior space map mentioned previously working controller found two variants directly searching space overall analysis trials shows critical component intelligent trial error algorithm algorithm reduces search space produces map behaviors space comparing variants searching map space variants search original space motor parameters bayesian optimization critically improves search initialized performance obtained simulation map creation step initialization without initialization check whether variants might perform better allowed number evaluations typically given previous continued experiments trials robot conducted extended data fig although results variants improved intelligent trial error still outperforms intelligent trial error random search bayesian optimization policy search results consistent previously published optimize dimensions hundred trials nevertheless used run algorithms original dimensions evaluations bayesian optimization policy gradient perform much worse bayesian optimization policy gradient results shows powerful method reduce dimensionality search space learning algorithms addition providing helpful priors search space speed bayesian optimization overall additional experiments demonstrate three main components intelligent trial error algorithm substantially improves performance results also indicate intelligent trial error significantly outperforms previous algorithms damage gait therefore considered state art supplementary experiments robustness environmental changes methods map creation algorithm uses undamaged robot flat terrain main experiments show algorithm provides useful priors damage recovery flat terrain supplementary experiments evaluated simulation map created flat terrain also provides useful starting point discovering gaits sloped terrains first evaluated effect slopes undamaged robots extended data fig launched replicates eight maps increment total experiments arxiv preprint supplementary experiments roughly simulate distribution noisy odometry measurements real robot perturbed performance values multiplicative gaussian noise centered standard deviation results results show slope negative descending intelligent trial error approach finds fast gaits fewer trials reference classic tripod gait section falls slopes degrees slope positive ascent intelligent trial error finds slower behaviors expected even gait learned intelligent trial error outperforms reference gait flat ground overall every slope angle controller found intelligent trial error faster reference controller evaluated damage recovery performance slopes setup experiments damage conditions launched replicates damage condition independently generated maps increment degrees therefore replicates increment total experiments intelligent trial error critically affected variations slope extended data fig slopes damage conditions intelligent trial error finds fast gaits less tests robot despite slope expected finds faster gaits negative slopes descent slower gaits positive slopes ascent slopes algorithm performs worse requires trials results likely caused constraints placed controller limited sensors robot rather inabilities algorithm specifically controller kept simple make science clearer intuitive reproducible constraints course prevent performing complex behaviors necessary deal highly sloped terrain example constraints prevent robot keeping legs vertical sloped ground would substantially reduce slippage nevertheless median intelligent trial error compensatory gait still outperforms median performance reference gait slope angles supplementary experiments comparison random sampling methods algorithm stochastic search algorithm attempts fill discretized map cully clune tarapore mouret performing solution point map explained main text point map represents different type behavior defined behavioral dimension map generates new candidate points randomly selecting location map changing parameters controller stored saving controller appropriate map location better current occupant location intuitively generating new candidate solutions best solutions found far better generating multitude controllers randomly keeping best one found location map section report experiments confirm intuition understand advantages random sampling compared two algorithms generating data simulated hexapod experiments virtual robot environment controller performance function behavioral descriptors main experiments see methods analyzed number cells solution found indication diversity behavior types algorithms generate average performance behaviors map maximum performance discovered replicated experiment times included million evaluations simulated robot results results show algorithm outperforms random sampling measures extended data fig million evaluations cells median percentiles filled percent map whereas random sampling filled approximately cells percent map extended data fig difference two algorithms large appears early extended data fig million evaluations mapelites filled cells map whereas random sampling filled cells map solutions discovered numerous also outperform found random sampling extended data fig million evaluations average performance filled cells whereas random sampling similar performance obtained reference controller damaged robot fig two results demonstrate much better algorithm random sampling find map diverse elite performers search space addition better optimization algorithm measured performance best single solution produced performance best solution map million evaluations compared random sampling arxiv preprint number behaviors number evaluations mean performance maximum performance number evaluations number evaluations random sampling extended data figure comparing random sampling generating maps number points map behavior discovered mean performance behaviors map maximum performance behaviors map figures middle lines represent medians independently generated maps shaded regions extend percentiles even variance distribution small difficult see see supplementary experiment methods analysis supplementary experiments alternative behavioral descriptors methods create map one define dimensions behavioral space behavioral descriptors main experiments show using predefined behavioral descriptor proportion time leg hexapod robot contact ground duty factor creates map provides useful priors damage recovery section describes tested simulation performance affected alternative behavioral descriptors including descriptors different number dimensions also evaluated performance affected behavioral descriptors randomly selected large list potential descriptors test simulates algorithm performance behavioral descriptors chosen without insight problem domain behavioral descriptors tested follows duty factor descriptor default one main experiment corresponds proportion time leg contact ground numtimesteps numtimesteps denotes boolean value whether leg contact ground time contact contact orientation behavioral descriptor characterizes changes angular position robot walking measured proportion intervals pitch roll yaw angles robot frame positive three dimensions negative three additional dimensions cully clune tarapore mouret denote pitch roll yaw angles respectively robot torso hence end interval denotes number intervals seconds simulated movement unit step function returns argument exceeds returns otherwise discount insignificant motion around rad orientation angles defined positive exceed rad similarly orientation angles defined negative less rad displacement behavioral descriptor characterizes changes postion robot walking measured proportion intervals robot positively negatively displaced along axes denote linear displacement meters robot interval denotes number intervals seconds simulated movement unit step function returns value argument exceeds returns value otherwise ignore insignificant motion linear displacements defined positive exceed defined negative less arxiv preprint total energy expended per leg behavioral descriptor captures total amount energy expended move leg seconds movement denotes energy utilized leg robot seconds simulated movement measured maximum amount energy available according servo model simulator seconds relative energy expended per leg behavioral descriptor captures amount energy expended move leg relative energy expended legs seconds simulated movement denotes energy utilized leg robot seconds simulated movement measured deviation descriptor captures range deviation center robot frame versus expected location robot traveled straight line constant speed max final max final max denotes ground reaction force grf leg generates averaged seconds simulated movement maximum force leg apply relative ground reaction force per leg behavioral descriptor corresponds amount force leg applies ground relative legs denote position robot center time xfinal yfinal zfinal denote final position seconds robot task move along starting position deviation along axes computed maximum difference robot position dimensions point seconds axis yfinal corresponds average speed robot distance covered divided total time therefore yfinal expected position timestep robot moving constant speed deviation axis computed respect theoretical position obtain values range final behavioral descriptors multiplied divided values determined empirically total ground reaction force per leg behavioral descriptor corresponds amount force leg applies ground measured fraction total possible amount force leg could apply ground specifically measurement denotes ground reaction force grf leg generates averaged seconds simulated movement pitch angle descriptor captures pitch angle respect ground global coordinate frame averaged seconds pitch angle hence contact ground time number touches ground foot pitch angles range leg penetrate ground normalized roll angle descriptor captures roll angle respect ground global coordinate frame averaged seconds roll angle hence contact ground time number touches ground foot roll angles range leg penetrate ground normalized yaw angle descriptor captures yaw angle respect ground global coordinate frame averaged seconds yaw angle hence contact ground time number touches ground foot yaw angles range normalized cully clune tarapore mouret arxiv preprint random random behavioral descriptor differs intentionally chosen descriptors consist one type knowledge instead randomly selected subset variables previously described behavioral descriptors descriptor intended simulate situation one little expectation behavioral descriptor perform well one quickly picks different descriptor dimensions without consideration experimentation instead generating one list fashion randomly sample large set find average performance approach many different possible choices random descriptor selected random without replacement available behavior descriptor dimensions described previous descriptors descriptors denotes ith dimension descriptor randomly selected uniformly without replacement available dimensions behavior descriptors necessary compare behavioral descriptors simulation physical robot would required months experiments would repeatedly worn broken robot modified simulator main experiments section emulate different possible damage conditions involved removing different leg algorithm run million iterations used create maps behavioral descriptors using simulation undamaged robot generation maps behaviors stored map cells discretizing dimension behavioral descriptor space five values behavioral descriptors twenty equidistant values behavioral descriptor adaptation phase behaviors used actual values thus discretized independently generated eight maps intentionally chosen behavioral descriptors twenty independently generated maps generated random behavioral descriptor launched ten replicates descriptor maps eight intentionally chosen behavioral descriptors twenty random behavioral descriptor six damage conditions therefore replicates intentionally chosen descriptors replicates random descriptor simulated experiments roughly simulate distribution noisy odometry measurements real robot simulated performance values randomly perturbed multiplicative gaussian noise centered standard deviation analyze fastest walking speed achieved behavioral descriptor two different numbers trials first case trials second case trials results following results include trials simulated robot maximum number trials required intelligent trial error find compensatory gait supplementary experiment performance cully clune tarapore mouret achieved alternative intentionally chosen behavioral descriptors numbers similar original duty factor behavioral descriptor number extended data fig alternative intentionally chosen descriptors numbers led median performance within duty factor descriptor performance difference performance effectively nonexistent deviation descriptor total grf descriptor roll angle descriptor lowest performance discovered relative grf descriptor lower duty factor descriptor terms statistical significance performance achieved duty factor descriptor different deviation total grf descriptors remaining descriptors difference performance statistically significant exceed additionally compensatory behaviors discovered alternative intentionally chosen descriptors always faster reference gait damage conditions check whether alternative intentionally chosen behavioral descriptors lead better performance allowed higher number evaluations extended experiments trials robot extended data fig trials difference performance duty factor behavioral descriptor alternative behavioral descriptors reduced three alternative intentionally chosen descriptors displacement total grf yaw angle median performance within duty factor descriptor difference performance orientation total energy relative energy deviation relative grf pitch angle lowerleg roll angle descriptors three remaining behavioral descriptors displacement total grf yaw angle performance respectively difference duty factor descriptor three cases terms statistical significance performance achieved duty factor descriptor barely statistically significantly different deviation descriptor remaining descriptors performance difference statistically significant larger random behavioral descriptor also performed similarly duty factor descriptor trials performance maps generated random descriptor lower duty factor descriptor performance difference statistically significant difference performance negligible difference performance reduced trials random descriptor performance duty factor description performance moreover intentionally chosen behavioral descriptors compensatory behavior discovered random descriptor also faster reference gait experiments show selection behavioral dimensions critical get good results indeed tested behavioral descriptors even randomly generated perform well median trials hand robot designers prior knowledge dimensions variation likely reveal different types behaviors algorithm benefit knowledge improve results duty factor descriptor arxiv preprint caption supplementary videos video video viewed https damage recovery robots via intelligent trial error video shows intelligent trial error algorithm action two robots introduced paper hexapod robot degrees freedom robotic arm fig video shows several examples different types behaviors produced map creation step classic hexapod gaits unexpected forms locomotion shows hexapod robot uses map deal leg lost power fig finally video illustrates intelligent trial error algorithm applied second robot different damage conditions video video viewed http map containing many different types walking gaits map creation step algorithm produces collection different types walking gaits video shows several examples different types behaviors produced classic hexapod gaits unexpected forms locomotion supplementary references eiben smith introduction evolutionary computing springer mouret clune illuminating search spaces mapping elites arxiv preprint lizotte wang bowling schuurmans automatic gait optimization gaussian process proceedings international joint conference artificial intelligence ijcai vol brochu cora freitas tutorial bayesian optimization expensive cost functions application active user modeling hierarchical reinforcement learning arxiv preprint snoek larochelle adams practical bayesian optimization machine learning algorithms advances neural information processing systems nips griffiths lucas williams kalish modeling human function learning gaussian processes advances neural information processing systems nips booker dennis frank serafini torczon trosset rigorous framework optimization expensive functions surrogates structural optimization forrester keane recent advances optimization progress aerospace sciences jin evolutionary computation recent advances future challenges swarm evolutionary computation simpson mauery korte mistree comparison response surface kriging models multidisciplinary design optimization american institute aeronautics astronautics jones schonlau welch efficient global optimization expensive functions journal global optimization sacks welch mitchell wynn design analysis computer experiments statistical science calandra seyfarth peters deisenroth experimental comparison bayesian optimization bipedal locomotion proceedings ieee international conference robotics automation icra cully clune tarapore mouret spatial variation stochastic models application problems forest surveys sampling meddelanden fran statens skogsforskningsinstitut stein interpolation spatial data theory kriging springer fiacco mccormick nonlinear programming sequential unconstrained minimization techniques vol siam dryanovski valenti xiao fast visual odometry mapping data proceedings ieee international conference robotics automation icra ieee quigley conley gerkey faust foote leibs wheeler ros robot operating system proceedings icra workshop open source software sproewitz moeckel maye ijspeert learning move modular robots using central pattern generators online optimization international journal robotics research yosinski clune hidalgo nguyen zagal lipson evolving robot gaits hardware hyperneat generative encoding parameter optimization proceedings ecal clune stanley pennock ofria performance indirect encoding across continuum regularity ieee transactions evolutionary computation clune beckmann ofria pennock evolving coordinated quadruped gaits hyperneat generative encoding proceedings ieee congress evolutionary computation lee yosinski glette lipson clune evolving gaits physical robots hyperneat generative encoding benefits applications evolutionary computing springer wilson insect walking annual review entomology saranli buehler koditschek rhex simple highly mobile hexapod robot international journal robotics research schmitz dean kindermann schumm cruse biologically inspired controller hexapod walking simple solutions exploiting physical properties biological bulletin ding wang rovetta zhu locomotion analysis hexapod robot proceedings conference climbing walking robots clawar steingrube timme manoonpong adaptation simple neural circuit enables complex robot behaviour nature physics delcomyn locomotion cockroach pariplaneta americana journal experimental biology thrun burgard fox probabilistic robotics mit press cambridge dissanayake newman clark csorba solution simultaneous localization map building slam problem ieee transactions robotics automation tesch schneider choset using response surfaces expected improvement optimize snake robot gait parameters proceedings international conference intelligent robots systems iros ieee kohl stone policy gradient reinforcement learning fast quadrupedal locomotion proceedings ieee international conference robotics automation icra vol ieee erden free gait generation reinforcement learning robot robotics autonomous systems christensen larsen stoy gait learning morphology optimization polymorphic walking robot evolving systems mahdavi bentley innately adaptive robotics embodied evolution autonomous robots koos cully mouret fast damage recovery robotics algorithm international journal robotics research hornby takamura yamamoto fujita autonomous evolution dynamic gaits two quadruped robots ieee transactions robotics barfoot earon eleuterio experiments learning distributed control hexapod robot robotics autonomous systems koos mouret doncieux transferability approach crossing reality gap evolutionary robotics ieee transactions evolutionary computation arxiv preprint
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jul competing risks model covariates estimation inference jue jelena ronghui department mathematics department family medicine public health university california san diego abstract purpose paper construct confidence intervals regression coefficients model competing risks data random censoring number covariates larger sample size despite strong motivation biostatistics applications highdimensional model attracted relatively little attention among methodological theoretical literatures fill blank proposing first consistent regularized estimator confidence intervals based estimator able generalize partial likelihood approach model random censoring despite many technical difficulties lay methodological theoretical framework estimator partial likelihood independent identically distributed entries also handle theory approximation error inverse probability weighting ipw proposing novel concentration results time dependent processes addition theoretical results algorithms present extensive numerical experiments application study mortality among prostate cancer patients using linked data key words survival analysis inference estimator introduction many applications want use data draw inferences effect covariate specific event presence many risks competing event examples include medical studies effect medical treatment health outcomes chronically ill patients studies unemployment duration transitions employment labor market programs evaluations environmental determinants child mortality studying internetwork competition risk strategic gridlock study firms use alliances respond alliance networks rivals historically datasets small meaningfully explore heterogeneity different risk factors beyond considering models recently however explosion experimental data sets potentially feasible develop estimates full competing risks models regression attracted increased interest statistical analysis provided useful tool modern biomedical ecological astrophysical economics data pertaining setting number parameters greater number samples see van geer overview regularized methods fan tibshirani provide straightforward interpretation resulting estimators allowing number covariates exponentially larger sample size considerable research effort devoted developing regularized methods handle various regression settings ravikumar belloni chernozhukov obozinski meinshausen basu michailidis cho fryzlewicz including data sun bradic guilloux johnson lemler bradic song huang among others however regression studied competing risks setting scenario frequently encountered practice random censoring covariates illustration project information contained patients electronic medical records harvested purposes precision medicine consider data set linking surveillance epidemiology end results seer program database national cancer institute federal health insurance program medicare database prostate cancer patients age older restricted patients diagnosed seermedicare database excluding additional patients missing clinical records total patients information available relevant clinical variables age psa gleason score ajcc stage ajcc stage respectively demographical variables race marital status metro registry year diagnosis plus binary insurance claim codes december end data total deaths due cancer deaths unrelated cancer goal paper develop methodology model many covariates number events used appropriately flexibly evaluate impact risk factors versus cancer mortality reflected clinical demographical claim codes indirectly describe events occur surgical procedures hospitalization outpatient activities understanding turn aid clinical decision making whether pursue aggressive therapy presence comorbidities least three major challenges addressing competing risks regression model directly associates risk factors cumulative incidence function particular cause structure score function related partial likelihood rather subtle issue many unobserved factors ruining simple gale representation shrinkage effects regularization methods add bias component inference primary interest additionally structure sample information matrix prevents naive usage wald score type hypothesis testing methods basing theoretical analysis hessian matrix renders problematic implementations attempts tackle inference problems regression model along direction would also undesirably require implementation bootstrap ideas however given known problems bootstrap setting approach longer applicable development highdimensional inferential methods competing risks data model particular may hampered considerations paper propose natural sensible formulation inferential procedure highdimensional competing risks regression first step formulate regularized estimator parameter interest derive properties interplay sparsity ambient dimension sample size directly seen note results easily generalizable number penalties due simplicity presentation present details regularization second step formulate estimator formulating new pragmatic estimator variance allows broad dependence structures within model step compensates potential bias first estimator arises due variables may weakly correlated risk scaues important due correlation risk interest find second step estimator effective capturing strong signal well weak signals combination leads effective estimator model many features setup notation subject study let event time event type cause use two words interchangeably following model consider without loss generality assume event type interest code event types without specifying presence potential time observed time type event also observed let vector covariates possibly assume observed data independent identically distributed since cumulative incidence function cif often quantity interest estimated data fine gray proposed proportional subdistribution hazards model cif exp coefficient unknown parameter interest baseline subdistribution hazard model corresponding subdistribution hazard throughout paper assume exists note define improper random variable subdistribution hazard seen conditional hazard given fine gray proposed estimate based partial likelihood principle complete data censoring censoring complete data censoring times always observed even general random censoring observed inverse probability weighting ipw used obtain consistent estimating equations denote counting process type event observed counterpart nio also denote counting process censoring time nic let note risk indicator like classic cox model note always observable even though may let estimator assume independent let following notation fine gray call ipw process words weight subject one zero censored failure due failure due causes modified log partial likelihood cause gives rise weighted score function fine gray log dnio maximal time following vector let define organization paper paper organized follows section provide estimation inference methodology developed model bounds prediction error lasso estimator presented section also discuss related result whose rate matches linear models see theorem section studies sampling distribution newly develop test statistics allowing ultra parameter space examine regularized estimator bias estimator thorough numerical examples section real data study section estimation inference competing risks parameters samples regularized estimator natural estimator consider estimator particular loss function interest would modified partial defined consider argmin suitable choice tuning parameter whenever possible suppress denote estimator paper quantify nonnotation use asymptotic oracle risk bound estimator allowing minimal set assumptions theoretical studies kind novel since context competing risks martingale structures typically utilized work ruined new techniques needed developed particular show inverse probability weighting finitesample effect separates model classic cox model example see comments theorem also establish certain tighter bound established whenever hazard rate bounded see theorem finally results presented therein easily broadened sparsity encouraging convex penalty function corrected estimator purposes constructing confidence intervals testing significance certain covariates utilizing naive regularized estimation appropriate example construction confidence intervals coefficients shrunk zero impossible established restrictive hand firm guarantees correct variable selection set assumptions including limited assumption minimal signal strength true parameter wasserman roeder fan meinshausen verified practice therefore interest develop inferential tools depend assumptions yet able provide theoretical guarantees quality estimation testing example inspired work zhang zhang van geer propose estimator defined estimator asymptotic precision matrix defined score function derivative modified log partial likelihood evaluated dnio construction estimator inspired first order taylor expansion notation indicates equivalence approximate higher order error terms omitted negative hessian limit denoted inverse hessian matrix would naturally good candidate however inverse necessarily exist therefore denoting aim find good candidate matrix identity matrix although construction inspired early works linear models specifics theoretical analysis remain challenge following elucidate construction construction inverse hessian matrix start writing negative hessian modified log partial likelihood dnio define dnio dnio regularity conditions specified later asymptotic negative hessian sense maximal norm kmax converges zero probability goal estimate inverse rows positive matrix also second moment random vector dnio defined expectation zero hence may draw inspiration works inverting matrix zhou estimate inverse consider minimizers expected loss functions argmin jth element dimensional vector created dropping jth element note also alternatively written convexity target function must satisfy first order conditions kkt applying define vector satisfies without loss generality may define accordingly satisfying matrix satisfies therefore inverse using empirical version propose consistent estimator due matrix inversion negative hessian difference two matrices inside integral easy work instead derive sample version alternative form written sample second moment advantage define dimensional vector obtained dropping jth element define nodewise lasso context jth element argmin construct accordingly use choosing first order kkt condition kmax goes zero estimator achieve hence converges true coefficient approximately rate proposed equivalent illustrated proposed estimator innovative several aspects given difficulty imposed model make inference simply inverting covariate matrix modified log partial likelihood dependent entries covariates allowed nevertheless identify model key element inference observation contribution score solution generalizes matrix inversion way complex models likelihoods weighting confidence intervals construct confidence intervals components need asymptotic distribution first establish asymptotic distribution score restrict space want establish asymptotic distribution general impossible establish convergence distribution jointly gaussian random variable due exploding dimensions asymptotic distribution established following sense matrix obtaining result technically challenging mentioned earlier apart modified log partial likelihood dependent summands addition ipw creates additional dependency estimator construct following estimator defined follows dmi dmi dmi illustrated dnio dnjo cic dnic dnjc asymptotically equivalent using sandwich estimator may estimate variance therefore confidence interval standard normal quantile proposed approach addresses various practical questions special cases first construct confidence interval chosen coordinate end one needs consider natural basis apply result generally construct confidence interval linear contrasts potentially dimension example confidence intervals linear predictors covariate also sparse may assume bounded dual problem may use wald test statistic test hypothesis theory confidence intervals described present theory estimators previous section first introduce additional notations unlike situation fine gray various empirical process results applicable none general results directly applied big challenge theory convergence empirical average processes common expectation needed various places generalize empirical process results relying heavily martingale theory elsewhere counting process martingales essentially helpful tools establishing theory dependent partial likelihoods unfortunately uncensored counting processes always observable observable counterpart nio known martingale related observed filtration nio compensator submartingale nio observed filtration involves nuisance distribution utilize martingale structure theory define censoring complete filtration nio martingale defined dmi relate martingale modified log partial likelihood define proxy measurable integrand log dnio define processes related derivatives also seen proxies processes fact compute following expectation first conditioning observed filtration weight true censoring distribution denote expectations differs proxies precisely targets weighted samples summands observed events require mild additional assumptions control approximation error note estimator written dnic study convergence denote martingale related nic counting process observed censoring let censoring hazard defined log censoring filtration nic martingale mic nic use arguments murphy argument page random martingale measures proof rigorous sense specify element probability space apply deterministic measure solve integral equations involving total variation defined sup sup since decomposed nondecreasing counting process minus another nondecreasing compensator bound total variation similar conclusion also applies mic bound total variation mic nic convention hereon suppress notation keep simple oracle inequality based following first establish oracle inequality initial estimation error regularity conditions constants assumed independent conditions design suppose independent exists finite differentiable hazard function probability equals one covariates satisfy sup sup kzi exist positive inf min continuity conditions estimation may jumps kzi minimal gap jumps bounded away zero two consecutive jumps elements lipschitz continuous lipschitz constant rest elements constant baseline cif differentiable baseline subdistribution hazard log lower bound restricted eigenvalue smallest eigenvalue matrix min min least remark conditions minimal sense appear oracle inequality cox model huang page theorem page remark due missing censoring times among observed events make additional assumptions control weighting errors although weighted processes asymptotically unbiased approximation errors tail poor finite avoid unnecessary complications set final administrative censoring time following conventional design literature andersen gill result partial likelihood becomes integral finite support let infinity delicate assumptions martingale representation approximation error see lemma supplement decide replace straightforward assumptions obscured ones harder verify remark condition equivalent apparently weaker assumption see example huang equation sup sup kzi seen noting cox model formulation invariant subtracting deterministic remark condition interpreted two statements first rate type events bounded away zero second arbitrarily close zero needs truncated finite necessary order rule irregular situation expectation relative risks dominated diminishing proportion excessively large relative risks argument applies lower bound restricted eigenvalue negative hessian bickel defined remark since continuity condition may appear obscure offer extra explanation fact subjects observed type events remain indefinitely risk sets type events seen definitions subject proportional hazard model one would expect type relative risks excessively large use fact establish slow growing rate maximal relative risks among subjects longer case unless linear predictor processes certain continuity property propose taking account likely practical scenarios covariates either measured baseline otherwise finitely many discrete time points note coordinate wise continuity insufficient grows infinity concentration empirical average processes around expectation one major challenge establishing theory covariates survival data prove two widely applicable lemmas appendix lemma produces concentration results empirical average processes certain continuity property around mean independently generated random grid use establish empirical bounds assumed population bound control approximation errors lemma produces uniform concentration results counting process martingales use establish sharper concentration results martingales regular deterministic argument huang van geer van geer remains valid model let indices set elements compliment define cone set kboc compatibility factor negative hessian cone set sup kbo defined event estimation error lasso estimator bound smaller solution probabilistic condition decaying zero however straightforward presence competing risks censoring greatest challenge stated beginning section lack martingale property even use martingale proxy error amplified product relative risks following first show relative risks among subjects observed type events tails achieved argument cif arbitrarily close one otherwise subjects would probability close one experiencing type event contrary observed fact cif monotone increasing relative risks also unlikely observe excessively large relative risks among subjects observed type events around zero use lemma establish concentration across observed type event time state result probabilistic condition lasso estimation error following lemma lemma denote ecz log log log kcn log assumptions lemma directly translates bound bias estimation theorem small set lemma let satisfying smaller solution assume log regularity conditions occurs probability less lemma comes bound maximal relative risk among observed type events contrary natural upper bound eso order log bound approximation error sup kmax ordered observed type event times total number unique observed type event times focusing leading terms order log order log log hence obtain diminishing tail bound order log log despite complicated expression result differs rate log established cox linear model factor log comes weights among terms magnitude oracle inequality may directly assume compatibility factor bounded away zero regularization parameter choosen order log log obtain log log conclusion theorem involves compatibility factor obtain converging zero probability assume converges zero sequence bounded away zero sample size goes infinity alternatively may achieve equivalent conclusion condition used later asymptotic distribution following lemma show negative hessian lower bound converging positive definite matrix whose eigenvalues assumed bounded away zero thus obtain lower bound restricted eigenvalue negative hessian cone using connection compatibility factor restricted eigenvalue van geer show compatibility factor cone bounded away zero probability tending lemma denote log solution exp assumptions rate condition log log obtain asymptotically therefore small neighborhood true parameter regularized estimator interest able establish theory statistical inference using local quadratic approximation asymptotic normality estimator honest coverage confidence intervals honest coverage establish asymptotic normality estimator confidence intervals based slightly stronger regularity conditions bound linear predictors true linear predictors uniformly bounded probability one sup sup conditions design suppose independent exists finite differentiable hazard function probability one covariates satisfy kzi exists continuity conditions inference generated random processes dzi counting process niz dniz dzi uniformly bounded uniformly bounded accordance accordance log log common cap number jumps bounded intensity function baseline cif differentiable baseline subdistribution hazard log exists bounded invertibility negative hessian smallest eigenvalue asymptotic hessian matrix least rows asymptotic precision matrix sparse sparsity constants smax also rate condition inference dimensions sparsity parameters satisfy smax log make notably stronger assumptions required oracle inequality previous section explain inevitability following discussion like works inference beyond linear regression normality assumed restriction linear predictor becomes unavoidable van geer fang case asymptotic normality depends fundamentally necessary condition predictable quadratic asymptotic tightness variation filtration martingale must finite bound independent dimension covariates requires magnitude summands either bounded light tails hence allow relative risk grow arbitrarily large additionally able achieve improved estimation error initial estimator extra log rate theorem bounding large probability longer necessary theorem choose log log addition use condition derive sharper bound needed purposes asymptotic normality mation error naturally conclusion requires extra smoothness compared lipschitz continuity one step make sure derivatives also lipschitz finally standard assumption validity nodewise penalized regressions define population versions nodewise components defined argmin true parameters uniquely define inverse negative hessian described section prove statement following lemma lemma moreover lemma also shows grow use lemma control processes normality lemma extends uniform concentration countable grid lemma entire domain result useful assess approximation error two processes integral absolutely continuous measures like apply lemma bound errors involving martingale integrals diminishing measurable integrands preparation start estimation error worth noting techniques van geer rely entries linear regression model unlike nodewise lasso model dependent definitions hence proof adopts additional common proxies eventually establish error rates approximation dependent following lemma leads error rate lemma log lemma choosing log supj smax log supj log thus remark total error product error initial estimator order error nodewise lasso order smax dimensions factor log compared linear regression case zhang zhang van geer affected estimation error initial estimator therefore makes sense extra rate compared generalized linear model glm case van geer involving initial estimator thus dependent error rate two different sources takes multiplicative form smax instead summation smax glm general consider rate optimal model proposed using lemma establish approximation condition satisfies approximation condition lemma estimator smax log next show asymptotic normality define asymptotic variance similar fine gray dmic mic defined lemma conditions directional vector proof uses approach initial result fine gray approximate sample average terms plus term note approach involves nontrivial techniques apply particular discover exploit martingale property term last piece convergence meat matrix sandwich variance estimator achieve following result repeatedly using lemmas lemma conditions vkmax log hence putting lemmas together obtain main result stated theorem theorem let theorem justifies proposed inference testing procedures section simulation experiments assess finite sample properties proposed methods conduct extensive simulation experiments various dimensions dependence structure among covariates setup first simulation setup follows closely one fine gray considers highdimensional covariates particular vectors consisting standard normal random variables cause cumulative incidence function exp exp cause exp consider four different combinations note setup considers sparsity cause cause effects model require modeling cause order make inference cause expect cause effects affect inference cause results presented table focus inference two coefficients well one arbitrarily chosen zero coefficient mean estimates average repetitions reported together quantities described see average estimates column presence many coefficients cause affect inference cause practice choice tuning parameters particularly challenging optimal value determined constant moreover theoretical results asymptotic nature together finite sample effects lead suboptimal performance many proposed correction estimators van geer fang survival models due nonlinearity loss function presence censoring require larger sample size order observe asymptotic statements finite samples following propose correction construction confidence intervals particular estimated standard error denote asymptotic standard error given section let bbj place bbj variance estimation based correction propose consider bbj replaced viewed another initial lasso estimate iteration formula resulting therefore estimator report coverage rate confidence intervals constructed correction table observe good coverage close nominal level note simulation runs margin error simulated coverage probability true coverage observed coverage range note coverage good three coefficients zero contrast results existing literature suffer coefficients last column table refers empirical rejection rate null hypothb nominal esis coefficient zero wald test bbj bbj used nominal level well preserved significance level see although bbj zero coefficient power high coefficients given sample sizes signal strength table simulation results independent covariates true mean est corrected coverage repeat simulations different values investigate power wald test results illustrated figure see power increases decreases expected figure power curve testing nominal level power setup setup second setup consider case covariates independent likely case practice high dimensional data consider block dependence structure also used binder consider rest zero rest zero covariates grouped four blocks size plus rest correlations equal four blocks separated horizontal lines table table shows inferential results coefficients well zero coefficients third correlated block also contains nonzero coefficients plus arbitrarily chosen zero coefficient initial lasso estimator tended select one every four coefficients correlated covariates data shown known block dependence structure particularly challenging lasso type estimators hand estimator performed remarkably well capturing coefficients compared results last part table block correlated although coverage remained high power also covariates led slightly bias remained high although third block mixed signal noise variables type error rates appeared slightly high table simulation results block correlated covariates true mean est corrected coverage data example linked database contains clinical information claims codes patients diagnosed clinical demographic information collected diagnosis insurance claim data year prior diagnosis clinical information contained psa gleason score ajcc stage year diagnosis demographic information included age race marital status data set considered hou emphasis variable selection prediction error focus testing construction confidence intervals binary claims codes data would like identify risk factors mortality using model kept claims codes least occurrences resulting dataset covariates following consider patients diagnosed year among died cancer deaths unrelated cancer center standardize covariates performing analysis determine penalty parameters used table present result selected set coefficients due space set variables presented full although computed presented variables larges lasso initial estimator returned zero initially also focused heart disease colon cancer potential significan mortality different prostate cancer well prostate cancer variables descriptions variables estimate given table coefficient report initial lasso estimate corrected constructed corrected wald test calculated using uncorrected table inference linked data mortality among prostate cancer patients variables age marital psa gleasonscore cpt cpt cpt cpt initial estimate estimate inference denotes significance bonferoni correction denotes significance bonferoni correction table description variables table code description age marital psa gleasonscore cpt cpt age diagnosis married race white race black white psa gleason score ajcc acute respiratry failure acute respiratory failure congestive heart failure nonhypertensive global cardiovascular stress test cardiac dysrhythmias diagnostic radiology diagnostic imaging procedures spine pelvis cpt cpt nodular prostate diagnostic radiology diagnostic imaging procedures abdomen delirium dementia amnestic cognitive disorders obstructive chronic bronchitis unspecified psychosis acute subacute forms ischemic heart disease endoscopy procedures rectum parkinsons disease table see claims codes cpt related heart disease significant level bonferonni correction variables included table heart attack indicator variable shows significant level correction multiple testing indicator disease abdomen cpt significant although initial lasso regularized method failed include variable similar result seen indicator fall cpt elderly person fatal indicator colon cancer cpt turns significant although lasso method set zero initially therefore method able recover important risk factors would missed initial lasso estimator contrast diseases expected significant mortality include parkinson psychosis bronchitis dementia table also note prostate cancer related variables psa gleason score ajcc large mortality consistent results hou competing risk models predictors second cause secondary importance predicting events due first cause discussion article focuses estimation inference model many covariates number events effective sample size survival data article proposes conditions lasso estimator performs well terms prediction error also develops new estimator utilized asymptotically optimal inference confidence intervals testing often overlooked restriction covariates model must observable even subject experiences type event practice either time independent external kalbfleisch prentice case continuity conditions easily satisfied majority elements time independent likely case practice theory apply studies involving longitudinal variables supposed truly measured continuously time illustrated method based regularization without bias correction might serious disadvantages many complex data situations example may potentially fail identify important variables associated response analysis data see variables like cpt related fall cpt related diagnostic imaging abdomen often suspected malignancies would discovered important risk factors mortality regularization alone reality life threatening events elderly patient estimate hand able detect therefore providing valuable tool practical applications estimator applicable long model sparse minimum signal strength required another important aspect makes estimator desirable practical use lasso type estimators acknowledgement would like acknowledge collaboration james murphy san diego department radiation medicine applied sciences linked data analysis project motivated work would also like thank group help preparing data set supplementary materials section provide details theoretical results preseding proof statement present needed preliminary lemmas numbered letters concentration inequalities processes lemma let pairs random processes counting process bounded denote jumps tiki let suppose ksi kmax almost surely tij tij max assume addition generated dni satisfying kds kmax kjs kmax rfor sup sup max log nkn lemma let mri counting process martint gales satisfying let dimensional processes sup sup kmax dmi kmax assume addition kmax kmax log proofs main results state related preliminary results proof main results proofs preliminary results given next section proof lemma without loss generality let first jump time assumption independent thus sequence martingale respect filtration bounded increment esj applying azuma inequality get kln kmax dropped first term also bounded get max since use simple union bound extend result tij whose number nkn define deterministic set union bound hoeffding inequality sup max combining result lemma obtain max log npq grid containing jumps need show variation sufficiently small inside bin created grid let consecutive elements order construction jump counting processes interval otherwise jump time another element consecutive elements order assumption lemma elements moreover deterministic along property obtain bound variation sup kmax sup sup ksi kmax bound variation kmax kds kmax kjs kmax arbitrary find corresponding bin contains putting results together kmax kmax kmax kmax log npq proof lemma summands integrals processes martingales martingale kalbfleisch prentice suppose jump times artificially set jump define rkn order statistics hence set ordered stopping times applying optional stopping theorem get discrete time martingale adapted frk increment comes either counting part compensator part bound separately construction bin satisfies two conditions one jump bin length bin increment martingale decomposed two integrals jump minus compensator dni assumed upper bound kmax almost surely jump bin bounded dni max additionally assumed upper bound compensator increases bin max obtain uniform concentration inequality azuma sup kmax remark uniform version azuma application doob maximal inequality durrett theorem page use bounded increment derived kmax max additional assumption kmax find every sup sup kmax apply lemma obtain event sup kmax log sup sup kmax occurs probability less lemma lemma let set nonnegative processes pni max max result maximal norms uniformly bounded lemma define min let observed events event sup occurs probability least lemma define event sup ecz log sup occurs probability least lemma define ipw weights true log event occurs probability least let lemma define observed events denote log log max sup max defined lemmas occurs probability least sup max proof lemma let observed events may decompose score martingale proxy plus approximation error recall counting process observed event written nio takes form cox model score counting process nio moreover process censoring complete filtration also equivalently generated nio thus may apply huang lemma notice inequality sharper lemma compensator part zero concentration result approximation error established lemma obtain concentration inequality adding bounds tail probabilities together proof lemma strategy lemma first show lower plus diminishing error since takes form cox model bounded hessian may apply results huang lemma huang similar result see van geer corollary kmax let observed events write bounded lemma kmax twice get apply lemma contains condition theorem huang hence may apply result log probability least bounded away zero observed events term absorbed proof theorem observe techniques huang apply see example lemmas therein structure partial likelihood cox model modular ipw weight functions following line proof easily obtain event proof completed utilizing lemma lemma proof theorem since assume relative risks bounded almost surely constants ekb may set ekb directly obtain also improve rate estimation error theorem log need let lemma grow proof lemma denote without loss generality set since define minimizer convex function must satisfy first order condition recall applying first order condition get construct vector satisfies hence directly bound minimal eigenvalue least obtain spectral decomposition maximal eigenvalue hence max max lemma denote kmax log kmax log define let differentiable operator adapted process uniformly bounded bound kmax whenever max sup iii sup sup proof lemma since implies thus equivalence dnio recall following calculation decompose dnio log lemma notice martingale hence apply lemma get nop log decompose terms one jump observed event time ekb elsewhere since set independent continuous random variables tie among probability one hence may modify integrand observed cenp soring times without changing integral replacing apply lemma get total variation max ekb lemma log log hence obtain log log similarly obtain log log besides one lemma another martingale representation denote estimator dnic martingale dmic lemma error exp discrepancy plus second order tailer expansion remainder shall show since sup second order remainder let observed second order remainder censoring time increment log log applying mean value theorem obtain hence log exp log uniformly consistent log log shown bounded away zero bounded probability tending obtain one putting together obtain define write sum plus error integration parts dmkc dmkc dmkc dmkc already sum shown hence uniformly bounded one jump kkh elsewhere hence apply log lemma get dmkc log notice martingales may apply lemmas obtain log log log log lemma bound norm finally write sum dmi dmic martingale structure show introducing martingale proxy first term zero martingale structure second term zero ipw weights satisfy mean zero bounded probability equaling one variance bounded limit hence satisfies lindeberg condition clt proof lemma define nio dnjo probability tending total variation one lemma difference dnjo log lemmas iii study bound kzi lemma measurable define new filtration martingale hence apply lemma get ekb log log hence taking union bound get log also take similar form likewise define recall lemmas iii sup log sup lemma log need find rate martingale applying lemma repeat trick mqi obtain kmqi sup kmqi log sup sup log hence log putting rates together directly obtain log lemma define dmi probability tending one lemma total variation using results far sup log remainder dmjc martingale put martingales rnp vector apply lemma sup log log log therefore get finally decompose vkmax max moreover sup shown lemmas addition observe average terms whose expectation defined lemmas uniform maximal bound sup kmax sup also finish thep proof applying hoeffding inequality last term decomposition max lemma event belongs cone estimation error kvojc kvoj compatibility factor sup kgoj lemma lemma log max log max log lemma setting inf proof lemma lemma may choose log defined lemma occurs probability establish oracle inequality lemma max min shown min tends one lemma hence smax log define according dnio lemma introduce zij dnio decompose log results theorem lemma first average terms expectation part lemma apparently summands defined hence finish proof applying hoeffding lemma prove previous results lemma smax log proof lemma decompose lemma smax log summand integral counting measure kkt condition minus weighted average theorem log putting together obtain smax log hence first term kkt condition theorem like proof lemma lemma applying mean value theorem get define log theorem lemma log along theorem lemma log proof theorem lemmas lemma shown kvkmax bounded ekb probability tending one lemma shown bounded apply lemmas get kvkmax max vkmax kvkmax max note use following fact proofs preliminary results proof lemma notice nonnegative hence apply inequality coordinate sup hence maximal norm bounded similar result achieved sum replaced expectation apply result set proof lemma since bernoulli random variable may apply hoeffding inequality lower tail exp find lower bounds probability may relax inequality lower bound summands uniformly bounded thus may apply lemma version sup expectation lower bound min relax inequality sup proof lemma since implies probability observing event conditioning upper bound exp exp hence may derive bound sup sup bound away zero certain whenever interval containing length jumps variation linear predictor bounded sup relative risk greater exp hence get lower bound exp finish proof taking union bound proof lemma recall mic counting process martingale adapted complete data filtration kalbfleisch prentice estimator martingale representation dmic able establish concentration result error max sup first obtain concentration result event integrated functions uniform bound hazard hence may apply lemma log obtain sup log proof lemma sharper inequality available may exploit martingale structure general covariates would decompose apb error proximation error two parts error estimate missingness among events define indicator since may alternatively write may use upper bound max lemma define error missingness among events since fine gray shown hence apply lemma applying tower property log sup log max finishes proof first result prove result decomposing differences terms weighted average maximal norm bounded event simply plug bounds tail probabilities proof lemma consider event expectation applying hoeffding get occurs probability apparently morevoer sup lower bounded max sup sup proof lemma max ekb thus terms involved bounded moreover jumps jumps niz define outer product arrays two consecutive jumps niz dzi max hence lzi dzi max satisfies continuity condition lemma like lemma first replace denote kmax lemma apply lemma mean zero process sup kmax log similarly kmax sup log finally extend results quotients decomposition denominators bounded away zero lemma choosing ekb first show related martingales mic nic observed event simplify notation define indicator denote type estimator censoring cumulative hazard tnx dnic let two consecutive define observed censoring times greater increment fact thus dmj dmi notice change beyond event observed since may modify integrand countable many points without changing integral hence gives first order linear integral equation general solution related homogeneous problem one unique solution thus need find one specific solution define integral operator solution written inductively using integration parts able calculate hence solution calculated series applying get martingale dmj dmi use martingale structure prove lemma denote martingale satisfies condition lemma hence kmg kmax log also define total variation lemma hence apply integration parts log shown assumption max result may replace log error since kmax log mean zero random variables hoeffding multiplying rates together get kmax log log expanded dmjc lemma integrand product term hence apply lemma get kmax log log may expand dzj dnjz assumption dzj bounded respectively log njz log log may replace mean zero error jumps ldz two consecutive jumps conditions applying lemma get sup hence kmax dzj max log log log applying lemma njz get jumps niz tik satisfy sup sup tik tik log nkz log hence kmax log log completes proof subscript means element iii define corespondent vector theorem since max weighted average max hence shown sup similar argument show event hence inf inf ekb event occurs probability tending one theorem lemma proof lemma simplify notation wherever possible use belong certain want prove differences convex cone follows kkt conditions sgn denote ojc follows kkt conditions event sgn ojc part convexity let direction satisfies kvoc relax inequality establish upper bound definition left hand side bounded xkvoj right hand side bounded using complete square kvoj kvojc voj kvoj kvoj combining bounds sides inequality get upper bound proof lemma define zij dnio log lemmas average vectors mean maximal bound get apply hoeffding matrix kmax log log max proof lemma define dnio total variation nio lemma log hence log log max average mean bounded maximal norm apply hoeffding union bound choosing log log alternatively use following form dnio lemma iii kmax also similar form dnio lemma kmax log finally use martingale property filtration integrands first two martingale terms bounded hence apply lemma obtain maximal norms log apply lemma integrand third term equivalently expressed max therefore obtain log put rates together triangle inequality proof lemma proof similar lemma define compatibility factor symmetric matrix sup kgoj notice apparently dropping jth row column lemma huang similar result see van geer corollary kmax let embedding defined may establish lower bound smallest eigenvalue inf inf hence inf using result lemma inf max smax max therefor must inf occurs probability tending one references andersen gill cox regression model counting processes large sample study annals statistics azuma weighted sums cerntain dempendent random variables mathematical journal basu michailidis regularized estimation sparse time series models annals statistics belloni chernozhukov quantile regression sparse models annals statistics bickel ritov tsybakov simultaneous analysis lasso dantzig selector annals statistics binder allignol schumacher beyersmann boosting highdimensional data competing risks bioinformatics bradic fan jiang regularization cox proportional hazards model annals statistics bradic song structured estimation nonparametric cox model electronic journal statistics van geer statistics data methods theory applications springer science business media cho fryzlewicz detection high dimensional time series via sparsified binary segmentation journal royal statistical society series statistical methodology durrett probability theory examples edition cambridge university press fan variable selection via nonconcave penalized likelihood oracle properties journal american statistical association fan selective overview variable selection high dimensional feature space statistica sinica fang ning liu testing confidence intervals high dimensional proportional hazards models appear journal royal statistical society series statistical methodology fine gray proportional hazard model subdistribution competing risk journal american statistical association guilloux additive hazards models lasso electronic journal statistics hoeffding probability inequalities sums bounded random variables journal american statistical association hou paravati murphy variable selection prediction competing risks application linked data arxiv huang xie regularized estimation accelerated failure time model covariates biometrics huang sun ying zhang oracle inequalities lasso cox model annals statistics johnson variable selection semiparametric linear regression censored data journal royal statistical society series statistical methodology kalbfleisch prentice statistical analysis failure time data john wiley sons hoboken new jersey lemler oracle inequalities lasso multiplicative aalen intensity model les annales institut henri arxiv preprint meinshausen graphs variable selection lasso annals statistics meinshausen recovery sparse representations highdimensional data annals statistics murphy consistency proportional hazards model incorporating random effect annals statistics obozinski wainwright jordan support union recovery highdimensional multivariate regression annals statistics ravikumar wainwright lafferty ising model selection using logistic regression annals statistics sun lin feng cox regression analysis genomic data statistica sinica tibshirani regression shrinkage selection via lasso journal royal statistical society series methodological van geer conditions used prove aracle results lasso electronic journal statistics van geer ritov dezeure asymptotically optimal confidence regions tests models annals statistics van geer deterministic technical report eth switzerland wasserman roeder high dimensional variable selection annals statistics zhang zhang confidence intervals low dimensional parameters high dimensional linear models journal royal statistical society series zhou covariance estimation based gaussian graphical models journal machine learning research
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survey visual question answering datasets techniques akshay kumar gupta indian institute technology delhi may abstract visual question answering vqa new exciting problem combines natural language processing computer vision techniques present survey various datasets models used tackle task first part survey details various datasets vqa compares along common factors second part survey details different approaches vqa classified four types learning models deep learning models without attention deep learning models attention models fit first three finally compare performances approaches provide directions future work introduction visual question answering task emerged last years getting lot attention machine learning community antol task typically involves showing image computer asking question image computer must answer answer could following forms word phrase answer choosing several possible answers fill blank answer visual question answering important appealing task combines fields computer vision natural language processing computer vision techniques must used understand image nlp techniques must used understand question moreover must combined effectively answer question context image challenging historically fields used distinct methods models solve respective tasks survey describes prominent datasets models used tackle visual question answering task provides comparison well models perform various datasets section covers vqa datasets section describes models section discusses results provides possible future directions datasets several datasets released past years vqa task discuss datasets table contains summary datasets daquar malinowski fritz dataset question answering realworld images daquar released first dataset benchmark released vqa task takes images nyudepth dataset contains images along semantic segmentations every pixel image labeled object class object possible classes images indoor scenes total images training test authors generated question answer pairs two ways automatically using question templates authors define templates questions whose answers taken existing dataset annotations example question template many object image using human annotations asked participants generate questions answers constraint answers must either colors numbers classes sets resultant dataset contains total daquar visual madlibs vqa coco vqa abstract number images number questions avg questions per image average question length chinese average answer length chinese generation human human human automatic human human human table vqa datasets answer pairs training test reduced dataset containing object classes also available authors propose two evaluation metrics dataset one simple accuracy good metric answers second wups score gives score generated answer based average match answer ground truth answers typically wups score thresholded wups score answer correct used generate questions separate set three workers used rate questions less two positive votes discarded multiple choice answers generated automatically amt workers amt workers also asked draw bounding boxes objects mentioned question image firstly resolve textual ambiguity image two red cars red car question could refer either secondly enable answers visual nature pointing object dataset contains images questions visual madlibs visual madlibs dataset well multiple choice dataset images collected descriptive questions generated automatically using templates object information question generated way answered group amt workers answer word phrase multiple choices blanks also provided additional evaluation benchmark dataset contains images questions multiple choice questions evaluated accuracy metric figure taken ren zhu visual dataset generated using images dataset lin image captioning recognition segmentation dataset gets name generating questions form workers amazon mechanical turk amt ren dataset another dataset based questions answers generated automatically using image captions broadly belong four categories object number color location one question per image answers dataset contains total images evaluation done using either accuracy wups score candidate responses evaluated answers machine generated answer normalized vqa evaluation system evaluated score min humans provided exact answer answer considered completely correct matches responses least three human annotators matches none candidate responses given score original vqa dataset images questions abstract images questions iteration vqa challenge bigger dataset total abstract images average questions per image exact number questions mentioned challenge website figure taken ren gao freestyle multilingual image question answering dataset takes images dataset uses baidu crowdsourcing server get workers generate questions answers answers words phrases full sentences pairs available chinese well english translations dataset contains images questions propose human evaluation visual turing test may one reason dataset gained much popularity vqa antol visual question answering vqa dataset widely used dataset vqa task dataset released part visual question answering challenge divided two parts one dataset contains images another dataset contains abstract clipart scenes created models humans animals remove need process noisy images perform high level reasoning questions answers generated workers answers obtained question unique workers answers typically word short phrase approximately questions yes answer evaluation answer generation well multiple choice formats available multiple choice questions figure taken antol models vqa task proposed deep learning approaches already gained wide popularity due performance various vision nlp tasks krizhevsky bahdanau result almost work vqa literature involves deep learning approaches opposed classical approaches like graphical models couple models use approach detailed first subsection addition several simple baselines authors use involve methods also described second describes deep learning models involve use techniques third details deep learning models vqa results models described summarized tables learning approaches answer type prediction atp kafle kanan propose bayesian framework vqa predict answer type question use generate answer possible answer types vary across datasets consider instance consider four answer types object color counting location model computes probability answer answer type given image question follows bayes rule marginalize answer types obtain denominator constant given question image thus ignored model three probabilities numerator three separate models second third probabilities modeled using logistic regression features used question vector representation kiros question use pretrained skip thought model first probability modeled conditional multivariate gaussian similar principle quadratic discriminant analysis original image features used model authors also introduced simple vqa baselines like feeding image features question features logistic regression classifier feeding image question features logistic regressor feeding features perceptron evaluated results daquar vqa datasets malinowski fritz paper models probability answer given question image latent variable corresponding semantic tree obtained semantic parser run question world representation image original image image along additional features obtained segmentation evaluated using deterministic evaluation function obtained training simple model model called swqa single world question answering authors extend swqa model scenario model uncertainty segmentation class labeling different labelings lead different worlds probability modeled set segments along distribution class labels segment sampling distribution segment give one possible world equation becomes intractable authors sample fixed number worlds model called mwqa multi world question answering models evaluated daquar dataset deep learning models deep learning models vqa typically involve use convolutional neural networks cnns embed image word embeddings mikolov along recurrent neural networks rnns embed question embeddings combined processed various ways obtain answer following model descriptions assume reader familiar cnns krizhevsky well like long short term memory units lstms hochreiter schmidhuber gated recurrent units grus cho approaches involve use rnns discussed first ibowimg zhou propose baseline model called ibowimg vqa use output later layer google net model image classification szegedy extract image features word embeddings word question taken text features text features simple words image text features concatenated softmax regression performed across answer classes showed model achieved performance comparable several rnn based approaches vqa dataset propose model refer use three different cnns image cnn encode image question cnn encode question join cnn combine image question encoding together produce joint representation image cnn uses architecture vggnet simonyan zisserman obtains vector layer network passed another fully connected layer get image representation vector size question cnn involves three layers convolution max pooling size convolutional receptive field set words kernel looks word along immediate neighbors joint cnn call cnn performs convolution across question representation receptive field size convolution operation provided full image representation final representation cnn given softmax layer predict answer model evaluated daquar datasets following models use cnns well rnns ask neurons ayn malinowski model uses cnn encode image obtain continuous vector representation image question encoded using lstm gru network input time step word embedding tth question word well encoded image vector hidden vector obtained final time step question encoding simple bag words baseline authors use encode question sum word embeddings answer decoded two different ways either classification different answers generation answer classification performed fully connected layer followed softmax possible answers ation hand performed decoder lstm lstm time point takes input previously generated word well question image encoding next word predicted using softmax vocabulary important point note model shares weights encoder decoder lstms model evaluated daquar dataset ren model similar ayn model model uses final layer vggnet obtain image encoding use lstm encode question contrast previous model provide encoded image first word lstm network question output lstm goes fully connected followed softmax layer call model authors also propose model uses bidirectional lstm instead backward lstm gets image encoding first input well outputs lstms concatenated passed fully connected softmax layer dynamic parameter prediction dppnet noh authors paper argue fixed set parameters powerful enough vqa task take architecture vgg net remove final softmax layer add three fully connected layers last followed softmax possible answers second fully connected layers fixed set parameters instead parameters come gru network gru network used encode question output network passed fully connected layer give small vector candidate parameter weights vector mapped larger vector parameter weights required second fully connected layer using inverse hashing function hashing technique included authors avoid predict full set parameter weights could expensive may lead dynamic parameter layer alternatively seen multiplying image representation question representation together get joint representation opposed combining linear swqa mwqa ayn dppnet atp san coatt ama daquar reduced accuracy wups wups daquar accuracy wups wups accuracy wups table results various models daquar reduced daquar full fashion model evaluated daquar vqa datasets deep learning techniques attention based techniques popular techniques used across many tasks like machine translation bahdanau image captioning etc vqa task attention models involve focusing important parts image question order effectively give answer look shih propose model henceforth referred wtl use vggnet encoding image concatenate outputs final two layers vggnet obtain image encoding question representation obtained averaging word vectors word question attention vector computed set image features decide region image give importance vector computed following way set image features question embedding importance region computed attention weights obtained normalising final image representation attention weighted sum different regions concatenated question embedding passed layer model evaluated vqa dataset loss function max margin based loss takes account vqa evaluation metric recurrent spatial attention zhu model step previous model two ways firstly uses lstms encode question secondly computes attention image repeatedly scanning word question concretely repeatedly compute attention weighted sum image features time step lstm goes additional input next time step lstm attention weights used obtain computed using layer previous hidden state lstm image thus intuitively read question repeatedly decide parts image attend parts attend depend current word well previous attention weighted image model evaluated dataset textual answering task well pointing task points region image answer softmax crossentropy loss actual predicted answer used textual answering task pointing task log likelihood candidate region obtained taking dot product feature representing region last state lstm loss used train model stacked attention networks san yang model similar spirit previous model repeatedly computes attention wups ibowimg dppnet wtl ayn san atp nmn coatt ama open ended number open ended number table results various models vqa dataset image get visual information predict answer however previous model word word model first encodes entire question using either lstm cnn question encoding used attend image using similar equation attention weighted image concatenated question encoding used compute attention original image repeated times question final image representation used predict answer authors argue sort stacked attention helps model iteratively discard unimportant regions image authors experiment report results daquar vqa datasets hierarchical coatt paper differs previous attention based methods addition modelling visual attention also models question attention part question give importance model two forms coattention parallel image question attend simultaneously done computing affinity matrix tanh learnable weight matrix cij represents affinity ith word question region image matrix used obtained image question attention vectors alternating iteratively attend image followed query followed image similar sans spirit one additional idea authors use encode question different levels abstraction word phrase question level question level representation obtained lstm word phrase level representation obtained cnns present results vqa cocoqa datasets models following models use ideas simply changing attend image question fit previous sections neural module networks nmns andreas model involves generating neural network fly individual image question done choosing various based question composing generate neural network modules five kinds attention computes attention map given image given dog instance attention dog try find dog classification outputs distribution labels belonging given image attention map color reattention takes attention map recomputes based means shift attention upward measurement outputs distribution labels based attention map alone combination merges two attention maps specified could decide modules compose together first parse question using dependency parser use dependency create symbolic expression based head word example paper standing field becomes stand symbolic forms used identify modules use whole system trained end end backpropagation authors test model vqa dataset also challenging synthetic dataset found vqa dataset require much high level reasoning composition incorporating knowledge bases present ask anything ama model tries leverage information external knowledge base help guide visual question answering first obtains set attributes like object names properties etc images based caption image image captioning model trained using standard image captioning techniques dataset possible attributes attribute generator trained using variation vgg net top five attributes used generate queries dbpedia database auer query returns text summarized using mikolov summary passed additional input decoder lstm generates answer authors show results vqa datasets discussion future work trend recent years deep learning models outperform earlier graphical model based approaches across vqa datasets however interesting note answer type prediction atp model performs better models proves simply introducing convolutional recurrent neural networks enough identifying parts image relevant principled manner important atp even competitive better attention models like look wtl stacked attention networks san significant improvement shown hierarchical networks coatt first attend question addition image may helpful especially longer questions harder encode single vector representation first encoding word using image attend important words helps model perform better neural module networks nmn uses novel interesting idea automatically composing pair performs similar coatt vqa dataset outperforms models synthetic dataset requiring high level reasoning indicating could valuable approach real world however investigation required judge performance model best performing model ask anything ama incorporates information external knowledge base dbpedia possible reason improved performance knowledge base helps answer questions involve world common sense knowledge may present dataset performance model good vqa dataset might many questions dataset require world knowledge model naturally gives rise two avenues future work first would recognizing external knowledge needed sort model hybrid coatt ama along decision maker whether access might provide best worlds decision might even soft one enable end end training second direction would exploring use knowledge bases like freebase bollacker nell carlson openie extractions schmitz seen novel ways computing attention continue improve performance task seen textual question answering task well xiong seo recent models space used guide vqa models study providing estimated upper bound performance various vqa datasets would valuable well get idea scope possible improvement especially automatically generated finally vqa tasks treat answering classification task vqa dataset allows answer generation limited manner would interesting explore answering generation task deeply dataset collection effective evaluation methodologies remain open question conclusion field vqa grown leaps bounds despite introduced years ago deep learning methods vqa continue models receiving attention showing results surveyed prominent models listed performance various datasets significant improvements performance continue seen many datasets means still plenty room future innovation task ryan kiros yukun zhu ruslan salakhutdinov richard zemel raquel urtasun antonio torralba sanja fidler vectors nips references lin michael maire serge belongie james hays pietro perona deva ramanan piotr lawrence zitnick microsoft coco common objects context eccv jacob andreas marcus rohrbach trevor darrell dan klein deep compositional question answering neural module networks cvpr stanislaw antol aishwarya agrawal jiasen margaret mitchell dhruv batra lawrence zitnick devi parikh vqa visual question answering iccv auer christian bizer georgi kobilarov jens lehmann richard cyganiak zachary ives dbpedia nucleus web open data semantic web pages dzmitry bahdanau kyunghyun cho yoshua bengio neural machine translation jointly learning align translate arxiv preprint kurt bollacker colin evans praveen paritosh tim sturge jamie taylor freebase collaboratively created graph database structuring human knowledge proceedings acm sigmod international conference management data andrew carlson justin betteridge bryan kisiel burr settles estevam hruschka tom mitchell toward architecture neverending language learning aaai kyunghyun cho bart van caglar gulcehre dzmitry bahdanau fethi bougares holger schwenk yoshua bengio learning phrase representations using rnn statistical machine translation arxiv preprint alex krizhevsky ilya sutskever geoffrey hinton imagenet classification deep convolutional neural networks nips quoc tomas mikolov distributed representations sentences documents icml jiasen jianwei yang dhruv batra devi parikh hierarchical coattention visual question answering nips lin zhengdong hang learning answer questions image using convolutional neural network arxiv preprint mateusz malinowski mario fritz multiworld approach question answering realworld scenes based uncertain input nips mateusz malinowski marcus rohrbach mario fritz ask neurons deep learning approach visual question answering arxiv preprint tomas mikolov ilya sutskever kai chen greg corrado jeff dean distributed representations words phrases compositionality nips hyeonwoo noh paul hongsuck seo bohyung han image question answering using convolutional neural network dynamic parameter prediction cvpr mengye ren ryan kiros richard zemel exploring models data image question answering nips michael schmitz robert bart stephen soderland oren etzioni open language learning information extraction emnlp haoyuan gao junhua mao jie zhou zhiheng huang lei wang wei talking machine dataset methods multilingual image question nips minjoon seo aniruddha kembhavi ali farhadi hannaneh hajishirzi bidirectional attention flow machine comprehension arxiv preprint sepp hochreiter schmidhuber long memory neural computation kevin shih saurabh singh derek hoiem look focus regions visual question answering cvpr kushal kafle christopher kanan answertype prediction visual question answering cvpr karen simonyan andrew zisserman deep convolutional networks image recognition arxiv preprint christian szegedy wei liu yangqing jia pierre sermanet scott reed dragomir anguelov dumitru erhan vincent vanhoucke andrew rabinovich going deeper convolutions cvpr damien teney peng wang chunhua shen anthony dick anton van den hengel visual question answering survey methods datasets arxiv preprint peng wang chunhua shen anthony dick anton van den hengel ask anything visual question answering based knowledge external sources cvpr caiming xiong victor zhong richard socher dynamic coattention networks question answering arxiv preprint kelvin jimmy ryan kiros kyunghyun cho aaron courville ruslan salakhutdinov richard zemel yoshua bengio show attend tell neural image caption generation visual attention icml zichao yang xiaodong jianfeng gao deng alex smola stacked attention networks image question answering cvpr licheng eunbyung park alexander berg tamara berg visual madlibs fill blank description generation question answering iccv bolei zhou yuandong tian sainbayar sukhbaatar arthur szlam rob fergus simple baseline visual question answering arxiv preprint yuke zhu oliver groth michael bernstein feifei grounded question answering images cvpr
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texturegan controlling deep image synthesis texture patches wenqi xian patsorn sangkloy varun agrawal amit raj jingwan chen fang fisher james hays dec georgia institute technology adobe research berkeley figure texturegan one generate novel instances common items hand drawn sketches simple texture patches fashion guru top row sketch texture patch overlaid bottom row results texturegan abstract eling rendering pipeline dates back least years rendering techniques techniques later graphics approaches focus image content database training images limited range image synthesis editing scenarios techniques allow nonexperts author photorealistic imagery last two years idea direct image synthesis without using traditional rendering pipeline gotten significant interest promising results deep network architectures variational autoencoders vaes generative adversarial networks gans however little investigation texture control deep image synthesis opposed coarse texture control style transfer methods paper introduce texturegan first deep image synthesis method allows users control object texture users drag one example textures onto sketched objects scene network realistically applies textures indicated objects texture fill operation difficult deep network learn several reasons existing deep networks particularly good synthesizing highresolution texture details even without user constraints typical results recent deep image synthesis methods low resolution texture prominent higher resolution relatively flat birds sharp boundaries paper investigate deep image synthesis guided sketch color texture previous image synthesis methods controlled sketch color strokes first examine texture control allow user place texture patch sketch arbitrary locations scales control desired output texture generative network learns synthesize objects consistent texture suggestions achieve develop local texture loss addition adversarial content loss train generative network conduct experiments using sketches generated real images textures sampled separate texture database results show proposed algorithm able generate plausible images faithful user controls ablation studies show proposed pipeline generate realistic images adapting existing methods directly introduction one grand challenges computer graphics allow anyone author realistic visual content traditional rendering pipeline produce astonishing realistic imagery hands talented trained artists idea traditional indicates equal contribution tails texturegan network must learn propagate textures relevant object boundaries undesirable leave object partially textured texture spill background accomplish network must implicitly segment sketched objects perform texture synthesis tasks individually difficult network additionally learn foreshorten textures wrap around object shapes shade textures according ambient occlusion lighting direction understand object parts handbag clasps textured occlude texture texture manipulation steps beyond traditional texture synthesis texture assumed stationary accomplish steps network needs rich implicit model visual world involves partial understanding fortunately difficulty task somewhat balanced availability training data like recent unsupervised learning methods based colorization training pairs generated unannotated images case input training sketches texture suggestions automatically extracted real photographs turn serve ground truth initial training introduce local texture loss networks handle diverse textures unseen ground truth objects make following contributions first demonstrate plausibility finegrained texture control deep image synthesis concert sketched object boundaries allows author realistic visual content network thus run interactively users modify sketch texture suggestions propose drag drop texture interface users place particular textures onto sparse sketched object boundaries deep generative network directly operates localized texture patches sketched object boundaries explore novel losses training deep image synthesis particular formulate local texture loss encourages generative network handle new textures never seen existing objects related work image synthesis synthesizing natural images one intriguing challenging tasks graphics vision machine learning research existing approaches fall parametric types one hand approaches history typically directly exploit borrow existing image pixels target tasks therefore approaches often excel generating realistic results limited generalization ability restricted limitation data examples data bias distribution hand parametric approaches especially deep learning based approaches achieved promising results recent years different methods approaches utilize image datasets training data fit deep parametric models shown superior modeling power generalization ability image synthesis hallucinating diverse relatively realistic images different training data generative adversarial networks gans type parametric method widely applied studied image synthesis main idea train paired generator discriminator networks jointly goal discriminator classify real images generated fake images generator aims fool discriminator generated images indistinguishable real images trained generator used synthesize images driven compact vector noise compared blurry outcome deep learning methods ganbased methods generate realistic results richer local details higher resolution controllable image synthesis conditional gans practical image synthesis tools require perceptually controllable interfaces ranging attributes object classes object poses natural language descriptions details segmentation masks sketches color scribbles images vanilla gan able generate realistic looking images noise controllable conditional gans models synthesize images based input modalities simple noise thus offering control generated results compared vanilla gans conditional gans introduce additional discriminators losses guide generators output images desired properties object category discriminator discriminator judge association simple loss generated images target images worth highlighting several recent concurrent works sketch deep image synthesis scribbler demonstrates image synthesis framework takes input user sketches short color strokes generates realistic looking output follows input sketch colorization schemes consistent color strokes similar system employed automatically painting cartoon images recently interactive image colorization system proposed offering users control color coloring recoloring input image distinct works system simultaneously supports richer user guidance signals including structural sketches color patches texture swatches moreover offer studies effect several variants improved loss functions show synthesized results various object categories texture synthesis style transfer texture synthesis style transfer two closely related topics image synthesis given input texture image texture synthesis aims generating new images visually similar textures style transfer two inputs content style images aims synthesize images layout structure content image texture style image texture synthesis style transfer methods typically given example images form output textureshop similar method aims texture object texture although textureshop used texture synthesis foreshorten texture appears follow object surface recently new deep style transfer method demonstrated correlations gram matrix features extracted deep neural network capture characteristics textures well showed promising results synthesizing textures transferring styles texture synthesis style transfer formalized optimization problem output image generated minimizing loss function two terms one measures content similarity input content image output measures style similarity input style output using gram matrix shortly many work improving aspects generalization efficiency controllability recently several texture synthesis methods used gans improve quality generated results uses adversarial training discriminate real fake textures based feature patch vgg network instead operating feature space apply adversarial training pixel level encourage generated results indistinguishable real texture proposed texture discriminator section differs prior work comparing pair patches generated ground truth textures instead using single texture patch intuitively discriminator tasked finegrained question texture rather general valid texture fooling discriminator difficult requires generator synthesize realistic texture also texture faithful various input texture styles similar texture synthesis image completion inpainting methods also showed promising results using gans task similarities image completion problem attempts fill missing regions image although missing area significantly larger partially constrained sketch color texture similar approach computes texture loss patches encourage inpainted region faithful original image regions however texture loss accounts similarity feature space approach similar spirit proposes using global local discriminators ensure results realistic consistent image context whereas texture discriminator instead checking consistency input texture patch output image texturegan seek image synthesis pipeline generate natural images based input sketch number texture patches users provide rough sketches outline desired objects control generation semantic content object type shape sketches contain enough information guide generation texture details materials patterns guide generation details want users somehow control texture properties objects scene elements towards goal introduce texturegan conditional generative network learns generate realistic images input sketches overlaid textures argue instead providing unanchored texture sample users precisely control generated appearance directly placing small texture patches sketch since locations sizes patches provide important information influence visual appearance setup user drag rectangular texture patches arbitrary sizes different sketch regions additional input network example user specify striped texture patch shirt dotted texture patch skirt input patches guide network propagate texture information relevant regions respecting semantic boundaries dots appear skirt legs major challenge network learning task uncertain pixel correspondence input texture unconstrained sketch regions encourage network produce realistic textures propose patchbased texture loss based texture discriminator gram matrix style loss helps generated texture follow input faithfully also helps network learn propagate texture patch synthesize new texture texturegan also allows users precisely control colors generated result one limitation previous color control gans input color constraints form rgb need fight network understanding semantics bags mostly black shoes seldom green address problem train network generate images lab color space convert groundtruth images lab enforce content texture adversarial losses channel enforce separate color loss channels show combining controls way allows network generate realistic photos closely following user color texture intent without introducing obvious visual artifacts figure shows training pipeline use network architecture proposed scribbler additional skip connections details network architecture included supplementary material use image input network channels support three different types controls one channel sketch two channels texture one intensity one binary location mask two channels color including limited texture color section describes method used generate input channel network first train texturegan reproduce shoe handbag clothes photos given synthetically generated input control channels generalize texturegan support broader range textures propagate unseen textures better network separate database aim propagate texture information contained small patches fill entire object scribbler use feature adversarial losses encourage generation realistic object structures however find losses alone reproduce texture details also scribbler uses pixel loss enforce color constraints fails input color rare particular object category therefore redefine feature adversarial losses introduce new losses improve replication texture details encourage precise propagation colors initial training derive network input channels photos objects computing losses compare generated images objective function consists multiple terms encourages network focus different image aspects feature loss shown previously features extracted middle layers neural network represent semantic information image given rough outline sketch would like generated image loosely follow object structures specified sketch therefore decide use deeper layer feature loss relu focus feature loss generating structures convert image generated image rgb color space lab generate grayscale images repeating channel values feed grayscale image extract features feature loss defined difference feature space back propagation gradients passing channel output image averaged three channels output adversarial loss ladv generative adversarial networks gans shown generate realistic images random noise seeds gans generator network discriminator network trained simultaneously minimax game discriminator tries distinguish generated images real photos generator tries generate realistic images tricking discriminator thinking real alternating optimization generator discriminator convergence generator would ideally generate images indistinguishable real photos recent work concept adversarial training also adopted context image image translation particular one attach trainable discriminator network end image translation network use constrain generated result lie training image manifold previous work proposed minimize adversarial loss loss discriminator network together standard losses pixel feature losses etc exact choice losses depends different applications work follows line use adversarial loss top feature texture color losses adversarial loss pushes network towards synthesizing sharp realistic images time constrains generated images choose among typical colors training images network understanding color sometimes conflicts user color constraints user provides rainbow color constraint handbag adversarial network thinks looks fake discourages generator producing output therefore propose applying adversarial loss ladv grayscale image channel lab space discriminator trained disregard color focus generating sharp realistic details gradients loss flow channel generator output effectively reduces search space makes gan training easier stable perform adversarial training using techniques proposed dcgan modification proposed lsgan lsgan proposed replacing cross entropy loss original gan linear least square loss higher quality results stable training style loss addition generating right content following input sketch would also like propagate texture details given input texture patch figure texturegan pipeline generative network trained directly transform input photo realistic texture details red arrows indicate losses active texture fine tuning see text detailed description various losses previous feature adversarial losses sometimes struggle capture details since focus getting overall structure correct similar deep learning based texture synthesis style transfer work use style loss specifically encourage reproduction texture details apply style loss channel adopt idea matching gram matrices feature correlations features extracted certain layers pretrained classification network gram mal trix gij rnl xnl defined gij fik fjk number feature maps network layer fik activation ith filter position layer use two layers network define style loss pixel loss find adding relatively weak pixel loss channel stabilizes training leads generation texture details faithful user input texture patch color loss losses applied channel output focus generating sketchconforming structures realistic shading sharp highfrequency texture details enforce user color constraints add separate color loss penalizes difference channels generated result combined objective function defined wadv ladv exact training details found section external texture one problem training images hard network focus reproducing lowlevel texture details due difficulty disentangling texture content within image example necessarily training examples object different textures applied might help network learn factorization structure texture also gram style loss dominated feature loss since optimized image much room network creative hallucinating texture details since tends focus generating structure color patterns finally many texture patches contain smooth color gradients without rich details trained solely network likely ignore hints unseen input texture patch test time especially texture hint conflicts information sketch result network often struggles propagate texture details results especially textures rarely seen training train network propagate broader range textures network reproduce propagate textures ground truth output introduce new local texture loss adapt existing losses encourage faithfulness texture rather faithfulness ground truth output object photo use losses introduced previous sections except global style loss keep feature adversarial losses ladv unchanged modify pixel color losses compare ated result entire input texture input texture patches extracted prevent color texture bleeding losses applied foreground object approximated segmentation mask section texture loss encourage better propagation texture propose texture loss applied small local regions output image handbags shoes utilize foreground background separation section locate foreground object apply texture loss within foreground clothes use detailed segmentation clothes items apply texture loss within semantic region input patch placed texture loss composed three terms wadv ladv local discriminator loss ladv introduce patchbased adversarial loss decides whether pair texture patches texture train discriminator recognize pair cropped patches texture positive example pair patches different textures one input texture one generated result negative example define ladv follows ladv dtxt use indicate fake indicate real local style loss pixel loss strengthen texture propagation also use gram style loss pixel loss cropped patches randomly sample two patches size generated result input texture separate texture database compute texture loss channel patches average propagate gradients texture loss corresponding patch region output performing texture network trying adapt understand propagate new types textures might forget learnt pretraining stage therefore training external textures mix iterations training fifty percent time final objective function becomes wadv ladv training setup train texturegan three datasets handbags shoes clothes photo collection contains large variations colors materials patterns domains also chosen demonstrate plausible product design applications supervised training need generate input output image pairs output network convert photos lab color space input network process photos extract images five channels include one channel binary sketch two channels texture intensities binary location masks two channels color controls section describe obtain segmentation masks used training generate input channels utilize separate texture database network finetuning also provide detailed training procedures parameters segmentation mask local texture loss hope encourage samples output texture match samples input texture output texture localized particular image regions interior objects want compare background patch input texture therefore sample patches fall inside estimated foreground segmentation mask handbag shoe datasets product images consistent white backgrounds simply set white pixels background pixels clothes segmentation mask already given dataset clothes segmentation mask process photos white background note segmentation masks used test time data generation sketch generation handbags shoes generate sketches using deep edge detection method used clothes leverage clothes parsing information provided dataset apply canny edge detection clothing segmentation mask extract segment boundaries treat sketch also apply xdog clothes image obtain variation training sketches finally mix additional synthetic sketches generated using methods proposed scribbler texture patches generate input texture constraints randomly crop small regions within foreground objects images randomly choose patch location within segmentation randomize patch size convert texture patch lab color space normalize pixels fall range image randomly generate one two texture patches clothes extract texture patches one following regions top skirt pant dress bag pass binary mask network indicate spatial support texture figure effect proposed local losses results model without local losses local pixel loss local style loss local texture discriminator loss local discriminator loss network tends produce consistent texture throughout object figure effect texture loss adversarial loss network trained using proposed losses effectively propagate textures foreground region removing adversarial loss leads blurry results removing texture loss harms propagation textures data generation encourage diverse faithful texture reproduction texturegan applying external texture patches texture dataset queried leather google manually filtered results high resolution leather textures clean dataset sampled roughly crops size image generate dataset textures train models textures since commonly seen materials handbags shoes clothes contain large appearance variations challenging network propagate training details use following parameters datasets wadv use adam optimizer learning rate optimize losses time use different weight settings wadv wadv also decrease learning rate train models input resolution except one clothes model resolution figure results discussions ablation study keeping settings train networks using different combinations losses analyze influence result quality figure given input sketch texture patch color patch first column network trained complete objective function second column correctly propagates color texture entire handbag turn texture loss fourth column texture details within area input patch preserved difficult textures fully propagated rest bag turn adversarial loss third column texture synthesized texture consistent input texture ablation experiment confirms style loss alone sufficient encourage texture propagation motivating local patchbased texture loss section external texture results train texturegan three datasets shoes handbags clothes increasing levels structure complexity notice object categories like shoes contain limited structure variations network able quickly generate realistic shading structures focus remaining capacity propagating textures texture propagation shoes dataset works well even without external texture sophisticated datasets like handbags clothes external texture critical propagation difficult textures contain sharp regular structures stripes figure demonstrates external texture proposed texture loss improves texture consistency propagation ground truth model faithful input texture patch output directly patch propagate throughout foreground region network texture examples enforcing local style loss local pixel loss local texture loss nudge network apply texture consistently across object local style loss column local texture discriminator loss column networks able propagate texture better without fine tuning column local pixel loss column using local texture discriminator loss tends produce figure results shoes handbags different textures odd rows input sketch texture patch even rows generated results resolution clothes dataset contains large variations structures textures image dataset contains multiple semantic regions network also handle multiple texture patches shown figure put different texture patches different parts clothes middle left bottom left network propagate textures within semantic regions sketch respecting sketch boundaries figure shows results handbags drawings differ synthetically generated training sketches results still high quality conclusion figure applying multiple texture patches sketch system also handle multiple texture inputs network follow sketch contours expand texture cover sketched object figure results sketches visually similar result input texture style loss figure shows results applying various texture patches sketches handbags shoes results typical result quality figure shows results clothes dataset trained presented approach controlling deep image synthesis input sketch texture patches system user draw object structure sketching precisely control generated details texture patches texturegan allows users see effect edits real time training texturegan local texture constraints demonstrate effectiveness sketch image synthesis texturegan also operates lab color space enables separate controls color content furthermore results fashion datasets show pipeline able handle wide variety texture inputs generates texture compositions follow sketched contours future hope apply network complex scenes references barnes shechtman finkelstein goldman patchmatch randomized correspondence algorithm structural image editing acm transactions graphicstog bergmann jetchev vollgraf learning texture manifolds periodic spatial gan arxiv preprint chen ming cheng tan shamir min internet image montage acm siggraph asia dosovitskiy springenberg brox learning generate chairs convolutional neural networks corr efros freeman image quilting texture synthesis transfer proceedings annual conference computer graphics interactive techniques pages acm efros leung texture synthesis nonparametric sampling computer vision proceedings seventh ieee international conference volume pages ieee fang hart textureshop texture synthesis photograph editing tool acm trans gatys ecker bethge texture synthesis using convolutional neural networks advances neural information processing systems pages gatys ecker bethge image style transfer using convolutional neural networks ieee conference computer vision pattern recognition cvpr pages june gatys ecker bethge hertzmann shechtman controlling perceptual factors neural style transfer arxiv preprint goodfellow mirza ozair courville bengio generative adversarial nets advances neural information processing systems pages van lier van gerven convolutional sketch inversion proceeding eccv workshop visart computer vision meets art hays efros scene completion using millions photographs acm transactions graphics tog volume page acm hertzmann jacobs oliver curless salesin image analogies proceedings annual conference computer graphics interactive techniques pages acm huang belongie arbitrary style transfer realtime adaptive instance normalization arxiv preprint iizuka ishikawa globally locally consistent image completion acm transactions graphics proc siggraph isola zhu zhou efros translation conditional adversarial networks arxiv preprint jetchev bergmann vollgraf texture synthesis spatial generative adversarial networks arxiv preprint johnson alahi perceptual losses style transfer european conference computer vision pages springer kingma adam method stochastic optimization arxiv preprint kingma welling variational bayes arxiv preprint lalonde hoiem efros rother winn criminisi photo clip art acm transactions graphics siggraph august larsson maire shakhnarovich learning representations automatic colorization european conference computer vision eccv lassner gehler generative model people clothing proceedings ieee international conference computer vision wand precomputed texture synthesis markovian generative adversarial networks european conference computer vision pages springer fang yang wang yang diversified texture synthesis networks arxiv preprint liang liu shen yang liu dong lin yan deep human parsing active template regression pattern analysis machine intelligence ieee transactions dec liang shen yang liu tang lin yan human parsing contextualized convolutional neural network proceedings ieee international conference computer vision pages liu qin luo wang cartoon image generation sketch using conditional generative adversarial networks arxiv preprint liu luo qiu wang tang deepfashion powering robust clothes recognition retrieval rich annotations proceedings ieee conference computer vision pattern recognition cvpr liu yan luo wang tang fashion landmark detection wild european conference computer vision eccv mao xie lau wang smolley least squares generative adversarial networks arxiv preprint mcmillan bishop plenoptic modeling imagebased rendering system proceedings annual conference computer graphics interactive techniques pages acm odena olah shlens conditional image synthesis auxiliary classifier gans arxiv preprint radford metz chintala unsupervised representation learning deep convolutional generative adversarial networks arxiv preprint reed akata yan logeswaran schiele lee generative adversarial text image synthesis proceedings international conference machine learning volume sangkloy fang hays scribbler controlling deep image synthesis sketch color computer vision pattern recognition cvpr simonyan zisserman deep convolutional networks image recognition proceedings international conference learning representations iclr ulyanov lebedev vedaldi lempitsky texture networks synthesis textures stylized images int conf machine learning icml wei levoy fast texture synthesis using treestructured vector quantization proceedings annual conference computer graphics interactive techniques pages acm publishing kyprianidis olsen xdog extended compendium including advanced image stylization computers graphics xie edge detection proceedings ieee international conference computer vision yang lin shechtman wang image inpainting using neural patch synthesis ieee conference computer vision pattern recognition cvpr july yoo kim park paek kweon pixellevel domain transfer european conference computer vision pages springer grauman visual comparisons local learning proceedings ieee conference computer vision pattern recognition pages zhang dana generative network transfer arxiv preprint zhang isola efros colorful image colorization eccv zhang zhu isola geng lin efros image colorization learned deep priors acm transactions graphics tog zhu shechtman efros generative visual manipulation natural image manifold proceedings european conference computer vision eccv
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feb minimax density estimation growing dimension daniel mcdonald department statistics indiana university bloomington dajmcdon version march abstract paper presents minimax rates density estimation data dimension allowed grow number observations rather remaining fixed previous analyses prove lower bound gives rate standard classes smooth densities show kernel density estimators achieve rate also give oracle choices bandwidth derive fastest rate grow maintain estimation consistency introduction convincing argument use sparsity structural priors machine learning statistics often begins discussion curse dimensionality donoho unmistakable evidence curse simply demonstrated fundamental scenario density estimation best estimator squared error order given independent observations dimensions striking contrast parametric rate even moderately large fixed accurate estimation requires significantly data small fact show allowed increase estimation accuracy degrades even quickly rate indicates first may seem allowing grow rather strange scenario use triangular array asymptotics exceedingly common theory estimation theoretical results lasso beginning least greenshtein ritov regularly adopt framework allowing number predictors grow van geer introduce idea beginning foundational text widely adopted literature regularized linear models belloni bickel meinshausen nardi rinaldo zhang framework marginal distribution predictors support whose dimension increasing scenario regression dimension increase quickly often order long dimensions irrelevant predicting response extension results linear models scenario studied mainly case generalized sparse additive models ravikumar yuan zhou allow predictor specific long final predictions merely additive across dimensions fully nonparametric regression without additivity assumption completely ignored outside framework although natural extension work presented another motivation appropriating triangular array framework density estimation burgeoning literature manifold estimation genovese talwalkar given data natural assumption data supported manifold embedded space estimating manifold possible may also wish estimate density regression function supported manifold recent work focused density estimation dimension manifold fixed known asta bhattacharya dunson hendriks pelletier extension results manifolds growing dimension missing extension presumes minimax framework present extended manifolds pointed reviewer short answer yes lower bound derive applies immediately modification need relates upper bound kernel depend metric given manifold rather euclidean distance use specific application setting would fmri data given sequence fmri scans patient researchers seek estimate dependence cubic centimeter voxels bullmore sporns scan contain order voxels number scans one individual smaller much estimate dependence voxels data averaged small number regions estimate dependence standard methods assume everything multivariate gaussian estimate covariance precision matrix gaussian assumption tested without density estimates using results could estimate smooth densities number scans grows would want increase number regions work illustrates quickly number regions grow remainder section introduces statistical minimax framework discusses specific data generating model examine details notation presents background estimator use achieves minimax rate gives short overview related literature section give main results discuss implications specifically obtaining fastest rate grow yield estimation consistency section gives proof lower bound possible estimators proof matching upper bound kernel density estimator given section finally discuss results section provide related results loss functions suggest avenues future research minimax framework order evaluate feasibility density estimation triangular array use statistical minimax framework situation framework begins specific class possible densities willing consider provides lower bound performance best possible estimator class bound hand quantified difficulty problem find estimator achieves bound possibly constants confident estimator performs nearly well possible given class densities thus minimax framework reveals gaps proposed estimators limits possible inference course bounds fail match know whether loose estimator poor model notation specify following setting density estimation triangular array suppose independent common density class define notational convenience generally suppress dependence clear specifying model assume relationship sequence densities rather seek understand limits estimation correspondence thus seek results behavior also employ following notation given characterize vectors let xsdd define let denote largest integer strictly less throughout use positive constants whose values may change depending context even fixed clear density estimation impossible allow reason restrict class densities willing allow definition nikol skii class let isotropic nikol skii class set functions sense adversary choose density give finite amount data estimators perform arbitrarily poorly iii partial derivatives exist whenever definition essentially characterizes smoothness densities natural way shown easily nikol skii class generalizes sobolev classes similar conditions see tsybakov kernel estimator given sample kernel density estimator point given nhd consider certain functions definition say isotropic kernel order satisfying standard case epanechnikov kernel satisfies conditions often default software gaussian kernel also member class kernel must take negative values possibly resulting negative density estimates although using estimator eliminates pathology without affecting results kernels constructed using orthonormal basis see tsybakov intuition estimator seen smooth generalization histogram density estimator uses local information rather fixed bins thus believe density smooth using smoothed version natural another way see observe kernel estimator convolution empirical density function defined implicitly via using empirical density unbiased estimator true density satisfies central limit theorem fixed adding bias kernel may able reduce variance achieve lower estimation risk densities match kernel certain way work chosen simplicity use isotropic kernels isotropic nikol skii class densities basically densities degree smoothness directions true kernels satisfy definition allowing anisotropic smoothness natural extension although notation becomes complicated quickly anisotropic case asymptotics see example goldenshluger lepski related work density estimation minimax framework problem many meaningful contributions last six decades pretend give complete overview recent advances tend build one four frameworks support smoothness whether loss adapted nature smoothness whether estimator adapt different degrees smoothness comprehensive overview concerns excellent resource goldenshluger lepski presents results adaptive estimators classes varying smoothness loss necessarily adapted smoothness also contextualizes compares existing work previous results similar present terms function classes losses see hasminskii ibragimov important work given devroye goldenshluger lepski juditsky kerkyacharian unlike density estimation setting related results information theory literature endeavor address limits estimation triangular array essentially work examines estimation joint distribution random variables observed sequence ergodic process supported finite set points marton shields show grows like log joint distributions estimated consistently extension results case markov random field embedded higher dimension given steif results slightly slower see corollary estimating continuous densities rather finitely supported distributions difficult main results main results give rates density estimation growing dimension generalizes existing results fixed recover usual rate deriving minimax rate density estimation requires two components finding risk best possible estimator hardest density class exhibiting estimator achieves risk results rate minimax upper lower bounds match constants may different first present lower bound proof given section theorem lower bound density estimation choose inf sup function infimum estimators result says exists triangular array densities best risk hope achieve possible estimators specific constant well minimum properties proof technique forms really relevant except independent specifically standard normal density standard deviation chosen small perturbation make explicit one could make choices worst case density result different values also note constant equation remainder paper quantifies smoothness class second result shows oracle choice bandwidth kernel density estimators achieve rate density risk kernel density estimator optimal proof given section theorem upper bound kernels let let isotropic kernel order satisfies take finally take constant large enough sup results far finite sample bounds nonetheless depend however also wish know quickly increase estimation risk still zero asymptotically estimation consistency clearly hope kernel density estimators consistent must increase quite slowly corollary log log sup fbh lambert function implicitly defined inverse exp large one show using series expansion log log log log log log essentially require grow slightly slower log information theoretic rate estimating finite distributions sample ergodic process see section stated main theorems terms expectations analogous bounds derived similarly without extra effort lower bound density estimation technique use finding lower bound rather standard idea convert problem density estimation one hypothesis testing proceeds first noting probability error exceeds constant lower bound risk reduce lower bound searching finite class rather possible densities finally ensure sufficiently many members class difficult distinguish true density relative previous techniques minimax lower bounds density estimation main difference proof must choose different members finite class right dependence construction make use divergence definition divergence divergence two probability measures log else derivatives respect dominating measure replace distributions densities integrate respect long divergence true density alternatives small average difficult discriminate therefore probability falsely rejecting true density large following lemma makes process explicit lemma tsybakov let monotone increasing let choose elements class show show log inf sup inf denotes infimum estimators constant depending use result first choose base density alternative densities show densities sufficiently take finally show alternatives uniformly small therefore small average proof use though discussed following proof choices monotone increasing functions simply modify conclusion proof order get right rate need choose base density series small perturbations create large collection alternatives getting perturbations right size allow sufficiently many main trick derive tight bounds case choice described effect multiplicative dependence turns necessary deviation existing lower bounds determining appropriate modification exercise even seemingly minor one enough compel complete overhaul proof densities define let satisfy standard gaussian density iii exist many functions satisfying conditions exp since infinitely continuously differentiable define integer let decreasing note finally take binary vector show densities density infinitely differen tiable choose also functions intervals form sup sup sufficiently smooth triangle inequality long density first remains show smallest value taken interval adding perturbations inf sufficient require smaller therefore require sufficient separation alternatives hamming distance binary vectors use collection alternatives need know many collection far enough apart following theorem tells size collection lemma tsybakov let subset densities exp restrict collection densities corresponding set constant likelihood ratio distributions density density log log therefore must choose large enough log equivalent requiring equivalent completing result combining results previous two sections gives following lower bound density estimators increasing dimensions proof theorem choose integer convenience define note following functions densities construction therefore log since therefore conditions lemma satisfied note lemma actually allows general lower bounds immediate consequences presented particular free choose distances may take powers norms apply functions example gives standard lower bound error pursue generalities however finding matching upper bounds often difficult requiring specific constructions combination deriving lower bounds also interest although requires complicated proof techniques case actually fairly straightforward extension discuss briefly section upper bound kernels prove theorem first use triangle inequality decompose loss bias component variance component efbh efh give two lemmas bound components separately bias need well known preliminary result lemma minkowski integral inequality let measure spaces let appropriate modifications lemma let let isotropic kernel order efbh bias proof technique depends smoothness density well smoothness kernel also holds proof taylor theorem hds since kernel order lower order polynomials equal applying lemma twice next find upper bound variance component result depend smoothness density properties kernel however depend strongly finally note result ignore outer expectation lemma let function satisfying probability density efh nhd proof proof easy generalization proposition masaon omitted intuition simply present case results hand prove theorem proof theorem applying lemma lemma gives sup fbh nhd taking balances terms gives result discussion paper developed first results density estimation triangular array asymptotics number observations ambient dimension allowed increase results generalize existing minimax results fixed rather increasing would recover previously known minimax rates lower upper bounds results also show kernel density estimators minimax optimal come surprise since minimax optimal fixed results presented paper say essentially large enough exist constants independent large enough inf sup sup fbh fbh kernel density estimator oracle result immediately result longer proofs generalize result case another extension case picks factor log numerator rate techniques used could also give results nonparametric regression triangular array asymptotics given pairs kernel regression written terms densities joint marginal densities respectively results kernel estimator gbh obtained similar proof techniques presented related extension would consider problem conditional density estimation directly using similar form qbh estimates conditional density estimator shown converge rate appropriate smoothness assumptions see hall results also suggest open questions wavelet density estimators projection estimators known fixed upper bounds match kernels though constants may larger smaller whether methods also match increasing remains seen class densities examined usually slightly different histograms also useful density estimators fixed minimax lipschitz densities slower rate kernels class allowable densities different upper bounds triangular array similar form presented shown mcdonald deriving minimax lower bounds class remains open problem extending results manifold setting mentioned obvious path toward fast rates large left future work acknowledgements material based upon work supported national science foundation grant institute new economic thinking grant author thanks anonymous referees program committee international conference artificial intelligence statistics insightful comments cosma shalizi comments early draft references asta nonparametric density estimation hyperbolic space neural information processing systems nips workshop modern nonparametric methods machine learning eds gretton kolar kpotufe lafferty liu belloni chernozhukov wang lasso pivotal recovery sparse signals via conic programming biometrika bhattacharya dunson nonparametric bayesian density estimation manifolds applications planar shapes biometrika bickel ritov tsybakov simultaneous analysis lasso dantzig selector annals statistics van geer statistics data methods theory applications springer new york bullmore sporns complex brain networks graph theoretical analysis structural functional systems nature reviews neuroscience devroye nonparametric density estimation view john wiley sons new york donoho data analysis curses blessings dimensionality ams conference math challenges century genovese verdinelli wasserman minimax manifold estimation journal machine learning research genovese verdinelli wasserman manifold estimation singular deconvolution hausdorff loss annals statistics goldenshluger lepski bandwidth selection kernel density estimation oracle inequalities adaptive minimax optimality annals statistics goldenshluger lepski adaptive minimax density estimation probability theory related fields greenshtein ritov persistence linear predictor selection virtue overparametrization bernoulli hall racine estimation conditional probability densities journal american statistical association hasminskii ibragimov density estimation view kolmogorov ideas approximation theory annals statistics hendriks nonparametric estimation probability density riemannian manifold using fourier expansions annals statistics juditsky minimax density estimation bernoulli kerkyacharian picard tribouley adaptive density estimation bernoulli marton shields entropy consistent estimation joint distributions annals probability correction annals probability masaon risk bounds kernel density estimators journal mathematical sciences mcdonald shalizi schervish estimating coefficients proceedings fourteenth international conference artificial intelligence statistics eds gordon dunson vol jmlr mcdonald shalizi schervish estimating coefficients via histograms electronic journal statistics meinshausen relaxed lasso computational statistics data analysis nardi rinaldo asymptotic properties group lasso estimator linear models electronic journal statistics pelletier kernel density estimation riemannian manifolds statistics probability letters ravikumar liu lafferty wasserman spam sparse additive models advances neural information processing systems eds platt koller singer roweis mit press cambridge ravikumar lafferty liu wasserman sparse additive models journal royal statistical society series statistical methodology steif consistent estimation joint distributions sufficiently mixing random fields annals statistics talwalkar kumar rowley manifold learning ieee conference computer vision pattern recognition ieee tsybakov introduction nonparametric estimation springer verlag zhang rate minimaxity lasso dantzig selector loss balls journal machine learning research yuan zhou minimax optimal rates estimation high dimensional additive models universal phase transition
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forward link interference mitigation mobile interactive satellite systems mar henarejos cocco centre telecomunicacions catalunya castelldefels barcelona spain universitat catalunya barcelona spain present results performance evaluation coding soft interference cancellation satellite systems interference realistic setups standard broadband global area network service bgan considered reference physical layer realistic interference channel models adopted work carried framework next generation waveform increased spectral ciency ngwise project founded european space agency esa introduction interactive satellite systems key communication solution huge potential market possible applications range provision data connectivity areas cellular connection pro table rural infeasible maritime aircraft scenarios backing emergency situations especially involving large geographical areas far limited broadcast transmissions mentioned applications rely multicast multiple unicast connections framework satellites provide increase spectral ciency respect global beam satellites especially case architectures low frequency reuse factor even higher throughput provided principle leverage polarization reusing adjacent beams transmission polarizations within single beam however adoption aggressive reuse schemes implies increase interference due satellite antenna low directivity user terminal antennas polarization mismatch due antennas imperfections atmospheric propagation moreover mobile terminals also keen propagation impairment shadowing fading large propagation delay typical satellite systems especially geo prevents availability channel state information csit time diversity largely exploited today interactive mobile systems standards overcome channel impairments mobile broadcast systems however new diversity techniques recently gained interest techniques based polarization spatial diversity allow apply multiple output mimo techniques precoding codes interference cancellation techniques also potential solution currently looked forward reverse link dual polarization transmission evaluated mobile broadcast scenario promising results joint outdated csit time variability channel makes cult use linear precoding applied previous works apart increase system diversity dual polarization transmission provide increase spectral ciency especially case low interference research engineer communication systems research engineer communication systems mavazquez researcher communication systems gcocco professor signal theory communications department upc american institute aeronautics astronautics senior researcher cttc present paper present part results obtained within next generation waveform increased spectral ciency ngwise project founded european space agency esa speci cally evaluate impact dual polarization transmission terms throughput presence cochannel interference satellite system evaluate possibility applying soft interference cancellation sic setups realistic channel model adopted scenarios considered namely maritime terrestrial standard adopted broadband global area network service bgan used reference standard physical layer rest paper refer bgan standard simplicity results show higher spectral ciency achieved considered techniques may lead increased overall system throughput also show frequency reuse scheme soft interference cancellation provides limited gain terms block error rate due low relative power high number interferers assimilated gaussian noise system model let consider forward link geostationary satellite communication system channel interference among adjacent beams mitigated frequency reusing four colors frequency reuse scheme considered following dual polarization transmission assumed satellite user terminal antennas transmits receive two orthogonal polarizations respectively received signal user terminal expressed block length code depends symbol assume general complex symbol mapping interference modelled matrix matrix accounts antennas characteristics terms gains channel matrix distribution depends scenario interference rejection vector hbc hbj represents received signal two time instants two polarizations transmitted power thermal noise taken account term whose entries zero mean gaussian random variables variance often case terrestrial communications also satellite context use mimo techniques combined channel coding leading called modulation figure depicts general block diagram scheme block picture represent coded bit interleaver deinterleaver respectively figure equivalent scheme considered dual polarization setup scenario main challenges faced two first mimo transmission scheme symbols transmitted two polarizations must decided secondly mimo demodulator symbols two polarizations detected needs also obtained reference signal signal received user terminal interfering signals bgan standard adopted bgan standard currently nition designed support voice broadband data services wide range scenario maritime land mobile channel code adopted bgan standard turbo code several possible con gurations terms code rate codeword length possible combinations channel code parameters modulation qpsk symbol rate physical layer characteristics ned iii transmission reception schemes present section consider several possible mimo modulation demodulation solutions previously proposed literature suited application considered setup one modulation one demodulation scheme selected based practical well theoretical considerations american institute aeronautics astronautics mimo transmission scheme alamouti objective code obtain maximum diversity gain constructively adding channel gains two polarizations rejecting interference isi case coding matrix cala detection scheme known low complexity reduces matrix multiplication set comparisons optimally decoding code symbol obtains diversity gain diagonal elements note equivalent siso gain obtained detection used polarization multiplexing polarization multiplexing obtains full multiplexing gain two time instants symbols transmitted channel use antenna coding matrix cmul case golden code full diversity technique still provides coding coding matrix constructed follows cgol demodulators low complexity detector mimo schemes presented hard decision detector decision rule consists solving following optimization problem arg min note detected symbol obtained via hard decision however channel code usually included mimo schemes thus mimo demodulator output ratios llr coded symbols passed decoder log llr coded symbol probability mass function coded bits conditioned channel output channel matrix order reduce complexity approximation applied little loss terms performance following consider schemes aim decreasing demodulator complexity order reduce number demodulators test refer recommendations several demodulators studied considering mutual information measurement low data rates mean square error mmse shown demodulator outperforms designs extension technique propose soft version receiver following brie describe schemes well optimal solution uncoded case linear equalizer obtained minimization mean square error mse expression gmmse american institute aeronautics astronautics detector soft version demodulator technique iteratively decodes subtracts isi order adapt setup substitute hard decision interference soft one resulting algorithm following find least faded polarization min components vector ski obtain mmse estimate signal diag hhh transmitted polarization gmmse subtract interference due symbol remove received signal time instant component column repeat polarization system design chose one scheme applied scenario among presented far choice trade theoretical considerations practical constraints either dictated bgan standard practical considerations complexity user terminal transmission scheme polarization multiplexing shown previous esa projects method outperforms alamouti one considering given transmit power another possible choice would use golden codes according preliminary results obtained shows improve around fer respect alamouti increase computational complexity however golden codes imply symbols mixed together two time instants makes detection computationally demanding keeping mind study case low complexity receiver asset considered enhancement terms fer justify increase complexity detection scheme described scheme outperforms demodulators presents good performance higher spectral ciency region matter facts observed gures demodulator performs slightly better low data rates higher data rates observed performs better soft interference cancelation gains potentially delivered schemes may interference coming beams depending interference strength rst approximation accurate number interferers large interference assimilated background noise dealt using mimo techniques presented section iii interference cancelation may help case rst classi cation interference cancelation methods done distinguishing hard hic soft interference cancelation sic hic one signals usually strongest one decoded treating others noise subtracted received waveform scheme relatively simple drawback signal subtracted decoded correctly error propagation severely limit performance system sic methods consist soft estimation transmitted signals followed decoding phase estimation taken account decoder examples found following consider iterative sic scheme depicted fig case two received signals one reference signal one interferer siso channel scheme received waveform fed soft estimator performs detection estimates transmitted channel symbols signals estimation performed using turbo decoder american institute aeronautics astronautics figure bler receiver center coverage maritime scenario figure bler receiver center coverage maritime scenario figure iterative sic scheme american institute aeronautics astronautics modi output soft estimates considered signal iterated number niter times decision taken symbols signals method easily extended dual polarization case especially case high cross polarization rejection numerical results section present performance evaluation results selected mimo scheme two scenarios namely maritime land mobile satellite intermediate tree shadowing bgan standard operating adopted reference physical layer consider maritime scenario rst channel model described following table fast fading rician rician factor doppler shift taps table channel parameters focus bearers types symbol rate speci cally focus bgan bearer types characterized qpsk modulation code rate described following table table code rate bearer types bearer name coding rate data rate kbps depending geographical location user terminal coverage beam may vary signi cantly due interference order take account rst derived noise value expression lkbt radiated power bandwidth antenna gain receiver boltzmann constant antenna noise temperature receiver array factor transmitter hence derive noise power bgan standard common user terminal parameters thus noise power khz remains dbm note simulation constant depends position user terminal changes benchmark system consider one single polarization considered siso system psiso assumed systems order evaluate advantage deriving using polarizations use normalized throughput ned follows total power per beam normalized throughput throughput dual polarization throughput siso throughput fer rate american institute aeronautics astronautics rst simulations assume correlation polarizations given interference known low band frequency reuse factor performance method assumed evaluate beams representing best worst case scenarios center edge beam coverage assuming user location center beam edge realistic beam patterns considered interference beams frequency reference one taken account indicate ratio power reference signal total interference power polarization beam gain values taken perfect channel state information receiver csir assumed figures show throughput proposed scheme maritime channel normalized throughput benchmark siso system seen given code rate scheme able double rate expenses incrementing transmit power impact system remarkable since considering beam end coverage power increment needed get twice siso throughput increased also notice bearers higher channel code rate succeed achieving gain order enhance throughput high rate bearers figure normalized throughput versus transmit power scheme receiver center coverage evaluated performance sic scheme considered section single polarization setup multibeam satellite system frequency reuse edge beam edge coverage using interference pattern gure considered best case scenario awgn perfect symbol alignment across signals strongest interferer relative taken account others much smaller power would determine increase complexity limited gain terms bler results simulations shown fig seen sic gives marginal gain even best case scenario due strong power unbalance reference signal interferers matter facts snr good signal starts decodable stronger interferer weak contribute decoding thus almost observed bler simulations carried reported matter space showed case relative strong interferer reference signal comparable power sic signi cantly enhance system bler certain bearers case instance systems aggressive frequency reuse factors thus conclude considered setup sic bring signi cant improvements hence rest paper consider sic techniques following present results obtained scenario mixed lms environment mix scenario scenario far challenging maritime one used real channel measurements obtained mix scenarios context esa mimosa project american institute aeronautics astronautics note figure normalized throughput versus transmit power scheme receiver edge coverage figure bler versus awgn case sic applied reference signal strongest interferer multibeam satellite system frequency reuse six strongest interferers considered three iterations used estimator bearer rate qpsk modulation used signals correlation implicitly taken account measurements fig part measured channel realization shown alternation periods moderate deep fading observed figures show normalized throughput mix channels seen certain throughput gain respect siso case achieved even considered channels loss terms tolerable note also interleaver included considered bearers inclusion time interleaver likely enhance performance system signi cantly allow exploit full potential dual polarization transmission following observations made given qpsk modulation determined code rate use polarization multiplexing jointly demodulator able double data rate expenses increasing transmit power best case terms maritime scenario worse situations considering challenging scenarios mix use dual polarization provide bgan standard includes use time interleaver msec depth higher order bearers american institute aeronautics astronautics figure channel realization real measurements components shown figure normalized throughput versus transmit power scheme receiver center coverage channel expected gain mainly due absence time interleaver fact results obtained low complexity demodulator makes application considered techniques appealing practical perspective system level perspective clear rate increase obtained also considering cient modcods higher channel code rates higher order modulations evaluated numerically concluded use dual polarization best option considered setup since increase transmit power obtain twice throughput siso case less demanding one certain level bler illustrate compare ways double throughput increase may achieved using dual polarization increasing coderate increasing modulation order techniques normalized baseline scenario remarkable aspect fact spatial multiplexing dual polarization powerless achieve rate american institute aeronautics astronautics figure normalized throughput versus transmit power scheme receiver center coverage mix channel figure doubling throughput techniques comparison conclusions future work presented results study application codes soft interference cancellation multibeam satellite systems adopted bgan standard used realistic channel models interference patterns future work plan extend simulations higher order modulations investigate use single user precoding described current high data rate terrestrial standards techniques require feedback receiver impact system delay also studied use demodulator must investigated also mobile broadcast standards use long interleavers might increase performance system yet maintaining low complexity receiver another possible research line study systems aggressive frequency reuse factors using joint sic mimo techniques likely provide good results terms throughput system availability american institute aeronautics astronautics acknowledgements present work carried artes programme founded european space agency view expressed herein way taken ect cial opinion european space agency references maral bousquet satellite communications systems systems techniques technologies john wiley sons ltd andrews interference cancellation cellular systems contemporary overview ieee wireless communications apr arapoglou burzigotti bertinelli bolea alamanac gaudenzi mimo mimo mobile satellite broadcasting systems ieee transactions wireless communications ibars serra del coso gomez caus mimo applicability satellite networks international workshop signal processing space communications pages arapoglou liolis bertinelli panagopoulos cottis gaudenzi mimo satellite review ieee communications surveys tutorials european space agency next generation waveform improved spectral ciency http etsi etsi component umts family satellite radio interface part physical layer speci cations physical layer interface technical report european telecommunications standards institute alamouti simple transmit diversity technique wireless communications selected areas communications ieee journal wolniansky foschini golden valenzuela architecture realizing high data rates wireless channel signals systems electronics issse ursi international symposium pages bel ore rekaya viterbo golden code times code nonvanishing determinants information theory ieee transactions coding approaches multiple antenna transmission fast fading ofdm signal processing ieee transactions sellathurai guinand lodge coding mobile satellite communications using channels vehicular technology ieee transactions fertl jalden matz performance assessment demodulators based mutual information signal processing ieee transactions collings butler mckay low complexity receiver design mimo coded modulation spread spectrum techniques applications ieee eighth international symposium pages seethaler matz hlawatsch cient demodulator mimo coded modulation global telecommunications conference globecom ieee volume pages verdu multi user detection cambridge university press wang poor iterative turbo soft interference cancellation decoding coded cdma ieee transactions communications colavolpe fertonani piemontese siso detection linear channels linear complexity number interferers ieee journal selected topics signal processing esa technical annex technologies models requirements work mss technical report european space agency american institute aeronautics astronautics
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graph konrad vadim nov department computer science durham university lower mountjoy south road durham united kingdom mathematics institute university warwick coventry united kingdom abstract daligault rao asked whether every hereditary graph class induced subgraph relation bounded lozin razgon zamaraev jctb gave negative answer question counterexample class characterised many forbidden induced subgraphs raises issue whether question positive answer hereditary graph classes apart two stubborn cases two induced subgraphs forbidden one two stubborn cases namely triangle case proving class triangle graphs bounded technique based special decomposition graphs also use technique prove class triangle graphs known bounded cliquewidth results enable complete graphs class triangle graphs introduction graph class containment relation infinite sequence graphs pair contained graph class bounded exists constant every graph bounded highly desirable properties graph classes areas discrete mathematics theoretical computer science illustrate let mention seminal project robertson seymour graph minors culminated proof wagner conjecture states set finite graphs minor relation algorithmic consequence given graph class possible test cubic time whether given graph belongs class algorithmic importance bounded follows fact many problems graph colouring hamilton cycle become solvable graph classes bounded follows combining results several papers result oum seymour courcelle proved class graphs obtained graphs via one edge contractions unbounded hence research supported epsrc leverhulme trust extended abstract paper appeared proceedings graph much smaller minors hand graph least induced subgraphs see example therefore focus hereditary classes graph classes closed taking induced subgraphs readily seen class graphs hereditary characterised unique set minimal forbidden induced subgraphs underlying research goal increase understanding relation hereditary graph classes start note hereditary class graphs degree induced subgraph relation contains class cycles form infinite antichain every graph degree cliquewidth bounded imply induced subgraph relation daligault rao asked reverse implication every hereditary graph class induced subgraph relation bounded lozin razgon zamaraev gave negative answer set minimal forbidden induced subgraphs infinite question daligault rao remains open finitely defined hereditary graph classes hereditary graph classes finite conjecture finitely defined hereditary class graphs induced subgraph relation bounded conjecture true finitely defined hereditary graph classes aforementioned algorithmic consequences bounded also hold property induced subgraph relation hereditary graph class defined single forbidden induced subgraph induced subgraph relation bounded induced subgraph see instance hence conjecture holds size consider case size say graph classes said bigenic graph classes case conjecture also known true except two stubborn open cases namely see results section prove conjecture holds class graphs showing class graphs bounded induced subgraph relation using general technique explained section technique based extending labelled version boundedness bipartite graphs hereditary graph class special subclass graphs crucial property graphs three vertices three different partition classes form clique independent set say graphs curious restricted version concept used prove graphs bounded section show generalise results curious graphs whole class graphs section also show apply technique prove class graphs already known class graphs bounded cliquewidth note results also imply class graphs previously known see see also fig pictures forbidden induced subgraphs mentioned paragraph dichotomies previously known graphs bipartite graphs bipartite graphs fig forbidden induced subgraphs considered results also shown bipartite graphs contains induced moreover every class graphs finite due ramsey theorem results lead following known dichotomy bipartite graphs theorem let graph class bipartite graphs induced subgraph relation induced subgraph combining aforementioned known results graphs bipartite graphs new results yields following new dichotomy graphs exactly one theorem theorem let graph class graphs induced subgraph relation induced subgraph besides technique based curious graphs also expect theorem useful ingredient showing results graph classes theorem already proven useful see following dichotomy known bipartite graphs theorem let graph class bipartite graphs bounded induced subgraph would interesting determine whether graphs allow dichotomy respect boundedness evidence far affirmative order answer question two remaining cases need solved namely see section definition graph cases turn highly particular class graphs contains class graphs class graphs contains classes graphs section give summaries boundedness bigenic graph classes include new results together overview missing cases problems include missing cases mentioned section preliminaries throughout paper consider finite undirected graphs without multiple edges define graph terminology disjoint union two graphs denoted disjoint union copies graph denoted complement graph vertex set edge two distinct vertices subset let denote subgraph induced vertex set edge set simplify notation may also write instead use denote graph obtained deleting every vertex write indicate induced subgraph graphs denote cycle complete graph star path vertices respectively graphs also called triangle claw respectively graph linear forest every component path least one vertex graph denotes subdivided claw tree one vertex degree exactly three leaves distance respectively observe let denote class graphs connected component either subdivided claw path set graphs graph induced subgraph isomorphic graph may write instead graph vertex partitioned possibly empty independent sets graphs also known bipartite graphs biclique complete bipartite graph bipartite graph sets partition size respectively every vertex one set adjacent every vertex set graph set denotes neighbourhood let set vertices vertex complete adjacent every vertex every vertex set vertices complete resp every vertex complete resp disjoint sets vertices graph say edges two sets form matching vertex one neighbour vice versa vertex exactly one neighbour say matching perfect similarly edges sets form vertex one vice versa say set dominates every vertex least one neighbour similarly vertex dominates every vertex adjacent vertex distinguishes neighbour set module vertex distinguishes module otherwise trivial graph prime trivial modules two vertices false twins neighbourhood note vertices must clearly prime graph least three vertices contain pair false twins pair vertices would form module use following structural result lemma let connected graph contain pair false twins either bipartite cycle graph minimum number labels needed construct using following four operations creating new graph consisting single vertex label taking disjoint union two labelled graphs joining vertex label vertex label renaming label class graphs bounded constant every graph otherwise unbounded let graph define following operations induced subgraph subgraph complementation operation acting respect replaces every edge present vice versa similarly two disjoint vertex subsets bipartite complementation operation respect acts replacing every edge one one vice versa state useful facts operations ones influence graph use facts throughout paper let constant let graph operation say graph class graph class following two conditions hold every graph obtained graph performing times every exists least one graph obtained performing times say preserves boundedness finite constant graph class graph class bounded bounded fact vertex deletion preserves boundedness fact subgraph complementation preserves boundedness fact bipartite complementation preserves boundedness lemma let graph let set induced subgraphs prime quasi order set reflexive transitive binary relation two elements comparable otherwise incomparable set elements chain every pair elements comparable antichain every pair elements incomparable infinite sequence elements contains pair equivalently infinite strictly decreasing sequence infinite antichain arbitrary set let denote set finite sequences elements defines follows sequence integers bij call subsequence relation lemma higman lemma define notion labelled induced subgraphs let consider arbitrary quasiorder say labelled graph vertex equipped element label given two labelled graphs say labelled induced subgraph isomorphic induced subgraph isomorphism maps vertex vertex clearly graph class contain infinite sequence labelled graphs respect labelled induced subgraph relation therefore say graph class labelled induced subgraph relation contains infinite antichains labelled graphs whenever class readily seen also induced subgraph relation daligault rao showed every hereditary class graphs labelled induced subgraph relation defined finite set forbidden induced subgraphs korpelainen lozin razgon conjectured hereditary class graphs defined finite set forbidden induced subgraphs induced subgraph relation labelled induced subgraph relation brignall engen vatter recently found class forbidden induced subgraphs counterexample conjecture induced subgraph relation labelled induced subgraph relation however far known results bigenic graph classes including paper verify conjecture bigenic graph classes similarly notion preserving boundedness say graph operation preserves labelled induced subgraph relation finite constant graph class graph class relation lemma following operations preserve labelled induced subgraph relation subgraph complementation bipartite complementation iii vertex deletion lemma hereditary class graphs labelled induced subgraph relation set prime graphs particular labelled induced subgraph relation set connected graphs lemma bipartite graphs labelled induced subgraph relation let define cartesian product order set lemma implies also labelling elements say construct combined labelling labels vertex label lemma fix least one element let wella class graphs fix labelling labelled induced subgraph relation every every labelling graphs order combined labelling obtained labellings also results set labelled graphs proof labelled induced subgraph relation definition labelled labels combined labellings obtained labelled induced subgraph relation must infinite set graphs whose vertices labelled elements graphs form infinite labelled antichain graph replace label vertex graphs labelled elements result infinite labelled antichain graphs labelled combined labellings completes proof graphs integer graph symmetric square matrix order graph vertices graph class defined follows let disjoint union infinitely many copies let subset containing vertex copy construct infinite graph vertex set applying subgraph complementation applying bipartite complementation pair thus two vertices adjacent hereditary class consisting finite induced subgraphs minimum uniformicity second next two lemmas follows directly definitions following result proved korpelainen lozin class disjoint unions cliques counterexample reverse implication lemma class graphs bounded uniformicity labelled induced subgraph relation following lemma follows directly definition definition graphs see also general result lemma every graph partitioning graphs section first introduce graph decomposition graphs show extend results bounded labelled induced subgraph relation bipartite graphs arbitrary hereditary class graphs graphs class decomposable way section give sufficient conditions graph decomposition decomposition let graph given partition vertex set three independent sets suppose set partitioned sets taking subscripts modulo vij complete let say graphs slices slices belong graph class say partitioned slices see fig example fig graph partitioned slices isomorphic lemma graph partitioned slices max proof since every slice constructed using labels applying relabelling operations necessary may assume end construction every vertex receives label modify construction use labels instead way points construction every constructed vertex label replace creation operations relabel operations join operations modified construction uses labels end every vertex labelled label may every slice independently show use constructed slices construct using six labels way every vertex labelled label induction done induction hypothesis construct way consider copy constructed earlier relabel vertices using operations copy every vertex labelled next take disjoint union obtained graph constructed apply join operations finally apply relabelling operations constructs way every vertex labelled induction follows max lemma let hereditary graph class containing class let set graphs partitioned slices labelled induced subgraph relation proof graph may fix partition independent sets respect graph partitioned slices let every pair distinct elements incomparable lemma need consider labellings graphs form belongs arbitrary suppose partitioned slices vertices labelled slices along labellings completely describe edges suppose another graph partitioned slices smaller subsequence relation induced subgraph result follows lemma curious graphs let graph given together partition vertex set three independent sets induced rainbow exactly one vertex set say curious respect partition contains rainbow vertex set partitioned way say curious partition respect curious section prove given hereditary class bipartite graphs labelled induced subgraph relation bounded true curious graphs linear order vertices independent set increasing implies decreasing implies monotone either increasing decreasing bipartite graphs also known bipartite chain graphs well known easy verify bipartite graph vertices independent set bipartition admit monotone ordering suppose curious graph respect partition say respect partition graph curious graph type exactly graphs contain induced curious graphs type note curious graph type respect partition without loss generality may assume lemma let curious graph respect vertices admit linear ordering decreasing increasing proof set use denote set vertices adjacent may choose linear order vertices according neighbourhood breaking ties according neighbourhood order clearly ordering decreasing suppose contradiction order increasing must indices property property follows means vertices adjacent rainbow rainbow contradiction implies order indeed decreasing completes proof lemma curious graph type respect partition partitioned slices bipartite proof let linear order satisfying lemma let let partition follows let let particular note vertices complete respectively following properties hold complete complete rainbow contradiction adjacent rainbow contradiction follows complete let properties edges sets vji show partitioned slices every slice bipartite completes proof curious graphs type lemma fix curious graph type respect partition partitioned slices type proof fix let curious graph type respect partition may assume contains induced claim given every vertex exactly two neighbours neighbours either lie lie let induce adjacent adjacent consider adjacent rainbow contradiction therefore one neighbour one neighbour rainbow contradiction therefore one similarly one therefore adjacent must must adjacent must similarly must adjacent must must adjacent hence claim follows consider maximal set sets induce copies say vertex distinguishes two graphs neighbours belong set group sets blocks distinguished vertex words every contains least one every vertex complete one sets let bji define relation blocks follows holds complete distinct blocks one hold claim let distinct blocks vertex differentiates complete see also fig complete definition blocks must vertex distinguishes without loss generality may assume complete anticomplete remains show adjacent rainbow contradiction rainbow contradiction therefore complete follows claim follows symmetry fig two blocks vertex claim relation transitive suppose since distinct must vertex distinguishes combining claim fact follows must complete suppose adjacent since adjacent choice must otherwise would rainbow therefore distinguishes claim follows completes proof claim combining claims find linear order blocks obtain following conclusion call chain property claim set blocks admits linear order complete exists complete complete next consider set vertices belong set let denote set note maximality set let claim neighbour complete neighbour complete suppose contradiction adjacent consider vertex distinguishes since claim implies complete adjacent rainbow rainbow follows neighbour complete claim follows symmetry claim allows update sequence blocks follows update procedure contains vertex neighj bour add sets bij remove claim applying update procedure preserves chain property see claim blocks assume chain property holds possibly applications update procedure without loss generality assume neighbour set case follows similarly show chain property continues hold adding removing recall every vertex neighbourhood definition first show neighbourhood vertices complete anticomplete rainbow contradiction similarly adjacent complete rainbow contradiction therefore must neighbourhood vertices claim means property chain property preserved apply update procedure vertex suppose must vertex differentiates claim vertex complete must otherwise would rainbow claim vertex complete must adjacent otherwise would rainbow conclude property chain property also preserved apply update procedure vertex symmetry induction completes proof claim claim may therefore apply update procedure exhaustively chain property continue hold procedure complete every remaining vertex either complete set fact claim know every vertex neighbour exists exists since complete set obtain following conclusion every vertex complete exists exists denote corresponding subset symmetry also obtain following every vertex complete exists exists denote corresponding subset also partition vertices subsets vertices complete complete complete complete claim complete complete suppose note vertices exist respectively empty first suppose choose arbitrary vertices note complete adjacent otherwise would rainbow must adjacent otherwise would rainbow suppose choose arbitrary vertices note complete must adjacent otherwise would rainbow must otherwise would rainbow completes proof claim let denote subgraph induced claims graph partitioned slices recall graph type contains induced since construction since original sequence maximal follows type furthermore since bipartite forms curious graph set empty type completes proof curious graphs ready state main result section theorem let hereditary class graphs set bipartite graphs labelled induced subgraph relation bounded cliquewidth property also holds set curious graphs proof suppose class bipartite graphs labelled induced subgraph relation bounded lemmas curious graphs type also property applying lemmas obtain conclusion curious graphs type applying lemmas obtain conclusion curious graphs type curious graphs completes proof applications technique section show class graphs class graphs bounded labelled induced subgraph relation first prove two structural lemmas lemmas consider case graph one classes contains induced subgraph isomorphic show use bounded number vertex deletions bipartite complementations transform graph disjoint union curious graphs graphs respective graph class use lemmas prove theorem main result first two lemmas implicit proofs lemma theorem without explicit upper bound number operations used number obtained curious graphs completeness give direct proof provide explicit bounds note bounds number vertex deletions bipartite complementations curious graphs lemmas tight upper bound numbers sufficient purposes lemma given connected graph contains induced apply vertex deletions bipartite complementation operations obtain graph disjoint union curious graphs proof let connected graph contains induced cycle vertices listed order along cycle aid notation remainder proof subscripts vertices sets interpreted modulo show use vertex deletions bipartite complementations partition graph disjoint union curious graphs end proof verify number operations used since every vertex cycle two neighbours two neighbours neighbours must nonconsecutive vertices cycle may therefore partition vertices eleven sets follows set vertices set vertices whose unique neighbour set vertices adjacent nonadjacent rest prove series claims claim may assume suppose contradiction two vertices neighbourhood set say adjacent note adjacent adjacent contradiction means every vertex neighbourhood every set every set since connected must vertex adjacent every vertex since must therefore independent set applying bipartite complementation vertices adjacent vertices disconnects rest graph since independent curious graph two three partition classes empty completes proof claim claim independent set indeed adjacent contradiction completes proof claim claim complete indeed contradiction claim follows symmetry claim complete suppose contradiction neighbour claim contradiction claim follows symmetry claim either complete suppose contradiction neighbour claim contradiction therefore every vertex complete suppose contradiction neighbour nonneighbour claim contradiction therefore every vertex complete completes proof claim claim every vertex complete either suppose contradiction claim contradiction completes proof claim claim partition set three possibly empty subsets follows set vertices dominate set vertices dominate set vertices dominate definition partition moreover every vertex vice versa thus vertices partitioned subsets may empty note sets pairwise claim furthermore therefore complete complete nearly every pair subsets either complete possible exceptions five disjoint pairs form next analyse edges subsets claim suppose contradiction neighbour definition must claim contradiction claim follows symmetry claim indeed contradiction completes proof claim claim neighbour complete neighbour complete suppose neighbour adjacent must otherwise would therefore complete note must adjacent otherwise would contradicting claim follows complete claim follows symmetry recall claim claim partition three possibly empty subsets follows set vertices complete set vertices complete set vertices therefore neighbours neighbours therefore therefore claim may assume first show remove edges vertices rest using three bipartite complementations claim claim definition claim complete claim either complete claim either complete either complete definition complete therefore applying one bipartite complementation remove edges one outside claim claim either complete claim complete definition complete definition complete claim complete definition complete therefore applying one bipartite complementation remove edges one outside symmetrically applying one bipartite complementation remove edges one outside thus apply three bipartite complementations disconnect rest graph claim independent sets therefore claim curious graph completes proof claim claim sets remain ones form prove following claim claim may assume first show remove edges vertices rest graph using finitely many bipartite complementations claim claim complete claim either complete either complete claim either complete complete claim definition complete therefore applying one bipartite complementation remove edges one outside symmetrically applying one bipartite complementation remove edges one outside either claim claim complete definition complete claim definition therefore applying complete one bipartite complementation remove edges one outside thus apply three bipartite complementations disconnect rest graph claim independent sets claim complete curious graph completes proof claim claim may assume contains five vertices cycle delete vertices complete proof remains verify number operations applied number obtained curious graphs claim apply one bipartite complementation obtain one curious graph claim apply three bipartite complementations obtain one curious graph claim apply three bipartite complementations obtain one curious graph finally apply five vertex deletions delete original cycle graph leads total five vertex deletions bipartite complementations obtained curious graphs prove second structural lemma lemma given prime graph contains induced apply vertex deletions bipartite complementation operations obtain graph disjoint union curious graphs one graph proof let prime graph contains induced cycle vertices listed order along cycle aid notation remainder proof subscripts vertices sets interpreted modulo show use vertex deletions bipartite complementations partition graph disjoint union curious graphs graphs end proof verify number operations used since every vertex cycle two neighbours two neighbours neighbours must nonconsecutive vertices cycle may therefore partition vertices eleven sets follows set vertices set vertices whose unique neighbour set vertices adjacent nonadjacent rest start showing use bounded number vertex deletions bipartite complementations disconnect bipartite induced subgraph containing rest graph enable assume aid prove series claims claim independent set indeed adjacent contradiction completes proof claim claim independent set indeed adjacent contradiction completes proof claim claim indeed induce would contradiction completes proof claim claim every vertex either complete indeed suppose contradiction neighbour claim follows contradiction completes proof claim let denotesthe set vertices neighbours let let show use five bipartite complementations separate rest graph note definition claim vertex neighbour outside therefore sufficient show disconnect vertices outside fact vertices prove stronger version claim claim indeed suppose contradiction adjacent let neighbour claim follows contradiction completes proof claim claim adjacent every vertex exactly one neighbour indeed suppose adjacent adjacent contradiction contradiction claim follows symmetry claim complete follows claim definition completes proof claim claim complete suppose contradiction vertex let neighbour claim therefore contradiction claim follows symmetry claim complete every vertex furthermore vertex neighbour definition every vertex complete claim claim complete claim claim complete definition every vertex claim definition completes proof claim recall start proof assumed graph prime claim apply bipartite complementation would remove edges leave edges unchanged however result might graph prime use claim disconnect rest graph stage wait later proof longer require property prime next analyse structure note independent sets claims claim implies every component complete bipartite graph fix consider component containing least one vertex let note since every vertex contains least one neighbour follows prove series claims claim first show module since vertex distinguish two vertices claim since independent sets claims respectively every component complete bipartite graph claim vertex distinguish two vertices claim vertex distinguish two vertices complete bipartite graph claim implies every vertex either complete complete distinguish two vertices follows module vertex since prime conclude completes proof claim claim show module suppose contradiction module since vertex distinguish two vertices claim since independent sets claims respectively every component complete bipartite graph claim vertex distinguish two vertices either complete bipartite graph claim implies every vertex complete complete vertex distinguish two vertices therefore must vertex distinguishes two vertices without loss generality assume vertex adjacent let vertex vertex adjacent note two vertices belong different connected components claim means must vertex must neighbour note claim follows adjacent respectively contradiction implies indeed module since prime conclude completes proof claim claim suppose contradiction contain least two vertices claims every connected component containing vertex consists single edge since contains least two vertices must least two components similarly must two components therefore find adjacent adjacent claim contradiction claim follows symmetry claim set either bipartite let union sets consider graph every set either contains zero least two vertices one set contains two vertices claims bipartite graph done remains consider case two sets contain two vertices case claim must consecutive vertices cycle sets contain vertices may assume show graph independent sets claims claims every vertex exactly one neighbour every vertex one neighbour one neighbour common neighbour must adjacent claim neighbour must adjacent claim therefore vertex adjacent vertex unique neighbour obtain graph applying bipartite complementation every component induced subgraph one vertex component therefore graph terms definition graphs matrix zero everywhere else completes proof claim claims would allow remove graph discussed earlier actually stage preserve property prime informally may think vertices dealt concern remainder graph vertices first look sets edges sets claim either complete suppose contradiction adjacent claim contradiction symmetry completes proof claim fact strengthen claim follows claim either complete empty claim follows claim suppose contradiction neighbour adjacent respectively contradiction completes proof claim claim contains say vertices contradiction completes proof claim since independent sets claim follows claim every component complete bipartite graph suppose independent sets every component complete bipartite graph say pair number components containing least one edge least two otherwise say pair simple note simple edges removed using one bipartite complementation claim complete furthermore case complete suppose suppose contradiction adjacent respectively contradiction implies adjacent conclude complete symmetry also complete therefore first part claim holds suppose vertex since claim implies applying first part claim find complete complete claim follows symmetry claim complete suppose claim claim complete suppose contradiction contains vertex claim since complete follows complete therefore complete since simple must adjacent vertices contradiction claim follows symmetry claim contains two pairs contains two must described claim suppose contradiction claim false must pairs claim claim complete claim follows complete since must adjacent vertices choosing arbitrary find contradiction completes proof claim pair let set vertices neighbours let set vertices neighbours simple pair set let next look edges sets remainder show could apply bipartite complementations remove sets graph claim indeed suppose contradiction vertex must pairs choose adjacent claim complete contradiction claim follows claim suppose pair complete similarly complete suppose contradiction since vertex adjacent nonadjacent respectively contradiction implies complete suppose contradiction neighbour since must neighbour previous paragraph must adjacent contradiction therefore claim follows symmetry claim suppose pair every vertex either complete similarly every vertex either complete claim suppose contradiction claim hold vertex must neighbour since may assume different components indeed component must vertex different component case replace adjacent nonadjacent respectively since must neighbour contradiction claim follows symmetry claim every vertex either complete either complete may assume otherwise empty claim holds trivially symmetry enough show every vertex either complete first note complete definition claim every vertex claim every vertex complete every vertex claim every vertex either complete claim every vertex claim every vertex either complete definition every vertex suppose contradiction vertex neighbour since claim implies claim complete since must vertex adjacent contradiction therefore claim follows symmetry claim every module independent set contains vertices recall prime contains modules let suppose contradiction contains module least vertices independent set claim two sets form empty two sets form empty claim every vertex either complete claim every vertex either complete every vertex therefore every vertex one neighbourhoods choose two vertices neighbourhood since module distinguished vertex since independent set distinguished vertex therefore module contradiction completes proof claim claim may assume claim may apply five bipartite complementations separate rest graph claim delete five vertices obtain graph either bipartite case curious suppose contains pair let claim shows remove edges vertices applying two bipartite complementations furthermore bipartite graph claim curious graph one part empty claim contains two pairs contains two claim hence claim implies applying four bipartite complementations separate rest separating curious graph pair turn may therefore assume completes proof claim assume simple note proof claim edit graph way may stop prime however claim still contain modules independent sets vertices property suffice remainder proof claim every set vertices two together dominate either let note independent set claim claim module must two vertices distinguished vertex outside without loss generality assume adjacent definition vertices cycle distinguish two vertices set show dominates either suppose contradiction nonadjacent claim vertices symmetry may therefore assume case must adjacent otherwise would belong otherwise would therefore adjacent otherwise would claim follows contradiction completes proof claim claim deleting vertices set may assume vertex dominates either note claim let denote set vertices consider say note empty done vertex adjacent exactly one say adjacent contradiction contradiction shows choose maximum suppose contradiction vertex note must adjacent furthermore since maximum must vertex adjacent contradiction follows every vertex dominates either claim delete vertices completes proof claim claim may assume every vertex dominates either therefore partition set three possibly empty subsets follows set vertices dominate set vertices dominate set vertices dominate definition partition moreover every vertex vice versa thus vertices partitioned subsets may empty note sets pairwise claim furthermore complete complete definition therefore nearly every pair subsets either complete possible exceptions five disjoint pairs form recall two independent sets vertices components complete bipartite graphs contains one component say pair simple otherwise recall simple remove edges applying one bipartite complementation claim every pair simple fifteen sets considered previous paragraph pairs ones form next analyse edges sets form sets form subsets use bipartite complementations partition graph induced subgraphs curious first recall claim also claim simple therefore sets connection claim indeed contradiction completes proof claim claim neighbour complete neighbour complete suppose neighbour adjacent must otherwise would therefore note must adjacent otherwise would contradicting claim follows complete claim follows symmetry claim partition three possibly empty subsets follows therefore set vertices neighbours complete set vertices neighbours therefore complete set vertices therefore recall sets connection therefore claim set nonsimple set must one following pairs recall among pairs sets form possible pairs ones form see also fig notice disjoint claim partition allows distinguish two kinds induced subgraphs form subgraph first kind empty let empty graph fig connections pairs sets form edge shown two sets possible pair sets edge shown two sets sets form simple pair therefore edges removed using one bipartite complementation let graph induced empty case also add similarly empty case also add say graph second kind next two claims show disconnect kinds graph hij rest graph claim graph separated rest graph using finitely many bipartite complementations show every set set simple three sets case separate applying one bipartite complementation every pair sets taking account claim recall simple every set except simple every set except simple every set except thus suffices show simple simple symmetry need show simple done otherwise choose neighbour note vertex assumption otherwise empty done claim complete therefore must otherwise would contradiction show complete suppose contradiction definition must let note adjacent claim contradiction shows neighbour complete therefore simple similarly simple applying bipartite complementations removes edges vertices vertices cycle every set simple every set applying one bipartite complementation pair sets remove remaining edges rest graph completes proof claim claim graph separated rest graph using finitely many bipartite complementations graph contains taking account claim recall set set set thus need concern case pair pair simple symmetry enough consider second cases graph done claim therefore may assume contained graph thus add empty symmetry follows use bounded number bipartite complementations disconnect sets form outside finally applying bipartite complementation sets neighbourhood cycle disconnects remainder graph completes proof claim next two claims show graphs first kind curious graphs second kind partitioned two curious induced subgraphs applying two bipartite complementations claim graph bipartite therefore curious sets independent claim claim complete independent sets bipartite complete every vertex neighbour one since hence bipartite since bipartite graphs curious graphs one partition classes empty completes proof claim claim apply two bipartite complementation operations obtain disjoint union two curious induced subgraphs sets independent claim contains vertices curious claim may therefore assume belong note independent sets claim therefore bipartite curious graph one part empty empty either complete claim taking setting complete find curious graph partition indeed suppose contradiction vertices therefore forming since follows contradicts claim contradiction shows complete indeed curious graph finally suppose claim may assume simple contains one component component exists must complete bipartite graph claim let resp set vertices resp neighbours resp since complete bipartite graph follows complete show disconnect rest applying two bipartite tations indeed suppose neighbour let adjacent must vertex definition claim contradiction therefore adjacent definition follows conclude symmetry suppose vertex adjacent therefore must conclude every adjacent adjacent since chosen tex either complete antifrom arbitrarily means every vertex complete therefore applying one bipartite complementation recall remove edges assumption complete claim therefore applying one bipartite complementation remove edges vertices vertices symmetry applying one bipartite complementation remove edges vertices vertices disconnects rest since bipartite graph curious may therefore assume set note independent sets suppose contradiction curious respect partition must vertices since follows claim symmetry may assume let arbitrary vertex must otherwise claim must therefore adjacent contradiction follows indeed curious graph respect partition completes proof claim applying claims remove vertices except maybe sets definition set complete claim delete five vertices original cycle apply bipartite complementation pair sets yield independent set curious graph two parts partition empty remains count number operations applied number obtained curious graphs explained proof claim claim apply five bipartite complementations separate rest graph delete five vertices obtain graph either curious next also explained proof claim separate rest applying bipartite complementations obtain two curious graphs next claim delete vertices deleting five vertices original cycle yields five vertex deletions recall partitioned partitioned yielding subsets vertices altogether claims apply bipartite complementations certain pairs subsets separate graphs kinds thus claims apply bipartite complementations claim says curious graph claim apply two bipartite complementations obtain two curious graphs finally apply five bipartite complementations sets yielding one curious graph gives total bipartite complementations vertex deletions yielding curious graphs one graph completes proof prove main result recall already known class graphs bounded known class theorem class graphs labelled induced subgraph relation bounded proof let lemmas need consider prime graphs class recall prime graph least three vertices contain two vertices false twins otherwise two vertices would form module therefore lemma since classes prime free graphs containing induced precisely graph may therefore restrict graphs since graphs class follows contain induced cycles eight vertices may therefore restrict prime free graphs either contain induced bipartite lemmas given prime contains induced apply constant number vertex deletions bipartite complementation operations obtain graph disjoint union curious graphs case graphs lemmas facts theorem sufficient consider bipartite graphs graphs bipartite graphs furthermore form subclass class bipartite graphs since class bipartite graphs labelled induced subgraph relation lemma bounded theorem completes proof summaries bigenic graph classes class graphs bigenic graph class left conjecture still needs verified see details claim also deduced theorems open problem conjecture true class graphs class graphs equivalence one six remaining bigenic graph classes still open one six bigenic graph classes boundedness still open refer details first claim details second claim make paper recall two theorems papers sum current knowledge bigenic graph classes including new results paper also include lists corresponding open cases summary bigenic graph classes theorem let class graphs defined two forbidden induced subgraphs labelled induced subgraph relation equivalent class graphs one following holds iii given four graphs classes graphs graphs said equivalent unordered pair obtained unordered pair combination operations complementing graphs pair one graphs pair replacing vice versa two classes equivalent one induced subgraph relation one similarly two classes equivalent one bounded one induced subgraph relation equivalent class graphs one following holds neither linear forest iii theorem cover six cases still open open problem class graphs induced subgraph relation summary boundedness bigenic graph classes theorem let class graphs defined two forbidden induced subgraphs bounded equivalent class graphs one following holds iii vii unbounded equivalent class graphs one following holds iii theorem cover six cases still open open problem class graphs bounded unbounded iii references allen lozin rao speed hereditary properties electronic journal combinatorics research paper atminas lozin labelled induced subgraphs order dabrowski johnson lozin paulusma zamaraev graph classes closed complementation proc mfcs lipics brignall engen vatter counterexample regarding labelled arxiv courcelle edge contraction information processing letters courcelle makowsky rotics linear time solvable optimization problems graphs bounded theory computing systems courcelle olariu upper bounds clique width graphs discrete applied mathematics dabrowski dross paulusma colouring graphs journal computer system sciences dabrowski lozin paulusma ordering graph classes proc lncs dabrowski lozin paulusma versus new results bigenic classes order press dabrowski paulusma classifying bipartite graphs discrete applied mathematics dabrowski paulusma graph classes two forbidden induced subgraphs computer journal daligault rao relabel functions order damaschke induced subgraphs journal graph theory ding subgraphs journal graph theory espelage gurski wanke solve graph problems cliquewidth bounded graphs polynomial time proc lncs higman ordering divisibility abstract algebras proceedings london mathematical society lozin recent developments graphs bounded discrete applied mathematics kobler rotics edge dominating set colorings graphs cliquewidth discrete applied mathematics korpelainen lozin bipartite induced subgraphs journal graph theory korpelainen lozin two forbidden induced subgraphs discrete mathematics korpelainen lozin razgon boundary properties sets graphs order lozin rautenbach graphs bounded vertex degree siam journal discrete mathematics lozin razgon zamaraev versus journal combinatorial theory series press oum seymour approximating journal combinatorial theory series ramsey problem formal logic proceedings london mathematical society rao msol partitioning problems graphs bounded treewidth theoretical computer science
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numerical implicitization oct justin chen joe kileel abstract present package numericalimplicitization allows computation basic invariants image polynomial map dimension degree hilbert function values package relies methods numerical algebraic geometry homotopy continuation monodromy introduction many varieties interest algebraic geometry applications usefully described images polynomial maps via parametrization implicitization process converting parametric description variety intrinsic implicit description classically implicitization refers procedure computing defining equations parametrized variety theory accomplished finding kernel ring homomorphism via bases practice however symbolic basis computations often even problems scale well respect size input despite one would often like know basic information parametrized variety even symbolic methods prohibitively expensive terms computation time best examples information discrete invariants dimension degree hilbert function values variety projective examples include boolean tests whether particular point lies parametrized variety goal package provide information words numerically implicitize parametrized variety using methods numerical algebraic geometry numericalimplicitization builds top existing numerical algebraic geometry software bertini phcpack used path tracking point sampling default native engine used notation following notation used throughout remainder article source variety defined ideal polynomial ring list polynomials specifying map zariski closure image target variety consideration projective closure respect standard embedding currently numericalimplicitization implemented integral reduced irreducible varieties equivalently ideal prime since numerical methods used always work complex numbers arithmetic moreover internally represented affine cone easier computers work points affine space time suffices find invariants sampling methods package rely crucially ability sample general points end two methods provided package numericalsourcesample numericalimagesample allow user sample many general points desired numericalsourcesample compute witness set unless taking numerical irreducible decomposition step avoided witness set known points sampled negligible time numericalimagesample works sampling points via numericalsourcesample applying map one way view difference computation time symbolic numerical methods upfront cost computing basis replaced upfront cost computing numerical irreducible decomposition used sample general points however sampling done generating random tuples essentially immediate thus unrestricted parametrization case upfront cost numerical methods becomes zero mathematics subject classification key words phrases implicitization homotopy continuation monodromy interpolation code documentation see https justin chen joe kileel dimension basic invariant algebraic variety dimension compute dimension image variety numerically use following theorem theorem let dominant morphism irreducible varieties zariski open subset induced map tangent spaces dfx surjective proof immediate corollary generic smoothness preceding setting since singular locus sing proper closed subset general dim dim dim dfx dim dim ker dfx kernel jacobian matrix given jac ker dfx kernel jacobian evaluated intersected explicitly ker dfx kernel matrix compute kernel dimensions numerically explained prior example get dim example let variety symmetric tensors border rank equivalently affine cone secant variety fourth veronese embedding naively one expects dim fact dim verified following code version needspackage numericalimplicitization sum flatten entries basis variables tolist time numericalimagedim ideal used seconds example largest exceptional case celebrated work note timing printed hilbert function turn problem determining hilbert function recall projective variety given homogeneous ideal hilbert function argument definition vector space dimension dth graded part hye dim counts maximum number linearly independent degree hypersurfaces containing compute hilbert function numerically use multivariate polynomial interpolation fixed argument let set general points consider interpolation matrix rows indexed points columns indexed degree monomials whose entries values monomials points vector kernel corresponds choice coefficients homogeneous degree polynomial vanishes large one expects form vanish entire variety following theorem makes precise theorem let set general points let interpolation matrix dim ker dim ker dim ker dim numerical implicitization proof identifying vector ker form degree coefficients suffices show ker vanishes particular ker converse ker consider universal interpolation matrices set min every minor lies ideal specialization points matrix rank moreover points general specialization rank exactly since irreducible particular rank rank dim ker dim ker implies follows specializing gives rank matrix hence every degree form ker evaluates every since reduced deduce ker follows theorem integers dim ker dim ker decrease exactly first instance fail decrease point stabilize dim ker dim ker stable value hilbert function dim ker hye particular suffices compute dim ker one may assume interpolation matrix square although may seem wasteful stabilization may occurred fewer rows indeed numericalhilbertfunction due algorithm used compute kernel dimension numerically precise kernel dimension found via singular value decomposition svd namely gap ratio consecutive singular values greater option svdgapthreshold default value observed list singular values taken indication singular values past greatest gap numerically zero example problems observed taking one additional row needed often reveal satisfactory gap singular values addition numerical stability improved via preconditioning interpolation matrices namely row normalized euclidean norm computing svd example let random canonical curve genus complete intersection random quadric cubic let projection random cubics plane curve degree deg ideal contains single form degree verify follows ideal random random tolist random numericalhilbertfunction sampling image points used seconds creating interpolation matrix used seconds performing normalization preconditioning used seconds computing numerical kernel used seconds hilbert function value numericalinterpolationtable output numericalinterpolationtable hashtable storing results interpolation computation described one obtain approximation basis done via command extractimageequations justin chen joe kileel extractimageequations experimental feature find equations may called option attemptexact true degree dimension degree basic invariant projective variety set dim general linear space complementary dimension intersection finite set reduced points degree definition cardinality independent general linear space thus one approach find deg fix random compute set points numericalimplicitization takes tack method used find obvious first foremost know equations solving must done secondly compute equations equations pulled back degree deg deg potentially much bigger deg instead monodromy employed find state technique consider map projection onto first factor nonempty zariski open subset restriction deg covering map namely equals complement hurwitz divisor fix generic basepoint fundamental group acts fiber action known monodromy key fact induced group homomorphism sym symdeg surjective irreducibility explicitly theorem let write height column vector linear forms fix another generic point consider following loop linear subspaces nonempty zariski open subset loop contained moreover classes loops generate full symmetric group sym proof let pencil linear subspaces generated via monodromy maps surjectively onto sym corollary topological space homeomorphic riemann sphere minus finite set points isomorphic free group finitely many letters explicit loops theorem statement miss finite set general moreover may chosen loop encloses exactly one point therefore classes loops generate visualize loops reader may consult proof lemma numericalimagedegree works first sampling general point manufacturing general linear slice moved around loop form described theorem loop pulls back homotopy use equations track endpoint track point numerically distinct loop learned new point otherwise discarded repeat process tracking points known point according loops theorem note random loop positive probability bounded away learning new points known thus carrying many loops theorem probability finding points approaches practice several numerical implicitization consecutive learn new points suspect calculated verify pass trace test see corollary provides characterization subset equals trace test failed replaced new random preimages known points tracked preimages points afterwards monodromy begins anew trace test failed maxtracetests default times total numericalimagedegree exits lower bound deg example let find deg using commands terms product apply tolist terms product apply tolist apply tolist time numericalimagedegree ideal maxrepetitivemonodromies sampling point source tracking monodromy loops points found points found points found points found points found points found points found points found points found points found points found points found points found points found points found points found points found running trace test degree image used seconds pseudowitnessset theorem proven via representation theory combinatorics prime ideal generated minors flattenings tensors confirm deg however naive attempt compute degree symbolically taking kernel ring map polynomial ring variables hope finishing reasonable amount time output pseudowitnessset hashtable stores computation numerical representation parameterized varieties introduced membership classically given variety point determine whether finding equations generate ideal radical testing satisfies equations pseudowitnessset available point membership instead verified parameter homotopy precisely isonimage determines lies constructible set follows fix general affine linear subspace complementary dimension passing suffices compute set pseudowitnessset specified option maxrepetitivemonodromies default value justin chen joe kileel provides general section preimages move theorem pulls back homotopy use equations track preimages applying endpoints track gives isolated points theorem since general proof corollary shows procedure computes entire set example let defined resultant three quadratic equations three unknowns words consists coefficients system admits solution hypersurface matrix formula defining equation derived using exterior algebra methods rapidly determine point membership numerically follows ideal tolist numericalimagedegree verboseoutput false degree numericalimagesample point random time isonimage isonimage used seconds true false acknowledgements grateful anton leykin encouragement luke oeding testing numericalimplicitization also thank david eisenbud bernd sturmfels helpful discussions comments work supported national science foundation references alexander hirschowitz polynomial interpolation several variables alg geom bates gross leykin rodriguez bertini bates hauenstein sommese wampler bertini software numerical algebraic geometry available https eisenbud commutative algebra view toward algebraic geometry graduate texts mathematics new york eisenbud schreyer resultants chow forms via exterior syzygies amer math soc grayson stillman software system research algebraic geometry available http gross verschelde interfacing phcpack softw algebra geom hartshorne algebraic geometry graduate texts mathematics new york hauenstein rodriguez numerical irreducible decomposition multiprojective varieties hauenstein sommese witness sets projections appl math comput leykin numerical algebraic geometry softw algebra geom leykin sottile trace test raicu secant varieties varieties algebra number theory sommese verschelde wampler symmetric functions applied decomposing solution sets polynomial sets siam numer anal sommese wampler numerical solution systems polynomials publishing pte hackensack sturmfels hurwitz form projective variety symb comput verschelde algorithm phcpack solver polynomial systems homotopy continuation acm trans math software available https department mathematics university california berkeley california address jchen jkileel
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tangent cones monomial curves feb feza arslan anargyros katsabekis melissa nalbandiyan abstract paper give necessary sufficient conditions tangent cone monomial curve affine space study particularly case gorenstein noncomplete intersection monomial curve introduction tangent cones monomial curves studied many authors see instance constitutes important problem since example cohenmacaulayness tangent cone guarantees hilbert function local ring associated monomial curve therefore reduces computation computation hilbert function artin local ring article first deal problem case monomial curve affine space field section using classification terms critical binomials given katsabekis ojeda study detail problem case classification give sufficient conditions tangent cone consider remaining cases appendix give cases necessary sufficient conditions tangent cone section consider problem intersection gorenstein monomial curves case bresinsky shown minimal generating set defining ideal monomial curve consisting five generators also given explicit form generators actually permutations generator set worth note theorem provides sufficient condition tangent cone four aforementioned cases paper generalize result provide necessary sufficient condition tangent cone six permutations finally use results give families gorenstein monomial curves corresponding local rings hilbert function thus giving partial answer rossi problem problem asks whether hilbert function gorenstein local ring dimension one recently shown many families monomial curves giving negative answer problem one note rossi problem still open gorenstein local rings associated monomial curves let set positive integers gcd let polynomial ring variables shall denote date february mathematics subject classification primary secondary key words phrases tangent cone monomial curve second author supported tubitak visiting scientists scientists sabbatical leave fellowship program arslan katsabekis nalbandiyan monomial xud stands set integers consider affine monomial curve defined parametrically tnd toric ideal denoted kernel homomorphism given tni grade semigroup setting degs monomial xud defined degs ideal generated binomials degs degs see example lemma let theorem monomial curve tangent cone origin integers exist gcd gcd note gcd every also gcd thus decide tangent cone suffices consider extra condition gcd computations paper performed using cocoa general case let set relatively prime positive integers definition binomial xai called critical respect least positive integer nnj critical ideal denoted ideal generated critical binomials support supp monomial set supp divides support binomial set supp supp support equals set say full support let minimal number generators ideal theorem permuting variables necessary exists minimal system binomial generators critical ideal following form case every xai xui case either case case xai xui tangent cones monomial curves xai xui case xui denotes appropriate monomial whose support cardinality greater equal two theorem union set binomials uij indispensable set binomials full support indispensable cases full support properly divides properly divides case minimal system generators permutation indices binomial called indispensable every system binomial generators contains corollary every indispensable binomial notation given monomial write deg rest section assume prove results make repeated use theorem proposition suppose given case let generating set let supp following cases tangent cone origin supp supp deg deg every binomial supp deg deg supp supp deg deg every monomial exists monomial supp also deg deg iii supp supp deg every monomial exists monomial supp also deg deg supp supp deg every monomial exists monomial supp also deg deg proof let thus exists monomial supp let consider monomial also deg deg deg arslan katsabekis nalbandiyan let consider monomial also deg deg deg let consider monomial also deg deg deg suppose cases exists binomial supp divides note deg deg condition also deg deg deg deg exists binomial divides recall generates also supp supp supp necessarily divides let also deg deg deg deg deg deg let let consider monomial also deg deg deg let consider monomial also deg deg deg suppose condition done iii let let consider monomial also deg deg deg let consider monomial also deg deg deg suppose condition done let let consider monomial also deg deg deg suppose condition done proposition suppose given case let supp folxv generating set let lowing cases tangent cone origin supp supp deg every binomial supp deg deg tangent cones monomial curves every monomial exists monomial supp also deg deg supp supp deg every monomial exists monomial supp also deg deg iii supp supp every monomial exists monomial supp also deg deg proof let thus exists monomial supp let consider monomial also deg deg deg let consider monomial also deg deg deg suppose done assume cases exists binomial supp divides note deg deg condition also deg deg deg exists binomial divides recall generates also supp supp necessarily divides let suppose divides let also deg deg deg deg deg deg suppose divides binomial belongs also degs thus exists monomial supp degs degs also deg deg consequently deg deg deg deg let let consider monomial also deg deg deg let condition done iii let let consider monomial also deg deg deg suppose assumption done arslan katsabekis nalbandiyan gorenstein case section study case intersection gorenstein monomial curve semigroup symmetric theorem let monomial curve parametrization semigroup symmetric intersection curve minimally generated set polynomials unique isomorphism aij remark bresinsky showed symmetric previous theorem gcd aij remark theorem implies intersection gorenstein monomial curve embedding dimension four variables renamed obtain generators exactly given form means six isomorphic possible permutations considered within three cases notation denote generators xai xakik xai xal thus given gorenstein monomial curve extra condition generator set exactly given one six permutations remark corollary toric ideal intersection gorenstein monomial curve generated indispensable binomials let lex inf total order monomials defined follows lex lex order lexicographic order largest variable respect lex proposition set reduced basis respect appropriate lex inf order tangent cones monomial curves proof suppose given case respect lex inf lex lex lex prove pair since relatively prime get similarly thus basis respect lex inf lex lex lex hard show basis respect lex inf lex lex lex case lex lex lex case lex lex lex case lex lex lex case lex lex lex case apery set semigroup relative defined using lemma get following corollary let set monomials polynomial ring divisible monomials set case case case case case case theorem suppose given case cohenmacaulay tangent cone origin proof case minimally generated set theorem curve tangent cone origin conversely suppose cohenmacaulay tangent cone origin since generated indispensable binomials every binomial indispensable particular arslan katsabekis nalbandiyan binomial indispensable exists monomial belongs replace binomials contradiction fact indispensable thus therefore theorem remark suppose given case holds binomial belongs proposition suppose given case cohenmacaulay tangent cone origin every monomial exists monomial supp also deg deg proof case minimally generated set suppose tangent cone origin since generated indispensable binomials every binomial indispensable particular binomials indispensable therefore inequalities hold theorem condition also true conversely proposition iii enough consider monomial property exists least one monomial supp also suppose let also deg deg since similarly let also deg deg inequalities hold condition implies exists monomial supp also deg deg suppose recall generates divided least one monomials divided integers also deg deg divided integers therefore binomial also deg deg assume neither divides necessarily divides divided leading monomial respect lex inf lex lex lex thus contradiction fact therefore proposition tangent cone origin proposition suppose given case assume tangent cone origin also tangent cones monomial curves proof first note since therefore since tangent cone origin monomial supp also deg arise term sum term sum therefore canceled another term sum distinguish following cases suppose note vided thus term arise sum canceled another term sum consequently monomial must divide therefore term arise sum note deg done otherwise canceled another term sum thus must divide consequently term arise sum done otherwise canceled another term sum continuing way finally reach contradiction thus suppose also note divided term arise sum canceled another term sum note divided term arise sum since canceled another term sum thus term arise sum done otherwise canceled another term sum continuing way finally reach contradiction thus either theorem suppose given case also tangent cone origin either proof proposition conditions true proposition condition also true proposition enough consider monomial property exists least one monomial supp also suppose let denote either monomial monomial let deg deg deg suffices consider case recall generates binomial belongs polynomials therefore term sum arslan katsabekis nalbandiyan note divided monomials monomial divided monomial term sum canceled another term sum remark thus divides contradiction proposition suppose given case also tangent cone origin proof proposition conditions true suppose first proposition deduce exists monomial supp deg since binomial belongs polynomials forp arise term sum term sum therefore canceled another term sum thus term sum canceled suppose another term sum contradiction thus also suppose note thus theorem suppose given case also assume tangent cone origin proof proposition conditions true proposition enough consider monomial property exists least one monomial supp also suppose let deg deg suffices assume since binomial belongs polynomials term sum term sum canceled another term sum remark thus divides contradiction example consider toric ideal minimally generated set note tangent cones monomial curves thus tangent cone origin remark suppose given case holds binomial belongs proposition suppose given case cohenmacaulay tangent cone origin every monomial exists monomial supp also deg deg proof case minimally generated set suppose tangent cone origin since generated indispensable binomials every binomial indispensable particular binomials indispensable therefore inequalities hold theorem condition also true prove converse statement proposition enough consider monomial property exists least one monomial supp also let also deg deg similarly let also deg deg conditions hold condition implies exists monomial supp also deg deg therefore proposition tangent cone origin proof next proposition similar proposition therefore omitted proposition suppose given case assume tangent cone origin also theorem suppose given case also tangent cone origin either proof proposition conditions true proposition condition also true proposition enough consider monomial property exists least one monomial supp also suppose arslan katsabekis nalbandiyan let denote either monomial monomial let deg deg deg suffices consider case since binomial belongs polynomials monomial divided monomial term sum canceled another term sum remark thus divides contradiction proposition suppose given case also tangent cone origin proof proposition conditions true suppose first proposition deduce exists monomial supp deg since binomial polynomibelongs als arise term sum term sum therefore canceled another term sum thus term sum suppose canceled another term sum contradiction thus also suppose note thus monomial supp deg polynomials arise term sum term sum therefore canceled another term sum thus term sum let suppose canceled another term sum contradiction thus also theorem suppose given case also assume tangent cone origin proof proposition conditions true proposition enough consider monomial property exists least one monomial supp also suppose let deg deg tangent cones monomial curves suffices assume since binomial belongs polynomials term sum canceled another term sum thus divides contradiction example consider toric ideal minimally generated set note thus macaulay tangent cone origin remark deg deg remark suppose given case holds binomial belongs proposition suppose given case cohenmacaulay tangent cone origin every monomial exists monomial supp also deg deg proof case minimally generated set suppose tangent cone origin since generated indispensable binomials every binomial indispensable particular binomial indispensable therefore inequality holds theorem condition also true conversely proposition iii enough consider monomial property exists least one monomial supp also degs degs suppose let also deg deg similarly let also deg deg conditions hold condition implies exists monomial supp also deg deg suppose divided least one monomials divided integers also deg deg divided integers therefore binomial also deg deg assume neither divides necessarily divides divided leading monomial respect lex inf lex lex lex arslan katsabekis nalbandiyan thus contradiction fact therefore proposition tangent cone origin proof following proposition similar proposition therefore omitted proposition suppose given case assume tangent cone origin also theorem suppose given case also tangent cone origin either proof proposition condition true proposition condition also true proposition enough consider monomial property exists least one monomial supp also suppose let denote either monomial monomial let deg deg deg suffices consider case since binomial belongs polynomials monomial divided monomial term sum canceled another term sum remark thus divides contradiction proposition suppose given case also tangent cone theorem suppose given case also assume tangent cone origin example consider toric ideal minimally generated set note consequently tangent cone origin theorem suppose given case cohenmacaulay tangent cone origin tangent cones monomial curves proof case minimally generated set theorem curve tangent cone origin conversely suppose tangent cone origin since generated indispensable binomials every binomial indispensable particular binomials indispensable exists monomial belongs replace binomials contradiction fact indispensable thus therefore theorem similarly get remark suppose given case holds binomial belongs proposition suppose given case cohenmacaulay tangent cone origin every monomial exists monomial supp also deg deg proof case minimally generated set suppose tangent cone origin since generated indispensable binomials every binomial indispensable particular binomial indispensable therefore inequality holds theorem condition also true conversely proposition enough consider monomial property exists least one monomial supp also let also deg deg similarly let also deg deg conditions hold condition implies exists monomial supp also deg deg therefore proposition tangent cone origin proof next proposition similar proposition therefore omitted proposition suppose given case assume tangent cone origin also theorem suppose given case tangent cone origin arslan katsabekis nalbandiyan either proof proposition conditions true proposition condition also true proposition enough consider monomial property exists least one monomial supp also suppose let denote either monomial monomial let deg deg deg suffices consider case since binomial belongs polynomials monomial divided monomial term sum canceled another term sum remark thus divides contradiction proposition suppose given case also tangent cone origin theorem suppose given case also assume tangent cone origin families monomial curves supporting rossi problem section give examples showing criteria given previous one used give families monomial curves supporting rossi problem example consider family given let corresponding monomial curve toric ideal generated set thus case remark sufficient consider binomial guarantees tangent cone fixed using technique given construct new family monomial curves tangent cone symmetric semigroup generated corresponding monomial curve minimally generated set method given semigroup generated integer symmetric tangent cones monomial curves whenever gcd moreover toric ideal corresponding monomial curve minimally generated set construction case binomial deduce corresponding monomial curve tangent cone case interchange get semigroup generated thus toric ideal corresponding monomial curve generated set case remark binomials guarantee corresponding monomial curve tangent cone way construct infinitely many families gorenstein monomial curves tangent cones words corresponding local rings hilbert functions supporting rossi problem literature examples intersection gorenstein monomial curve families supporting rossi problem although tangent cones next example gives family monomial curves property prove use following proposition proposition proposition let monomial ideal monomial ideal monomial let denote numerator hilbert series let denote total degree monomial example consider family integer let corresponding monomial curve toric ideal minimally generated binomials thus case remark consider binomial since theorem tangent cone cohenmacaulay enough show hilbert function ideal generated polynomials homogeneous summand least degree standard basis computation generated set let note apply proposition ideal case since isomorphic arslan katsabekis nalbandiyan obtain substituting recursively equation obtain hilbert series since numerator negative coefficients hilbert function way shown hilbert function local ring corresponding intersection gorenstein monomial curve appendix appendix provide results concerning cases theorem except case proofs similar section therefore omitted let positive integers gcd theorem suppose given case let xai xakk xuk generating set assume tangent cone origin every binomial supp deg deg every binomial supp monomial supp degs degs deg deg every monomial xvi xvkk exists monomial supp also deg deg theorem suppose given case let xai xakk xui generating set assume tangent cone origin every binomial supp deg deg every supp monomial supp degs degs deg deg every monomial xvi xvkk exists monomial supp also deg deg corollary suppose given case let xai xakk xui generating set assume every binomial supp deg deg every supp monomial supp degs degs deg deg tangent cone origin example consider toric ideal minimally generated binomials note binomial also binomial tangent cone corollary tangent cones monomial curves theorem suppose given case let generating set assume tangent cone origin every binomial supp deg deg every supp monomial supp degs degs deg deg every monomial xui xuk exists monomial supp also deg deg example consider toric ideal minimally generated binomials also let note also furthermore thus exists monomial supp also deg deg theorem monomial curve tangent cone origin remark case generated set xai xai xakk complete intersection also cohenmacaulay tangent cone origin theorem suppose given case let xai xakk xui generating set min tangent cone origin every binomial supp deg deg every monomial xvi xvkk exists monomial supp also deg deg remark suppose given case assume generating set either xai xai xui xai xai xui positive integers tangent cone origin every binomial supp deg deg every monomial exists monomial supp also deg deg theorem suppose given case let xai xai xakk generating set let cohenmacaulay tangent cone origin every monomial xvi xvkk exists monomial supp also deg deg arslan katsabekis nalbandiyan proposition suppose given case let xai xai xakk generating set let assume deg belongs support every binomial supp deg deg every monomial xvi xvkk exists monomial supp also deg deg tangent cone origin corollary suppose given case let xai xai xakk generating set let assume belongs support every binomial supp deg deg deg tangent cone origin theorem suppose given case let xai xai xakk generating set tangent cone origin every monomial xzi xzkk exists monomial supp also deg deg proposition suppose given case let generating set assume supp supp deg deg every binomial supp deg deg tangent cone origin theorem suppose given case let xai xakk generating set tangent cone origin every monomial xzi xzkk exists monomial supp also deg deg proposition suppose given case let xakk generating set assume supp supp deg deg every binomial supp deg deg every binomial supp monomial supp degs degs deg deg tangent cone origin references arslan tangent cones proc amer math soc tangent cones monomial curves arslan mete hilbert functions gorenstein monomial curves proc amer math soc bayer stillman computation hilbert functions symbolic comput bresinsky symmetric semigroups integers generated elements manuscripta math casares short resolution lattice ideal proc amer math soc cocoateam cocoa system computations commutative algebra available http garcia associated graded semigroup rings comm algebra gimenez srinivasan note gorenstein monomial curves bull braz math soc herzog generators relations abelian semigroups semigroup rings manuscripta math herzog regular sequence super regular nagoya math herzog stamate defining equations tangent cone numerical semigroup ring algebra jafari zarzuela monomial curves obtained gluing semigroup forum katsabekis ojeda indispensable classification monomial curves pac math kunz value semigroup one dimensional gorenstein ring proc amer math soc molinelli patil tamone associated graded rings monomial curves algebra geom molinelli tamone hilbert function certain rings monomial curves pure appl algebra oneto strazzanti tamone gorenstein local rings decreasing hilbert function arxiv patil tamone defect hilbert functions monomial curves journal pure applied algebra robbiano valla equations defining tangent cones math proc cambridge philos soc rossi hilbert functions local rings commutative algebra connections geometry contemporary math ams sapko associated graded rings numerical semigroup rings comm algebra sharifan minimal free resolution associated graded ring monomial curves generalized arithmetic sequences pure appl algebra shen tangent cone numerical semigroup rings embedding dimension three comm algebra sturmfels bases convex polytopes volume university lecture series american mathematical society providence department mathematics mimar sinan fine arts university istanbul turkey address department mathematics mimar sinan fine arts university istanbul turkey address katsabek department mathematics mimar sinan fine arts university istanbul turkey address
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question asking program generation anselm anselm nov brenden brenden todd department psychology center data science new york university abstract hallmark human intelligence ability ask rich creative revealing questions introduce cognitive model capable constructing humanlike questions approach treats questions formal programs executed state world output answer model specifies probability distribution complex compositional space programs favoring concise programs help agent learn current context evaluate approach modeling types questions generated humans attempting learn ambiguous situation game find model predicts questions people ask creatively produce novel questions present training set addition compare number model variants finding question informativeness complexity important producing questions introduction active machine learning learner able query oracle order obtain information expected improve performance theoretical empirical results show active learning speed acquisition variety learning tasks see review although impressive work active machine learning focused relatively simple types information requests often request supervised label contrast humans often learn asking far richer questions directly target critical parameters learning task human child might ask dogs long tails difference cats dogs long term goal artificial intelligence develop algorithms similar capacity learn asking rich questions premise make progress toward goal better understanding human question asking abilities computational terms end paper propose new computational framework explains people construct rich interesting queries within particular domain key insight model questions programs executed state possible world output answer example program corresponding john prefer coffee tea would return true possible world states correct answer false others questions may return different types answers example many sugars john take coffee would return number etc depending world state thinking questions syntactically programs recasts problem question asking one program synthesis show powerful formalism offers new approach modeling question asking humans may eventually enable question asking machines evaluate model using data set containing natural language questions asked human participants game given ambiguous situation context model predict questions human learners ask capturing constraints humans construct semantically meaningful questions method successfully predicts frequencies conference neural information processing systems nips long beach usa human questions given game context also synthesize novel questions present training set related work contemporary active learning algorithms query labels causal interventions lack representational capacity consider richer range queries including expressed natural language dialog systems designed ask questions yet systems still far achieving question asking dialog systems applied tasks booking table restaurant typically choose relatively small set canned questions help type food looking little genuine flexibility creativity deep learning systems also developed visual questions style tasks although models produce new questions questions typically take stereotyped form person glove question asking achieved systems trained large amounts natural language dialog recent progress demonstrated however approaches capture intentional forms human question asking recent work probed aspects question asking visual question generation vqg data set contains images paired interesting questions instance image car wreck might paired question caused accident deep neural networks similar used image captioning capable producing types questions extensive training however require large datasets images paired questions whereas people ask intelligent questions novel scenario limited practice shown task moreover human question asking robust changes task goals neural networks generalize flexibly ways question data set goal develop model question asking humans falls outside capabilities systems described focused analysis data set collected consists natural language questions asked human players resolve ambiguous game situation similar battleship players individually presented game board consisting grid tiles tiles initially turned could flipped reveal underlying color player goal identify quickly possible size orientation position ships objects composed multiple adjacent tiles color every board exactly three ships placed nonoverlapping otherwise random locations ships identified color blue red purple ships width length orientation horizontal vertical tile overlap ship displayed null water color light gray flipped extensive instructions rules purpose game number practice rounds see target contexts players presented partly revealed game board similar figure provided ambiguous information actual shape location ships given chance ask question configuration player goal use question asking opportunity gain much information possible hidden game board configuration rules given players questions must answerable using one word number color coordinate like row column number combination questions allowed questions recorded via html text box people typed wanted ask good question context figure purple red ship touch color tile helpful inferred revealed game board rules game ship sizes etc answer water see figure additional example questions player completed contexts presented different underlying game board partially revealed pattern since usefulness asking question depends context data https abcdef hidden gameboard abcdef abcdef partially revealed gameboard figure battleship game used obtain question data set rothe hidden positions three ships blue red purple game board players sought identify observing partly revealed board players allowed ask natural language question partly revealed board context set consists pairs questions per basic challenge active learning method predict question human ask given context overall rules game particularly challenging data set model subtle differences contexts determine question potentially useful along nature human question asking probabilistic model question generation describe components probabilistic model question generation section describes two key elements approach compositionality computability reflected choice model questions programs section describes grammar defines space allowable section specifies probabilistic generative model sampling relevant programs space remaining sections cover optimization program features alternative models sections compositionality computability analysis data set revealed many questions data set share similar concepts organized different ways example concept ship size appeared various ways across questions long blue ship blue ship tiles ships tiles blue ship less blocks ships size red ship blocks blue ship result first key element modeling question generation recognize compositionality questions words conceptual building blocks predicates like size plus put together create meaning questions plus size red size purple combining meaningful parts give meaning larger expressions prominent approach linguistics compositionality generally influential idea cognitive science second key element computability questions propose human questions like programs executed state world output answer example program executed looks number blue tiles hypothesized imagined battleship game board returns said number corresponds question long blue ship way programs used evaluate potential useful information question executing program set possible likely worlds preferring questions informative identifying true world state approach modeling questions closely although players asked question context small number questions excluded data set ambiguous extremely difficult address computationally see related formalizing question meaning partition possible worlds notion used previous studies linguistics psychology machine systems question answering also fruitfully modeled questions programs computational work cognitive science modeled various kinds concepts programs important contribution work tackles question asking provides method generating meaningful scratch grammar producing questions capture compositionality computability represent questions simple programming language based lambda calculus lisp every unit computation language surrounded parentheses first element function following elements arguments function using prefix notation instance question long blue ship would represented small program size blue examples discussed step abstracted question representation exact choice words maintaining meaning questions thought represented language thought programs language combined example size red size blue asking whether red ship larger blue ship compute answer first inner parentheses evaluated returning number corresponding number red blue tiles game board respectively numbers used arguments function returns either true false final property interest generativity questions ability construct novel expressions useful given context system generate expressions language designed grammar exceptions inspired grammar consists set rewrite rules recursively applied grow expressions expression grown rewrite rules applicable guaranteed interpretable program language create question grammar begins expression contains start symbol rewrites symbols expression applying appropriate grammatical rules symbol rewritten example applying rules size red arrive expression size red table supplementary materials shows core rewrite rules grammar set rules sufficient represent questions human data set enrich expressiveness conciseness language added lambda expressions mapping set operators table supplementary material use seen question ships size conveniently represented map size set blue red purple evaluation map sequentially assigns element set ultimately returns vector three ship sizes three ship sizes compared function course question could also represented size blue size red size purple probabilistic generative model artificial agent using grammar able express wide range questions decide question ask agent needs measure question usefulness syntactically programs informative useful instance program size blue size blue representing question blue ship larger syntactically coherent however useful question ask unlikely asked human answer always false matter true size blue ship propose probabilistic generative model aims predict questions people ask parameters model fit predict frequency humans ask particular questions particular context data set formally fitting generative model problem density estimation space programs space defined grammar define probability question probability question asked model first energy question weighted sum question features weight feature question describe features model variants differ features use second energy related probability exp exp exp vector feature weights highlighting fact probability dependent parameterization weights normalizing constant set possible questions generated grammar tables limit question length normalizing constant needs approximated since large enumerate optimization objective find feature weights maximize likelihood asking questions thus want optimize arg max log questions translated programs human data set optimize via gradient ascent need gradient respect given term expected average feature values given empirical set human questions term expected feature values given model thus gradient zero model perfectly matched data terms average values features computing exact expected feature values model intractable since large number possible questions normalizing constant equation use importance sampling approximate expectation create proposal distribution denoted use question grammar probabilistic context free grammar uniform distributions choosing rules details optimization follows first large set questions sampled order approximate gradient step via importance second run procedure given model training set ran iterations gradient ascent learning rate last purpose evaluating model computing importance sampler also used approximate normalizing constant via estimator question features turn describe question features considered equation namely two features informativeness one length four answer type informativeness perhaps important feature question informativeness model combination bayesian belief updating expected information gain eig compute informativeness agent needs represent several components belief current world state way update belief receives answer sense possible define set questions fewer functions remove rule draw grammar corresponding questions data set asked demonstration colored tile although straightforward represent questions rule probabilistic nature draw led exponentially complex computations set answers empirical question frequency context context context context context context context context context context context context context context context context negative energy figure model predictions regarding frequency asking particular question shows empirical question frequency shows model energy question based full model rank correlation shown context answers battleship game agent must identify single hypothesis hidden game board configuration space possible configurations possible board games agent ask question receive answer updating hypothesis space applying bayes rule prior specified first uniform choice ship sizes second uniform choice possible configurations given sizes likelihood valid output question program executed zero otherwise expected information gain eig value question expected reduction uncertainty true hypothesis averaged across possible answers question eig shannon entropy complete details bayesian ideal observer follow approach used figure shows eig scores top two human questions selected contexts addition feature feig eig added second feature eig zero otherwise provide offset linear eig feature note eig value question always depends game context remaining features described independent context complexity purely maximizing eig often favors long complicated programs polynomial questions size red size blue size purple although machine would problem answering questions poses problem human answerer generally speaking people prefer concise questions rather short questions data set reflect probabilistic context free grammar provides measure complexity favors shorter programs use log probability grammar fcomp log complexity feature answer type added four features answer types boolean number color location question program belongs exactly one answer types see table type orientation subsumed boolean horizontal true vertical false allows model capture differences base rates question types people prefer questions types relevance finally added one auxiliary feature deal fact grammar produce syntactically coherent programs reference game board thus really questions game filter feature marks questions assume agent goal accurately identify current world state general setting agent would require cost function defines helpfulness answer reduced distance goal refer battleship game board value see marker table alternative models evaluate features important question generation tested full model uses features well variants respectively lesioned one key property model use feig thus ignored informativeness questions model ignored complexity feature model ignored answer type features results discussion probabilistic model question generation table log likelihoods model variants uated two main ways first tasked averaged across held contexts ing distribution questions people asked novel scenarios evaluate quantitatively second model tasked generating genuinely novel questions present data set evaluate full qualitatively make predictions different date models fit contexts asked predict remaining one leave one results different model fits models fits first verify compositionality essential ingredient account human question asking given context human questions appear contexts model attempts simply past questions unable account productivity effectively achieving least without much larger training set questions grammar programs provides one account productivity human behavior second compared different models ability quantitatively predict distribution human questions table summarizes model predictions based questions asked contexts full model learned features informativeness complexity answer type relevance provides best account data case lesioning key components resulted lower quality predictions model performed far worse others highlighting important role complexity opposed pure informativeness understanding questions people choose ask full model also outperformed models suggesting people also optimize information gain prefer certain question types questions common values approximate bootstrapped estimate normalizing constant compared full model alternative full model loglikelihood advantage model held bootstrap samples model samples model third considered overall match model human question frequencies figure shows correlations energy values according predictions full model frequencies human questions often participants asked size red ship particular context results show strong agreement contexts along modest alignment others average spearman rank correlation coefficient comparison model achieved model achieved model achieved one limitation human data sparse many questions asked thus correlations features identical questions like size blue refer board zero eig size blue ship already known computational reasons drop contexts especially large hypothesis spaces however made sure grammar designed based full set contexts could express questions human question data set limited measure fit however surprisingly correlation question generation frequency eig alone suggesting key role question complexity features last model tasked generating novel questions part human data set figure shows five novel questions sampled model across four different game contexts questions produced taking five weighted samples set programs produced section approximate inference weights determined energy ensure novelty samples rejected equivalent human question training data set already sampled question equivalence two questions determined mutual information answer distributions partitions possible hypotheses programs differed arguments size blue equivalent size red generated questions figure demonstrate model capable asking novel clever questions useful respective contexts interesting new questions observed human data include ships horizontal context top left ship tiles context blue purple ships touching red purple touching vice versa context column top left tiles color bottom right corner board context four contexts selected illustrate creative range model complete set contexts shown supplementary materials conclusions people use question asking cognitive tool gain information world although people ask rich interesting questions active learning algorithms make focused requests supervised labels formalize computational aspects rich productive way people inquire world central hypothesis active machine learning concepts generalized operate complex compositional space programs evaluated possible worlds end project represents step toward capable active learning machines also number limitations current approach first system operates semantic representations rather natural language text directly although possible system interface recent tools computational linguistics bridge gap second aspects grammar specific battleship domain often said knowledge needed ask good question critics approach point model begins substantial domain knowledge special purpose structures hand many aspects grammar domain general rather domain specific including general functions programming constructs logical connectives set operations arithmetic mapping extend approach new domains unclear exactly much new knowledge engineering needed much preserved current architecture future work bring additional clarity extend approach different domains perspective computational cognitive science results show people balance informativeness complexity producing semantically coherent questions formulating question asking program generation provide first predictive model date human question asking acknowledgments thank chris barker sam bowman noah goodman doug markant feedback advice research supported nsf grant john templeton foundation varieties understanding project john mcdonnell foundation scholar award tmg mooresloan data science environment nyu figure novel questions generated probabilistic model across four contexts five model questions displayed next two informative human questions comparison model questions sampled equivalent training set natural language translations question programs provided interpretation questions lower energy likely according model references bordes weston learning dialog arxiv preprint chouinard children questions mechanism cognitive development monographs society research cognitive development fodor language thought harvard university press fodor pylyshyn connectionism cognitive architecture critical analysis cognition goodman tenenbaum gerstenberg concepts probabilistic language thought margolis laurence editors concepts new directions mit press cambridge groenendijk stokhof semantics questions pragmantics answers phd thesis university amsterdam gureckis markant active learning strategies spatial concept learning game proceedings annual conference cognitive science society gureckis markant learning cognitive computational perspective perspectives psychological science hawkins stuhlmuller degen goodman ask good questions provoke informative answer proceedings annual conference cognitive science society jacobson compositional semantics oxford university press jain zhang schwing creativity generating diverse questions using variational autoencoders arxiv preprint johnson hariharan van der maaten hoffman zitnick girshick inferring executing programs visual reasoning international conference computer vision lake salakhutdinov tenenbaum concept learning probabilistic program induction science lake ullman tenenbaum gershman building machines learn think like people behavioral brain sciences marcus algebraic mind integrating connectionism cognitive science mit press cambridge mostafazadeh misra devlin mitchell vanderwende generating natural questions image proceedings annual meeting association computational linguistics pages piantadosi tenenbaum goodman bootstrapping language thought formal model numerical concept learning cognition roberts information structure discourse towards integrated formal theory pragmatics working papers state university department linguistics pages rothe lake gureckis asking evaluating natural language questions papafragou grodner mirman trueswell editors proceedings annual conference cognitive science society austin serban sordoni bengio courville pineau building dialogue systems using generative hierarchical neural network models proceedings thirtieth aaai conference artificial intelligence settles active learning morgan claypool publishers strub harm mary piot courville pietquin optimization visually grounded dialogue systems international joint conference artificial intelligence ijcai vijayakumar cogswell selvaraju sun lee crandall batra diverse beam search decoding diverse solutions neural sequence models arxiv preprint wang berant liang building semantic parser overnight proceedings annual meeting association computational linguistics international joint conference natural language processing volume long papers pages young thomson williams statistical spoken dialog systems review proceedings ieee supplementary material supplementary material contains following game boards served contexts human question data set figure full set grammatical rules used simulations table five novel questions context produced computational model table trial trial abcdef trial trial trial trial abcdef abcdef abcdef abcdef trial trial trial abcdef abcdef abcdef trial abcdef abcdef trial abcdef trial trial abcdef trial trial abcdef abcdef abcdef trial trial trial abcdef abcdef abcdef figure partly revealed game boards serving contexts participants generated questions scratch rothe table part grammatical rules defining set possible questions based rewrite rules represent questions human question data set see text details rules marked reference battleship game board evaluation function orient looks orientation ship game board rules evaluated without access game board answer types true false setn setb setb touch issubset setl setl numbers setn setb size row col setsize setl colors water color blue red purple orientation orient locations topleft setl bottomright setl draw boolean number color orientation location true elements set numbers equal true element set booleans true true elements set booleans true true two ships touching diagonal count true first set locations subset second set locations number true elements set booleans size ship row number location column number location number elements set locations ship color color location lambda variable ships horizontal vertical orientation ship row column lambda variable locations left top location set locations right bottom location set locations sample location color table part grammatical rules see text details mapping setb map fyb setl setb map fxb sets setn map fxn sets setl map fxl sets map boolean expression onto location set map boolean expression onto ship set map numerical expression onto ship set map location expression onto ship set lambda expressions fyb fxb fxn fxl boolean expression location variable boolean expression ship variable numeric expression ship variable location expression ship variable sets sets set blue red purple setl set setl coloredtiles setl setdifference setl setl setl union setl setl setl intersection setl setl setl unique setl ships locations locations color remove second set first set combine sets elements exist sets unique elements set table part novel question programs generated probabilistic model model questions sampled filtered novelty meaning never appeared training set please see main text details sampling process context refers contexts figure energy scores reflect quality assigned model context program energy coll topleft coloredtiles water coll bottomright coloredtiles red rowl topleft coloredtiles red bottomright coloredtiles color purple color rowl bottomright coloredtiles purple setsize coloredtiles color purple color topleft coloredtiles color color topleft map lambda set blue red purple setsize coloredtiles color bottomright set setsize coloredtiles color bottomright coloredtiles color coll bottomright unique coloredtiles water coll bottomright coloredtiles color coll topleft coloredtiles water rowl bottomright coloredtiles red setsize coloredtiles color map lambda touch blue red coloredtiles water rowl bottomright coloredtiles color coll bottomright coloredtiles water rowl topleft coloredtiles purple map lambda orient set blue red purple coll bottomright coloredtiles color map lambda size set blue red purple rowl bottomright coloredtiles red coll bottomright coloredtiles red setsize coloredtiles color map lambda true coloredtiles color topleft set coll topleft coloredtiles color topleft coloredtiles blue table part novel question programs context program energy coll bottomright coloredtiles blue setsize coloredtiles color topleft setdifference set coloredtiles water color topleft coloredtiles color touch blue purple touch red purple rowl bottomright coloredtiles blue coll topleft coloredtiles purple rowl topleft coloredtiles purple setsize coloredtiles color bottomright coloredtiles color coll topleft coloredtiles red coll bottomright coloredtiles red topleft unique coloredtiles water topleft coloredtiles color color bottomright coloredtiles color coll bottomright coloredtiles water coll topleft coloredtiles water rowl bottomright coloredtiles water setsize coloredtiles color topleft coloredtiles color coll topleft coloredtiles water setsize coloredtiles color bottomright set rowl bottomright coloredtiles blue topleft coloredtiles color coll bottomright coloredtiles color coll bottomright coloredtiles red setsize coloredtiles color bottomright coloredtiles color topleft coloredtiles water topleft coloredtiles color coll bottomright coloredtiles color rowl bottomright coloredtiles red rowl topleft coloredtiles red setsize coloredtiles color coll topleft coloredtiles color bottomright set issubset coloredtiles water coloredtiles color setsize coloredtiles color topleft set coll topleft coloredtiles red setsize coloredtiles color topleft coloredtiles color color bottomright setdifference set coloredtiles water coll bottomright coloredtiles water coll topleft coloredtiles water setsize coloredtiles color bottomright set bottomright coloredtiles color topleft set map lambda orient set blue red purple setsize coloredtiles color topleft set bottomright coloredtiles color topleft set setsize coloredtiles color rowl bottomright coloredtiles color coll bottomright coloredtiles color
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bounds general formula region successive refinement problem tetsunao feb source symbol encoder encoder decoder reprod symbol decoder reprod symbol tomohiko region since codeword used decoders always optimize rates like case codeword used reconstruction separately however cases achieve optimum rates necessary sufficient conditions cases independently given koshelev equitz cover complete characterization region discrete stationary memoryless sources given rimoldi yamamoto also gave region special case general coding problem later effros characterized region discrete stationary ergodic sources recently asymptotic analysis rates blocklength becomes active target study especially successive refinement problem zhou gave lot results set rates discrete gaussian stationary memoryless sources considered separate criteria probability distortion exceeds given distortion level less given probability level separately reconstruction hand zhou considered joint criterion probability either distortions exceeds given distortion level less given probability level although also gave several nonasymptotic bounds set pairs rates mainly focus asymptotic behavior set hand paper consider bounds pairs rates finite blocklengths especially since rate easily calculated number codewords focus pairs two numbers codewords although adopt separate criteria result easily applied joint criterion give inner outer bounds pairs numbers codewords bounds characterized using smooth max divergence introduced warsi lossy source coding problem also used smooth max divergence characterize function minimum rate blocklength unlimited proof techniques similar employ several extended results successive refinement problem inner bound derived using extended version previous lemma lemma give lemma special case extended version previous generalized covering lemma lemma outer bound derived using extended version previous converse bound lemma paper also consider region figure successive refinement problem summary successive refinement problem sequence emitted information source encoded two codewords two encoders order give two reconstructions sequence one two reconstructions obtained one two codewords reconstruction obtained two codewords coding problem give inner outer bounds pairs numbers codewords two encoders probability distortion exceeds given distortion level less given probability level also give general formula region general sources region set rate pairs two encoders maximum value possible distortions less given distortion level key words general source information spectrum bound region successive refinement introduction successive refinement problem lossy source coding problem many terminals see fig coding problem sequence emitted information source encoded two codewords two encoders order give two reconstructions sequence one two reconstructions obtained one two codewords using decoder reconstruction obtained two codewords using decoder important parameter successive refinement problem pair rates two encoders distortion source sequence reconstruction less given distortion level set pairs length blocklength source sequence unlimited called portions paper presented symposium information theory applications ieice society conference tetsu uematsu authors dept information communications engineering tokyo institute technology tokyo japan general sources case adopt criterion maximum value possible distortion less given distortion level reconstruction using spectral information rate inner outer bounds give general formula region show ratedistortion region coincides region obtained rimoldi source discrete stationary memoryless furthermore consider mixed source mixture two sources show region intersection two sources rest paper organized follows section provide notations formal definition successive refinement problem section give several lemmas inner bound pairs numbers codewords region lemmas extended versions previous results lemma lemma section give outer inner bounds using smooth max divergence pairs numbers codewords section give general formula region section consider region discrete stationary memoryless sources mixed sources section conclude paper note numbers codewords two decoders decoder decoder represented functions respectively refer tuple encoders decoders code order measure distortions source symbol reconstruction symbols introduce distortion measures defined functions define two events exceeding given distortion levels follows define achievability criterion definition positive integers real numbers let source say exists code numbers codewords encoder encoder respectively follows constants often use simple notations setting consider set pairs numbers codewords excessdistortion criterion according set defined follows preliminaries let sets positive integers real numbers real numbers respectively unless otherwise stated use following notations pair integers set integers denoted finite countably infinite sets set probability distributions denoted respectively set conditional probability distributions given denoted probability distribution random variable denoted subscript notation conditional probability distribution given denoted cartesian product set denoted sequence symbols denoted denoted sequence rvs letter sequences probability distributions conditional probability distributions denoted letters respectively successive refinement problem fig let finite countably infinite sets represents source alphabet represent two reconstruction alphabets let represents single source symbol since source characterized also refer source consider cartesian product certain finite countably infinite set regard source symbol source sequence thus sake brevity deal single source symbol unless otherwise stated two encoders encoder encoder represented functions respectively positive integers definition source real numbers define basically paper deals coding single source symbol however section deal coding source sequence hence section abuse notation regard sets cartesian products respectively also regard source symbol source sequence call sequence source sequences general source required satisfy consistency condition use superscript code distortion measures numbers codewords make clear dealing source sequences length code define rates log hereafter log means natural logarithm introduce maximum distortions sequence codes end define limit superior probability definition limit superior probability arbitrary rvs define limit sequence superior probability sup inf lim introduce maximum distortions sup sup denotes indicator function proof define achievability maximum distortion criterion definition real numbers let general source real numbers say pair exists sequence codes satisfying sup sup remark show region closed set definition using diagonal line argument note regard sequence definition gives region pairs rates given finite blocklength covering lemma recalling independent coincides right side section introduce useful lemmas corollaries inner bound set next lemma basic important result sense subsequent results section given lemma lemma implies exact analysis error probability covering set terms given condition codewords random coding hence lemma regarded extended version theorem although lemma gives exact analysis difficult use characterizing inner bound pairs numbers codewords region instead use next convenient lemma lemma let arbitrary rvs pair independent integer let rvs independent distributed according integer let rvs independent distributed according set definition region general source real numbers define follows constants often use simple notation setting consider set rate pairs maximum distortion criterion according set usually called region defined follows lim sup lemma let arbitrary rvs rvs pair independent let function constant furthermore let function constant pbc proof max follows since comes fact importance lemma able change rvs arbitrary correlated rvs makes possible characterize inner bound pairs numbers codewords region lemma regarded extended version previous lemma lemma multiple correlated rvs hence like previous lemma changing functions constants gives many types bounds following two corollaries abc pbc comes follows since lemma comes since probability grater abc corollary real numbers integers exp exp log log log log exp exp proof let exp exp hand let constants exp exp abc log log log log inf easily check constants functions satisfy plugging functions constants desired bound corollary regarded bound terms information spectrum best knowledge type bound reported far although converse bounds lemma theorem hand next corollary gives bound terms smooth max divergence pkq defined pkq inf abc inf abc abc log sup thus satisfy plugging functions constants max corollary real numbers integers exp exp exp exp log sup pkq simply defined pkq log sup inf definition may negative value depending pkq since case meaningless study adopt also note since real valued functions holds lemma sup remark original definition smooth max divergence follows log sup pkq inf abc abc use inequalities since arbitrary completes proof proof arbitrarily fixed let functions abc abc pkq pkq follows since inner outer bounds set pairs numbers codewords remark proof valid even without auxiliary introduced merely consistency outer bound section give outer inner bounds using smooth max divergence first show bound probability two events successive refinement problem follows let arbitrary set use next notation sake simplicity definition rvs define abc theorem source let rvs independent real numbers exists code numbers codewords encoder encoder respectively also define following set probability distributions given source constants puy using theorem give inner bound theorem inner bound source real numbers puy proof generate independently subject probability distribution define set generate independently subject probability distribution define set denote given set given symbol choose puy min puy log log log log log log exists pair set pair define encoders tuple rvs probability distribution puy hand define decoders proof show puy exp log exp log taking average random selection average probability follows end let rvs independent marginal distribution puy according theorem exists code numbers codewords encoder encoder respectively denote randomly selected sequences noting generated given theorem follows lemma setting max lemma function let kgkc denote size image one output fixed hand according corollary kgkc exp log sup kgkc log exp proof let supc kgkc define subset function min since pbc exists thus follows fact puy comes implies remark proof also valid restrict distribution however sake simplicity consider restricted case outer bound given next theorem kgkc theorem outer bound source real numbers set puy using lemma easy see abc puy puy log log log thus log proving theorem show necessary lemmas log lemma suppose pair rvs satisfies log sup sup log sup abc log log comes definition completes proof proof since lemma proved similar manner lemma omit proof give proof theorem proof theorem let size image encoder since assumption exists injective function next lemma extended version lemma gives bound size image function function let uid image uid inverse function uid suppose exists code since arbitrary imply puy recalling satisfy pair puy thus setting puy remark employ role fixing certain codeword bound thus proof introducing quite important constant value let according lemma log log log remark gave inner outer bounds using information order infinity information generalized version mutual information paper however use smooth max divergence compatible information spectrum quantity well studied useful analyze rates code hand uid general formula distortion region last inequality follows since size uid combining log log log sup log sup log sup hand uid smooth max divergence related spectral information rate shown corollary next lemma lemma consider two sequences rvs set definition sequence rvs define section deal coding source sequence give general formula region first introduce spectral conditional information rate let according lemma log completes proof lim lim sup kpy sup log real numbers holds proof sequence lim lim thus according note uid holds combining log log kpy lim lim sup lim lim sup kpy direct part section show according lemma holds kpy lim lim sup sup log let suppose every sufficiently large furthermore according lemma right side thus combining lemma corollary sequence rvs lim lim sup lim lim sup hence according theorem exists sequence codes sufficiently large proof since corollary immediately follows lemma definition omit proof let sequence conditional probability distributions define sup sup log log exp log log log exp log thus lim sup lim sup sequence rvs induced general source main result section next theorem gives general formula region lim sup lim sup log exp log lim sup log exp lim sup lim sup log exp theorem general source real numbers set remark show right side closed set using diagonal line argument remark remark sure whether sequence auxiliary rvs really necessary may possible characterize region without proof theorem presented subsequent two sections sections code denote lim sup means according theorem exists sequence conditional probability distributions nonwhere comes corollary fact increasing function follows since using usual diagonal line argument construct sequence codes log lim sup lim sup log lim sup lim sup lim sup lim sup since holds lim lim sup lim log lim lim sup implies lim sup sup log lim lim sup sup comes corollary hand since must satisfy combining conclude pair thus implies converse part implies section show discrete stationary memoryless sources section show region given theorem coincides region rimoldi source discrete stationary memoryless source let finite sets since discrete stationary memoryless source assume sequence independent copies also assume distortion measures additive two functions distortion measures represented suppose exists sequence codes satisfying sup sup lim sup log thus lim define lim implies exists sequence sequence rvs independent tuple rvs induced conditional probability distribution given next theorem theorem discrete stationary memoryless source additive distortion measures set pxi pxi pxi marginal distribution according lemma thus introducing set probability distributions independent rvs pyi remark right side written pyz function gives boundary given defined see corollary min min note convex continuous functions triple see remark lemma proof left side right side thus hand way exists prove theorem two parts separately respectively come fact sequence rvs induced given source last equality comes since convex continuous functions triple see remark holds let exist sufficiently small defining becomes sequence independent copies rvs thus combining use fact rvs hence noting sup according remark completes proof proof left side right side since exists injective function function let uid image uid inverse function uid let symbol uid define puy puy puy implies holds exists otherwise since uid pxuy pxy otherwise uid otherwise according pair included gion hand since holds due pair also included region therefore since holds completes proof remark unlike region rimoldi region includes sequence auxiliary rvs comes fact argument employed holds general furthermore since otherwise mixed sources section give region mixed sources mixed source defined max max max next lemma shows fundamental property information spectrum mixed sources max lemma sequences rvs let defined comes lemma lemma fact max max proof since lemma proved way lemma using lemma omit details completes proof conclusion next theorem shows region mixed source intersection two sources paper dealt successive refinement problem gave inner outer bounds using smooth max divergence set pairs numbers codewords bounds obtained extended versions previous covering lemma converse bound using bounds also gave general formula using spectral information rate region showed special cases region discrete stationary memoryless sources mixed sources theorem mixed source defined real numbers proof define acknowledgment work supported part jsps kakenhi grant number definition implies exist hand trivially exists thus references matsuta uyematsu bounds numbers codewords successive refinement problem proc symp inf theory apps matsuta uyematsu general formula achievable rate region successive refinement problem proc ieice society conference koshelev estimation mean error discrete scheme problemy peredachi informatsii thus according theorem kostina tuncel successive refinement abstract sources arxiv preprint july koshelev hierarchical coding discrete sources problemy peredachi informatsii equitz cover successive refinement information ieee trans inf theory mar rimoldi successive refinement information characterization achievable rates ieee trans inf theory yamamoto source coding theory triangular communication system ieee trans inf theory may effros bounds variablerate multiresolution source codes ieee trans inf theory ingber weissman strong successive refinability tradeoff ieee trans inf theory june zhou tan motani moderate deviations asymptotics successive refinement ieee trans inf theory may warsi bounds various information theoretic problems using smooth min max divergences proc ieee inf theory workshop uyematsu matsuta revisiting theory using smooth max divergence proc ieee inf theory workshop matsuta uyematsu new bounds numbers codewords lossy compression ieice trans fundamentals han methods information theory springer kostina lossy compression finite blocklength regime ieee trans inf theory june cover thomas elements information theory wiley new york kostina tuncel function successive refinement abstract sources proc ieee int symp inf theory june information inf theory apps workshop kanlis narayan error exponents successive refinement partitioning ieee trans inf theory
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training testing object detectors virtual images dec yonglin tian xuan kunfeng wang member ieee wang fellow ieee area computer vision deep learning produced variety models rely massive labeled data however collecting annotating images real world great demand labor money investments usually passive build datasets specific characteristics small area objects high occlusion level framework parallel vision paper presents purposeful way design artificial scenes automatically generate virtual images precise annotations virtual dataset named paralleleye built used several computer vision tasks training dpm deformable parts model faster detectors prove performance models significantly improved combining paralleleye publicly available datasets training phase addition investigate potential testing trained models specific aspect using intentionally designed virtual datasets order discover flaws trained models experimental results conclude virtual dataset viable train test object detectors index vision virtual dataset object detection deep learning ntroduction atasets play important role training testing computer vision algorithms however datasets usually satisfactory due insufficient diversity labeling images real world especially largescale complex traffic systems moreover highly subjective work annotate images manually example different people may different annotation results image result labeling result deviate extent ground truth even seriously affect performance computer vision algorithms work supported national natural science foundation china yonglin tian department automation university science technology china hefei china also state key laboratory management control complex systems institute automation chinese academy sciences beijing china tyldyx xuan school automation beijing institute technology beijing china kunfeng wang corresponding author state key laboratory management control complex systems institute automation chinese academy sciences beijing china also qingdao academy intelligent industries qingdao china wang state key laboratory management control complex systems institute automation chinese academy sciences beijing china also research center computational experiments parallel systems technology national university defense technology changsha china feiyue existing datasets originate real world kitti pascal voc coco imagenet datasets many advantages also shortcomings kitti dataset world largest computer vision dataset automatic driving scenarios including one hundred thousand labeled cars however kitti lacks common types objects bus number trucks small pascal voc dataset serves benchmark classification recognition detection visual objects pascal voc contains categories images per category average less one thousand imagenet dataset world largest database image recognition including categories however semantic segmentation labeling information pictures classes objects coco dataset task annotating dataset onerous example takes hours determine object categories present images coco generally speaking real datasets confronted many problems small scale tedious annotation setting dataset precise annotations real world means great labor financial investments let alone building dataset specific features like diverse areas objects occlusion levels however latter occupies significant position addressing problems visual perception understanding work wang proposed theoretical framework parallel vision extending acp approach elaborated significance virtual data acp methodology establishes foundation parallel intelligence provides new insight tackle issues complex systems framework parallel vision depicted fig obvious see great advantage virtual world produce diverse labeled datasets different environmental conditions texture change usually regarded important image features object detection work take specifically designed virtual datasets resources train object detectors also tool produce feedback performance trained models testing phase choose dpm deformable parts model faster object detectors work dpm one effective object detectors based hog histogram oriented gradient resurgence deep learning faster currently state art approach widely used object detection based fast ren introduced region proposal fig appearance artificial scene left map information exported osm right final artificial scene fig basic framework architecture parallel vision network rpn share convolutional features whole image detection network work greatly reduced time cost generate region proposals improved qualities well faster lays foundation many detection models recent years paper present efficient way construct virtual image datasets advanced computer graphics techniques proves flexible feasible method build training datasets greatly satisfy needs diversity scale specific occlusion level basis study effectiveness virtual dataset train test object detectors elated ork many attempts use virtual world carry scientific researches bainbridge investigated feasibility utilizing second life world warcraft sites research social behavioral economic sciences well computer science virtual living lab prendinger conducted several controlled driving travel studies area computer vision early works involved training pedestrian detectors based hog linear svm pedestrian detector virtual datasets generated video game besides training models data also used explore invariance deep features dcnns missing cues domain adaptation issues semantic segmentation research richter presented way build virtual datasets via modern video game got corresponding annotations using outside graphics hardware without access source code game approaches rely video games rather setting virtual world scratch resulting bad flexibility research process recently ros set virtual world collected images semantic annotations virtual cameras different weather conditions observing angles generated virtual dataset named synthia flexible way used training dcnns semantic segmentations driving scenes however synthia lacks annotations computer vision tasks object detection tracking similar way gaidon proposed world cloning method released video dataset called virtual kitti multiobject tracking analysis basically dataset clone real kitti overall framework layout constricted real kitti dataset nowdays generative adversarial networks gans widely used produce photorealistic synthetic images however images lack corresponding annotations work flexible approach building artificial scenes scratch proposed intend set virtual datasets specific features diverse annotations train test object detectors iii irtual dataset parallel construction artificial scene imitate layout urban scene real world exported map information area interest zhongguancun area beijing platform openstreetmap osm based raw map generated buildings roads trees vegetation fences chairs traffic lights traffic signs static entities using cga computer generated architecture rules cityengine finally imported scene game engine cars buses trucks added using scripts controlled vehicles move according certain traffic rules virtual world also help shaders weather lighting condition adjusted needed artificial scene shown fig annotations virtual images build dataset need get annotations corresponding images data labeling always headache machine learning researchers however simple efficient get ground truths virtual images artificial scene via components like meshfilter shader achieved simultaneous ground truth generation including depth optical flow bounding box semantic segmentation scene running fig shows annotations different vision tasks setting virtual dataset order increase diversity virtual dataset configured different weather cloudy sunny rainy foggy illumination sunrise sunset conditions artificial scene shown fig changes deemed significant effect performance object detectors real world placed virtual camera moving car used capturing images scene produce obvious change object appearance fig diversity illuminations weather conditions top virtual images took left right bottom virtual images weather foggy left rainy right fig framework constructing virtual dataset cityengine set static city including buildings roads using cga rules based map information osm import models interesting objects cars people animals trees static city thus forming several static scenes activate static scenes controlling virtual objects move using scripts control virtual camera move capture images using scripts compute annotations bounding box semantic segmentation using scripts shaders fig virtual camera artificial scene fig annotations different vision tasks top depth left optical flow right bottom bounding box left semantic segmentation right set different parameters camera including height orientation field view virtual camera illustrated fig sight distance camera adjusted much longer practice placed several cars buses trucks lanes move following instruction scripts vehicles put near roads different manners sparse dense create diverse occlusion levels purpose testing improving performance object detectors objects distinct colors poses achieved color pose change interesting objects artificial scene virtual camera collecting images based techniques built virtual dataset named paralleleye composed three sub datasets paralleleye first part virtual dataset set camera looking five directions degree respect moving direction therefore camera long sight distance capture small far objects paralleleye orientation camera adjusted degree respect moving direction set vehicles rotate around axes occlusion intentionally introduced get better understanding effect pose change trained models paralleleye designed investigate influence color occluded condition performance object detector placed vehicles crowdedly changed colors every frame camera set look forward sample images three parts fig sample images three virtual sub datasets first row paralleleye second row paralleleye third fourth rows paralleleye annotations workstation efficient compared manual labor example usually takes dozens minutes finish labeling work single image containing various categories used segmentation task virtual dataset mainly built researches intelligent vehicles included three common types objects objects person motorbikes yet contained due less appearance view virtual camera shown fig future add objects virtual scene framework shown fig build dataset containing categories specifically add models person motorbikes animals virtual scene step take account scripts steps able capture images containing objects diverse annotations also worth mentioning ways reduce computational complexity generalize virtual scene bigger city environments importing corresponding maps step models step fig great computational capacity demanded scale virtual world increases action taken practice two methods handle problem one take account objects visible virtual camera instead objects scene via tricks like occlusion culling besides also effective way replace intricate textures structures rougher ones objects far camera measures help decrease workload cpu gpu platform number vertex compute thus leading acceptable frame rate even scene running virtual dataset shown fig dataset properties paralleleye includes images annotations three classes objects car bus truck voc format numbers images objects three sub datasets recorded table bar graphs fig depict object occurrence object geometry statistics well occlusion level three sub datasets objects labelled small whose areas smaller pixels large whose areas larger pixels rest labelled medium occlusion level objects labelled slightly occluded whose occlusion rates less largely occluded whose occlusion rates partly occluded rest clear see paralleleye paralleleye crowded paralleleye means less objects one image paralleleye compared sub datasets importantly paralleleye small objects paralleleye higher occlusion level discussion experiments platform work fps frames per second produce virtual images different table umbers images objects three virtual sub datasets sub dataset number images number cars number buses number trucks paralleleye paralleleye paralleleye xperiments experiments images annotations stored form pascal voc faster weights initialized imagenet weights experiments dpm used model category ratio positive negative examples set training experiments paralleleye obtain real dataset containing three common traffic objects car bus truck selected images including car bus pascal voc images containing truck coco transformed voc style coco includes images containing truck chose images excluded images fig object occurrence object geometry occlusion level virtual datasets figure shows left right distribution number instances within image object area distribution occlusion level truck shares fairly small proportion images corresponding annotations voc coco combined together randomly divided training set testing set proportion firstly trained faster model real training set initial learning rate set decreased factor iterations chose one image batch size momentum factor weight decay factor randomly picked images virtual dataset paralleleye combined real training data mixed training set train another model setting used faster respectively experiments finally carried experiments dpm using datasets models evaluated real testing set generated pascal voc coco followed steps standard evaluation procedure pascal voc calculate average precision category intersection union iou set faster rcnn dpm results shown table examples detection results faster based architecture shown fig noticed bus dpm higher faster may caused fact shape bus flexible car truck easier learn also number bus less two types objects adverse faster deep learning models usually require data learn traditional ones also noticed introducing truck category coco car bus decreased compared models trained purely pascal voc dataset faster may interpreted fact coco dataset challenging object detection contains difficult images objects partially occluded amid clutter etc table erformance models evaluated model training dataset car bus truck dpm real mixed faster real mixed faster real mixed training experiment kitti paralleleye images annotations containing car object picked kitti dataset divided real training set testing set ratio one one first trained faster detector purely real training set next images annotations containing car object randomly selected paralleleye dataset used faster model model using real training set performed experiments dpm real mixed data experiments executed setting experiments trained models tested real kitti testing set average precision recorded table iii fig depicts examples detection results faster based architecture table iii erformance models evaluated kitti model training dataset test kitti kitti mixed faster kitti paralleleye kitti faster kitti paralleleye kitti dpm testing experiments kitti paralleleye section investigate potential using virtual dataset test object detector trained real dataset kitti chosen real dataset train faster rcnn models due diversity object area occlusion condition images annotations containing car object picked kitti dataset divided training set testing set ratio one one named full training set train full avoid confuse train full set possessed images cars first trained faster detector train full detector used reference detectors evaluate impact small objects model performance deleted objects train full whose areas smaller pixels according annotations standard got images containing cars called devised training set train large used fig examples object detection results upper one every couple objects detected model trained purely real training set lower one every couple objects detected model trained real virtual datasets train second faster detector third detector trained train visible kept objects labeled fully visible train full images cars experiments carried setting table valuation purposefully designed virtual datasets training dataset test paralleleye test paralleleye test paralleleye train full train large train visible three detectors tested paralleleye characterized small area objects average paralleleye marked high level occlusion well paralleleye larger objects lower occlusion level calculated average precision manner pascal voc results recorded table purpose making results explicit also calculated rate descent removed small objects occluded objects respectively training set model trained train full set regarded reference results shown table table ate descent virtual datasets training dataset rate descent paralleleye rate descent paralleleye rate descent paralleleye train large train visible discussion one hand results show virtual datasets viable improve performance object detector used training together real dataset hand conclude purposefully designed virtual datasets potential tools assess performances trained models specific aspect results table table show small objects removed training set performance model became worse sub datasets bigger rate descent occurred paralleleye testing phase may result smaller average area object paralleleye paralleleye witnessed huger drop rate deleted occluded objects training set paralleleye higher occlusion rate onclusion paper presents pipeline build artificial scenes virtual datasets possessing specific characteristics desire like occlusion level area objects framework parallel vision prove mixing virtual dataset several real datasets train object detector helps improve performance also investigate potential testing trained models specific aspect using intentionally designed virtual datasets work may help deep learning researchers get better understanding models especially areas autonomous driving eferences kaneva torralba freeman evaluation image features using photorealistic virtual world ieee international conference computer vision ieee liu wang shen visual tracking based dynamic coupled conditional random field model ieee transactions intelligent transportation systems vol gou wang yao vehicle license plate recognition based extremal regions restricted boltzmann machines ieee transactions intelligent transportation systems vol liu wang wang tracklet visual object tracking state art beyond acta automatica sinica vol geiger lenz urtasun ready autonomous driving kitti vision benchmark suite ieee conference computer vision pattern recognition ieee everingham van gool williams winn zisserman pascal visual object classes voc challenge international journal computer vision vol deng dong socher imagenet hierarchical image database ieee conference computer vision pattern recognition ieee lin maire belongie hays perona ramanan zitnick microsoft coco common objects context european conference computer vision springer wang gou zheng rehg wang parallel vision perception understanding complex scenes methods framework perspectives artificial intelligence review vol wang gou wang parallel vision approach intelligent vision computing acta automatica sinica vol wang wang xiong wang parallel imaging new theoretical framework image generation pattern recognition artificial intelligence vol wang parallel system methods management control complex systems control decision vol parallel control management intelligent transportation systems concepts architectures applications ieee transactions intelligent transportation systems vol parallel control method computational control acta automatica sinica vol wang zhang zheng wang yuan dai zhang yang alphago thesis alphago thesis beyond journal automatica sinica vol wang zhang wei zheng pdp parallel dynamic programming journal automatica sinica vol wang wang steps toward parallel intelligence journal automatica sinica vol lin zheng wang parallel learning perspective framework journal automatica sinica vol wang yao vehicle detection approach datadriven adaptive networks international journal pattern recognition artificial intelligence vol felzenszwalb girshick mcallester ramanan object detection discriminatively trained models ieee transactions pattern analysis machine intelligence vol ren girshick sun faster towards realtime object detection region proposal networks advances neural information processing systems girshick fast proceedings ieee international conference computer vision bainbridge scientific research potential virtual worlds science vol prendinger gajananan zaki fares molenaar urbano van lint gomaa tokyo virtual living lab designing smart cities based internet ieee internet computing vol marin learning appearance virtual scenarios pedestrian detection ieee conference computer vision pattern recognition ieee ponsa learning pedestrian detector virtual world ieee transactions intelligent transportation systems vol peng sun ali saenko learning deep object detectors models proceedings ieee international conference computer vision sun saenko virtual reality fast adaptation virtual object detectors real domains proceedings british machine vision conference vol richter vineet roth koltun playing data ground truth computer games european conference computer vision springer ros sellart materzynska vazquez lopez synthia dataset large collection synthetic images semantic segmentation urban scenes proceedings ieee conference computer vision pattern recognition gaidon wang cabon vig virtual worlds proxy tracking analysis proceedings ieee conference computer vision pattern recognition goodfellow pougetabadie mirza wardefarley ozair courville bengio generative adversarial networks advances neural information processing systems vol wang gou duan lin zheng wang generative adversarial networks introduction outlook journal automatica sinica vol wang huang tian wen measuring driving behaviors live video ieee intelligent systems vol wang liu gou wang learning approach foreground detection traffic surveillance applications ieee transactions vehicular technology vol zhang wang wang advances perspectives applications deep learning visual object detection acta automatica sinica vol wang tian yan wang paralleleye dataset constructing artificial scenes traffic vision research ieee international conference intelligent transportation systems published yonglin tian received bachelor degree university science technology china currently master student department automation university science technology china well state key laboratory management control complex systems institute automation chinese academy sciences research interests include computer vision pattern recognition xuan received degree changsha university science technology changsha china currently working toward degree control theory control engineering key laboratory intelligent control decision complex systems beijing institute technology beijing china research interests include computer vision pattern recognition intelligent transportation systems printing kunfeng wang received control theory control engineering graduate university chinese academy sciences beijing china joined institute automation chinese academy sciences became associate professor state key laboratory management control complex systems december january visiting scholar school interactive computing georgia institute technology atlanta usa research interests include intelligent transportation systems intelligent vision computing machine learning member ieee wang received computer systems engineering rensselaer polytechnic institute troy new york joined university arizona became professor director robotics automation lab ral program advanced research complex systems parcs founded intelligent control systems engineering center institute automation chinese academy sciences cas beijing china support outstanding oversea chinese talents program state planning council talent program cas appointed director key lab complex systems intelligence science cas became state specially appointed expert director state key laboratory management control complex systems wang current research focuses methods applications parallel systems social computing knowledge automation founding international journal intelligent control systems founding eic ieee magazine eic ieee intelligent systems eic ieee transactions currently eic china journal command control since served general program chair ieee informs acm asme conferences president ieee society chinese association science technology cast usa american zhu kezhen education foundation vice president acm china council since vice president secretary general chinese association automation wang elected fellow ieee incose ifac asme aaas received class national prize natural sciences china awarded outstanding scientist acm work intelligent control social computing received ieee outstanding application research awards ieee smc norbert wiener award
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dec decidable fragment second order logic applications synthesis madhusudan umang mathur shambwaditya saha mahesh viswanathan university illinois urbana champaign usa abstract propose fragment second order logic esmt show checking satisfiability sentences fragment decidable logic quantifier prefix conducive modeling synthesis problems moreover allows reasoning using combination background theories provided decidable satisfiability problem linear arithmetic decision procedure reduces satisfiability esmt formulae satisfiability queries background theories allowing use existing efficient smt solvers theories hence procedure seen effectively smt esmt reasoning keywords second order logic synthesis decidable fragment introduction goal program synthesis automatically construct program satisfies given specification problem received lot attention research community recent years several different approaches proposed address challenge see one approach program synthesis reduce problem satisfiability problem decidable logic constructing sentence whose existentially quantified variables identify program synthesized formula expresses requirements program needs meet paper furthers research program identifying decidable secondorder logic suitable encoding problems program synthesis get useful results one needs constrain semantics functions relations used encoding synthesis problem therefore logic set background theories background theories assumed independently axiomatized equipped solver finally leverage advances made logic solvers aim develop algorithm satisfiability logic makes calls decision procedures background theories goal mind let describe logic multisorted logic roughly described uninterpreted combination theories uct uct universe special sort declared foreground sort sorts declared background sorts assume fixed signature functions relations constants individual background sort purely sort furthermore assume background sort comes associated background theory arbitrary even infinite constrained formulas involving functions relations constants involve background sort main result decidability result satisfiability problem modulo background theories boolean combinations sentences form set existentially quantified first order variables variables take values sorts background foreground set existentially quantified relation variables whose arguments restricted foreground sort set existentially quantified function variables take arguments elements foreground sort return value background sorts set universally quantified first order variables sorts set universally quantified relation variables whose arguments could sorts set universally quantified function variables whose arguments sort could return values sort thus logic sentences prefix allowing quantification first order variables variables relational functional obtain decidability carefully restrict types variables existentially universally quantified described decidability result proceeds follows crucially exploting disjointness universes background theories series simple observations transformations like ackermanization decision procedures similar epr logic class exploit finite model property reduce satisfiability problem logic satisfiability pure logic formulas individual background theories consequently background theories admitted individually decidable satisfiability problem fragment satisfiability logic decidable examples background theories include presburger arithmetic theory fields theory linear arithmetic rationals algorithm satisfiability therefore makes finitely many calls engines individual background theories salient aspects logic decidability result expressing synthesis problems primary design principle logic express synthesis problems synthesis problems typically expressed fragments ask whether exists object wish synthesize using block existential quantifiers object satisfies certain properties expressed universally quantified formula instance synthesizing program snippet required satisfy condition encode asking whether program snippet values variables modeling input snippet verification condition corresponding hoare triple holds context existentially quantified variables first order second order used model program snippets furthermore allowing universal quantification functions allows model aspects program state require uninterpreted functions particular pointer fields model heap design decidability logic also defined carefully avoid undecidability looms logic power first note quantifierfree decidable logics combined get decidable logics using combinations combining quantified logics notoriously hard design choice forces communication theories using foreground sort keeping universes different sorts disjoint allows decidable combination theories illustrative examples synthesis illustrate applicability logic two classes synthesis problems first class involves synthesizing recursive programs work inductive given precise condition program synthesized show model recursive program synthesis synthesizing program output recursive calls provided inputs straightline program show correctness condition recursive program modeled using universally quantified verification condition validated using logic theory combines uninterpreted functions presburger arithmetic using technique literature called natural proofs show synthesis programs possibly integers program unbounded modeled logic modeling universal quantification functions plays crucial role modeling pointers heaps modeling uninterpreted predicates capture inductive predicates list lseg illustrate class examples showing modeling synthesis recursive program searches key singly linked list second class synthesis involves taking recursive definition function synthesizing iteration free function equivalent modeling existential quantification foreground sort well background sort integers utilized synthesized function involves integers show synthesize function equivalent generalizations recursively defined mccarthy function crux contribution therefore providing decidable logic express synthesis problems succinctly believe logic useful researchers working practical applications synthesis implementing decision procedure effectively engineered efficiently researchers working logic engines motivating esmt synthesis applications program synthesis problems conveniently encode esmt define formally section esmt allows existential universal quantification variables relations functions signatures restrictions types existentially quantified relations functions order ensure decidability sentences esmt initial block existential quantifiers followed universal quantifiers section show constituent background theories admit decision procedures satisfiability respective fragments fragment esmt examples background theories include presburger arithmetic linear arithmetic rationals theory real closed fields program synthesis goal search programs bounded size satisfy given specification equation used express search programs terms existential quantification specification candidate program expressed universally quantified formula encode equation one salient features fragment esmt ability quantify functions relations often specifications programs manipulate heaps involve universal quantification uninterpreted functions esmt aptly provides functionality still remaining within boundaries decidability let throw light aspect esmt aspect particularly useful synthesis programs manipulate objects wide variety types integers arrays lists strings sets heaps logic esmt allows existential quantification variables many different sorts thus allowing synthesis richer class programs believe unique aspects esmt useful modeling synthesis general models computation reactive hybrid systems proceed give concrete example synthesis problem demonstrate usability esmt specification function mthree slight variant classical mccarthy function mthree mthree mthree mthree otherwise interested synthesizing straight line program implements mthree expressed term grammar fig term ite pred term term pred fig grammar mthree represent respectively constants variables sort ite stands fig program skeleton expressions grammar trees let fix maximum height tree looking say also node tree children maximum arity function grammar corresponding ite skeleton expression tree show fig every node tree labeled according path root node synthesis problem encoded formula esmt ite input symbols add addition ite input input consti constanti represent allowed operations programs existentially quantifying variables sort allow program numeric constants variables represent choice operator node labeled formula constrains interpretations add ite pairwise different add ite add imposes constraint operators internal nodes one add ite leaf nodes one symbols constraint encodes specification program case definition function mthree universally quantifies values returned operators value required satisfy mthree purpose evaluating effectiveness logic encoded esmt formula smt solver synthesized following expression ite indeed function satisfies specification function mthree thus implying effectiveness approach section show synthesize large class programs amongst others second order logic esmt fragment briefly recall syntax semantics general second order logic present esmt fragment second order logic logic signature tuple fun rel nonempty finite set sorts fun rel respectively sets function symbols relation symbols first order variables function variables relation variables variables associated type one sorts represented function symbol function variable also associated type relation symbol relation variable type assume set symbols either finite countably infinite fun rel countably infinite constants modeled using functions say unsorted singleton sort terms signature associated sort inductively defined grammar fun formulae inductively defined relation variables function variable appropriate types note equality allowed terms sort formula said use function relation variables semantics many sorted logics described using structures tuple collection pairwise disjoint indexed universes interpretation function maps variable element universe function symbol function variable function appropriate type underlying universe similarly relation symbols relation variables also assigned relations appropriate type underlying universe interpretation standard use denote interpretation maps otherwise identical function variable relation variable defined analogously interpretation terms model usual one obtained interpreting variables functions function variables using underlying interpretation model skip details satisfaction relation also defined usual sense skip details theory tuple set possibly infinite sentences theory complete every sentence negation entailed either every model satisfying satisfies every model satisfying satisfies theory consistent case sentence entailed logic esmt describe esmt fragment second order logic prove decidable paper show model synthesis problems let fun rel many sorted signature pure signature type every function symbol every relation symbol single sort however function variables relation variables allowed mix sorts special sort call foreground sort sorts called background sorts function relation symbols involving fragment esmt set sentences defined pure signature foreground sort background sorts following grammar defined grammar quantifier free formulas consist existential quantification block followed universal quantification block existential block variables sort relation variables foreground sort function variables map tuples foreground sort background sort inner universal block allows forms quantification variables function variables relation variables possible types inner formula retrict attention sentences logic assume variables quantified denote resp set existentially resp universally quantified first order variables sort every problem problem consider deciding satisfiability esmt background theories background sorts first introduce concepts uninterpreted combination theories uct pure signature sorts union theories multitheory signature sentence sorted structure satisfies sentences satisfiability problem esmt background theories following given uct sentence logic determine show decidable problem furthermore decision procedure uses finite number calls satisfiability solvers underlying theories check satisfiability esmt sentences rest paper technical convenience assume boolean theory tbool one background theories means bool constants bool bool set sentences tbool abool bool note checking satisfiability sentence tbool decidable decision procedure esmt section present decidability result sentences esmt presence background theories let first state main result paper theorem let pure signature foreground sort background sorts let uct checking satisfiability sentences decidable problem checkk esmt sentences decidable ing prove theorem showing given esmt sentence uct signature transformed using sequence three steps satisfiability preserving transformations finally satisfiability formulae individual theories give brief overview sequence transformations steps step replace occurrence every relation variable quantified universally existentially sort function variable sort bool note outer existentially quantified relation variables keeps within syntactic fragment step eliminate function variables existentially quantified crucially relies small model property foreground universe similar epr process however adds existential variables universally quantified function variables step eliminate universally quantified function variables using standard ackermann reduction adds universally quantified variables steps result sentence combined background theories empty theory foreground sort step show satisfiability formula reduced finite number satisfiability queries sentences individual theories since step eliminating relation variables idea introduce every relational variable function variable corresponds characteristic function let esmt formula transform esmt formula signature every occurrence atom form replaced every quantification replaced bool correctness transformation captured following lemma iff lemma step eliminating existentially quantified function variables first note property respect foreground sort esmt sentences property crucially relies fact existentially quantified function variables ranges foreground sort lemma property let esmt sentence foreground sort background sorts let number existentially quantified variables sort iff structure elements fact proof sketch shows model model foreground universe contains elements interpretations variables foreground sort hence bounded consequently instead existentially quantifying function arity foreground sort background sort instead quantify variables sort capture image functions combination let esmt sentence occurrence relation variables let quantifier free part matrix define let sentence obtained replacing matrix correctness transformation noted iff lemma one eliminate existentially quantified function variables one let background sort every set introduce variable sort let set variables number existential first next introduce fresh function variable order variables sort sort quantify universally used emulate define obtained replacing occurrences define sentence following lemma states correctness guaranteek transformation iff lemma step eliminating universal function variables recipe perform ackermann reduction every universally quantified function variable arity introduce fresh variable range sort every occurrence term augment quantifier free part formula ensures consistency transformation let quantifier free part let every term form introduce fresh first order variable sort replace every occurrence let collection term get newly introduced variables let define ytf ytf transformed formula correct following sense lemma iff decomposition black box calls theory solvers esmt sentence obtained sequence steps first order sentence sentence however may possibly contain occurrences variables foreground sort objective step roughly decompose sentences single sort alone use decision procedures respective theories decide satisfiability decomposed single sorted sentences since decomposition result sentences foreground sort must ensure indeed decision procedure achieve purpose let define empty theory checking satisfiability sentences decidable also satisfiability preserved presence following sense iff lemma first transform quantifier free part equivalent cnf formula let obtained replacing let conjunction clauses disjunction atoms since first order formula pure signature atoms either form possibly leading negation equality atoms restricted terms sort also since pure argument terms relation applications sort means every atom unique associated sort denote sort clause let atoms set ofwatoms let atoms sort let identity lemma state decomposition lemma iff mapping suchvthat formula contract identifies clause atomic constraint makes clause true thus order decide satisfiability straightforward decision procedure involves enumerating contracts contract sort construct sentence make call theory solver contract calls return satisfiable thus original formula satisfiable otherwise unsatisfiable undecidability results logic defined carefully chosen avoid undecidability satisfiability problem show natural generalizations removal restrictions logic renders satisfiability problem undecidable believe results hence simple generalize one restriction functions existentially quantified range sort related restriction universal quantification block quantify uninterpreted function symbols otherwise must existentially quantified outside block let consider fragment logic formulas form fact even background theory since formula single sort droped sort annotations variables hard see logic undecidable theorem consider signature single sort background sorts satisfiability problem sentences following form undecidable theorem simple proof theorem new fact even restrictive logics known undecidable see another important restriction foreground sort various background sorts pariwise disjoint requirement also negotiable decidability desired easy show following theorem consider signature single sort let theory presburger arithmetic satisfiability problem sentences form undecidable stepping back subclasses logic equality decidable satisfiability problem standard class admits prefixes class see results seen extension class background theories background theories admit locally decidable satisfiability problem fragment applications synthesis synthesis validity satisfiability though argued section synthesis problems modeled using satisfiability esmt sentences one subtlety would like highlight synthesis problems asked find expression expression satisfies specification expressed logic assuming specification modeled universally quantified formula background theories would like know holds synthesized expression however logical setting qualify holds means natural way phrasing valid underlying background theories holds models satisfy background theories however existential block models existence expression clearly best seen satisfiability problem asks whether foreground model captures expression requiring holds foreground models including might one element would unreasonable summarize synthesis problem naturally modeled logical problem ask whether foreground model background models satisfy respective background theories inner formula expressing synthesized expression captured foreground model satisfies specification strictly speaking neither satisfiability problem validity problem resolve considering complete consistent background theories hence validity formula background theory equivalent consequently synthesis problems using theories seen asking whether foreground universe modeling expression synthesized background models specification holds expression hence model synthesis purely satisfiability problem esmt described section many background theories used smt solvers complete theories like presburger arithmetic fol reals one incomplete theory often used verification theory uninterpreted dropped sort annotations since single sort functions however case notice since functions sort uninterpreted validity formulas modeled using universal quantification functions logic supports adjustment ensure background theory infinite models choose background theory theory decidable satisfiability problem various scenarios modeling pointers heaps arrays formulation using uninterpreted functions done domain naturally second issue modeling synthesis problems satisfiability problems esmt synthesis need construct expression rather know one exists easy see individual background theory solvers support finding concrete values existentially quantified variables pull back values back across reductions give values existentially quantified variables sorts existentially quantified function variables well existentially quantified relation variables expression synthesized constructed expressing synthesis problems esmt illustrate applicability result solving synthesis problems modeling synthesis recursive problems want model problem synthesizing recursive programs manipulating datastructures given contract seek program meet contract assuming recursive calls smaller satisfy contract though programs seek recursive model certain classes programs using simply programs let take example synthesizing program finds particular key linked list instead ask whether program takes additional input models return value possible recursive call made tail list program must work head list additional input assumed satisfy contract produce output meets contract problem modeled program synthesized using existential quantification grammar generates bounded length programs described section pointer next recursive data structures list lseg verification condition modeled using universal quantification function variables relation variables respectively moreover order tractable verification condition used technique natural proofs soundly formulates condition decidable theory though implementation decision procedure several steps manually finally encoded problem sygus format used enumerative guided synthesis cegis solver tool solved problem within second reported solution manually verified correct convinced recursive program synthesis bounded size separation logics specifications expressed using natural proofs handled using logic modeling synthesis programs equivalent given recursive programs second class examples turn synthesizing programs given recursive function specification example consider knuth generalization recursive mccarthy function otherwise every integer usual mccarthy function consider problem synthesizing equivalent expression programs consider may statements nesting depth conditionals linear expressions unbounded constants existential quantification background arithmetic sort allowed model synthesizing unbounded constants specification demanded value expression satisfy recursive equations given though implementation logic engine modeled foreground model inside arithmetic converted synthesis problem sentence presburger arithmetic booleans experimented several values interestingly solutions synthesized given knuth result closed form expression involves taking remainder modulo expression since taking remainder syntax turns simple expressions exist otherwise also whenever solution found matched expression given knuth see theorem instance found solution ite also verified following values took respectively solve solutions agreed knuth closed form solution also modeled tak function takeuchi given specification otherwise modeled asking expression function using grammar generating finitely many programs got solution ite ite related work several logics known literature express synthesis problems decidable foremost example monadic theory trees express church synthesis problem reactive synthesis problems finite data domains decidability rabin theorem one celebrated theorems logic applicable computer science reactive synthesis well studied applied computer science see example work reported tad closer program synthesis done today synthesizes syntactically restricted programs recursion work finite domains caulfield considered decidability synthesis sygus problems synthesized expressions constrained belong grammar operators usual semantics axiomatized standard theory arithmetic satisfy universally quantified constraint show problem undecidable many cases identify class asks expressions satisfying regular grammar uninterpreted function theory constraints decidable fragment pure predicate logic without function symbols shown decidable bernays without equality ramsey equality often called effectively propositional reasoning epr class one fragments logic known decidable epr class used program verification efficient smt solvers supporting epr developed work extends epr stratified typed logics similarity restriction universes foreground background disjoint however logic therein allow background smt theories unlike epr simple liner arithmetic without addition shown decidable recent paper develops sound complete reasoning safe fragment uninterpreted combination theories however logic undecidable general also support quantification sygus format recently proposed language express syntax guided synthesis problems several synthesis engines developed various tracks sygus however syntax typically allows unbounded programs hence synthesis problem decidable furthermore unlike logic grammar allow arbitrarily large constants like integers synthesized part program conclusions future work logic esmtdefined herein meant decidable logic communication researchers modeling program synthesis problems researchers developing efficient logic solvers liaisons extremely fruitful verification smt solvers served purpose shown logic decidable efficacy modeling synthesis problems however decision procedure several costs paid front practical synthesis tool ways curb costs known literature building efficient synthesis tools particular searching foreground models similar epr efficient engines developed search also guided cegis approaches exponential caused guessing contracts solvers step procedure similar arrangements agreed upon theories combined using method efficient solvers developed hope researchers working logic engines engineer efficient decision procedure esmtthat solve synthesis problems references abadi rabinovich sagiv decidable fragments logic symb comput ackermann ackermann solvable cases decision problem alur bodik dallal fisman garg juniwal madhusudan martin raghothaman synthesis bernays zum entscheidungsproblem der mathematischen logik mathematische annalen bloem galler jobstmann piterman pnueli weiglhofer interactive presentation automatic hardware synthesis specifications case study proceedings conference design automation test europe date gurevich classical decision problem springer science business media buchi landweber solving sequential conditions strategies transactions american mathematical society caulfield rabe seshia tripakis decidable synthesis christof loding madhusudan foundations natural proofs quantifier instantiation tech http horbach voigt weidenbach combination fragment simple linear integer arithmetic itzhaky banerjee immerman nanevski sagiv effectivelypropositional reasoning reachability linked data structures international conference computer aided verification knuth textbook examples recursion artificial intelligence mathematical theory computation papers honor john mccarthy madhusudan synthesizing reactive programs computer science logic csl international annual conference eacsl manna mccarthy properties programs partial function logic tech stanford univ calif dept computer science nelson oppen simplification cooperating decision procedures acm transactions programming languages systems toplas padon mcmillan panda sagiv shoham ivy safety verification interactive generalization acm sigplan notices pek qiu madhusudan natural proofs data structure manipulation using separation logic acm sigplan notices piskac moura deciding effectively propositional logic equality tech technical report microsoft research pnueli rodeh strichman siegel small model property small information computation qiu garg madhusudan natural proofs structure data separation acm sigplan notices rabin decidability theories automata infinite trees transactions american mathematical society proofs section proof lemma proof sketch present interesting direction consider model let interpretation function extends inner universally quantified subformula let restriction foreground universe interpretations variables clearly let first show first see every extension must quantifier free part thus also clearly equation must also hold extensions map universal variables set maps universally quantified function variables range sort function interpretations whose ranges limited set thus must also case restrict universe set every universal extension also projection one interpretations proof lemma proof sketch let consider skolem norm form obtained replacing existential variables skolem constants propositional boolean structure use notation ith clause consider structure suppose contrary clause every sort means every sort interpretation extends valuations either leads falsity clause let values assigned universal variables construct interpretation extending variables interpreted interpretation shown either violate one theory axioms formula case contradiction proofs section proof lemma proof sketch show mild modification standard proofs undecidability logic existentially quantify variable zero function succ demand element succ zero every succ succ establishes infinite model distinct elements succn zero every proceed encode problem machine using relation stands state time counters respectively easy see done using universal quantification relation modeled function easily proof lemma proof sketch use similar proof theorem except use successor function available presburger arithmetic reduce turing machines machines satisfiability formulas
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dec sandwich structures arbitrary functions group theory ian hawthorn abstract functions groups property function conjugates inverse preserving called sandwich morphisms maps preserve structure within group known sandwich structure sandwich structures left distributive idempotent left involutary magmas provide generalisation groups call sandwich paper explores sandwiches relationship groups introduction group homomorphisms maps groups preserve group structure earlier papers author looked arbitrary functions groups general homomorphisms sense said partially preserve group structure study functions closely related study generalisations groups given generalisation groups functions groups morphisms generalisation give interesting collection conversely given collection natural look properties groups preserve generalisation groups may obtained considering algebraic structures properties interesting sets arbitrary functions give group generalisation fashion done may lead generalisations groups might otherwise considered paper take fairly interesting class functions groups namely function conjugates inverse preserving use set functions obtain generalisation groups function conjugation inverse preserving functions notion function conjugation introduced arbitrary function finite groups define new function call conjugate clearly group homomorphism consider example inverse function hence function conjugation generalises usual conjugate note hence conjugation maps set functions onto set identity preserving ones furthermore function conjugation defines group mathematics subject classification primary thanks tim stokes advice introducing ian hawthorn action set identity preserving functions mapping since homomorphisms precisely functions invariant action function action defined left similar action defined right introduce temporary notation action left define action right indeed action might expect actions left right related fact equivalent intertwining map given note hence order two particular bijection one easily check defines equivalence left right actions two actions equivalent reasonable look left actions via initial less cumbersome notation use refer right action necessary left right actions identical words function conjugation commutes inverse map function property called inverse preserving preserves relationship inverse inverse preserving functions easy construct furthermore odd collection functions closed map must contain inverse preserving function note inverse preserving general need inverse preserving however case inverse preserving function conjugates inverse preserving say strongly inverse preserving strongly inverse preserving functions first introduced although little done paper property strongly inverse preserving interesting surprisingly strong constraint shall show next section strongly inverse preserving inverse preserving however need inverse preserving left multiplication function strongly inverse preserving function inverse preserving usual consider identity preserving functions working function actions case hence strongly identity preserving implies identity preserving case identity preserving strongly inverse preserving left right conjugation actions conversely inverse preserving function left right actions strongly inverse preserving sandwich morphisms structures call function groups sandwich morphism initial examples homomorphisms inverse function sandwich structures arbitrary functions group theory proposition function finite groups strongly inverse preserving sandwich morphism proof strongly inverse preserving hence rearranging obtain substituting gives sandwich morphism property conversely sandwich morphism conclude strongly inverse preserving claimed sandwich morphisms functions preserve structure group specified binary operation call binary operation sandwich product structure imposes group called sandwich structure wish study sandwich product binary operation facilitate denote simply using denote usual group operation hence proposition sandwich product finite group following properties left distributivity idempotency left involutary left symmetry proof properties directly checked expanding terms group product note mention identities inverses clearly group identities inverses distinguish elements using sandwich product sandwich product relate however know identity element recover inverses sandwich product defining obvious next step throw away group leads make following definition definition sandwich set binary operation denoted satisfies properties proposition subsandwich sandwich subset sandwich case closed sandwich operation technical terms sandwich left involutary left distributive left symmetric idempotent magma properties stated independent sense none proved others counterexamples small order demonstrate generated using program ian hawthorn figure cayley tables counterexamples sandwich properties except indicated one described matrices specify cayley tables binary operation elements labelled etc listed figure proposition following identities hold sandwich proof follow directly definition sandwich first identity left cancellative property proved left involutary property raises question whether converse also true magmas satisfying sandwich properties case demonstrated non example figure since sandwiches left cancellative multiplication left transitive sandwich useful sandwiches usually right cancellative however particular sandwiches contain right zero elements indeed sandwich consist right zero elements show right zero semigroup set multiplication sandwiches one directly check call right zero sandwiches unique right zero sandwich every order right zero sandwich least elements right cancellative since left distributive property tells multiplication left sandwich automorphism since moreover involution since automorphisms sandwich form group like automorphisms algebraic object hence natural map sandwich automorphism group defined mapping element left multiplication function properties natural map furthermore since write hence natural map sandwich homomorphism sandwich structure automorphism group call sandwich group sandwich sandwich structures arbitrary functions group theory sandwich structure group call sandwich group subsandwich subsandwich group sandwich hence left multiplication defines sandwich homomorphism maps arbitrary sandwich onto group subsandwich congruences map sandwich equivalence classes relation iff means particular true gives hence congruence class subsandwich thus congruences map right zero sandwiches proved theorem every sandwich group subsandwich right zero sandwiches consequence theorem sandwich right zero subsandwiches must group subsandwich hence condition sandwich prevents right zero subsandwiches result group subsandwich corollary right cancellative sandwich group subsandwich natural map look like apply group let group sandwich product consider congruence defined natural map sandwich automorphism group setting gives also let must setting gives hence follows thus equivalent element conversely element properties equivalent proves following proposition congruences natural map sandwich structure group sandwich structure sandwich automorphism group cosets subgroup consisting central elements order corollary central elements order natural map sandwich automorphism group natural map gives sandwich isomorphism onto image image natural map consists elements form sandwich automorphism group however elements order since interesting effectively sandwich isomorphism onto subsandwich group elements subsandwich group elements order course subset elements order group need constitute subgroup however subsandwich proposition let group let subsandwich proof enough show closed sandwich product true ian hawthorn natural map takes sandwich subsandwich sandwich automorphism group next consider question whether sandwiches must arise sandwich structures groups proposition right zero sandwiches group subsandwiches proof consider right zero sandwich order set elements product one algebraic structure isomorphism let elementary abelian order sandwich product thus sandwich structure right zero sandwich order every subset right zero sandwich right zero subsandwich since subset hence right zero subsandwich order follows right zero sandwich order group subsandwich claimed proposition right zero sandwich group sandwich order proof sandwich structure elementary abelian right zero sandwich order group sandwich conversely let group order sandwich product elements group order follows elementary abelian must therefore order corollary sandwiches group sandwiches proof right zero semigroup order three provides counterexample question whether sandwiches group subsandwiches remains open theorem proposition suggest true may easy construct counterexamples references ian hawthorn yue guo arbitrary functions group theory new zealand journal mathematics vol derek robinson course theory groups edition springer ian hawthorn nil series arbitrary functions group theory accepted publication cmuc http
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ideals generated two generic quadratic forms exterior algebra mar veronica crispin samuel lundqvist gleb nenashev abstract based structure theory pairs matrices give conjecture hilbert series exterior algebra modulo ideal generated two generic quadratic forms show conjectured series upper bound sense determine majority coefficients also conjecture series equal series squarefree polynomial ring modulo ideal generated squares two generic linear forms introduction let let homogeneous ideal longstanding conjecture due minimal hilbert series namely equals tdr denotes degree form means truncate first term clear minimal series attained generic forms forms mutually algebraically independent coefficients conjecture proved special cases including anick stanley let denote exterior algebra generators natural believe hilbert series generic forms even degree equal tdr snellman showed true principal ideal generated even element however theorem showed two generic quadratic forms hilbert series equals starting group connected stockholm problem solving seminar working problem determining minimal hilbert series quotients exterior algebras paper case two quadratic forms considered results principal ideals generated element odd degree presented separate paper outline follows section use structure theory skewsymmetric matrices order reduce problem finding minimal series among forms certain pairs canonical forms mathematics subject classification primary secondary key words phrases hilbert series exterior algebra generic forms crispin lundqvist nenashev section combine results previous section combinatorial methods show result tip iceberg able determinine first coefficients last coefficient generic series conjecture series whole conjecture let generic quadratic forms exterior algebra hilbert series equal number lattice paths inside rectangle bottom left corner top right corner moves two types show theorem conjectured series upper bound gives situation opposite conjecture commutative setting case lower bound generic series known section turn hilbert series generic linear forms known hilbert series equal see fact special case result loc cit every artinian monomial complete intersection strong lefschetz property hilbert series equal first published counterexamples given last part paper present surprising connection hilbert series generic quadratic forms conjecture let generic linear forms hilbert series equal conjecture although conjectures give hilbert series different algebras conjecture weaker conjecture theorem following equivalent conjecture holds even conjecture holds prove theorem section canonical forms exterior algebra recall structure theory pairs matrices complex field apply quadratic forms exterior algebra let pair quadratic matrices matrix pencil sum parameter two matrix pencils said strictly nonsingular block diagonal matrix diagonal blocks called direct sum denoted ideals two generic quadratic forms exterior alg canonical forms complex matrix pencils author presents canonical matrices complex matrix pencils congruence transformation description based result matrices algebraically closed field characteristic present three canonical forms derived section slightly different setting another notation obtained section provided comments rank theorem let complex quadratic matrices matrix pencil strictly direct sum possibly zero matrix followed diagonal blocks three canonical types type may occur finite number times possibly zero may different sizes diagonal block type sum rank matrices aii type sum rank respectively rank matrix aiii finite root necessarily elementary divisor pair crispin lundqvist nenashev diagonal block type iii sum matrix rank nonzero matrix rank quadratic forms complex numbers tany quadratic formn written product aij vertical vector matrix aji nonsingular matrix change variables theorem let two generic quadratic forms proper change variables write nonzero proof let matrices correspond theorem discussion write direct sum canonical diagonal blocks ranks well ranks know matrix corresponding generic quadratic form maximal rank hence examine diagonal sums canonical pairs property additional condition maximaze rank well odd even case differ due fact rank matrix always even odd matrix pencil always singular even case consider matrix pencil strictly regular nonsingular let see matrix pencil maximal rank well included terms hence matrices corresponding generic quadratic forms maximal rank strictly thus result follows let full rank must strictly every finite root elementary divisor pair det moreover two regular pencils strictly congruent elementary divisors see determinant matrix square pfaffian every elementary divisor det even multiplicity show every root pfaffian therefore also simple let finite root consider different nonzero simple roots generic matrix pencil also must simple roots since every root simple elementary divisors det det multiplicity exactly two theorem matrix pencil strictly congruent diagonal sum matrices type distinct ideals two generic quadratic forms exterior alg notice rank unless equal case rank thus result follows hilbert series ideal generated two generic quadratic forms let theorem follows question minimal hilbert series quadratic generic forms equivalent determining hilbert series get immediate result proposition conjecture correct odd proof see coefficients table obtained calculations first sight one might suspect possible determine basis ideal respect suitable monomial ordering determine hilbert series however success approach yet instead turn combinatorial methods mentioned introduction order bound hilbert series remark case even one two canonical forms contains generic coefficients draw similar conclusions odd case however due results derive proposition proposition determine almost coefficients small odd see table mention although case odd explicit description generators conclusions draw strong even case odd case bounds hilbert series upper bound part consists proof following result theorem let exterior algebra two generic quadratic forms dimension graded component proof upper bound need couple lemmas work certain kind lattice paths bijection paths monomials degree generally admissible path size consists steps numbered step going either right move left move moreover last step goes always right exactly left steps given path define monomial xjs indices left steps see figure clearly correspondence set paths size monomials degree denote crispin lundqvist nenashev table coefficients odd agree values hilbert function modulo ideal generated two generic quadratic forms see proposition table coefficients even bold numbers means equality hilbert function two generic quadratic forms according proposition proposition non bold numbers upper bounds hilbert function let subset respectively consist monomials degree whose corresponding paths always cross line respectively note paths cross line well inevitable example lemma cardinalities given proof natural bijection paths one see considering central symmetry point hence rest proof work simple cases left moves exactly one left move path cross vertical line case follows easily path must cross restrictions left moves may occur use induction remaining second case first fixing cases even already handled respectively let consider last two steps ideals two generic quadratic forms exterior alg figure red path corresponds monomial blue one monomial two possibilities either clear number paths first type calculate number paths second type replace step goes left step right follows number paths second type hence obtain formula know induction assumption get finally last equality well known recursive formula binomial coefficients lemma let quadratic form dim min crispin lundqvist nenashev let graded reverse lexicographical ordering degrevlex induced set leading monomials graded component remark generic quadratic form canonical form satisfies condition lemma see proof first want prove monomial corresponding path form degree let index path contains step least one crosses define part path note even degree set zero variables index larger assumption dim min means contains everything degree hence since degree hence respect degrevlex obtain define proved subset leading monomials equality follows dim min dim lemma conclude set leading monomials similar way get following result lemma let quadratic form dim min let degrevlex ordering induced set leading monomials graded component remark generic quadratic form canonical form satisfies condition lemma proof apply lemma change variables ready prove main statement ideals two generic quadratic forms exterior alg proof theorem key idea proof consider two different monomial orderings leading monomials first ordering leading monomials second ordering correspond monomials whose paths intersect respectively let quadratic form satisfying condition lemma choose subset dim span dim hence span respect degrevlex lemma vector assign form monomial xil number til consider product define matrix size rows consist coefficients define matrix similarly products given see equal multiplied hence equation main minors namely subset det det order proceed first define sufficiently large constant uses constant defined end proof let max det det consider vector monomials know implies since subset either det together det det yields inequality det det let quadratic form satisfying condition lemma similarly subset respect degrevlex lemma span matrix defined corresponding way know main minor det thus subset size determinant minor vanish let submatrix corresponding crispin lundqvist nenashev construct big matrix size first rows come last rows since enough show main minor det mln vanish rewrite minor det det det mln due equation possible choose sufficiently large det det det det det det altogether large main minor mln vanish means dimension component ideal generated least thus dimension quotient algebra concludes proof lower bound recall following result exterior version lower bound used commutative setting lemma let generic quadratic forms proof know multiplication map rank min dim dim thus dim max dim dim coefficients hilbert series using bounds derived previous section determine majority coefficients hilbert series proposition let two generic quadratic forms odd equal dim dim ideals two generic quadratic forms exterior alg even equal dim dim proof definition numbers follows immediately proving second statement case concentrate first statements begin odd case theorem assume easy see dim hand know theorem dim one path inside rectangle proceed manner even case theorem assume observe monomial lies outside dim theorem enough show possible paths inside rectangle hence requested number indeed proposition let generic quadratic forms let integers dimension graded component equal proof since bound upper bound enough show number paths crossing line hence principle yields number paths cross ending path crosses line line length least hence length path get paths cross cross line line path chose first point first point reflect path step step change dirrection steps change steps type steps converse get new path point point possible reverse procedure number exactly crispin lundqvist nenashev number without restrictions case number paths equal squares generic linear forms square free algebra turn commutative setting recall conjecture iarrobino stating lrd hilbert series generic linear forms generic forms degree except cases focus hilbert series simplest exceptional case linear change variables enough consider generic linear forms bounds hilbert series theorem let polynomial ring variables let two generic linear forms dimension graded component quotient proof use proof theorem indeed references lemma lemma valid since generic linear hilbert series equal see proof theorem let dimensions sgraded components generic quadratic forms generic linear forms remark fact algebra isomorphic algebra thus define coefficients lemma let positive integers proof theorem assume subset let ideals two generic quadratic forms exterior alg denote subalgebra generated notice form subset define monomial since follows basis element basis mod product regard element dim dim since possibilities choose commutative algebra isomorphic degree two dim lemma let positive integers proof path consider pair steps one step right left another opposite note inside rectangle deleting two steps new path inside furthermore path inside insert two steps odd place new paths inside path inside rectangle collect ycoordinate steps set steps set deleting pairs get path inside obtain crispin lundqvist nenashev number paths inside first step left last right steps pairs right left note odd indeed let number left pairs follows equal number paths inside steps doubled hence equal number paths inside concluding proof proof theorem conjecture holds number variables lemma lemma follows conjecture hold even number variables assume conjecture holds even number variables lemma hand lemma holds assumption gives theorem know get equality hence pair natural ask might connections series quotients quotients end giving question direction whose formulation based upon computer experiments ideals two generic quadratic forms exterior alg question let two generic linear forms let two generic quadratic forms two algebras hilbert series remark case question weaker version conjecture acknowledgements authors want thank participants stockholm problem solving seminar many fruitful discussions progress work references anick thin algebras embedding dimension three journal algebra inequality hilbert series graded algebras math hollman hilbert series ideals generated generic forms journal symbolic computation koszul homology lie algebras application generic forms points homology homotopy applications gantmacher theory matrices vol chelsea new york inverse system symbolic power iii thin algebras fat points compositio math grayson stillman software system research algebraic geometry available lancaster rodman canonical forms real matrix pairs strict equivalence congrunce linear algebra appl lundqvist nicklasson generic principal ideals exterior algebra arxiv snellman conjectures hilbert series generic ideals exterior algebra homology homotopy applications stanley weyl groups hard lefschetz theorem sperner property siam algebraic discrete methods thompson pencils complex real symemtric skew matrices linear algebra appl crispin department mathematics uppsala university uppsala sweden address lundqvist nenashev department mathematics stockholm university stockholm sweden address samuel nenashev
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annals probability vol doi institute mathematical statistics sep hafnians perfect matchings gaussian matrices mark alex ofer university hebrew university weizmann new york analyze behavior barvinok estimator hafnian even dimension symmetric matrices nonnegative entries introduce condition barvinok estimator achieves subexponential errors show condition almost optimal using hafnians count number perfect matchings graphs conclude barvinok estimator gives polynomialtime algorithm approximate subexponential errors evaluation number perfect matchings introduction number perfect matchings bipartite graph given permanent bipartite adjacency matrix graph since computing permanent generally computationally hard various algorithms proposed compute approximately mention particular mcmc algorithm rescaling algorithm denoted lsw sequel algorithm analysis latter algorithm subject previous work general hence hard combinatorial problem computing number perfect matchings graph even number vertices let denote adjacency matrix graph vertices relevant combinatorial notion hafnian defined received september revised june supported part nsf grant usaf grant supported part grants binational science foundation israel science foundation supported part grant israel science foundation herman taubman professorial chair mathematics ams subject classifications primary secondary key words phrases hafnian perfect matching random gaussian matrices electronic reprint original article published institute mathematical statistics annals probability vol reprint differs original pagination typographic detail rudelson samorodnitsky zeitouni haf denotes symmetric group immediate check see perfect matchings haf thus interest efficient computation haf permanent exact computation haf computationally expensive problem estimating hafnian seems harder attack corresponding problem permanent since many algorithms known permanent approximation break extended hafnians particular lsw rescaling algorithm transforms adjacency matrix graph almost doubly stochastic one yet nontrivial lower estimate hafnian doubly stochastic matrix impossible see also contrast computation permanent points proof convergence mcmc algorithm breaks approximate computation hafnian unless minimal degree least see consider paper computation haf symmetric matrices nonnegative entries note diagonal entries play role computation haf therefore rest paper always assume aii seminal paper discussing estimator permanent barvinok also introduces probabilistic estimator haf symmetric matrix possessing nonnegative entries let real skew symmetric matrix independent centered normal entries wij diagonal satisfying aij words let gskew denote skew symmetric matrix independent entries main diagonal let denote skew symmetric matrix wij gij aij write gskew denotes aij aij haf det thus det easily computable quantity consistent estimator haf barvinok proceeds prove matrix haf det haf high probability euler constant approaches computing hafnian include however apply adjacency matrices nontrivial graphs hafnians perfect matchings gaussian matrices deterministic algorithm subexponential complexity constructed analyzed random algorithm analyzed precision algorithm depends complicated way number perfect matchings goal paper analyze performance barvinok estimator hafnian establishing concentration random determinant hinges bounding singular values gaussian matrix crucial step however essentially differs thus less independence unrestricted matrix handling dependences required different arguments smallest intermediate singular values first case employ conditioning argument tailored take account structure graph lemmas fact entries real thus main minors degenerate plays central role hand instead developing estimate intermediate singular values difficult task due use fact matrix hermitian allows use estimates recent work local law formulate results introduce notion strong expansion graphs notion strengthens standard notion vertex expansion assuming sets many connected components expand faster set vertices denote con set connected components boundary edge definition let let say graph strongly expanding parameter level set vertices definition use following notational convention important parameters appear definitions theorems denoted greek letters unimportant constants whose value may change line line denoted etc simplest form results case adjacency matrix graph theorem fix let adjacency matrix graph assume strongly expanding level haf log det rudelson samorodnitsky zeitouni fix addition assumptions matrix possesses spectral gap haf log det assumption means modulus eigenvalues either smaller theorem immediate consequence general theorem discuss definition strongly expanding graphs remark extension theorem irregular graphs requires notion doubly stochastic scaling matrices also need notion spectral gap stochastic matrices definition matrix nonnegative entries said possess doubly stochastic scaling exist two diagonal matrices positive entries psuch matrix doubly stochastic call doubly stochastic scaling definition symmetric stochastic matrix said possess spectral gap exist eigenvalues show see corollary adjacency matrix strongly expanding graph appropriate lower bound minimal degree possesses unique doubly stochastic scaling use fact following theorem required lower bound minimal degree satisfied theorem fix let adjacency matrix graph whose minimal degree satisfies assume strongly expanding level doubly stochastic scaling satisfies maxi bij haf log det hafnians perfect matchings gaussian matrices fix addition assumptions possesses spectral gap haf log det condition readily checked polynomial time applying lsw scaling algorithm stopped error bounded indeed time lsw algorithm output matrix almost doubly stochastic sense denoting doubly stochastic scaling one maxij maximal entry least implies maximal entries order note also spectral gap condition point theorem depends eigenvalues also checked polynomial time note given exist stronger expansion conditions graph ensure maximal element doubly stochastic scaling adjacency matrix size satisfies stronger properties condition automatically satisfied refer section proposition details conditions play different roles proof first one needed establish lower bound smallest singular value second one guarantees singular values greater remark definition strongly expanding graphs definition reminds one vertex expander yet stronger two senses first strong expansion property takes account geometry set requiring rapid expansion spread sets second want expansion property hold sets size relatively close classical expanders corresponding property required sets vertices may look unnatural first glance however one may construct example graph strong expansion property level arbitrary close yet matrix corresponding may degenerate probability proposition construct graph whose adjacency matrix barely misses condition definition yet det haf high probability appropriate proposition let exists graph vertices rudelson samorodnitsky zeitouni det det constants depending theorems adjacency matrices based general result pertaining doubly stochastic symmetric matrices nonnegative entries consider matrices many relatively large entries formulate requirement precisely introduce notion large entries graph definition let symmetric matrix nonnegative entries parameter define large entries graph connecting vertices whenever aij matrix variances entries skew symmetric matrix also refer large variances graph formulate two theorems concentration hafnian skew symmetric matrix whose large variances graph satisfies strong expansion condition theorem fix let symmetric stochastic matrix even size nonnegative entries let replacing let denote large variances graph assume minimal degree vertex least level max bij exists haf log det somewhat tighter bounds available matrix possesses spectral gap theorem assume conditions theorem addition assume matrix spectral gap haf log det constant depends relevant parameters hafnians perfect matchings gaussian matrices structure paper follows section consider unit vectors close vectors small support derive uniform small ball probability estimates images action estimates used section obtain lower bound smallest singular values section provide local estimates empirical measure eigenvalues section devoted proof theorems section devoted proof combinatorial lemma concerning doubly stochastic scaling adjacency matrices strongly expanding graphs used proof theorem section also present sufficient conditions ensure holds finally section present construction graph discussed proposition provide proof latter compressible vectors establish concentration determinant matrix bound smallest singular value usual context view smallest singular value matrix minimum norms images unit vectors min bounding minimal norm whole sphere let consider behavior fixed begin small ball probability estimate valid unit vector lemma let matrix independent symmetry restriction normal entries assume exist least numbers var wij depend proof let choose coordinate set var wij condition entries matrix except jth row column conditioning ith coordinate vector normal random variable variance var since coordinates vector conditionally independent elementary estimate gaussian density yields assumption lemma integration respect variables completes proof next lemma rough estimate norm random matrix rudelson samorodnitsky zeitouni lemma let matrix independent symmetry restriction normal entries assume var wij lemma follows estimate side sum squares independent centered normal variables whose variances uniformly bounded course estimate lemma rough disregard constant power argument lemma allows extend lower bound small ball probability single vector neighborhood subspace formulate precisely recall definition compressible incompressible vectors definition denote sparse comp sparse incomp comp next lemma uses standard net argument derive uniform estimate highly compressible vectors lemma let matrix satisfying conditions lemmas comp depends proof let number chosen later set constant lemma exists sparse cardinality lemma union bound hafnians perfect matchings gaussian matrices provided appropriately chosen assume let comp comp choose next goal show small ball probability estimate propagates strongly compressible vectors moderately compressible ones step assumption large variances graph strongly expanding plays crucial role strong expansion condition guarantees matrix enough independent entries derive small ball estimate single vector despite dependencies introduced skewsymmetric structure next simple lemma instrumental exploiting independence still present lemma let finite tree root assume corresponds random variable variables independent assume also corresponds event depends suppose connected numbers proof prove lemma induction depth tree assume first tree depth statement lemma follows fact events independent assume statement holds trees depths smaller let tree depth let sets vertices edges connected root tree events conditioned independent therefore note vertices form forest roots since events independent different trees forest statement lemma follows applying induction hypothesis tree rudelson samorodnitsky zeitouni using lemma strong expansion property large variances graph establish small ball probability bound image incompressible vector lemma let let centered gaussian matrix assume large variances graph satisfies strong expansion condition parameter level let incomp depends proof define event let since incomp indeed let vector containing largest absolute value coordinates dist sparse choose subset set whenever otherwise set indeed normal random variable variance least var previous inequality follows bound maximal density prove lemma use lemma end construct forest consisting con trees vertices assume forest already constructed events independent different trees forest hence used lemma last inequality since strong expansion condition last quantity less equal required hafnians perfect matchings gaussian matrices proceed construction forest first step construct spanning tree tel connected component set trees obviously disjoint add vertices leaves trees induction let smallest number add vertices connected tree containing descendants let smallest number used process add vertices connected already added tree containing descendants since vertex connected vertex whole set added end process denote trees obtained way construction guarantees trees disjoint finishes construction forest proof lemma similarly lemma extend small ball probability result lemma uniform bound using net argument lemma let gaussian matrix assume large variances graph satisfies strong expansion condition parameter level exists constant depending incomp comp proof proof repeats lemma sketch set constant lemma choose sparse incomp cardinality depends union bound rudelson samorodnitsky zeitouni provided using appropriately defined depending approximation points derive previous inequality incomp sparse complete proof notice vector incomp vector incomp sparse lemma follows using approximation lemmas combined treat compressible vectors statement fixed large enough universal positive integer proposition let gaussian matrix assume large variances graph minimal degree satisfies strong expansion condition parameter level exists constant depending one comp remark proof shows enough take log proof proposition set max constants lemmas let smallest natural number definition implies log log log hafnians perfect matchings gaussian matrices define induction constant lemma definition implies thus comp comp comp comp comp lemmas combined union bound imply comp applying lemma derive estimate comp proposition follows previous inequality lemma smallest singular value main result section following lower bound smallest singular value gaussian skewsymmetric matrix strongly expanding large variances graph theorem let even number let skewsymmetric matrix denote large variances graph assume var minimal degree vertex least log expanding level positive rudelson samorodnitsky zeitouni remark tracing proof theorem using remark one show enough take log proof theorem prove theorem use negative second moment identity let matrix columns let vector orthogonal columns except jth one htj hence min khs let proposition argument shows use matrix place define unit vectors theorem would follow inequalities hold indeed theorem follows assumption union bound establish inequality cases proved way let block consisting rows columns matrix large variances graph subgraph containing vertices therefore properties slightly relaxed parameters indeed property remains unchanged property valid replaced property satisfied parameter place since boundary differs one vertex recall matrix odd size matrix degenerate exists allows define vector orthogonal columns matrix define event comp hafnians perfect matchings gaussian matrices graph strongly expanding level proposition condition matrix conditioning may assume incomp set least elements since degree vertex large variances graph least means exists var therefore conditionally normal random variable variance var var bound density normal random variable implies finally completes proof since proof values proves theorem immediate corollary theorem following corollary satisfies let adjacency matrix graph minimal degree vertex least log expanding level possesses unique doubly stochastic scaling dad graph possesses perfect matching proof proof begin showing perfect matching exists assume otherwise since det haf latter equality contradicts theorem show possesses doubly stochastic scaling choose edge create graph erasing edges attached graph satisfies assumptions statement slightly smaller constants thus possesses perfect matching implies edge exists perfect matching containing edge bregman theorem theorem implies rudelson samorodnitsky zeitouni possesses unique doubly stochastic scaling fact follows strict convexity relative entropy characterization doubly stochastic scaling minimizer see equation local bound eigenvalues density section prove general bound crowding eigenvalues class hermitian matrices whose variance matrix results somewhat general needs rest paper may independent interest therefore introduce new notation let denote matrix hermitian sense xij entries xij independent zero mean random variables application xij variables gaussian following set sij xij sij sij assume variables possess uniformly bounded moments finally denote eigenvalues matrix use empirical measure eigenvalues assume avoid trivialities matrix sij irreducible otherwise matrix decomposed blocks due symmetry let maxij sij assume following assumption one assumption following proposition proposition notation assumptions setup fix assumption holds every exists one proof use theorem simplified form quote introducing notation following notation let denote stieltjes transform semicircle law set hafnians perfect matchings gaussian matrices note one equation universal constant log introduce similarly equation parameter min min since need relation note use note bounded universal constant hence using get log universal constant given chose whenever denote stieltjes transform empirical measure eigenvalues following theorem theorem exists uniformly fix let denote complement event assume occurs using uniform boundedness inequality obtain log choosing small enough guarantee side last display uniformly bounded choice dln provided occurs means derive previous inequality one use union bound rudelson samorodnitsky zeitouni better estimate obtained one assumes spectral gap first following lemma value exactly one eigenvalue one proof claim concerning eigenvalue theorem check claim eigenvalues consider may reducible blocks indeed suppose disjoint blocks disjoint subsets irreducibility one path odd length connecting similarly path odd length connecting hence path even length connecting contradiction block disjointness claim follows applying theorem blocks lemma spectral gap eigenvalues later exists unique isolated proposition notation assumptions setup fix assumption holds possesses spectral gap every exists one proof identical proposition using theorem instead theorem omit details concentration hafnian random matrix section prove theorems results follow concentration gaussian measure lipschitz functions end consider gaussian vector gij use form skewsymmetric matrix gskew however function log det gskew bij lipschitz overcome obstacle write skew log det log gskew use theorem proposition obtain lower bounds singular values valid probability close event hafnians perfect matchings gaussian matrices replace function log truncated version makes lipschitz controlled lipschitz constant application gaussian concentration inequality yields concentration new truncated function expectation expectation close log det gskew recall instead concentration value want establish concentration log haf log det gskew words swap expectation logarithm estimate error incurred process achieved due fast decay tail concentration inequality proof theorem proof proceeds section argument traced back without loss generality may assume appears proposition indeed choose constant formulation theorem appropriately large case theorem follows barvinok theorem fix statement theorem theorem replaced proposition introduce events theorem proposition let det note det log det log set log det log det next derive concentration results map log det lipschitz constant therefore standard concentration gaussian distribution see using variance entries bounded universal constant det log det exp therefore exp exp rudelson samorodnitsky zeitouni particular obtain log edetw log detw log detw first inequality follows jensen inequality second one complete proof theorem markov inequality log det log det log log hand note det edet therefore log det log det log log det log edet log log det log edet log used last display using upper bound get log det log det log log det log det log using applying conclude log det log det log together yields det log det log obtain statement theorem prove previous inequality instead choose log proof theorem similar theorem however exploit tighter bounds intermediate singular values provided proposition use different truncation redefining function log det estimate lipschitz constant accurately proof theorem fix proof theorem may assume introduced proposition inequality rewritten hafnians perfect matchings gaussian matrices inequality used bound lipschitz constant truncated logarithm let number chosen later define log det log denote moment vector consider skew symmetric matrix skew whose entries main diagonal equal corresponding entries let matrix whose entries square roots corresponding entries note function defined gskew log det composition three functions gskew whose lipschitz constant exceed defined defined inequality klip klip therefore klip rudelson samorodnitsky zeitouni applying standard gaussian concentration lipschitz functions obtain det log det exp exp replaces formula proof theorem arguing proof theorem obtain log det log edet log det inequality let theorem set let theorem imply det log det log plays role arguing proof theorem show log det log det log log det log edet log det log det exp first inequality follows log det log edet second one upper bound third one select optimal inequality since condition holds sufficiently large log det log det combining bound log det log det following markov inequality complete proof hafnians perfect matchings gaussian matrices doubly stochastic scaling proof theorem prove theorem scale adjacency matrix graph order apply theorems existence scaling already established corollary show smallest nonzero entry scaled adjacency matrix least polynomial crucial step proof theorem allows conclude large entries graph scaled matrix coincides original graph proposition fix let adjacency matrix graph whose minimal degree satisfies assume strongly expanding level exists constant possesses doubly stochastic scaling dad min dii particular assumptions proposition bij whenever bij describing proof proposition let state complementary claim says stronger expansion conditions guarantee entries scaled adjacency matrix polynomially small ensures stronger expansion property condition theorem automatically satisfied follows set vertices graph denotes edges graph connecting vertices proposition fix exists constant depending following holds let adjacency matrix graph whose minimal degree satisfies assume subset vertices satisfying holds denotes set external neighbors least neighbors possesses doubly stochastic scaling dad max dii rudelson samorodnitsky zeitouni particular assumptions proposition bij whenever bij prove proposition argue contradiction assume one diagonal entries say smaller chosen end proof double stochasticity scaled matrix implies exists neighbor corresponding entry scaling matrix large fact prove one entry lemma construct set vertices cardinality least corresponding entries scaling matrix greater use base induction lemma show exists set vertices cardinality containing entries scaling matrix corresponding still polynomially large proceeding induction construct increasing sequence sets diagonal entries corresponding vertices greater number induction steps able perform depend chosen large enough get reaching desired contradiction proof proposition similar assume toward contradiction say dnn larger double stochasticity scaled matrix exists set neighbors corresponding entries scaling matrix small using double stochasticity scaled matrix produces set vertices cardinality least corresponding entries scaling matrix greater use induction base lemma show exists set vertices cardinality containing entries scaling matrix corresponding still large proceeding induction construct increasing sequence sets diagonal entries corresponding vertices greater number induction steps able perform depend chosen large enough get reaching contradiction proof proposition without loss generality assume throughout constants small enough corollary possesses doubly stochastic scaling dad diag without loss generality assume hafnians perfect matchings gaussian matrices note since doubly stochastic need simple lemmas lemma let assume exists subset cardinality least proof assume otherwise least elements therefore next lemma quantifies following intuition given large set indices corresponding small entries scaling matrix find large set indices neighbors corresponding large entries scaling matrix lemma let exists subset cardinality least proof denote bij doubly stochastic scaling pfor let bij bij lemma set indices let next lemma base inductive construction lemma let exists subset cardinality least rudelson samorodnitsky zeitouni proof therefore least one index let set neighbors proof completed application lemma lemma used inductive step lemma let let subset indices exists subset indices disjoint cardinality least satisfies proof clearly two vertices connected otherwise would entry size least scaling therefore set disconnected vertices since contains perfect matching since disconnected imply proving first claim lemma let note show second claim lemma find subset indices disjoint let recall doubly stochastic scaling since bij bij bij bij bij bij hafnians perfect matchings gaussian matrices second inequality used pthat set bij lemma set least indices call set completing proof lemma ready perform inductive procedure proving proposition let log sufficiently large assume reach contradiction use lemma construct set cardinality least assuming may apply lemma construct set disjoint cardinality least define set apply lemma assuming small continue process obtain increasing sequence sets since number steps upper bounded log absolute constant hand definition large enough number steps larger reaching contradiction proof proposition assume contradiction sake since least half neighbors holds let set neighbors assumption minimal degree rudelson samorodnitsky zeitouni applying lemma gives subset cardinality least holds induction base inductive step provided following lemma lemma fix constant let exists subset cardinality least proof since holds dij hence assumptions graph holds applying lemma produces set satisfying requirements lemma ready perform inductive procedure proving proposition let log sufficiently large assume reach contradiction start constructing sequence starting set constructed applying lemma iteratively clearly stop steps however definition large enough would able make steps reaching contradiction combine bound scaled matrix theorems derive theorem proof theorem recall dad denotes doubly stochastic scaling gskew denotes skew symmetric matrix independent entries main diagonal note det gskew det gskew det hafnians perfect matchings gaussian matrices denotes matrix whose entries square roots entries therefore enough consider concentration det gskew proof theorem follows applying theorems strong expansion condition noted introduction strong expansion condition stronger classical vertex expansion condition might desirable replace strong expansion property weaker natural classical vertex expansion condition proposition introduction shows latter condition insufficient guarantee subexponential error barvinok estimator fact example graph associated random matrix barely misses strong expansion property barvinok estimator yields exponential error high probability provide proof proposition proof proposition without loss generality assume let set define graph vertices follows vertices form clique called center vertices called peripheral connected vertices center addition vertices connected see figure adjacency matrix block shape adjacency matrix matrix main diagonal everywhere else matrix whose entries equal matrix right lower block contains matrices main diagonal rudelson samorodnitsky zeitouni fig graph matrix similar form gaussian matrix independent gaussian matrices independent random variables recall det matchings number perfect matchings graph vertex matched vertex center done ways hence vertex matched peripheral neighbor done unique way thus matchings hafnians perfect matchings gaussian matrices consider det let constant chosen later simple pigeonhole argument shows det homogeneous polynomial degree entries hence det det exp det det exp exp first term smaller exp chebyshev inequality second term also exceed exp constant chosen small enough proves part proposition related error barvinok estimator remains check condition satisfied let set cardinality contains vertex center con condition holds assume con also therefore since choose completes proof proposition acknowledgment thank alexander barvinok many helpful discussions references ajanki local semicircle law imprimitive variance matrix electron commun probab rudelson samorodnitsky zeitouni barvinok polynomial time algorithms approximate permanents mixed discriminants within simply exponential factor random structures algorithms barvinok samorodnitsky random weighting asymptotic counting inverse isoperimetry israel math barvinok samorodnitsky computing partition function perfect matchings hypergraph combin probab comput bayati gamarnik katz nair tetali simple deterministic approximation algorithms counting matchings stoc proceedings annual acm symposium theory computing acm new york properties nonnegative matrices permanents soviet math dokl knowles yau yin local semicircle law general class random matrices electron probab friedland rider zeitouni concentration permanent estimators certain large matrices ann appl probab godsil gutman matching polynomial graph algebraic methods graph theory vol szeged eds amsterdam guionnet zeitouni concentration spectral measure large matrices electron commun probab jerrum sinclair approximating permanent siam comput jerrum sinclair vigoda approximation algorithm permanent matrix nonnegative entries acm ledoux concentration measure phenomenon mathematical surveys monographs amer math providence linial samorodnitsky wigderson deterministic strongly polynomial algorithm matrix scaling approximate permanents combinatorica minc permanents encyclopedia mathematics applications reading rudelson vershynin problem invertibility random matrices adv math rudelson vershynin smallest singular value random rectangular matrix comm pure appl math rudelson zeitouni singular values gaussian matrices permanent estimators random structures algorithms appear available valiant complexity computing permanent theoret comput sci hafnians perfect matchings gaussian matrices rudelson department mathematics university michigan church street ann arbor michigan usa rudelson samorodnitsky school engineering computer science hebrew university jerusalem givat ram jerusalem israel salex zeitouni faculty mathematics weizmann institute pob rehovot israel courant institute new york universirty mercer street new york new york usa
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jammali research aligning coding sequences frameshift extension penalties apr safa esaie ayoub michelle correspondence informatique des sciences sherbrooke sherbrooke canada full list author information available end article abstract background frameshift translation important phenomenon contributes appearance novel coding dna sequences cds functions gene evolution allowing alternative amino acid translations gene coding regions frameshift translations identified aligning two cds gene homologous genes accounting codon structure two main classes algorithms proposed solve problem aligning cds either amino acid sequence alignment simultaneously accounting nucleotide amino acid levels former allow account frameshift translations latter exclusively accounts frameshift translation initiation considering length translation disruption caused frameshift results introduce new scoring scheme algorithm pairwise alignment cds accounting frameshift translation initiation length simultaneously considering nucleotide amino acid sequences main specificity scoring scheme introduction penalty cost accounting frameshift extension length compute adequate similarity score cds alignment second specificity model search space problem solved set feasible alignments two cds previous approaches considered restricted search space additional constraints decomposition alignment algorithm described paper asymptotic time complexity classical algorithm conclusions compare method cds alignment methods based application comparison pairs cds homologous human mouse cow genes ten mammalian gene families database results show method particularly robust parameter changes compared existing methods also appears good compromise performing well presence absence frameshift translations implementation method available https keywords coding dna sequences pairwise alignment frameshifts dynamic programming background biological sequence alignment cornerstone bioinformatics widely used fields phylogenetic reconstruction gene finding genome assembly accuracy sequence alignments similarity measures directly jammali page lated accuracy subsequent analysis cds alignment methods many important applications gene tree protein tree reconstruction fact useful cluster homologous cds groups orthologous splicing isoforms combine partial trees orthology groups complete protein tree gene family aligning measuring similarity homologous cds requires account frameshift translations detected amino acid level lead high similarity nucleotide level functionnaly different translation consists alternative translations coding region dna using different translation frames important phenomenon resulting different scenarios insertion deletion nucleotide sequence whose length multiple cds alternative splicing evolutionary genomic indels programmed ribosomal frameshifting sequencing errors recent studies reported role translations appearance novel cds functions gene evolution translation also found linked several diseases crohn disease computational detection translations requires alignment cds accounting codon structure classical approach aligning two cds used alignment tools consists method cds first translated sequences using actual coding frame sequences aligned finally alignment cds alignment approach account alternative translations two cds leads incorrect alignment coding regions subject translation opposite problem aligning protein sequences recovering hypothetical nucleotide cds sequences accounting translation also studied several papers consider problem aligning two cds accounting translation simultaneously accounting nucleotide sequences problem recently regained attention due increasing evidence alternative protein production translation eukaryotic gene families problem first addressed hein proposed model score alignment two cds length combination score nucleotide level score level described algorithm later improved algorithm computing optimal score alignment constraint search space problem restricted arvestad later proposed cds alignment scoring model alignment algorithm accounting codon structures translations based concept generalized substitutions introduced model score cds alignment depends decomposition concatenation codon fragment alignments codon fragment cds defined substring length decomposition codon fragment alignments allows define score cds alignment level recently ranwez proposed simplification model arvestad limiting maximum length codon fragment model cds alignment algorithm described jammali page extended context multiple sequence alignment models arvestad ranwez several scores may computed alignment based different decompositions codon fragment alignments corresponding algorithms aligning two cds consist computing optimal score decomposition alignment two cds optimal score exclusively accounts translation initiations translation alignment penalized adding constant cost penalizes initiation accounting length translation however taking account translation lengths important order increase precision cds alignment scores lengths induce less disruptions protein sequences paper propose first alignment algorithm accounts initiation length translations order compute similarity scores cds alignments remaining paper organized follows motivation section illustrate importance accounting translation length aligning cds preliminaries section give preliminary definitions introduce new cds alignment scoring model definition score alignment penalizing initiation extension translations method section dynamic programming algorithm computing optimal score alignment two cds described finally results section present discuss results comparison method cds alignment methods pairwise comparison cds homologous genes ten mammalian gene families motivation importance accounting translation length two main goals aligning biological sequences evaluate similarity identify similar regions sequences used thereafter realize molecular analyses evolutionary functional structural predictions practice cds alignment used exhaustively identify conserved features set proteins thus definition cds similarity must account sequence conservation disruptions nucleotide protein levels figure illustrates importance accounting translations translation length order compute adequate similarity score cds alignment describes example three cds length cds length obtained deleting nucleotide position adding nucleotides end cds length obtained deleting nucleotide position adding nucleotides end looking translations observe similarity higher similarity protein level share longer prefix amino acids black characters alignments however pairwise cds alignment algorithms account length translations would return score two following optimal alignments penalizing initiation one illustrates importance accounting translations translation length order compute adequate similarity score cds alignment describes example three cds length cds length obtained deleting nucleotide position adding nucleotides end cds length obtained deleting nucleotide position adding nucleotides end jammali jammali jammali atgaccgaatccaagcagccctggcataagtgggggaacgattga atgaccgaatccaagcagccctggcataatgggggaacgattgaagtaggaacgatttaa atgaccgaatccaacagccctggcataagtgggggaacgattgaagtaggaacgatttaa page page page page optimal alignment optimal alignment mwhen looking translations wof observe similarity sbetween isw higher nthan similarity atgaccgaatccaagcagccctggcataagtgggggaacgattga protein level share longer prefix amino acids black gcharacters ein alignments hows kcds alignment algorithms gthattdo inot eaccount gforttheilength ever tthe epairwise jammali translations would return two following optimal alignoptimal alignment andfor pbetween optimal alignment saa penalizing ments andw one initiation score translation kbothqp positions marked cases symbol kin alignments atgaccgaatccaagcagccctggcataagtgggggaacgattga level extension atgaccgaatccaagcagccctggcataagtgggggaacgattga penalizing sequence disruptions cost gap open therefore good scoring evaluating similarity two cds model forsan therefore good scoring model evaluating similarity two cds figure example three cds middle optimal alignment translations initiation figure top example athree cds penalize middle anan optimal alignment translationshould region penalize length bottom optimal alignment translations initiation translation region length bottom optimal alignment also length translations extension amino acids gray characters translation region length also length translations extension amino acids gray characters translation region length score score score alignments alignment would higher similarity alignments alignment would higher similarity score alignment alignment length score alignment pos preliminaries score cds alignment translation cases positions marked symbol alignments preliminaries score cds alignment section formally describe new definition score cds alignment pthe protein level ofga cds alignment penalizing thewe sequence disruptions section formally describe new definition score penalizes initiation extension translations atgaccgaatccaag cagc cctggcc tga penalizes initiation extension translations anthat evolutionary point view good evaluating atg definition coding dna sequence cds penalize similarity two cds presence translations dna esequence dna cds isequence sis dna sequence alphabet nucleotides definition coding cds coding initiation also length translations extension amino coding dna sequence cds dna sequence alphabet nucleotides whose length multiple coding sequence composed figure alignment bwhose definitions ofalignments length two cds codons acids gray characters alignment would notations multiple coding sequence composed following frameshifts used ending set concatenation codons areofthe length thecolumns sequence codons number arrays indicate positions thewords consecutive alignment codons concatenation codons words length sequence ending fsinit codons resp divided two sets set fsinit positions higher similarity score alignment andat colored set belong codonswith blue color fsext according translation cds protein sequence positions translation cds protein sequence codons contains codons grouped codons inat red color indel green color fsinit codons black color mfs caused length alphabet codon cds translated nucleotides contained fsinit codons underlined length alphabet codon cds translated alignment aligned one two nucleotides cds symbolof protein sequence preliminaries score cds alignment symbols proteinthe sequence two orsymbol one gap set fsinit codons caused insertions section formally describe new definition score cds align contains codons alignment note practice entire cds begins withgrouped start codon atg ends note inlines practice entire cdsthe begins start codon atg ends ment thatfile penalizes initiation extension translations additional additional tables stop codon taa tag tga pdf file containing lines taa tables tag needleprot additional stop codon tga needlenuc results section algorithm definition dna alignment dna sequences definition coding sequence cds additional file pairwise alignments thebetween benchmark definition alignment dna sequences alignment two dna sequences afor band aspace pair bbenchmark indna section describe time complexity algorithm zip coding file containing pairwise alignment files fasta format sequence cds asequences dna sequence alphabet two dna bon athe pair alignment nucleotides two sequences length derived inserting gap symbols considered inb results section five methods parameter configuration solves problem finding maximum score alignment two cds areg two length gap symbols sequences whose coding composed derived similarly sequence sequence position lengths alignment algorithms words bof length position ending concatenation codons sequence programming called column alignment wethe usealignment dynamic tables indexed pairs prefixes alignment called column alignment positions translation cds protein sequence ofalignments length onbetween two cds theaa table stores maximum scores given alignment length two cds let alphabet anletaa symbol prefixes asuch andthat table dfthe iscds usedistotranslated account forb potential cases given codon sequencean denote bylength two cds theasubstring ofss going extensions counted subsequently protein sequence sequence orkbto denote substring going position position denotes number letters position position denotes number letters codon definition dynamic programming tables given cds gap symbol example gap symbol codon example input algorithm uses two dynamic programming tables orinbpractice groupedan entire alignment athree appearand three note cds begins startnucleotides codon atg endsf size grouped alignment three nucleotides appear three cell contains maximum score consecutive columns theoralignment example codon acc appears inalignment stop codon taa tag tga consecutive columns alignment codon acc prefixes thea table filled onlyinfor values alignment acc grouped example appears appears acc grouped grouped appears ofinialignment mod mod mod resp following give definition score alignment following give definition score alignment bedefinition alignment dna sequences cds andthe onj contains alignment tween cell based partition ofthe score codons aan resp tween twodepending cdstwo based partition resp alignment dna sequences aofaand see aofpair codons ofan sets prefixes ithe mod resp four onb alignment codons figure illustration four thethe alignment codons seeas figure illustration depending mod table filled follows two sequences length derived inserting gap symbols mod mod mod mod mod mod contains maximum score alignment aligned together half jammali page alignment called column alignment position given alignment length two cds let sequence denote substring going position position denotes number letters different gap symbol example accat gtag tacgtag codon grouped alignment three nucleotides appear three consecutive columns alignment example first codon acc grouped first codon act grouped following give definition score alignment two cds based partition codons resp four sets depending alignment codons see figure illustration set matching codons contains codons grouped alignment aligned codon cds set frameshift extension codons fsext contains codons jammali grouped alignment aligned concatenation three nup saa cleotides overlaps codons cds score two saa contains codons set codons indel fsthat extension cost grouped alignment aligned concatenation gap symbols gapwith codons constitutes frameshift initiation codons open fsinit san set matching nucleotides fsinit codons mfs contains nucleotides belonging fsinit codons aligned nucleotide score cds score score alignment length pos atgaccgaatccaag cagccctggccag atg length two cds codons codons figure alignment athe following definitions notations frameshifts used set number arrays indicate positions consecutive alignment columns codons fsinit codons ato resp beimdivided setinof fsinit colored according set belong codons ininto blue two color sets fsextthe codons caused color deletions contains themfs codons grouped red color codons indel codons green fsinit codons black color nucleotides containedthe fsinit codonsand underlined alignment aligned one two nucleotides cds two one gap symbols set fsinit codons caused insertions contains theconventions codons thatare areused grouped following notationsall definition toalignment denote different sets codons nucleotides set codons resp algorithm denoted resp set fsext codons resp denoted fsext set indel codons resp resp section describe time space complexity algorithm denoted indel set mfs alignment nucleotidesbetween resp resp solves problem finding score two cds denoted sets sequence codons algorithms bbyofmfs lengths andmfs alignment resp use dynamic programming tables indexed thealignment pairs prefixes simply identified position column last nucleotide two cds table dbstores maximum scores alignments case always imbthe example mfs prefixes table used account potential cases nucleotides also identified positions alignment extensions counted subsequently definition dynamic programming tables given two cds input algorithm uses two dynamic programming tables size cell contains maximum score alignment jammali page example alignment depicted figure composition different sets fsext indel mfs alignment scoring model described definition substitutions fsext codons scored using scoring function aligned codons silent nucleotide mutations get score identity fixed extension cost denoted extend cost added fsext codon indel codons scored adding fixed gap cost denoted gap cost indel codon alignment mfs nucleotides scored independently using nucleotide scoring function insertions deletions nucleotides fsinit codons responsible initiation translations scored adding fixed opening cost denoted open cost fsinit codon note convention values penalty costs gap gap cost open cost extend cost negative note also scoring scheme assumes nucleotide scoring functions symmetric definition score alignment let alignment length two cds score alignment defined score extend cost gap cost open cost extend cost gap cost open cost method section describe time space complexity algorithm solves problem finding maximum score alignment two cds lengths similarly classical sequence alignment algorithms use dynamic programming tables indexed pairs prefixes two cds table stores maximum scores alignments prefixes table used account potential cases extensions counted subsequently definition dynamic programming tables given two cds input algorithm uses two dynamic programming tables size cell contains maximum score alignment prefixes table filled values jammali page mod mod mod resp mod cell contains score alignment prefixes mod resp mod table filled follows mod mod mod mod mod mod contains maximum score alignment aligned together half score aligning subtracted mod mod mod mod contains maximum score alignment aligned together half scores aligning subtracted lemma filling table mod mod max open open open cost open open open cost open cost san open cost open cost open cost gap cost open cost san open cost open cost open cost gap cost open cost cost cost cost cost mod mod max extend cost san san mod open cost open cost mod open cost san mod open cost open cost open cost san mod open cost open cost gap cost open cost mod mod equation symmetric previous case mod mod max jammali page jammali page proof lemma given additional file figure illustrates configurations alignment considered lemma computing cases case mod mod case mod mod figure illustration configurations alignment considered lemma computing figure thethe configurations alignment considered inillustration cases nucleotides sequencesinalemma represented using character nucleotides areofcolored accordinga codon cases nucleotides sequences type andofb belong codons blue color fsext codons red color indel codons green represented using character colored according color type codon color fsinit codons black nucleotides appear gray belong codons whose type yet decided case table usediningreen belonging codons blue color fsext codons red color indeldcodons order decide type codons subsequently adjust score accordingly color fsinit codons black color nucleotides appear gray color belonging codons whose type yet decided case table used order decide type codons subsequently adjust score accordingly lemma filling table mod mod jammali page mod mod max extend cost open cost san open cost open cost open cost mod mod equation symmetric previous case mod mod max extend cost san open cost san open cost mod mod equation symmetric previous case proof lemma proof follows lemma mod mod case trivial mod mod mod mod five cases follow application lemma case computing keeping cases aligned together cases among cases however cases must subtract half score aligns ing score added subsequently mod mod proof symmetric previous case mod mod mod mod three cases follow application lemma case computing keeping cases aligned together cases among cases however cases must subtract half scores aligning aligning theses scores added quently mod mod proof symmetric previous case alignment algorithm using lemma described next theorem theorem computing maximum score alignment given two cds lengths maximum score alignment computed time space using following algorithm algorithm align loor gap cost loor gap cost san open cost open cost mod mod jammali page san open cost open cost compute using lemma compute using lemma mod mod mod mod proof theorem proof relies two points algorithm computes maximum score alignment algorithm runs time space complexity validity algorithm fact fills cells tables according definition follows five points initialization tables direct consequence definition lemmas couples prefixes need considered algorithm possible couples couples mod mod see cases table used lemmas cases cases couples prefixes considered increasing order length computed cases mod mod backtracking algorithm allows find maximum score alignment time space complexity algorithm direct consequence number cells tables cell filled constant time exact formula computational complexity algorithm computed calls case lemma calls cases lemma calls case lemma calls case lemma calls cases lemma calls cases lemma total results discussion implemented present cds alignment algorithm affine gap penalty scheme penalty concatenation inserted resp deleted codons gap open cost gap cost gap open cost negative penalty cost gap initiations done adding two dynamic programming tables cell resp contains maximum score alignment prefixes codon resp indel codon data evaluated algorithm applications mammalian dataset containing cds sequences ten gene families obtained database jammali page compara version first gene family named three cds three paralogous human genes shown share common region translated three different frames three cds see figure illustration multiple alignment three cds nine families nine smallest term overall length cds fifteen gene families listed shown display one translation region pairs cds gene family cds human mouse cow genes belonging family satisfying definition downloaded overall number distinct pairs cds within ten gene families table gives details content size ten gene families cds ten gene families provided additional file table detailed description ten gene families mammalian dataset gene family human gene genes cds length iii vii viii total number pairs cds gene family family identifier used ensembl identifier human gene member family number human mouse cow genes family total number cds genes total sum lengths cds number distinct pairs cds given evaluation strategies compared accuracy five pairwise global alignment methods including present method computing cds alignments presence absence translation compared cds five methods vary according alignment algorithm used either present cds alignment algorithm called fsepsa allowing penalize translation initiation extension cds alignment algorithm called macse penalizing translation initiation sequence alignment algorithm penalizing neither table summarizes alignment algorithm values parameters used five methods present cds alignment algorithm used two five methods namely fse two methods differ according value given parameter extend cost either extend cost method fse penalizing translation extension extend cost method penalizing translation extension pairwise version macse used method called macse alignment algorithm used last two methods method called needlenuc computing scores alignments nucleotide level method called needleprot level methods using amino acid nucleotide scoring functions fixed overall score consecutive nucleotide identities alignment scores less smallest identity jammali page alignment seaview wed may atggcgcccg aggagaacgc ggggagcgaa ctcttgctgc agagtttcaa gcgccgcttc ctggcagcgc atggcgcccg aggagaacgc ggggaccgaa ctcttgctgc agggttttga gcgccgcttc ctggcggtgc gcgccctgcg ctccttccgc tggcagagct tagaagcaaa gttaagagac tcatcagatt ctgagctgct gcacactgcg ctccttcccc tggcagagct tagaggcaaa gttaagagac tcatcagatt ctgagctgct gcgggatatt ttgcagaaga ctgtgaggca tcctgtgtgt gtgaagcacc cgccgtcagt caagtatgcc cacgaggctg tccacacaga gcctttggat gagctgtacg tggtgctttc tctcagaact catcaaaaag tctcagaact catcaaaaag aggtgctggt ggagaccctg atggccaagg agtccaccca gggccaccgg agctatttgc tcctcaggag tcctcaggag actctccaag agcacagcca tcatctccca cggtaccaca ggcctggtca catgggatgc gctcagtcac actctccaag agcacagcca tcatctccca cggtaccaca ggcctggtca catgggatgc tgtattgccc cgccctctac cttgcagaat gggccatcga gaacccggca gccttcatta acagacgtgc tgtattgccc cgccctctac cttgcagaat gggccatcga gaacccggca gccttcatta acagacgtgc tgtattgccc agaagccatc gtgtcgctgg tcggggtcct gcggaggctg gctgcctgcc gggagcacca gcgggctcct agaagccatc gtgtcgctgg tcggggtcct gcagaggctg gctgcctgcc gggagcacaa gcgggctcct agaagccatc gtgtcgctgg tcggggtcct gcagaggctg gctgcctgcc gggagcacaa gcgggctcct caattctaca tggcccttac cgtctgcaac ccagagatgt gccagctgtt caccaccgag ctatgctgga tggcctttac cgtccgcaac ccagagacat gccagctgtt caccaccgag ctaggccggg ctgggatcag atgggaagcg gaagctcatc atgaccagaa actgtttccc tacagagagc acttggagat atgggaagcg gaagctcatc atgaccagaa actgtttccc tatggagagc acttggagat ggcaaagctg ggcaatgctg aacctcacac tgtag roughrepresentation representationofofthe thereal realalignment alignment cds cds figure rectangular colored portions represent concatenations nucleotides rectangular colored portions represent concatenations nucleotides alignment blank portions represent concatenations gap symbols lengths thein alignmentwhile portions areportions given atrepresent bottom colors ofofthe indicate alignment blank concatenations gapnucleotide symbols regions lengths coding frame translated taking frame cds reference alignment portions given bottom colors nucleotide regions indicate example nucleotide region length shared three cds translated coding frame translated takingofthe frame cds different coding frames bottom real alignment three cdsof figure obtained using thereference visualization software nucleotides structure ofin example seaview nucleotide region lengthare shared according three cdscodon translated first cds coding frames bottom real alignment three cds figure obtained using visualization software seaview score parameters shared several methods given value methods particular three methods fse macse penalizing translation initiation parameter open cost given values parameters fixed default values algorithm implementation ncbi blast nucleotide levels used five methods compute pairwise alignments pairs cds within ten gene families dataset yielding alignments total five methods absence available benchmarks direct evaluation accuracy cds alignments base evaluation four indirect strategies jammali page table description five methods considered experiment method fse macse needleprot alignment approach specific parameters present approach extend cost present approach extend cost ranwez stop cost level initiation cost open cost parameters gap open cost gap cost matrix applicable gap open cost gap cost needlenuc level applicable method alignment approach values specific common parameters given first strategy consider cds multiple alignment gene family obtained using macse benchmark strategy exploits fact multiple alignments usually accurate pairwise alignments assumes macse multiple alignments closer reality pairwise alignments obtained using five methods note pairwise alignment methods included comparison extended multiple sequence alignment methods using classical strategies thus accurate pairwise alignment methods lead accurate multiple alignment methods focus comparison pairwise versions methods second strategy consider six composition criteria cds pairwise alignment called identity identity gap init gap length init length definitions criteria given used compare five methods third strategy manually build use benchmark real multiple alignment three cds three paralogous human genes gene family fourth strategy generate use set three cds splicing orthology groups group containing seven existing putative cds seven genes gene family based results experiments discussed following best compromise default values fsepsa parameters open cost extend cost discussion first strategy using macse multiple alignments benchmark macse used default parameters open cost stop cost matrix gap open cost gap cost compute cds multiple alignment ten gene families macse muln tiple alignment cds consider induced pairwise alignments benchmark total obtained benchmark composed pairwise alignments order compare alignment obtained one five methods corresponding alignment benchmark computed number nucleotides aligned partner benchmark alignment table shows overall percentage nucleotides aligned partners benchmark compared methods varying jammali page open cost extend cost shows different versions fse method method best scores greater followed needleprot method score opposite needlenuc macse method open return worst scores respectively results also show fse method robust open cost parameter changes compared macse method whose scores show large variation note needlenuc needleprot account open cost parameter method identity identity gap init gap length init length fse macse needlenuc needleprot fse macse needleenuc needleprot fse macse needlenuc needleprot varying values parameter open cost number cds pairs dataset given values criteria reference method needleprot indicated bold characters methods fse macse needlenuc variations criteria values compared reference values given criteria method number cds pairs closest value reference needleprot value given parentheses open cost cds pairs table values six criteria nofs dataset variations compared needleprot open cost fse fse fse macse needlenuc needleprot percentage nucleotides aligned partner benchmark alignments induced macse multiple alignments method varying open cost extend cost case number cds pairs alignment presents highest similarity corresponding benchmark alignment compared methods given parenthesis best results indicated bold table comparison macse multiple alignments benchmark jammali page jammali page second strategy using six composition criteria cds pairwise alignment six criteria defined used compare five pairwise alignment methods given pairwise cds alignment first criterion identity counts number columns alignment containing nucleotide match second criterion identity counts number fsext codons alignment aligned triplet nucleotides yielding amino acid third criterion gap init number columns alignment either insertion deletion columns preceded different type column fourth criterion gap length overall number columns alignment fifth criterion init number translation segments found alignment last criterion length overall number columns alignment intersecting fsext codon note definitions six criteria exploit definitions codon sets used definition independent alignment scoring scheme example alignment depicted figure identity counting columns except five columns positions containing nucleotide mismatch identity counting fsext codons except two codons aag aat ending position yielding two different amino acids fsext codon aat ending position yielding amino acid different amino acid yielded triplet aag gap init counting positions init counting positions two last criteria values gap length length nine cases obtained combining values parameters open cost extend cost considered pairs cds ten gene families dataset partitioned three sets case first set called nofs dataset composed pairs cds pairwise alignments obtained using fse macse methods criteria init second set called dataset composed pairs cds alignments obtained using fse macse methods criteria init third set called ambigufs dataset composed remaining pairs cds note nine cases set cds pairs init macse method strictly included set cds pairs init fse method nine cases computed overall value six criteria method fse macse needlenuc needleprot dataset nofs ambigufs tables present results results nofs datasets nofs datasets assume real alignments contain translations needleprot method likely computes accurate alignments since allow translation alignments indeed computes maximum score alignment level alignment nucleotide level take needleprot result reference nofs dataset cases construction nofs dataset fixed value parameter open cost fse methods necessarily return two alignments similarity jammali page score pair cds dataset indeed observed value open cost alignments obtained using methods fse varying values parameter extend cost unchanged table summarizes results open cost presenting results varying versions fse single line cases shows results fse methods closest reference six criteria cases however slightly overestimate underestimate criteria tendency overestimating identity criteria particularly accentuated macse method compared fse methods cases opposite needlenuc method always largely underestimates identity overestimating criterion results datasets datasets assume real alignments must contain translations needleprot method longer produce accurate results contrary likely underestimates identity criterion indeed correctly aligns cds regions free translation translation regions either leads several mismatches case high mismatches scores overestimation gap init criterion expected observed value identity needleprot method always lowest data shown additional file focus four methods table summarizes results nine cases considered identity identity criteria differences values four methods negligible main differences results reside values gap init init criteria particular init criterion useful compare accuracy methods correctly identifying real translation regions family families one translation region detected manually validated pair cds ten gene families expected number translation regions per alignment data table observe cases fse methods methods average numbers init close standard error values smaller macse method especially needlenuc method overestimate number translation regions per alignment large standard error values cases results ambigufs datasets ambigufs datasets methods agree presence absence translation regions pairs cds note needlenuc method reports translations pairs cds highest average number translation regions per alignment cases data shown additional file needlenuc already shown perform poorly absence presence translation regions focus four methods table summarizes results observe criteria macse higher values fse needleprot similar values significant difference results resides values init length criteria fse method always reports null small number regions average init equals expected cases macse methods overestimate number translation regions per alignment table values six criteria dataset open extend cost identity identity gap gap init method cost init length length cds pairs avg fse macse needlenuc fse macse needlenuc fse macse needlenuc fse macse needlenuc fse macse needlenuc fse macse needlenuc fse macse needlenuc fse macse needlenuc fse macse needlenuc varying values parameters open cost extend cost number cds pairs dataset given values criteria fse macse needlenuc methods indicated method average number init per alignment corresponding standard error values also indicated jammali page extend cost cds pairs method identity identity gap gap init init length avg length fse macse needleprot fse macse needleprot fse macse needleprot fse macse needleprot fse macse needleprot fse macse needleprot fse macse needleprot fse macse needleprot fse macse needleprot varying values parameters open cost extend cost method number cds pairs displaying translation given values criteria method indicated method average number init per alignment corresponding standard error values also indicated open cost table values six criteria ambigufs dataset jammali page jammali page third strategy using benchmark manually built real pairwise alignments three cds three paralogous human genes gene family cds coding protein coding protein coding protein real multiple alignment three cds roughly depicted detailed figure figure observe shares nucleotide region length translated frame nucleotide region length translation shares nucleotide region length entirely translation clear cds similar figure also shows pair cds shares single translation region table shows normalized pairwise similarity scores number translation regions computed five alignment methods pairwise alignments computed five methods varying open cost extend cost given additional file shows needleprot fse cases extend two methods allow infer similar table also illustrates fact needlenuc macse strongly overestimate number translation regions per alignment cases fse method parameters open extend method allows infer similar detect single translation region alignment table pairwise similarity scores number translation regions computed methods method fse fse fse macse fse fse fse macse fse fse fse macse needlenuc needleprot normalized pairwise similarity scores number translation regions computed five methods benchmark composed cds similarity scores normalized dividing lengths alignments open cost fourth strategy inferring cds splicing orthology groups protein phylogenies based three cds used previous strategy cds human gene human gene human gene generated dataset jammali page three cds splicing orthology groups composed homologous cds group contains one three initial cds six splicing orthologs following set seven genes gene family human genes denoted denoted denoted containing one initial cds chimpanzee gene denoted mouse gene denoted rat gene denoted cow gene denoted cds splicing orthologs predicted based spliced alignment tool splign follows initial cds gene gene different aligned putative existing cds ortholog splicing structure inferred resulting cds given additional file computed normalized pairwise similarity scores cds using five alignment methods pairwise alignments computed five methods varying open cost extend cost given additional file method constructed phylogeny using upgma algorithm based computed cds similarity matrix upgma algorithm used classify cds three groups infer similarity relationships groups independently rate evolution algorithm used reconstruct phylogeny inside group table summarizes results three splicing orthology groups denoted containing cds containing cds containing cds methods allow correctly classify cds three initial splicing orthology groups however needleprot fse methods methods allow infer correct similarity relationships groups confirming results third evaluation strategy methods cds phylogeny reconstructed inside group inducing evolution seven genes speciation event root gene tree phylogeny reconstructed groups inducing evolution genes duplication event root phylogeny comparing running times table shows running times five methods three first gene families dataset processor ram needleprot method fastest followed macse needlenuc fse slowest methods note fse needlenuc needleprot used implementations python used java implementation macse provided authors explains fact macse unexpectedly faster fse even needlenuc indeed five methods share asymptotic time complexity exact complexity dependent number calls main recurrence formulas execution number cases considered recurrence formula exact computational complexity five methods terms lengths two compared cds jammali page table similarity relationships groups five methods method fse fse fse macse fse fse fse macse fse fse fse macse needlenuc needleprot similarity relationships splicing orthology groups computed using similarity matrices five methods dataset open cost fse shown proof theorem macse needlenuc needleprot table running time seconds method gene family fse macse needlenuc needleprot iii method gene families iii running time calculated computer processors ram parameters open extend conclusions paper introduce new scoring model alignment cds accounting frameshift translation length motivation new scoring scheme increasing evidence protein divergence frameshift translation eukaryotic coding gene families calling automatic methods able compare align classify cds accounting codon structure aim paper validate necessity accounting frameshift translation length comparing cds show computing maximum score pairwise alignment new scoring scheme possible quadratic time complexity results comparing five cds alignment methods pairwise alignment cds ten eukaryotic gene families show method best compromise sets cds pairs cds display translations future work make use benchmarks cds alignments generated manually simulation order confirm experimental results also defer future work extended study model robustness parameter changes calibration parameters using real data benchmarks perspectives work also include design heuristic algorithm using local alignment achieve scalability large datasets keeping high accuracy extension method toward multiple alignment finally plan apply algorithms discovery frameshifts evaluation extent frameshifts eukaryotic gene families jammali page availability supporting data implementation pairwise alignment method python available https dataset used section results available additional files list abbreviations cds coding dna sequence frameshift nucleotide amino acid declarations author contributions wrote program documentation conceived study design ran experiments analyzed interpreted data wrote manuscript critically revised manuscript authors read approved final manuscript acknowledgements scholarship faculty science sherbrooke funded canada research chair computational biological complexity sherbrooke competing interests authors declare competing interests ethics approval consent participate applicable consent publication applicable funding research funded canada research chairs crc crc grant sherbrooke author details informatique des sciences sherbrooke sherbrooke canada biochimie des sciences sherbrooke sherbrooke canada references zambelli pavesi gissi horner pesole assessment orthologous splicing isoforms human mouse orthologous genes bmc genomics irimia pan xiong gueroussov lee slobodeniuc kutter watt evolutionary landscape alternative splicing vertebrate species science christinat moret transcript perspective evolution transactions computational biology bioinformatics tcbb kuitche lafond ouangraoua reconstructing protein gene phylogenies extending framework reconciliation appear proceedings international conference bioinformatics computational biology bicob arxiv preprint pruitt harrow harte consensus coding sequence ccds project identifying common gene set human mouse genomes genome research okamura feuk navarro scherer frequent appearance novel sequences frameshift translation genomics barmak christopher genomic view alternative splicing nature genetics stoffers zinkin stanojevic clarke habener pancreatic agenesis attributable single nucleotide deletion human gene coding sequence nature genetics ikuo yuichi hisao yoshifumi shuji yoshitaka nobuo yutaka nucleotide deletion resulting frameshift possible cause complete globulin deficiency six japanese families nature genetics robin programmed ribosomal frameshifting alternative proteomes front genet wei valerie jonathan mary frank john claudia natalia alexander robert john analysis sequencing error rate error sources artifact recombination detection drug resistance mutations dna retrovirology raes van peer functional divergence proteins frameshift mutations trends genetics ogura bonen inohara nicolae chen ramos britton moran karaliuskas duerr achkar brant bayless kirschner hanauer nunez cho frameshift mutation associated susceptibility crohn disease nature abascal zardoya telford translatorx multiple alignment nucleotide sequences guided amino acid translations nucleic acids research morgenstern dialign multiple dna protein sequence alignment bibiserv nucleic acids research suppl jammali page kucherov discovering distant protein homologies presence frameshift mutations algorithms molecular biology moreira maass tip protein backtranslation aided genetic algorithms bioinformatics ranwez harispe delsuc douzery macse multiple alignment coding sequences accounting frameshifts stop codons plos one danny catherine cyntia guillaume julie xavier overlapping reading frame coding sequence encodes novel interacting protein journal biological chemistry hein algorithm combining dna protein alignment journal theoretical biology pedersen hein comparison coding dna combinatorial pattern matching springer arvestad aligning coding dna presence errors combinatorial pattern matching springer sankoff kruskal time warps string edits macromolecules theory practice sequence comparison reading publication edited sankoff david kruskal joseph needleman wunsch general method applicable search similarities amino acid sequence two proteins journal molecular biology altschul erickson optimal sequence alignment using affine gap costs bulletin mathematical biology cunningham amode barrell ensembl nucleic acids research gouy guindon gascuel seaview version multiplatform graphical user interface sequence alignment phylogenetic tree building molecular biology evolution johnson zaretskaya raytselis merezhuk mcginnis madden ncbi blast better web interface nucleic acids research suppl kapustin souvorov tatusova lipman splign algorithms computing spliced alignments identification paralogs biology direct additional files additional file proof lemma pdf file containg detailed proof lemma additional file cds ten gene families zip file containing cds files fasta format ten gene families considered results section additional file additional lines tables pdf file containing additional lines tables needleprot needlenuc results section additional file pairwise alignments benchmark zip file containing sequence file pairwise alignment files fasta format benchmark considered results section five methods parameter configuration additional file pairwise alignments dataset zip file containing sequence file pairwise alignment files fasta format benchmark considered results section five methods parameter configuration
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cautious nmpc gaussian process dynamics miniature race cars nov lukas alexander melanie paper presents adaptive high performance control method autonomous miniature race cars racing dynamics notoriously hard model first principles addressed means cautious nonlinear model predictive control nmpc approach learns improve dynamics model data safely increases racing performance approach makes use gaussian process takes residual model uncertainty account chance constrained formulation present sparse approximation dynamically adjusting inducing inputs enabling implementable controller formulation demonstrated simulations show significant improvement respect lap time constraint satisfaction compared nmpc without model learning ntroduction control autonomous cars challenging task attracted considerable attention recent years one particular case autonomous driving autonomous racing goal drive around track fast possible potentially race competitors avoid collisions order achieve high performance extreme conditions racing teams today spend significant amount time effort modeling challenging especially near limits tire adhesion control methods proposed address challenge show great potential towards improving racing performance however often suffer poor model accuracy performance transient learning phases lead violation critical constraints related keeping car track avoiding collisions compromising performance success entire race addition iteratively learning racing task basis considered suffers poor generalization typically allow maintaining high performance dynamic racing tasks obstacle avoidance overtaking paper addresses challenges learning dynamics model data considering model uncertainty ensure constraint satisfaction nonlinear model predictive control nmpc approach offering flexible framework racing control recently number autonomous racing control methods presented rely nmpc formulations nmpc racing approach miniature race cars proposed work supported swiss national science foundation grant institute dynamic systems control eth zurich zurich switzerland lhewing mzeilinger institute automatic control eth zurich zurich switzerland aliniger uses contouring control formulation maximize track progress finite horizon enables obstacle avoidance extended stochastic setting order take model uncertainty account using model learning mpc framework allows generalizing collected data improving performance varying racing tasks instance demonstrated using mean estimate gaussian process dynamics model nmpc method based furthermore mpc approach recently proposed applied problem autonomous racing model improved iterative parameter estimation technique method presented paper makes use regression improve dynamics model measurement data since gps inherently provide measure residual model uncertainty integrated cautious nmpc controller end extend approach presented learning module reformulate controller stochastic setting key element differentiating approach available results stochastic treatment model nmpc controller improve performance constraint satisfaction properties derive tractable formulation problem exploits improved dynamics model uncertainty show chance constraints states approximated deterministic form framework thereby allows specifying minimum probability satisfying critical constraints track boundaries offering intuitive systematic way defining desired aggressive driving safety terms collision avoidance use gps mpc offers many benefits poses computational challenges use fast sampled larger scale systems race car problem since evaluation complexity gps generally high directly scales number data points considered various approaches address limitation presented literature one class methods relies approximation finite number basis functions sparse spectrum approximation also used nmpc present approach predictive control based sparse approximation using inducing inputs selected according approximate trajectory space enables local approximation currently relevant control given measured state facilitates implementability presented controller finally evaluate proposed cautious nmpc troller simulations race results demonstrate provides safe high performance control sampling times computationally par nmpc schemes without model learning improving racing performance constraint satisfaction furthermore demonstrate robustness towards process noise indicating fitness hardware implementation reliminaries following specify notation used paper briefly introduce regression sparse approximations based inducing inputs relevant presented control approach notation two matrices vectors use vertical concatenation use refer element vector similarly column matrix normal distribution mean variance denoted use kxk vector diag express diagonal matrix elements given vector gradient function rnz rnf respect vector rnx denoted rnz rnf gaussian process regression consider input locations collected matrix corresponding measuret ments arising unknown function rnd following statistical model gaussian noise zero mean diagonal variance diag assuming prior output dimension measurement data normally distributed kzz kzz gram matrix data points using kernel function input locations kzz choice kernel functions parameterization determining factor inferred distribution typically specified using prior process knowledge optimization based observed data throughout paper consider squared exponential kernel function exp rnz positive diagonal length scale matrix however straightforward use differentiable kernel function joint distribution training data arbitrary test point output dimension given kzz kzz kzz kzz kzz kzz kzz similarly kzz resulting conditional distribution gaussian kzz kzz kzz kzz kzz kzz call resulting approximation unknown function diag evaluating cost mean variance respectively thus scales number data points many data points fast applications limits use model overcome issues various approximation techniques proposed one class sparse gaussian processes using inducing inputs briefly outlined following sparse gaussian processes sparse approximations understood using concept inducing targets yind inputs zind inducing conditional distribution approximate joint distribution assuming test points training data conditionally independent given yind yind yind dyind yind yind yind dyind numerous options selecting inducing inputs heuristically subset original data points treating hyperparameters optimizing location letting coincide test points paper make use fully independent training conditional fitc approximation approximate distribution reduce computational complexity given selection inducing inputs zind using shorthand notation kzaind zind kza approximate posind ind terior distribution given qazz qazz kzz qazz qazz qzz diag kzz qazz concatenating output dimensions similar arrive approximation several matrices used precomputed evaluation complexity becomes independent number original data points using inducing points computational complexity evaluating sparse test point reduced predictive mean variance respectively model uncertainty well process noise affect velocity states system use mpc formulation finally discretize system using euler forward scheme sampling time resulting following description normally distributed process noise diag together uncertain dynamics function inferred measurement data fig schematic car model iii ace odeling section presents race car setup nominal modeling car dynamics serve base model control approach largely based material presented provides detailed exposition car dynamics consider following model structure describe dynamics miniature race cars nominal system dynamics car modeled first principles reflects unmodeled dynamics considered nominal dynamics obtained bicycle model nonlinear tire forces shown figure resulting cos sin sin cos sin mvy cos mvx cos state system position orientation longitudinal lateral velocities yaw rate inputs system motor duty cycle steering angle furthermore mass moment inertia distance center gravity rear front tire respectively difficult components model tire forces drivetrain force tires modeled simplified pacejka tire model drivetrain using motor model combined friction model exact formulations forces refer order account model mismatch due inaccurate parameter choices limited fidelity simple model integrate capturing unmodeled dynamics well additive gaussian white noise due structure nominal model since dynamics first three states given purely kinematic relationships assume race track constraints consider race track given centerline fixed track width centerline described piecewise cubic spline polynomial parametrized path length given evaluate corresponding centerline position orientation letting correspond projection centerline constraint car stay within track boundaries expressed half track width additionally system subject input constraints steering angle limited maximal angle duty cycle lie zero one earning based ontroller esign following first present model learning module subsequently used cautious nmpc controller briefly state contouring control formulation serving basis controller integrate dynamics using stochastic model afterwards introduce suitable approximations reduce computational complexity render control approach feasible model learning apply gaussian process regression infer function system dynamics previously collected measurement data states inputs training data generated deviation nominal system model specific data point pseudoinverse note form directly apply derive model data resulting stochastic model state obtained model used predictive controller given form stochastic distribution contouring control nmpc controller makes use contouring control formulation introduced shown provide good racing performance objective optimal contouring control formulation maximize progress along race track approximation car position along centerline introduced optimization variable including integrator dynamics position along track time step incremental progress progress along centerline horizon maximized means overall incremental progress order connect progress variable race car position linked projection car centerline achieved minimizing lag error contouring error defined cos sin sin cos small contouring error lag error approximates distance projection car position small lag error ensures good approximate projection stage cost function formulated lreg term encourages progress along track using relative weighting parameter parameters weights contouring lag error respectively lreg regularization term penalizing large changes control input incremental progress lreg kui kvi corresponding weights based contouring formulation define stochastic mpc problem integrates learned minimizes expected value cost function finite horizon length min current system state corresponding position centerline state constraints formulated centerline position approximation projection car position form chance constraints guarantee track constraint violated probability less solving problem computationally demanding especially since distribution state generally gaussian first prediction time step addition fast sampling times considered race car setting pose significant challenge realtime computation following subsections present sequence approximations reduce computational complexity nmpc problem autonomous racing eventually provide feasible approximate controller still leverage key benefits learning approximate uncertainty propagation time step evaluates stochastic distribution according residual model uncertainty propagated forward time rendering state distributions order solve therefore approximate distributions state prediction step gaussian dynamics equations gaussian distributions found sigma point transform first order taylor expansion detailed appendix make use taylor approximation offering computationally cheap procedure sufficient accuracy resulting following dynamics mean variance star denotes corresponding element symmetric matrix simplified chance constraints gaussian approximation state distribution allows simplified treatment chance constraints approximated deterministic constraints mean variance state using following lemma lemma let random variable set quantile function distribution degrees freedom maximum eigenvalue proof let epx confidence region level epx epx dynamically choosing inducing inputs sparse approximation dynamic sparse fig planned trajectory active chance constraints shown mean trajectory car confidence level perpendicular car mean orientation outer approximation confidence region using direction largest variance implies bcx means bcx epx using lemma formulate bound probability track constraint violation enforcing marginal variance joint distribution procedure similar constraint tightening robust control amount tightening related approximate confidence region deviation mean system state constraint well cost require variance dynamics next section proposes simplification reduce computational cost considering approximate evolution state variance approximation variance dynamics variance dynamics require additional variables optimization problem increase computation time drastically trade accuracy system description computational complexity evaluating system variance around approximate evolution state input trajectory typically chosen reference tracked shifting solution mpc optimization problem earlier time step denoting point approximate stateaction trajectory approximate variance dynamics given variance along trajectory thus depend optimization variable computed state measurement becomes available sampling time precomputed variance used satisfy chance constraints approximately replacing resulting set denoted figure shows example planned trajectory active chance constraints according formulation following use similar ideas reduce computational complexity required evaluations sparse approximations outlined section considerably speed evaluation little deterioration prediction quality fast applications highdimensional spaces computational burden however still prohibitive therefore propose select inducing inputs locally sampling time relies idea mpc area interest sampling time typically lies close known trajectory space similar approximation presented previous subsection inducing inputs selected along approximate trajectory according solution computed previous time step illustrate procedure using example figure showing dynamic approximation simple double integrator shown contour plot posterior variance two input dimensions additionally two trajectories generated mpc shown solid red line corresponds current prediction trajectory dashed line shows previous prediction used local approximation figure illustrates full sparse approximation close correspondence along predicted trajectory system dynamic selection local inducing points receding horizon fashion allows additional computing successive approximations adding removing single inducing points means rank updates applied reformulation offers better numerical properties avoids inversion large matrix qazz kzz kzz qazz kzz ind kzaind zind kzaind kzz substitution ind single inducing points corresponds single line column changing corresponding cholesky factorizations thus efficiently updated resulting control formulation autonomous racing integrate approximations presented previous sections mpc problem resulting following approximate optimization problem min full sparse fig contour plots posterior variance full top left dynamic sparse approximation top right solid red line trajectory planned mpc dashed red line trajectory previous time step used approximation inducing points indicated black circles bottom respective variances along planned trajectory given reducing learned model mean dynamics considering approximate variance dynamics simplified chance constraints problem reduced deterministic nonlinear program moderate dimension presented form approximate optimization problem still requires optimization large spline polynomial corresponding entire track since evaluation polynomial derivative computationally expensive one apply additional approximation step quadratically approximate cost function around shifted solution trajectory previous sampling time expected value equivalent cost mean similarly fixed using previous solution evaluating state constraints spline evaluated separately optimization procedure done imulation finally evaluate proposed control approach simulations race race car simulated using system resulting random perturbation parameters nominal dynamics original value compare two approaches one using full available data points one dynamic sparse approximation baseline nmpc controller makes use nominal part model well reference controller using true system model knowledge simulation setup generate controllers using formulation full dynamic sparse approximation inducing inputs along previous solution trajectory fig prediction dynamic sparse inducing inputs race lap shown black dots error yaw rate process noise encountered time step blue line shows dynamics error predicted shaded region indicates confidence interval including noise mpc problem inducing points placed exponentially decaying density along previous solution trajectory putting additional emphasis current near future states car prediction horizon chosen formulate chance constraints guarantee feasibility optimization problem implement chance constraint using linear quadratic soft constraint formulation specifically use slack variables incur additional costs ksi sufficiently large soft constrained formulation exact feasible reduce conservatism controllers constraints tightened first prediction steps applied mean remainder prediction horizon similar method used system simulated one lap race starting zero initial velocity point centerline white noise power spectral density diag resulting measurements one lap baseline controller used generate controllers hyperparameters process noise level found likelihood optimization see exemplify learned deviations nominal system figure shows encountered dynamics error predicted error lap sparse controller overall learned dynamics good correspondence true model uncertainty predicted matches residual model uncertainty process noise well note apparent volatility plot correspond overfitting instead due fast changes input matches validation data solvers generated using forces pro sampling time number maximum solver iterations limited sufficient guarantee solution required accuracy simulations carried laptop computer ghz cpu ram table simulation results without process noise kek reference baseline process noise controller controller kek reference baselineb requires baseline reference eight fig resulting trajectories race track simulations without process noise baseline reference sparse controller results quantify performance proposed controllers compare lap time average squared slack realized states corresponding violations furthermore state average solve times nmpc problem percentile simulation run demonstrate learning performance also evaluate average error system dynamics kek difference mean state one prediction step realized state direct comparison first evaluate controller performance simulations without process noise evident figure baseline controller performs visually suboptimally unable guarantee constraint satisfaction even absence process noise reference controller sparse controller perform similarly table summarizes results simulations without process noise see full controller gpfull matches performance reference controller also displays small constraint violations reference controller exhibits corner cutting behavior leading constraint violations due unmodeled discretization error also evident dynamics error reference controller discretization error partly learned gps leading lower error even reference controller overall sparse controller demonstrates performance close full controller terms lap time constraint satisfaction able significantly outperform baseline controller table shows averaged simulation different process noise realizations values averaged additional sparse approximation outliers removed runs except percentile solve times qualitatively observations noisefree case carry simulations presence process noise strikingly baseline nmpc controller displays severe constraint violations noise eight cases even causes car completely lose track runs subsequently removed outliers table formulations tolerate process noise well achieve similar performance case reference controller achieves slightly faster lap times formulations however come expense higher constraint violations shaping allowed probability violation chance constraints formulations allow aggressive racing safety simulations underline capabilities sparse controller full formulation excessive computational requirements relative sampling time dynamic sparse formulation solved similar time baseline formulation however require successive update sparse formulation implementation took additional average note computation done directly previous mpc solution whereas mpc problem solved receiving state measurement sample time step computation sparse approximation thus affect time input applied system state times separately solve times computed input applied within sampling time leaving enough time subsequent precomputation sparse approximation results demonstrate presented controller significantly improve performance maintaining safety approaching performance reference controller using true model furthermore demonstrate controller implementable able tolerate process noise much better initial baseline controller overall indicates fitness hardware implementation onclusion paper addressed challenge automatically controlling miniature race cars mpc approach model inaccuracies lead dramatic failures especially high performance racing environment proposed control approach able learn model mismatch adapt dynamics model used control subsequently improve controller performance considering residual model uncertainty furthermore enhance constraint satisfaction thereby safety vehicle using dynamic sparse approximation demonstrated capability resulting controller finally showed simulations gpbased approaches significantly improve lap time safety learning one example lap ppendix ncertainty propagation nonlinear systems let denote mean variance respectively using law iterated expectation law total variance exi exi exi varxi exi bdt varxi first order expansions around approximated bdt eferences buehler iagnemma singh darpa urban challenge autonomous vehicles city traffic springer vol kritayakirana gerdes using centre percussion design steering controller autonomous race car vehicle system dynamics vol guiggiani science vehicle dynamics handling braking ride road race cars springer kolter plagemann jackson thrun probabilistic approach mixed control application extreme autonomous driving int conf robotics automation akametalu fisac gillula kaynama zeilinger tomlin safe learning gaussian processes conf decis control kapania gerdes path tracking highly dynamic autonomous vehicle trajectories via iterative learning control american control liniger domahidi morari autonomous racing scale cars optim control appl methods vol carrau liniger zhang lygeros efficient implementation randomized mpc miniature race cars european control liniger zhang aeschbach georghiou lygeros racing miniature cars enhancing performance using stochastic mpc disturbance feedback american control niekerk damianou rosman online constrained reinforcement learning conf uncertainty artificial intelligence rosolia borrelli learning model predictive control iterative tasks control ieee trans automat vol rosolia carvalho borrelli autonomous racing using learning model predictive control american control rasmussen sparse spectrum gaussian process regression mach learning vol rasmussen herbrich unifying view sparse approximate gaussian process regression mach learning vol rasmussen williams gaussian processes machine learning mit press rasmussen williams approximation methods gaussian process regression kernel machines snelson ghahramani sparse gaussian processes using adv neural information process tresp bayesian committee machine neural computation vol pacejka bakker magic formula tyre model vehicle system dynamics vol faulwasser kern findeisen model predictive pathfollowing constrained nonlinear systems conf decis control lam manzie good model predictive contouring control conf decis control girard rasmussen prediction uncertain input gaussian processes relevance vector machines application ahead forecasting danish tech technical report deisenroth efficient reinforcement learning using gaussian processes dissertation kit karlsruhe hewing zeilinger cautious model predictive control using gaussian process regression ostafew schoellig barfoot robust constrained nmpc enabling reliable mobile robot path tracking int robotics research vol seeger low rank updates cholesky decomposition epfl technical report seeger peters model learning local gaussian process regression adv robotics vol kerrigan maciejowski soft constraints exact penalty functions model predictive control ukacc int conf control domahidi jerez forces professional embotech gmbh http jul
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framework rate efficient control distributed discrete systems apr jie ren solmaz torabi john maclaren walsh key issue control distributed discrete systems modeled markov decisions processes often state system directly observable single location system participants control scheme must share information one another regarding state system order collectively make informed control decisions information sharing costly harnessing recent results information theory regarding distributed function computation paper derive several information sharing model structures minimum amount control information must exchanged enable local participants derive control decisions imaginary omniscient controller full knowledge global state incorporating consideration amount information must exchanged reward enables one trade competing objectives minimizing control information exchange maximizing performance controller alternating optimization framework provided help find efficient controllers messaging schemes series running examples wireless resource allocation illustrate ideas design tradeoffs ntroduction framework markov decision processes mdp provides principled method optimal design controllers discrete systems solving bellman equation example either value policy iteration one derives control mapping assigning possible state system control action take manner maximizes long run discounted expected reward increasing interest however discrete systems decentralized distributed sense single participant system full access global system state rather global state concatenation series local states directly observed different locations instance series agents may observe local state set local actions choose desire may design individual local controllers fact global system state available location either requires sufficient information exchanged either ordinary communications system state remedy situation modification control framework one way address part distributed knowledge network state use framework partially observable markov decision processes pomdp synthesize controllers general optimal control pomdp requires controller maintain probabilistic current system state based previous control actions previous observations assigning control action based belief states thus key issue pomdps problem structures enable simple forms control action even complex solving general decentralized multiagent mdps rich literature many others addressed finding approximate solutions key issue literature decentralized distributed mdps role amount communication coordination relationships communication control established several contexts literature information theoretic limits incorporated classical case single observer remotely controlling linear system noisy channel focus determining minimum rate required achieve control objectives vein proposes notion anytime capacity gives necessary sufficient condition communication reliability needed noisy channel stabilize unstable scalar linear process observer access noiseless feedback channel output building ideas provide explicit construction anytime reliable codes efficient encoding decoding noisy channels shifting decentralized control context deep connections communications controllers system state network coding recently investigated despite long standing interest relationships communication control role information theoretic limits state observations mdp distributed less well developed literature literature studying communication coordination made use related ideas multiterminal information theory compute communication cost instance considers coordination agent decisions made environment discusses computational complexity finding optimal decisions communicative team decision problem commtdp along dimension communication cost similarly recognize communication incurs cost global reward function show whether communicate also decision make however none models make use relevant information theoretic limits computing communication cost part reason information theoretic limits fully brought bear distributed mdp limits relevant models instance distributed interactive function computation somewhat recently derived bearing mind paper aims harness information theoretic limits coding designs approach help synthesis efficient control schemes distributed mdp particular section consider mdp model state considered vector local states directly observed different location goal distributed mdp operate manner omniscient mdp full access global state natural way achieve exchange state information participants information exchange however costly furthermore order achieve performance simulate omniscient controller truly necessary system make decisions desirable determine minimal amount information must exchanged order enable distributed system learn decisions centralized omniscient controller would take lower bounds minimal amount control information necessary provided section iii using recent results information theory regarding quantization interactive decentralized function computation alternatively described section one may wish incorporate cost communication reward enabling control designer trade cost communication performance controller case messaging scheme controller must designed simultaneously order provide candidate solutions associated optimization present section alternating optimization based method guaranteed yield sequence rewards converges messaging scheme control map selected scheme converges prove must lie nash equilibrium total reward combines reward controller cost communication several examples drawn design downlink resource controllers wireless networks throughout manuscript demonstrate ideas mniscient ontrol istributed arkov ecision rocess consider distributed discrete stochastic control system modeled markov decision process mdp global system state time vector composed series local states observed one network node global network state evolves according markov chain whose transition matrix time selected control action additionally reward function indicating payment obtained global system state transitions action taken omniscient controller access series global states would select actions maximizing total discounted expected reward min rat argument solution optimization mapping assigning state optimum action take optimal bellman equation states solution optimization must solve following system equations arg max solution simultaneous system equations associated optimal control mapping found first determining limit following value iteration max solving control policy via arg max alternatively one utilize policy iteration performs recursion first arg max solved followed solution linear system pck rck control mapping selected remain update point iteration ceases see note many problems given one choice achieving maximum instance one derive set candidate omniscient control functions obeying constraints arg max achieves maximum long run expected reward downlink wireless resource allocation example practical problem structure outlined distributing resources wireless downlink time slotted basestation shared buffer containing individual information must sent series users amount waiting user time slot denoted collectively buffer sizes form local state basestation user channel state indicating much information reliably transmitted user present timeslot evolves independently users channel states according markov chain transition distribution time slot random amount additional traffic arrives destined user basestation buffer independently time slots previous arrivals basestation buffer accommodate traffic dropped time slot must decided users give resource thus forms action mdp additionally users basestation must know outcome decision user transmit selected amount traffic minimum capacity slot amount traffic waiting buffer successfully transmitted removed buffer yielding markov chain dynamics amount new traffic arriving time slot user vector nth element min bat otherwise function performs package dropping process arriving traffic accommodated buffer note assumption amount information successfully transmitted received users basestation need know schedule consistent assumption physical layer scheduler uses rateless code feedback closely approximated hybrid arq making decision schedule several important metrics considered thereby combined reward natural metric throughput measures much information transmitted summed users gives reward function min metrics average maximum delay users traffic experiences also reasonable metrics incorporated reward function instance adding together rates trade one another selected metrics included reward function mdp framework gives formal way deciding allocate resources time slot series examples throughout rest paper find optimal controller investigate properties much information must exchanged order perform simulating omniscient mdp via information exchange however system distributed single node given access global network state control must performed nodes learning action take sort communication enabled strategy additionally introduce constraint observer given access local states rather accessing control messages nodes share one another must able infer action taken order enable system easily monitored require user observing state able learn optimal action selected exclusively information shared time slot omniscient control action must taken strategy key question much information must shared order enable optimal control action omniscient controller would taken selected based shared information words much information must shared order enable system simulate omniscient controller sense every node including one access local state observations observing shared control messages time slot learn action controller take thereby enabling distributed system obtain expected discounted long run reward omniscient system answer question depends course model way information exchanged control map particular characteristics transition kernels clearly problem designing communication transformed one distributed function computation node observing local state must convey message time slot learned messages additionally capability select still achieve maximum reward enables amount information exchanged minimized downlink wireless resource allocation continued assumptions made associated problem minimizing overhead special practical significance downlink wireless resource allocation setup example basestation direct access buffer observes amount information arrived destined various users users observe downlink channel qualities yet sufficient information must exchanged system make informed decision regarding schedule downlink much information send omniscient controller access state distributed throughout network could make series decisions solving associated mdp present instance lte wimax standards decisions made basestation requests receives channel quality statistics users schedules users much information send transmitting decisions downlink additionally desirable minimize amount control information efficient design type control measurement decision information together reference signals reached roughly quarter third time frequency footprint lte lte advanced furthermore essential messages exchanged time slot must overhead order learn control decision action nodes come network essential nodes arrived current slot able determine control decisions finally note evident problem description example various local states channel state user buffer state basestation evolve according independent markov chains given control actions iii inimal oordination ommunication equired istributed imulation mdp general models arbitrary dependence observations multiterminal information theory yet determine minimum sum rate required distributed function computation however limits known handful special cases including local observations independent independent case transition kernel initial state distribution admit factorization factorization turn implies local states evolve independently one another action specified quantities available encoded messages nodes independent one another fundamental limits special case subdivided based upon whether messages must sent parallel manner interaction users multiple rounds communication one time slot allowed one shot distributed simulation omniscient mdp case assumptions made regarding monitoring node minimum required distributed function computation given control map independent sources case independent states given sum graph entropies characteristic graphs user characteristic graph user set nodes possible local states user edge exists characteristic graph values since local states independent probability distribution positive product support edge possible values local states transitive property complement characteristic graph namely edge edge characteristic graph also edge therefore maximal independent sets characteristic graph overlap form partition set vertices graph owing transitivity property complement characteristic graphs graph entropies fact chromatic entropy hgn min represent colorings characteristic graph minimum expected rate achieved within one bit huffman coding coloring characteristic graph achieving minimum entropy achieved assigning different colors different maximal independent sets selecting omniscient control map requiring minimum rate gives minimum rate min min note searching minimum entropy colorings graph calculate chromatic entropy suffices consider exclusively obtained iteratively removing maximal independent sets following two examples describe control rate required form sharing quantized local states two particular distributed mdps minimum control information arg max let assume example buffer size infinite user infinitely many backlogged packets destined buffer control decision made regarding users channel qualities objective let basestation learn control decision user occupy resource block observing messages sent users let local states independent identically distributed downlink channel qualities known distribution discrete support set basestation wishes maximize system throughput control decision becomes finding one users best channel quality arg max problem shown characteristic graphs obey properties also shown minimum information required determine control action computed bits saved relative scheme users simply send channel qualities basestation rate required simulating omniscient wireless resource controller interaction return case finite buffer size without backlogged packets example assume channel qualities independently uniformly distributed support addition let amount additional traffic arrives destined user independent across users time distributed support probabilities additionally let packet dropping function operate according algorithm manner consistent total buffer size bumax let controller aim maximize throughput reward means users basestation system optimal control decisions solving mdp problem discounting factor give maximal total discounted reward system starts initial state meanwhile expected amount system throughput per expected amount data dropped per timeslot calculating characteristic graphs determining huffmann codes associated minimum entropy colorings find associated optimal control decision learned via quantization local states channel states two users buffer size basestation minimum rate bits encoder mappings respect minimum rate given users basestation control mapping decided deterministically given optimal control encoders interactive collocated network simulation omniscient mdp interactive communication case natural lower bound rate obtained via collocated network result update next round buffer status input current buffer status buffer size bumax new arriving packages remaining bumax new remaining new remaining remaining new new end mod end output algorithm packet dropping process buffer full messaging model model communication happens multiple rounds users consecutively take turns sending message overheard users particular round node index sends message based observation sent messages previous rounds time rounds communication finishes optimum omniscient control action must completely determined ishwar shown minimum block codes obtained strategy lower bounded solution following repeated convex geometric calculation solution written respect rate reduction functional maps coordinates marginal probability distributions local observations conditional entropy marginal distribution given rate required function computed rounds communication expressed pst evaluating rate reduction functional marginal probability distribution psr rate reduction functional turn found via following iterative convex program constant supp otherwise concavify fixed context operator concavify computing upper concave envelope function viewing restriction function minimum control information arg max interactive consider infinite packet buffer backlog throughput maximization variant wireless resource allocation described example added ability users basestation interact one another sending messages particular users take turns sending messages one time rounds participants including basestation sends messages overhear message goal enable anyone overhears messages learn index least one user whose channel quality maximum channel quality users curve labelled fundamental limit fig calculates fundamental limit lower bounding total number bits must exchanged order perform calculation case channel qualities uniformly distributed set explained fundamental limit general achievable vector quantization schemes problem setup hand demands scalar quantization schemes must used additionally second curve fig labelled optimal scalar hete gives rate required best possible scalar quantization scheme followed huffman coding problem seen quite close vector quantization limit curve found via exhaustive search scalar quantization schemes addition presenting problem detail describing curves also considers reduced complexity restricted smaller quantizer search spaces finally curve homo scalar interactive indicates rate required users must send messages parallel messages received send another series parallel interactivity model considered count form interaction requiring users send messages parallel substantially increases rate required fundamental limit optimal scalar hete homo scalar interactive ceo sum rate note problem hand lower bound may achievable scalar quantization coding strategies required assumptions made distributed mdp setup lower bound may general achieved limit vector quantization schemes particular scalar quantization available problem consideration repeated observations available use larger added constraint user overhearing exclusively messages time slot must able learn nonetheless demonstrate following example often best scalar quantization based interaction schemes still yield rate close fundamental limit number rounds fig rate required select user attaining max users observing independent support users take turns sending messages one time fundamental limit rate given compared obtained best scalar quantizer use collocated network interactive scheme optimal scalar hete substantially less rate required three users must send message parallel hear collectively sent messages send message parallel etc interactive model employed ncorporating ommunication ost eward unction previous discussion assumed reward function completely given many problems cost communicating network may subtract reward decisions manner may desirable design mdp consider cost explicitly incorporating weighted term reward function particular suppose total number bits communicated partial state sharing scheme messages state vector denoted form augmented reward function including communication cost reflecting number bits transmitted order enable system learn action goal shifts solving optimization problem max constraint set defined also equivalently set rewritten function exclusively observe observation encodings form effectively observation partial observed markov decision process pomdp requirement made able determine controller running full state knowledge mdp action decisions exclusively observation time step implies different problem structure pomdp memory past observations action decisions determining state distribution must neglected select map given controller define transition matrix whose element define row vector objective function optimization rewritten size limit basestation bumax packet dropping process described example system started initial state buffer empty channel qualities let transition reward function term represents system throughput choosing action state consider noninteractive control information sharing model node user base station sends quantized representation state everyone else enabling learn control action directly messages finding optimum solution via exhaustive search control mappings different enables control overhead versus throughput tradeoff control overhead versus packet dropping cost tradeoff plotted fig traced observe fig least bits control overheads required guarantee expected system throughput achieves limit presence constraint makes joint optimization problem substantially difficult combinatorial optimization problem ordinary mdp small problems set control maps enumerated control map component max expected reward associated minimum control information overhead calculated using results section messaging scheme interactive communications enabled results section used cases probability distribution selected multiplicatively proportional ensure reward maximized encoding determining inpthis manner expected discounted reward maximized encoding schemes control map control map yielding expected maximum reward particular selected furthermore tradeoff control overhead expected reward traced varying optimization overhead performance tradeoff wireless resource allocation tiny model consider setup distributed wireless resource allocation described example mobile users observing local channel quality time instant uniformly distributed support additionally let amount additional traffic arrives destined user uniformly distributed support let buffer finding candidate quantizations alternating optimization however many problems sort exhaustive search approach solving combinatorial optimization described nowhere near computationally feasible number possible control maps search case alternating optimization approach yields lower complexity search method well suited finding candidate solutions optimization problem reasonable goal alternating optimization method alternate optimizing control map optimizing quantizer let iteration index algorithm control map local state encoders iteration denoted respectively given iteration algorithm control map minimizing augmented value function among control maps determined present encoding could selected solving arg max next local encodings achieve minimum expected sum rate enabling distributed computation new control map selected arg max admits form alternating maximization sequence expected values monotone increasing sequence bounded via global optimum sequence must converge limit general limit may may global optimum guaranteed limit associated nash equilibrium particular limit iteration property unilateral change individually control map packets dropped per packet drop cost throughput packets throughput communication cost bits communication cost bits fig control overhead versus expected throughput tradeoff control overhead versus expected packet dropping cost tradeoff augmented reward function quantizer yield higher expected reward although may possible modify together achieve higher reward solve communications structure used results section equation used fairly tight close bound associated close achievability scheme within one bit interactive communications enabled results section equations used bound solving however quite complicated direct search solution combinatorial optimization complexity simplify matters control map update attacked alternating optimization overall iteration update complexity proportional simplest form alternating minimization cycles different possible quantizations updating associated order determined selected bijection according mod arg max lim wherein alternatively greedy form alternating optimization selected replaces arg max max putting pieces together overall low complexity alternating optimization algorithm find candidate solutions consists algorithm alternating optimization individual dimensions optimization sequence expected rewards achieved update monotone increasing sequence expected rewards bounded global maximum must converge limit depending initialization limit may may global maximum sequence control maps quantizations also converges must least nash equilibrium summarized following theorem thm iterative method solving constrained mdp described yields monotone increasing sequence expected rewards converges additionally sequence control maps quantizations converges convergent pair nash equilibrium sense unilaterial deviation axes yield increase expected reward overhead performance tradeoff wireless resource allocation larger model apply aforementioned alternating optimization algorithm solve following example consider wireless resource allocation setup examples mobile greedy scheme scheme communication coefficient communication cost bits throughput greedy scheme scheme communication coefficient cost throughput greedy scheme scheme communication cost bits greedy scheme scheme communication cost bits fig expected system throughput control overheads per respect weighting coefficient control overheads versus expected throughput tradeoff control overheads versus expected packet dropping cost tradeoff computed greedy alternating optimization algorithm augmented reward function users observe local channel quality distributed support probabilities additionally amount additional traffic arrivals destined user distributed support probabilities basestation observes buffer state buffer limit bumax let packet dropping process one described example refer algorithm consisting alternating optimization algorithm algorithm consisting greedy alternating optimization algorithm quantizers information sharing strategies learn action assume scheme must used furthermore algorithms initialized trivial quantizers simply relay full local state note observed thm alternating optimization methods always yield sequence rewards monotone converges associated control map quantizer converge general nash equilibrium possibly global optimum local convergence indeed observed experiments multipliers lead quantization control mappings negative expected discounted rewards sending control information expected reward lower bounded hence quantization control schemes presented remainder example guaranteed nash equilibria chance globally optimal nonetheless highly optimized thus tradeoffs yield varying expected throughput control overhead quite interesting applying greedy algorithm find local optimal quantizations control mappings different choice expected throughput communication cost computed shown fig based local optimal solutions found also plot throughput versus control overheads tradeoff packet dropping cost versus control overheads tradeoff fig observe fig expected system throughput maximized case communication cost involved transition reward function optimal control mapping becomes pick user maximize instant throughput optimal encoders designed minimize control overheads guarantee optimal control mapping learned node observing control messages also observe fig control overheads become grows larger system realizes cost pays encode local states weights reward could earn sending data traffics destined user hence optimal decision encode local state blindly schedule user occupy resource block finally observe fig expected system throughput per goes system decision blindly pick user although basestation may pick users send data traffics first however long run global state must absorbed one recurrent classes instant throughput always remain buffer fills traffic destined users stays full fact given blind control mapping local observation global states buffer local state satisfying bumax form recurrent class result expected system throughput becomes system decision blindly pick user indicates system randomly picking users perform better deterministic control mapping system matches conclusion important note however would precluded present model would require user would transmitted known deterministically participant arriving network case would observed anything since control information sent additional control rate savings increased rewards enabled randomization require assumption synchronized common randomness participants scheme important direction future work onclusion paper analyzed markov decision process state composed series local states observed different location network using recent results multiterminal information theory regarding distributed interactive function computation minimum amount control information would necessary exchange order system simulate centralized controller access global state determined next information theoretic cost communication incorporated reward function mdp problem simultaneously designing controller messaging scheme maximize associated combined reward formulated creating tradeoff communication performance provide candidate solutions associated optimization problem alternating optimization method presented produces sequence rewards always converges associated messaging scheme controller map converges converges nash equilibrium problem series running examples downlink wireless resource allocation illustrated ideas throughout important directions future investigation involve allowing time varying messaging control schemes use historical observations messages pomdp like framework use rate distortion theory aid derivation tradeoffs communication control reward present decentralized mdp context eferences bellman markovian decision process dtic document tech putterman markov decision processes discrete stochastic dynamic programming new york john wiley sons mar hsu marcus decentralized control finite state markov processes ieee transactions automatic control vol april chu team decision theory information structures optimal control problems part ieee transactions automatic control vol february sandell varaiya athans safonov survey decentralized control methods large scale systems ieee transactions automatic control vol april park sahai network coding meets decentralized control equivalence decision control european control conference ieee conference ieee network coding meets decentralized control network linearization equivalence arxiv preprint online available http krishnamurthy partially observed markov decision processes filtering controlled sensing cambridge university press bernstein zilberstein immerman complexity decentralized control markov decision processes proceedings sixteenth conference uncertainty artificial intelligence boutilier planning learning coordination multiagent decision processes proceedings conference theoretical aspects rationality knowledge march nair tambe yokoo pynadath marsella taming decentralized pomdps towards efficient policy computation multiagent settings international joint conference artificial intelligence august chades scherrer charpillet heuristic approach solving assessment pursuit problem proceedings acm symposium applied computing march szer francois zilberstein maa heuristic search algorithm solving decentralized pomdps conference uncertainty artificial july olienhoek spaan vlassis optimal approximate functions decentralized pomdps journal artificial intelligence research jair vol may seuken zilberstein formal models algorithms decentralized decision making uncertainty autonomous agents systems vol october tatikonda sahai mitter stochastic linear control communication channel ieee transactions automatic control vol september tatikonda mitter control communication constraints ieee transactions automatic control vol july borkar mitter lqg control communication constraints communication computation control signal processing tribute thomas kailath longo norwell kluwer wong brockett systems finite communication bandwidth contraints state estimation problems ieee transactions automatic control vol september sahai mitter necessity sufficiency anytime capacity stabilization linear system noisy communication link part scalar systems information theory ieee transactions vol sukhavasi hassibi anytime reliable codes stabilizing plants erasure channels decision control european control conference ieee conference ieee linear error correcting codes anytime reliability information theory proceedings isit ieee international symposium ieee linear anytime codes control noisy channels ieee transactions automatic control vol park sahai algebraic theorem information theory proceedings isit ieee international symposium ieee gmytrasiewicz durfee rational coordination multiagent environments autonomous agents systems vol december pynadath tambe communicative multiagent team decision problem analyzing teamwork theories models journal artificial intelligence research vol june xuan lesser zilberstein communication markov decision processes multiagent systems proceedings fourth international conference ieee xuan lesser policies centralized ones decentralized ones proceedings first international joint conference autonomous agents multiagent systems part xuan lesser zilberstein communication decisions cooperation model experiments proceedings fifth international conference autonomous agents doshi shah medard jaggi functional compression graph coloring ieee transactions information theory vol sefidgaran tchamkerten distributed function computation rooted directed tree ieee trans inform theory vol february prabhakaran tse ramchandran rate region quadratic gaussian ceo problem proc int symp inform theory isit jun ishwar interactive source coding function computation collocated networks ieee trans inform theory vol july bertsekas dynamic programming optimal control athena scientific luby codes proceedings symposium foundations computer science verdu shamai channel capacity ieee transactions information theory vol jun erez trott wornell rateless coding gaussian channels ieee transactions information theory vol gwanmo john maclaren walsh resource allocation link adaptation lte lte advanced tutorial ieee communications surveys tutorials vol online available http ren boyle weber walsh overhead performance tradeoffs resource allocation perspective ieee trans inform theory vol february cardinal fiorini joret minimum entropy coloring journal combinatorial optimization vol torabi walsh practical interactive scheme extremum computation distributed networks international symposium information theory isit jul appear online available http isit bradford boyle jie ren john maclaren walsh steven weber interactive scalar quantization distributed resource allocation ieee transactions signal processing vol mar online available http torabi walsh interactive quantization extremum computation collocated networks data compression conference dcc mar page paper presentation online available http dcc murphy survey pomdp solution techniques univ berkeley tech september online available http myerson game theory harvard university press singh jaakkola jordan learning without partially observable markov decision processes proceedings eleventh international conference machine learning jie ren received degree electrical engineering tsinghua university china degree telecommunication widener university chester since september pursuing drexel university philadelphia within adaptive signal processing information theory research group interning wireless big data research team huawei joined futurewei bridgewater software engineer wireless access solmaz torabi received degree electrical engineering sharif university technology iran working toward degree since within adaptive signal processing information theory research group department electrical computer engineering drexel university philadelphia research interests include areas information theory interactive communication john maclaren walsh received magna cum laude degrees electrical computer engineering cornell university ithaca respectively september joined department electrical computer engineering drexel university philadelphia currently associate professor drexel directs adaptive signal processing information theory research group
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dec supersoluble residual mutually permutable products monakhov january abstract prove group mutually permutable product supersoluble subgroups supersoluble residual coincides nilpotent residual derived subgroup keywords finite groups supersoluble subgroup mutually permutable product groups paper finite formation group smallest normal subgroup quotient group called mutually permutable product subgroups groups studied see also prove following theorem theorem let mutually permutable product supersoluble subgroups respectively formation nilpotent groups formation supersoluble groups need following lemmas lemma let product two subgroups hereditary formations according product also hereditary formation lemma let formations group gfh subgroup group denotes smallest normal subgroup containing lemma let subnormal subgroup group belongs fitting class particular nilpotent also nilpotent also lemma let product supersoluble subgroups lemma lemma subgroups nilpotent normal lemma nilpotent lemma normal view baer theorem supersoluble hence fitting class also formation called fitting formation class abelian groups denoted lemma let fitting formation let product normal subgroups theorem lemma since lemma verify reverse inclusion since nilpotent supersoluble view theorem thus lemma subgroups subnormal theorem nilpotent therefore normal nilpotent lemma view lemma get corollary let mutually permutable product supersoluble subgroups nilpotent supersoluble class groups coincides product class class group chief factors order divisible prime exactly order derived subgroup group class groups denoted clear theorem let mutually permutable product subgroups gpu lemma gpu verify reverse inclusion quotient group mutually permutable product subgroups derived subgroup corollary consequently gpu thus gpu lemma subgroups subnormal group theorem hence normal lemma view lemma get gpu corollary let mutually permutable product subgroups references asaad shaalan supersolubility finite groups arch math alejandre cossey permutable products supersoluble groups algebra cossey products supersoluble groups rev mat iberoamericana beidleman heineken mutually permutable subgroups group classes arch math asaad york walter gruyter gruyter expositions mathematics monakhov introduction theory finite groups classes minsk vyshejshaja shkola russian doerk hawkes finite soluble groups york walter gruyter huppert endliche gruppen berlin heidelberg new york springer baer classes finite groups properties illinois math monakhov department mathematics gomel scorina state university gomel belarus address
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jordan property algebraic groups projective varieties feb sheng meng zhang abstract century ago camille jordan proved complex general linear group gln jordan property jordan constant every finite subgroup gln abelian subgroup index show every connected algebraic group necessarily linear jordan property jordan constant depending dim full automorphism group aut every projective variety jordan property introduction work algebraically closed field characteristic zero unless explicitly stated otherwise group jordan group constant called jordan constant satisfies following jordan property every finite subgroup abelian subgroup index definition equivalently may even require normal though require paper indeed consider action left multiplication yields homomorphism symmetric group letters kernel homomorphism normal subgroup index dividing contained define smallest jordan constant hence jordan family groups uniformly jordan constant denoted serving jordan constant every group family question asked professor popov question problem let projective variety dimension full automorphism group aut jordan group projective surfaces question affirmatively answered popov except birational elliptic curve zarhin remaining case see references therein higher dimensions prokhorov shramov theorem proved jordan property birational automorphism group bir assuming either mathematics subject classification key words phrases automorphism groups projective varieties jordan property groups sheng meng zhang vanishing irregularity conjecture boundedness terminal fano varieties affirmatively confirmed birkar see theorem approach towards jordan property full automorphism group aut projective variety arbitrary dimension theoretical use classification projective varieties projective variety classical result grothendieck says neutral component aut algebraic group conversely brion theorem every connected algebraic group dimension isomorphic smooth projective variety dimension general let connected algebraic group necessarily linear classical result chevalley unique maximal connected linear algebraic normal subgroup abelian variety fitting following exact sequence classical result camille jordan jordan group course also jordan group however extension two jordan groups may jordan group nevertheless would like ask question let algebraic group jordan group clearly suffices consider connected algebraic groups positive answer question implies positive answer question however every connected algebraic group isogenous product abelian variety connected linear algebraic group diminishing hope give positive answer question using fact latter two groups jordan groups lemmas remark conversely positive answer question implies positive answer question virtue lemma lemma jordan property group related bounded rank property finite abelian subgroups exists constant every finite abelian subgroup generated elements denote rkf smallest constant see definition details similarly may define uniformly bounded rank property finite abelian subgroups family groups state main results positively answer questions jordan property theorem fix integer let family connected algebraic groups dimension uniformly jordan finite abelian subgroups uniformly bounded rank see theorem upper bounds rkf functions theorem fix integer let projective variety dimension uniformly jordan finite abelian subgroups uniformly bounded rank see theorem upper bounds rkf functions remark theorem follow theorem directly dim bounded terms dim example theorem hirzebruch surface degree unipotent radical additive group dim bounded order prove theorem remove influence unipotent radical key lemma let connected algebraic group denote gaff largest connected affine normal subgroup note gaff every subgroup connected contained center particular every subgroup normal denote gant largest subgroup main ingredient proof old classical decomposition theorem gaff gant connected algebraic group due rosenlicht corollary see also brion modern elaborations also use effective optimal upper bound brion proposition dimension part see lemma last main result immediate consequence theorem lemma theorem let projective variety aut jordan group remark abelian variety rational variety bir jordan group corollary hence jordan constant theorem general depends birational invariant refer related results acknowledgement authors heartily thank friends colleagues especially referees paper providing remark improving lemma clarifying corollary removing inaccuracies many suggestions improve paper author supported arf nus sheng meng zhang preliminary results paper denotes identity element group denote centre necessarily linear algebraic group use denote neutral component projective variety denotes group nsq definition given group introduce following constants sup rkf sup rkf abelian rkf minimal number generators finite abelian group define rkf similarly may define constants family groups easy observations frequently used lemma consider exact sequence groups finite rkf rkf rkf rkf proof clear refer proof lemma lemma important constants jordan every general linear group gln jordan group hence every linear algebraic group jordan group denote gln known theorem minkowski gln field finitely generated bound depends theorem rkf dim algebraic torus rkf gln see proposition use rkf dim abelian variety jordan property projective variety normalization aut lifts may assume normal theorem following lemma theorem imply theorem lemma let normal projective variety exists constant depending finite subgroup aut proof take aut hence equivariant projective resolution action aut induces natural representation nsq consider exact sequence kernel representation note glm dimq nsq picard number lemma constant thus find subgroup acts trivially nsq particular preserves ample divisor class hence aut proposition aut finite constant proposition identify thus lemma follows setting remark alternatively one may show directly following statement without taking resolution let projective variety ample line bundle aut subgroup aut fixes class nsq aut finitely many components see identify aut open subscheme hilbert scheme hilb associating automorphism graph theorem also ample line bundle identified numerically equivalent aut replacing positive multiple may thus assume algebraically equivalent hilbert polynomial independent aut let polynomial aut contained hilbp note hilbp projective scheme chapter theorem aut closed connected component aut hence aut scheme yields assertion lemma works need results algebraic group theory proofs easy give proofs convenience reader sheng meng zhang lemma let isogeny two semisimple linear algebraic groups lemma let connected almost simple linear algebraic group rank dim rank rank rank equals dim every maximal algebraic torus contained proof isogeny simply connected almost simple lemma replacing may assume simply connected isomorphism correspondence simply connected almost simple linear algebraic groups dynkin diagrams given following table also shows centres ranks lemma follows table dim rank lemma let connected semisimple linear algebraic group dimension proof let minimal closed connected normal subgroups positive dimension set trivial natural product map gives isogeny kernel contained centre domain map lemmas remark well known isomorphism finitely many ndimensional semisimple linear algebraic groups thus function every connected semisimple linear algebraic group dimension embedded jordan property gln denote supremum jordan constants connected semisimple linear algebraic groups dimension clearly proof theorems section prove theorems precise versions theorems jordan constant uniformly bounded rank theorems could made optimal expense complicated expressions done give notations facts first let group denotes commutator subgroup let connected algebraic group use conventions facts see also gaff gant denote respectively affine part antiaffine part connected normal rosenlicht decomposition gaff gant gant gant gant gaff abelian variety albanese variety gaff gaff gant largest affine quotient group let connected algebraic group denote unipotent radical gaff levi subgroup gaff gaff levi subgroups gaff one fix let connected reductive linear algebraic group solvable radical algebraic torus semisimple connected see let connected linear algebraic group closed normal subgroup quotient map levi subgroup levi subgroup every nontrivial unipotent element linear algebraic group infinite order ground field characteristic zero sheng meng zhang lemma let connected reductive linear algebraic group dim rank proof product map gives isogeny since semisimple isogeny connected almost simple hence isogeny take maximal torus note algebraic torus rank rank lemma algebraic torus rank dim rank dim dim rank rank dim rank rank rank lemma let connected reductive linear algebraic group dim rkf proof refer lemma give proof reduction consider exact sequence rkf rkf rkf since algebraic subtorus lemma implies rkf dim note semisimple connected dimension dim remark embedded gln lemma rkf combining two inequalities ranks get lemma via display lemma let connected reductive linear algebraic group dim proof product map gives isogeny thus lemma commutativity imply remark jordan property corollary let connected linear algebraic group dim closed normal subgroup proof first consider exact sequence unipotent radical since lemmas note dim dim see hence also holds remark lemma slightly extends theorem key lemma proof crucially utilizes rosenlicht decomposition gaff gant recall gaff lemma let connected algebraic group dim let finite subgroup exists subgroup index proof consider natural homomorphism ker gaff gant gant since abelian corollary gaff gaff gant exists subgroup index abelian note connected semisimple thus natural injective homomorphism following commutative diagram gaff gaff gaff gaff gaff gant gant surjective group homomorphism restriction group homomorphism gaff gaff sheng meng zhang ker gaff note ker ker since finite injective since abelian imply thus ker diagram since dimension lemma lemma let connected algebraic group dim dim gant finite subgroup rkf proof refer lemma give proof reduction first diagram lemma gaff gaff claim note write since inductively equals hence thus proves claim clearly normal since consider exact sequence gaff abelian variety lemma rkf rkf gaff rkf note gant gant rkf dim dim gant lemma gaff levi decomposition gaff gaff imply gaff thus rkf gaff rkf rkf since unipotent group rkf also connected reductive group dim dim lemma rkf lemma follows display two inequalities ranks obtained theorem precise version theorem theorem let connected algebraic group dimension defined remark rkf jordan property proof rkf straightforward lemma let finite subgroup lemma exists subgroup note normal closed finite abelian subgroup connected algebraic group dimension lemma rkf applying lemma find abelian subgroup clearly discussed remark giving proof theorem need following result proved proposition lemma let algebraic torus acting faithfully projective variety acts generically freely stabilizer subgroup trivial general point particular dim dim dim lemma let connected reductive linear algebraic group acting faithfully projective variety dim dim proof let maximal torus dim dim lemma lemma dim dim also need following effective optimal bound proposition lemma let algebraic group acting faithfully projective variety dim dim theorem precise version theorem theorem let projective variety dimension rkf defined remark proof let levi subgroup gaff lemmas dim dim gant done argument proof theorem corollary let projective variety dimension connected algebraic group contained bir setting defined remark rkf sheng meng zhang proof theorem exists variety birational acts biregularly since smooth locus may assume smooth covered open subsets theorem may assume admits embedding projectivization vector bundle abelian variety theorem particular admits embedding projective let closure also projective birational dim result follows theorem references birkar singularities linear systems boundedness fano varieties brion basic results actions nonaffine algebraic groups symmetry spaces progr math boston brion geometry algebraic groups homogeneous spaces algebra brion automorphism groups fiber bundles publ mat urug brion connected automorphism groups algebraic varieties ramanujan math soc brion automorphisms endomorphisms projective varieties automorphisms birational affine geometry springer proc math springer cham curtis reiner representation theory finite groups associative algebras pure applied mathematics vol interscience publishers division john wiley sons new demazure rang maximum groupe cremona french ann sci norm sup fantechi illusie kleiman nitsure vistoli fundamental algebraic geometry grothendieck fga explained mathematical surveys monographs american mathematical society providence humphreys linear algebraic groups graduate texts mathematics volume springer mundet riera finite subgroups ham symp rational curves algebraic varieties ergebnisse der mathematik und ihrer grenzgebiete folge series modern surveys mathematics berlin lieberman compactness chow scheme applications automorphisms deformations manifolds fonctions plusieurs variables complexes iii norguet lecture notes springer berlin maruyama automorphism groups ruled surfaces math kyoto univ mostow fully reducible subgroups algebraic groups amer math jordan property popov jordan groups automorphism groups algebraic varieties automorphisms birational affine geometry springer proc math springer cham prokhorov shramov jordan property groups birational selfmaps compos math rosenlicht basic theorems algebraic groups amer math serre bounds orders finite subgroups group representation theory epfl press lausanne zarhin theta groups products abelian rational varieties proc edinburgh math soc issue zarhin jordan groups elliptic ruled surfaces transform groups department mathematics national university singapore lower kent ridge road singapore address department mathematics national university singapore lower kent ridge road singapore address matzdq
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applications graded methods mar cluster variables arbitrary types thomas thesis submitted degree doctor philosophy department mathematics statistics lancaster university june applications graded methods cluster variables arbitrary types thomas thesis submitted degree doctor philosophy june abstract thesis concerned studying properties gradings several examples cluster algebras primarily infinite type start considering two classes finite type cluster algebras type give number cluster variables occurring degree verify grading balanced results complete classification finite type cluster algebras consider gradings cluster algebras generated matrices show matrices give rise gradings occurring degrees positive finitely many associated cluster variables excepting one particular case matrices prove occurring degrees infinitely many variables give direct proof gradings balanced provide condition graded cluster algebra generated quiver infinitely many degrees based presence subquiver mutation class use study gradings cluster algebras quantum coordinate rings matrices grassmannians show contain cluster variables degrees next consider finite list given quivers correspond triangulations marked surfaces show grading two degrees infinitely many cluster variables infinitely also show gradings arising many variables certain degrees finally study gradings arising triangulations marked bordered surfaces see adapt definition define space valuation functions surface prove combinatorially space isomorphic space gradings associated cluster algebra illustrate theory applying family examples namely annulus marked points show standard grading mixed type finitely many variables degrees infinitely many others also give alternative grading degrees infinitely many cluster variables acknowledgements firstly would like thank supervisor jan grabowski generous support guidance throughout course studies extremely grateful time gave many useful discussions regarding research well knack finding pertinent results literature helped make progress thank also constant willingness help provide interesting research topics made phd experience enjoyable also thank examiners anna felikson mark macdonald making many useful suggestions improved thesis many people grateful met time lancaster thank phd students met many become good friends group made social life great experience also grateful friends made reminders inappropriate group phd students repeatedly make lame mathematical jokes presence additionally thank staff maths stats department help make exceptionally friendly positive place owe enormous debt gratitude parents moshe lesley always believed supported everything done also thank rest family including sister maddy love support last least owe special thanks faye love encouragement given especially towards end studies thank also providing open problem far deeper could encounter phd consistently make cup tea vile grateful chocolate shared places visited together wonderful times always putting rambling iii declaration thesis work submitted substantially form award higher degree elsewhere contents introduction thesis overview notation preliminaries definition cluster algebra graded cluster algebras quivers denominator vectors gradings finite type cluster algebras review background gradings type gradings type gradings rank case structure problem finitely many degrees mixed case matrices cluster algebras generated acyclic matrices singular cyclic case degree subquivers growth subquivers growing arrows growing degrees new degree quivers old gradings cluster algebra structure coordinate algebras matrices grassmannians graded cluster algebra structure matrices infinitely many degrees occurring degrees corollaries grassmannians gradings finite mutation type quivers arising surface triangulations initial information finite list quivers grading gradings grading grading grading gradings quivers arising surface triangulations graded cluster algebras associated surfaces annulus marked points odd annulus marked points even appendix chapter introduction cluster algebras commutative unital subalgebras generated certain iterative process endows particular combinatorial structure introduced fomin zelevinsky series papers third berenstein attempt capture certain combinatorial patterns observed number algebras associated particularly nice classes geometric objects authors founding papers realised though combinatorial patterns appear complex arise relatively simple rules iterated original goal cluster algebras provide algebraic framework study dual canonical bases related notion total positivity semisimple lie groups quantum analogues since introduction however quantum analogues appeared many areas mathematics algebraic symplectic geometry noncommutative algebra mathematical physics others iterative process produces generators cluster algebra starts small finite subset generators eventually obtain generators known cluster variables finite subset start called initial cluster initial cluster clusters produced replacing one old variable new one process called mutation mutation initial cluster carried controlled matrix called initial exchange matrix whose entries fact determine entire cluster algebra initial exchange matrix used mutate initial cluster mutates obtain new exchange matrix along new cluster together called seed turn mutated process seed mutation iterated repeatedly union clusters thus obtained set generators algebra consequence method construction often end many generators require indeed finitely generated algebras two important examples coordinate rings matrices grassmannians cluster algebra structures infinitely many generators hand relations generators always restricted relatively simple form exchange matrix dictates exactly mutation carried new cluster variable obtained given cluster via single mutation direction always satisfies exchange relation form monomials variables common divisors definition two cluster variables two clusters obtained via sequence mutations thus cluster variable may expressed rational function variables given cluster one fundamental behaviours cluster algebra exhibits laurent phenomenon theorem says cluster variable fact laurent polynomial variables given cluster property remarkable mutating cluster obtain new cluster variable requires dividing laurent polynomial general many terms numerator fact cancellation must always occur therefore highly surprising another property arises form exchange relations cluster variable always represented rational function variables given cluster reflects connection cluster algebras theory total positivity cluster algebras either finite type infinite type corresponding whether set cluster variables finite infinite cluster algebra finite type iterative process mutation exhausts possible clusters finite number steps infinite type otherwise cluster algebras finite type fully classified classification incarnation classification simple lie algebras theorem theorem let cluster algebra following equivalent finite type every seed entries matrix bij satisfy bji iii mutation equivalent matrix whose cartan counterpart cartan matrix finite type hand cluster algebras infinite type make vast majority much less understood one reasons lack good tools control infiniteness present thesis concerned studying one first tools namely gradings cluster algebras introduction contains brief historical overview use cluster algebra gradings literature first notion grading cluster algebras introduced fomin zelevinsky proposition rank cluster algebra number cluster variables cluster notion graded quantum seed defined berenstein zelevinsky definition gives rise module grading algebra grading another notion grading due gekhtman section via language toric actions definition given gekhtman equivalent one eventually settle appropriate geometric perspective setting arises definition grading form interested introduced grabowski launois authors use prove quantum grassmannians admit quantum cluster algebra structure definition generalised also studies fully classifies gradings finite type cluster algebras without coefficients classification types due work appears chapter thesis general attempting endow algebra grading structure assign degrees generators check relations homogeneous cluster algebra gradings essentially however instead assigning degrees generators initially assign degrees generators initial cluster according certain grading condition degrees cluster variables determined recursively initial cluster turns exchange relations mean cluster variables homogeneous prior thesis almost nothing known properties gradings infinite type cluster algebras almost question one could ask open goal thesis study several classes examples graded cluster algebras including matrix grassmannian cases mentioned discover structure gradings particular interested describing cluster variables terms degrees precisely given cluster algebra aim answer following questions grading balanced bijection variables degree degree ask applicable sometimes end exist cluster variables infinitely many different degrees finitely many degrees occur variables distributed among finitely many variables occurring degree infinitely many variables occurring degree mixture degrees associated infinitely many variables others finitely many questions posed loosely increasing order difficulty though real determinant difficulty question rank cluster algebra asking rank two cluster algebras nontrivial gradings increasing rank one immediately find gradings complex structures rich variety behaviours cases able answer questions rank three cluster algebras generated matrices even setting though questions require significant effort answer one particular case remain open rank continues grow complexity cluster algebras graded structures increases extremely rapidly way difficult control higher rank cases therefore ambitions must much modest even examples look great deal additional structure exception occurs cluster algebras associated triangulated marked surfaces see background associated structure allows describe cluster variables geometrically proves powerful tool applied gradings results would otherwise extremely difficult obtain essentially read associated geometric combinatorial objects anticipated heavy use representation theoretic techniques would required order make progress questions turns surprising amount said using direct combinatorial methods though combinatorial facts use rely underlying representation theory thesis persisted elementary approach attempted see far pushed obtaining useful results additional tool use times calculation results strictly rely calculations contained chapter practice several results conjectures mainly chapter would possible obtain otherwise code developed carrying research along accompanying documentation available along electronic version thesis thesis overview thesis organised follows chapter summarise basic theory graded cluster algebras associated structures collect notation definitions need throughout particular say means two exchange matrices equivalent term essentially equivalent perspective mutation directions chapter study gradings associated finite type cluster algebras generated matrices whose cartan counterparts types using formula corollary arises known bijection theorem cluster variables cluster algebra finite type almost positive roots root system corresponding type classification results obtain complete classification finite type cluster algebras without coefficients presented chapter turn first nontrivial infinite type case gradings rank matrices goal classify much entire family start noting mutation essential equivalence every matrix form similarly note integer scaling one grading vector matrix entries grading vector contained matrix give conjectured classification grading behaviours table turn working table remainder chapter starting showing division finitely many infinitely conjectured next many degrees consider mixed type case prove grading behaves next section give algorithm similar one determines whether matrix gives minimal representative matrix mutation class prove aside one particular case matrices give rise gradings finitely many cluster variables degree achieved showing growth degrees mutation fast allow finitely many variables per degree penultimate section chapter considers matrices require work classify prove corresponding graded cluster algebras infinitely many variables occurring degree crucial result use corollary concerns connectedness subgraphs important object associated cluster algebra exchange graph another result helpful classification exchange graphs acyclic matrices whose structures identified theorem finally last section consider singular cyclic case matrices form fast growth behaviour matrices show infinitely many occurring degrees contain infinitely many variables conjecture fact degrees contain infinitely many variables briefly indicate proving difficult suggest possible approaches chapter collects results provide condition identifying graded cluster algebra infinitely many degrees general apply chapter involve inferring information degree quiver diagrammatic way representing matrix along compatible grading vector information subquivers first results kind line research remains largely undeveloped although generalising results likely approachable goal final section discuss detail introduce way sum degree quivers together give another degree quiver gives one potential way viewing gradings cluster algebras made smaller gradings define notion irreducible degree quiver give example restricted setting using sums degree quivers useful inferring results summands mentioned coordinate rings matrices matrix matrices mutation class correspond quivers cyclic consist oriented cycle general matrix mutationacyclic matrix mutation class corresponding acyclic quiver one without oriented cycles grassmannians along quantum counterparts important examples algebras turn cluster algebra structures chapter consider gradings cluster algebras specific grading chosen one defined first cluster algebras encounter frozen vertices coefficients vertices exchange quiver equivalently variables present every cluster never allowed mutate frozen vertices needed order allow quantum versions graded cluster algebras nontrivial gradings quantum cluster algebras without coefficients admit gradings also first setting encounter goals decide whether cluster algebras variables infinitely many different degrees whether includes every positive degree properties suggested properties considered graded algebras traditional sense show also case cluster algebra gradings using results chapter mostly reduces finding appropriate subquivers present degree quiver mutation class simple principle high rank means subquivers sparsely distributed amongst mutation class one situations benefited calculation known theorem matrices finite mutation class either adjacency matrices triangulations marked bordered surfaces one finite list eleven exceptional matrices chapter investigate said gradings cluster algebras generated exceptional matrices use calculation find size degree quiver class set occurring degrees corresponding several matrices one prove grading gives rise infinitely many variables degree would usually difficult cluster algebra high rank possible due show large amount symmetry quiver certain occurring degrees contain infinitely many cluster variables conjecture degrees one finite number variables considered matrices adjacency matrices triangulated surfaces turn attention chapter class matrices process cluster algebra arises bordered marked surface described authors define arc complex simplicial complex whose ground set set arcs marked points whose maximal simplices ideal triangulations collections arcs fully triangulate surface show arc complex surface isomorphic cluster complex associated cluster algebra dual graph arc complex isomorphic exchange graph particular means cluster variables bijection arcs seeds bijection ideal triangulations section see also section muller gives notion endpoint grading assigns degrees arcs based values marked points endpoints notion match definition grading make adjustment obtain one compatible gradings sense define space valuation values marked points according certain compatibility marked surface prove using combinatorics surface theory isomorphic space gradings associated cluster algebra allows translate grading questions questions degrees arcs noted lets capture great deal information grading structure much less effort would expected lastly indicate promising possibilities research involving theory chapter notation preliminaries chapter summarise basic definitions results notation need throughout fix throughout chapter definition cluster algebra definition let set cardinality let bijection call pair labelled set labelling call element labelled often refer labelled set may write labelled set form way think labelled set tuple needed usually convenient labelled sets working clusters order elements important seed mutation terms defined hereafter labelled set written without reference assumed written form written tuple understood shorthand notation let permutation let labelled set write mean labelled set cluster algebra several associated structures define giving definition cluster algebra definitions remainder section excluding associated mutation paths essential equivalence follow original source found seminal papers introducing cluster algebras definition coefficient group free multiplicative abelian group rank generators let denote field fractions group ring ambient field field rational functions indeterminates coefficients principal part definition given matrix obtained deleting rows skewn submatrix symmetrisable multiplied invertible diagonal matrix obtain also property matrix say notation retain notation bij rather using ebij refer also use commas indices situations entry aid readability definition cluster extended cluster seed notions cluster seed defined iteratively though require notion mutation briefly suspend definition allows neater presentation initial cluster labelled set cardinality transcendence base recall means algebraically independent set field extension algebraic elements algebraic given fixed initial cluster cluster corresponding cluster algebra either labelled set satisfies requirements initial cluster obtained mutation elements cluster called cluster variables elements initial cluster called initial cluster variables given cluster define extended cluster labelled set generators defined extend labelling setting may refer variables called coefficient variables described stable frozen variables whereas cluster variables called mutable extended cluster initial seed pair integer matrix given fixed initial seed seed corresponding cluster algebra either pair satisfies requirements initial seed matrix obtained seed mutation given seed called extended exchange matrix principal part called exchange matrix matrix initial seed called initial extended exchange matrix extended clusters cluster algebra share stable variables usually replace extended cluster cluster associated notation similarly context clear may simply refer extended cluster cluster keep mind needed carrying mutation define stable variables present seed let definition cluster mutation let labelling cluster mutation direction labelled set defined exchange relation empty product taken equal otherwise definition indices equation correspond stable variables corresponding parts monomials formed called coefficients precisely define call respectively positive negative coefficient associated exchange relation definition matrix mutation given define mutation direction integer matrix given matrix otherwise define otherwise fundamental theory cluster algebras skewsymmetrisable straightforward show also easy see mutation definition seed mutation given seed direction seed straightforward show often wish make several mutations row make following definitions definition mutation path list written right left length subpath mutation path form rooted subpath subpath form reverse path path empty path path consisting mutation directions remark notation mutation paths standard notation found literature uses need write mutation paths frequently quite long use notation reduce clutter improve readability notation concatenation two paths denoted use notation mean list repeated times use denote first terms infinitely repeated list example path equal matrix mutation notation let matrix obtained path cluster denote first mutate direction resulting applying matrix direction etc denote varbe newest cluster variable clear context obtained applying seed may omit similarly denote seed cluster obtained applying sdbe clbe respectively though initial cluster often omit notation alternatively may use definition two clusters matrices seeds mutation equivalent exists mutation path latter respective object obtained former mutation class cluster matrix seed set consisting clusters matrices seeds respectively mutation equivalent original object matrix mutation class infinite otherwise ready give definition cluster algebra definition cluster algebra let extended exchange matrix let tended cluster let unital subring contains coefficients every seed mutation equivalent cluster algebra generated cluster variables ones occur entry cluster seed mutation equivalent ring called ground ring number mutable variables rank cluster algebra said finite type set cluster variables finite infinite type otherwise cluster algebras geometric type take see proposition choice indeed contain required coefficients valid one remark alternative way defining cluster algebra generated set mutation path priori cluster variable rational function ambient field however turns cluster variables possess following remarkable property theorem theorem cluster variable laurent polynomial variables initial cluster coefficients known laurent phenomenon case cluster algebras geometric type even stronger result proposition proposition cluster algebra geometric type cluster variable laurent polynomial initial cluster whose coefficients polynomials stable variables remark let mutation path let clae var var think var rational function variables coefficients varae think tuple makes sense cluster algebra often contains multiple seeds whose clusters equal relabelling respective sets produce set exchange relations mutated possible directions seeds sense respective mutation permutation direction integer definition let essentially equivalent exists matrices say permutation varae varbe similarly say two seeds write essentially equivalent exists may write example let seed essentially equivalent permutation explicitly varb varb varb varb vara vara varb varb vara remark defined essential equivalence terms mutation directions literature matrices seeds known equivalent defined slightly different perspective saying holds differ sign simultaneous permutation rows saying columns precisely let permutation objects define whose principal part simultaneously function takes matrix permutes rows columns favour perspective since preserves intuition mutating one direction mutating another let check essential equivalence matrices transitive useful lets write explicit permutation involved matrices lemma let essentially equivalent essentially suppose permutations equivalent essentially equivalent proof since varbe varce varbe varce varbe varce varae varce essentially equivalent shows notation situation may write note remark equation equal required lemma essentially equivalent lemma suppose seeds let mutation path following essentially equivalent permutation essentially equiv seeds clae alent proof part easy see length one full result follows induction length one result immediate using part otherwise suppose length assume result true rooted subpath length path clae clae clae clae clbe clbe clbe clbe wish show clae clbe essentially equivalent true since established paths length one since assumption essentially equivalent since thus follows induction lemma immediately implies following formula useful relating mutation paths involving essentially equivalent seeds essentially equivalent matrices corollary suppose varae notice matrix trivially essentially equivalent essentially equivalent identity permutation thus reversing sign matrix nothing far variables concerned therefore may times liberal signs exchange matrices context essentally equivalent matrices initial proposition generate isomorphic algebras seeds proof write usual let cluster variable rational function corresponding varae path let let permutation path respect corollary rational function function corresponding path therefore corresponds cluster variable cluster variable obtained replacing every occurrence initial variable since cluster variables generate algebras follows isomorphic matrix mutation definition let directions called essentially equivalent respect mutation direction yields cluster variable mutation direction latter swapped example let directions essentially equivalent respect since mutation directions produces elements respectively lemma let permutation sends mutation direction essentially equivalent direction respect matrix var var proof immediate definition example example varb varb often want repeatedly apply mutation path leaves mutated degree seed essentially equivalent initial one trivially case need permute path along way ensure mutating new seed analogous way following gives notation notation let mutation path permutation define finally three important structures associated cluster algebra definition exchange tree cluster algebra tree vertex associated seed two vertices connected edge labeled usually denote vertex associated initial seed definition exchange graph exchange tree modulo essential equivalence seeds longer makes sense label edges mutation direction may instead choose label relevant exchange relation right hand side equation definition cluster complex denoted simplicial complex whose ground set set cluster variables whose maximal simplices clusters graded cluster algebras several notions gradings cluster algebras literature definition gradings follow definition let cluster algebra rank graded seed cluster integer triple matrix satisfying ith column degree matrix called grading degree homogeneous rational function variables given graded cluster possibly coefficients frozen variables defined obvious way condition ensures exchange relations therefore cluster variables homogeneous given cluster variable denote degree respect degg mutation graded seed direction defined way definition addition entry replaced degree new cluster variable alternatively may directly mutate replace variables equation degrees replace multiplication division taking powers variables addition subtraction multiplication respectively degrees following objects useful information need cluster variables pertains degrees degree seed pair graded seed occurring graded cluster degree cluster pair form tuple respectively defined similar way finally graded cluster algebra mean cluster algebra generated initial graded seed stipulate entries integers reason principle could elements arbitrary abelian group provided satisfies see discussion along define set lines may wish refer grading space matrices field satisfying nullspace think vector space often interested working grading matrices whose rows form basis grading space possibly subspace gradings space call rows grading vectors practice scale basis vectors consist integer entries always possible entries notation sometimes use notation mean matrix tuple whose entries columns matrix though clear context version referring given time precisely let columns write refer entries thought tuple containing degrees variables corresponding cluster fit better overloaded notation transposed matrix say tuple integer degrees extend notation graded setting natural way aside wish denote degree variable degree seed cluster add prefix deg respective notation may simply mean graded seed write deg mean deg var obtained applying mutation path graded seed proposition proposition let graded cluster algebra algebra every cluster variable homogeneous respect grading definition grading matrix called standard grading rows form basis grading space seed degree seed containing called standard graded seed degree seed mentioned often choose grading matrix rows form basis grading space since understanding behaviour grading essentially lets understand gradings given cluster algebra next two results show case lemma lemma mutation standard graded seed standard degree seed standard standard graded seed lemma lemma let exists integer matrix let grading cluster variable degh degg essentially equivadefinition two graded seeds essentially lent underlying seeds two degree seeds holds degrees rather equivalent variables course essential equivalence degree seeds much weaker essential equivalence graded seeds two graded seeds essentially equivalent may underlying degree seeds indeed often interested finding infinite sequences graded seeds whose underlying degree seeds essentially equivalent allows prove existence infinitely many different cluster variables particular degree set degrees finally note simple relationship mutation degree cluster negative degree seed mutation path lemma let deg clbe deg clbe proof immediate equation full result follows inductively quivers extended exchange matrix whose principal part quiver directed weighted graph loops represent working quiver often easier working corresponding matrix lets represent whole seed one diagram identifying cluster variables degrees vertices quiver background quivers see example unless otherwise stated assume quiver refer connected definition let matrix quiver associated vertices labelled arrow weight bij vertex vertex negative entry counts arrow opposite direction weight arrow counts arrow extended exchange matrix also assign quiver quiver vertices last correspond frozen variables arrows assigned way frozen vertices denoted drawing square around quiver acyclic contains oriented cycles rank quiver cyclic oriented cycle subquiver quiver whose vertices subset vertices edge weight direction define subquiver often called full subquiver literature definition quiver mutation mutation quiver direction quiver corresponding equivalently obtain follows reverse arrow incident vertex directed path form bkj add bik bkj weight arrow vertex may involve creating arrow bik reversing deleting present arrow general use quiver corresponding matrix interchangeably make couple specific definitions working corresponding objects definition graded quiver seed mean graded seed identify vertex quiver associated extended exchange matrix cluster variable position cluster represent diagram placing cluster variables corresponding vertices would wish emphasise vertex numbering place cluster variable inside parentheses corresponding vertex number subscript terms quivers equivalent condition matrix vertex balanced mean bik deg bki deg bki bik left sum runs weighted arrows vertex right sum arrows sometimes refer left right hand side weight denote degree quiver mean degree seed identify vertex associated quiver ith entry corresponding diagram ith vertex replaced though sometimes may omit subscript needed degree quiver positive degree corresponding vertex extend definition essentially equivalent objects obvious way two graded quiver seeds essentially equivalent underlying graded seeds two degree quivers essentially equivalent underlying degree seeds alternatively two quivers essentially equivalent relabelling vertices taking opposite quiver reversing arrows similarly graded quiver seeds degree quivers example let let graded seed strictly accordance notation degree seed tuple matrices corresponding matrix quiver seed corresponding seed degree vertex deg deg shows vertex balanced two degree seed essentially equivalent denominator vectors denominator vector cluster variable invariant make frequent use denominator vectors introduced fomin zelevinsky take definition definition let cluster cluster algebra theorem express given cluster variable uniquely xdnn polynomial coefficients divisible vector called denominator vector respect ith entry referred cluster respect writing denominator vector clear omit subscript write specified assume initial cluster generating cluster algebra working denominator vector may write tuple rather column vector example respect cluster variable denominator vector definition let cluster rank denote corresponding denominator cluster whose ith entry respect initial cluster refer ith entry also use clb denominator vector cluster variable cluster respectively obtained mutation path denominator quiver mean quiver corresponding given seed identify ith vertex corresponding diagram ith vertex replaced sometimes omitting subscript clear context definition define partial order set denominator vectors setting entry greater equal corresponding entry also say least one entry strictly greater corresponding entry sometimes interested finding mutation paths produce growing denominator vectors proving denominator vectors growing becomes cumbersome work entire cluster denominator vectors single one entry denominator vector simplifies calculations showing entry grows enough distinguish variables motivates following definition definition denominator slice vector whose ith entry entry denote using angle brackets given seed denominator quiver mean quiver corresponding identify ith vertex corresponding diagram ith vertex replaced sometimes omitting subscript example consider seed denominator cluster respect clx denominator quiver first denominator slice first denominator quiver next formula lets mutate denominator vectors directly lemma equation new cluster variable obtained mutation seed direction satisfies max bik otherwise max taken chapter gradings finite type cluster algebras chapter classify gradings cluster algebras type work completed prior publication results reported paper classification finite type cases admit nontrivial gradings namely also found review background definition section let bij matrix cartan counterpart aij defined setting aij otherwise theorem theorem matrix gives rise finite type cluster algebra mutation equivalent matrix whose cartan counterpart finite type irreducible principal minors strictly positive definition matrix bipartite bij bik lemma theorem let bipartite matrix finite type cluster algebra bijection almost positive roots positive roots along negation simple roots lie algebra corresponding cluster variables write linear combination simple roots cluster variable expressed terms initial cluster polynomial constant term bijection negative simple roots correspond initial cluster variables note types almost positive roots following allows calculate degree cluster variable terms corresponding root lemma corollary let graded cluster algebra finite type bipartite let almost positive root write linear combination simple roots deg turns gradings cluster algebras type indeed finite type cluster algebra must balanced mean bijection cluster variables degree degree explanation fact see corollary remark following verify types gradings type appropriate bipartite matrix type matrix given corresponding whether odd even respectively one checks even full rank grading odd rank grading space spanned one dimensional vector two cases grading vector assume final results obtain cases take following description root system found function selects entry diagonal matrix taken denote simple roots description positive roots one following forms find degrees remaining variables apply equation lists along negative simple roots tells degree cluster variable corresponding almost positive root counting number variables degree summarise results table degree type type type negative simples total total table thus expected grading balanced gradings type appropriate bipartite matrix type matrix given corresponding whether odd even respectively even full rank grading odd rank grading space spanned one dimensional vector two cases grading vector case distribution degrees obtain time description root system taken find positive roots one following forms applying equation counting number variables degree obtain following distribution variables degree also degree variables degree grading balanced expected chapter gradings rank case goal chapter classify graded cluster algebras generated matrices classification terms cardinality set occurring degrees cluster variables distributed respect degrees consider matrices since give rise trivial gradings reason also assume matrix consider corresponds connected quiver cluster algebras generated quivers connected direct sums matrices easily understood terms cluster algebras generated corresponding connected subquivers thus matrices corresponding connected quivers give rise nontrivial grading behaviours example consider matrices structure problem graded cluster algebra classified following criteria cardinality set degrees occur degrees cluster variables may finite infinite degrees occur may infinitely many cluster variables degree finitely many variables degree case finitely many degrees means cluster algebra finite type mixed case degrees finitely many variables others infinitely many first note purposes classifying graded cluster algebras immediate reduction number cases generating trices need considered proof proposition see essentially equivalent matrices generate isomorphic graded algebras simply relabel variables initial cluster need consider matrices essential equivalence also clear need consider matrices mutation equivalence see let suppose mutation path set cla since initial seed cluster algebra chosen arbitrarily mutate cluster along get back clearly isomorphic graded algebra given turns one class matrices needs studied note mean need look detail cluster algebras generated classes matrices process rather classified corresponding class matrices done definition let matrix column called entries sign otherwise column mixed sign called acyclic associated quiver acyclic also called cyclic associated quiver cyclic call every matrix mutation class cyclic matrix mutation class acyclic remark easy see acyclic essential equivalence form hand column must also one two forms essential equivalence proposition matrix either essentially alent matrix mutation equivalent matrix essentially equivalent proof let given matrix write easy check directly one following essential equivalence classes note classes indexed entries appear columns mixed sign also strictly sets full equivalence classes sign however noted previously sign exchange matrix consider enough representatives say mutating direction obtain matrices clearly essentially equivalent matrix form definition see proof proposition every matrix essentially equivalent least one define standard form denoted first element list essentially equivalent say standard form remark let sign integer scaling degree vector uniquely determined graded seed namely thus degree vector uniquely determined sign scaling though scale must match initial degree vector every graded seed cluster algebra corresponding matrix consequence following lemma lemma let mutation path without repetition let vara cluster variable deg entry deg proof equation easily checked directly suppose true path length let path assume even proof similar odd write integers corresponding degree vector mutating matrix degree vector directions deg cla deg cla deg cla thus true paths length particular lemma implies following corollary graded cluster algebra generated matrix infinitely many degrees finally note condition lets infer grading balanced theorem corollary let quiver cluster algebra generated admits cluster categorification every grading balanced cover background cluster categorification see note following relevant situation lemma finite connected acyclic quiver admits cluster categorification fact follows work palu quivers deal finite connected whenever consider grading arising acyclic quiver quiver mutation equivalent one automatically know balanced main result chapter following theorem partial classification graded cluster algebras form given table noted earlier restrictions exclude graded cluster algebras generated matrices far questions classification concerned proposition matrices give rise finitely many degrees finite type mixed infinitely many variables per degree matrices give rise infinitely many degrees finitely many variables per degree mixed infinitely many variables per degree table entries table cover possible cases except one call singular cyclic case cases covered follows see section lemma shows results either mutation acyclic singular cyclic matrix lemma shows also need corollary says precisely three matrices top row matrices conjecture singular cyclic case placed lower right cell table conjecture singular cyclic case graded cluster algebra infinitely many variables occurring degree remainder chapter prove table correct note covered chapter gives rise markov type cluster algebra infinitely many variables degree address said singular cyclic case section finitely many degrees mixed case let consider associated cluster algebra show graded ter algebra mixed type finitely many degrees specifically exactly one variable degree one degree infinitely many variables degree degrees occur proposition occur graded cluster algebra degrees proof one way find occurring degrees attempting compute exchange tree closing branch whenever obtain degree seed essentially equivalent sign degree cluster one already occurred case lemma subsequent degrees obtained particular branch already occurred sign process terminates recorded diagram black box indicates found degree seed essentially equivalent sign previously occurring one circle degree cluster indicates newly obtained degree mutation see degrees occur figure notice negating degree clusters degree seeds obtained mutation paths essentially equivalent one obtained mutating initial degree seed direction obtain either new degree seeds negative degree encountered far thus degree must occur diagram also shows matrices mutation class matrices whose corresponding quivers acyclic therefore lemma grading cluster algebra considering balanced infinitely many cluster variproposition ables degree degree proof show vara distinct denote ith entry cluster cla cluster variables rational functions evaluate claim strictly increasing simplify notation write mean rational functions easy show following exchange relations claim follow induction base case note cla cla induction hypothesis assume implies implies wish show first using using thus result since strictly increasing must distinct since deg vara infinitely many variables degree next note deg vara find infinitely many variables degree let var show strictly increasing since shown strictly increasing also case thus infinitely many variables degree proof shown cluster variables distinct considering numerators future usually instead use efficient method considering via denominator complete classification mains prove one variable degrees make use example authors consider cluster algebra generated matrix show corresponding exchange graph given figure graph initial cluster vertex cluster general vertex given permutation three variables three regions adjacent vertex proposition exactly one variable degree one variable degree proof since essential equivalence degree seed mutation equivalent inferred results concerning behaviour grading figure may alternatively obtained considering grading thus enough show latter graded cluster algebra one variable degrees referring example following exchange relations cluster variables equations may calculate recursively degrees cluster variables start var degree know obtained mutation initial cluster direction since corresponding cluster obtained replacing second element initial cluster similarly find variable degree need check variables degree degree also degree similarly find deg deg deg enough infer deg deg therefore cluster variable degree variable degree corollary mixed type curring degrees graded cluster algebra one variable degrees infinitely many variables degrees grading also balanced figure figure matrices according table need able determine whether given matrix section give simple algorithm allows also prove excluding singular cyclic case matrices give rise graded cluster algebras finitely many variables degree stage mention concerned parametrising testing matrices covers material results section derived particular algorithm similar spirit algorithm section detects whether input matrix however general treat material slightly different perspective algorithm motivated observation mutating matrix rearranging standard form one chance obtain matrix form smaller absolute values new values corresponding entries happen mutate direction never happen therefore must reason use slightly weaker condition rather used detect whether matrix mutationcyclic algorithm allows possibility detecting property step earlier means may perform additional mutation want obtain minimal matrix addition detecting mutationcyclicity algorithm gives output standard representative mutation class algorithm gives output mutationcyclic necessarily otherwise lets determine whether two matrices mutation equivalent tells table input matrix placed particular input matrix tells classification corresponding graded cluster algebra modulo conjecture singular cyclic cases matrix output also tells case next section mutation class falls idea heart algorithms simple input matrix fork defined next section fork repeatedly mutate point return get either minimal fork means matrix fork means finally mutation class need determined gives condition allows read whether entries theorem theorem let acb markov constant triple remark note following fact mutation degree seed form deg sda deg sda mutation directions yields degree seed essentially equivalent one form degree least large strictly larger however deg sda may equivalent degree seed form algorithm take input matrix assume form perform algorithm instead write set counter value let say passed algorithm step note means next columns say failed step exists matrix essentially equivalent form let matrix say failed step say passed step say passed failed step increment counter repeat step note reach step define output algorithm denote follows passed algorithm matrix computed recently passed otherwise continue computing higher values way find acyclic show later must eventually proofs lemma lemma let first check algorithm terminates finitely many steps noted algorithm reaches step conclude terminate need check passes fails finite number iterations suppose pass fail step fail step consider since satisfy must either mutation direction produced degree strictly smaller subdiagonal entries larger absolute value corresponding entry least one strictly smaller either pass algorithm must strictly smaller entry provided fails continuing way either must pass computing showing must eventually obtain matrix least one entry strictly smaller point immediately determine whether passes fails definition call output algorithm minimal mutation representative denoted remark algorithm allows determine two skewsymmetric matrices mutation class follows see unique essential equivalence passes algorithm fails algorithm may three minimal representatives equivalence however share set entries case easy work representatives equations next section algorithm tests whether show fails algorithm definition let matrix call also call matrix essentially essentially cyclicpreserving respectively note lemma passes algorithm may write either essentially either proof immediate considering possible forms required order passes algorithm lemma let let let uniquely determined chosen proof follows considering beginning subsection lemma singular cyclic case proof follows forms corollary suppose let mutation path without repetitions suppose either rooted proper subpath empty path suppose proof assume without loss lemma let permutation exists lemma serves base case let denote rooted subpath length assume induction exists matrix essentially equivalent write assume also mutating direction obtain using part lemma second equality since know otherwise must contain repetitions direction lemma essentially equivalent say shows lemma may write finally since know lemma chosen second part use lemma may replace inequalities involving strict inequalities may also since mutation property preserved corresponding entries mutated matrix standard form matrix one mutation away minimal matrix following sense proposition suppose let mutation path without repetitions suppose either rooted proper subpath empty path particular suppose also proof starts direction result holds corollary suppose preserving need show subpath obtained deleting first mutation rightmost entry indeed case corollary since first mutation either direction strict inequality suppose result holds corollary assume inequality corresponding holds thus obtain mutating matrix form along path start therefore corollary list results tell whether certain matrices lemma let proof since holds proposition lemma let proof lemma let acyclic since proof result holds proposition suppose case infer using theorem however give alternative proof shows directly mutate obtain acyclic wish able part algorithm three cases occur acyclic weakly column lemma strict inequality since matrix form next consider matrix either immediately seen acyclic way continuing process must eventually point may conclude proposition input matrix passes algorithm proof passes algorithm essentially mutation equivalent suppose either first case must lemma second must lemma suppose fails essentially mutation equivalent matrix either column form first case essentially mutation equivalent acyclic matrix second must lemma apply graded cluster algebras generated mutationcyclic matrices theorem let let suppose singular cyclic matrix graded cluster algebra arising finitely many variables occurring degree also grading balanced proof since essentially mutation equivalent enough consider since passes algorithm proposition lemma let mutation path length second part proposition implies must degm since degm bounded smallest entry thus finitely many possible mutation paths result given degree finitely many possible cluster variables degree grading clearly balanced since occurring degrees positive cluster algebras generated acyclic matrices gradings cluster algebras generated acyclic matrices property infinitely many variables occurring degree reason one may expect behaviour following cluster algebras associated acyclic matrices special mutation sequence gives rise infinitely many seeds equal essentially equivalent initial seed whose corresponding degree seeds given degree exists mutation path applied initial seed results cluster variable degree accounting essential equivalence mutation path applied different seeds gives infinitely many different ways obtaining variables degree purpose section prove variables obtain way fact distinct turns nontrivial rely property exchange graphs acyclic cluster algebras order prove result exchange graphs identified warkentin chapter following theorem crucial property need result appears conjecture proved setting acyclic matrices corollary theorem let cluster algebra arising matrix cluster variable seeds whose clusters contain form connected subgraph exchange graph show certain variables distinct showing define different subgraphs regions exchange graph definition let cluster variable cluster algebra call subgraph formed set seeds whose clusters contain region defined denoted theorem acyclic matrix connected cluster variable clear regions bijection cluster variables also acyclic cluster algebras rank region must either loop polygon infinite line vertices may see follows region loop let vertex let list vertices encountered far initially containing exactly two vertices connected say also since mutation exactly two three possible directions replace add list exactly two neighbours must also since loop new vertices must already precise true contains loops rather loop clear must loop contains one continuing way see infinite line let fix representative acyclic matrices may restrict attention assume matrix form may assume essential equivalence satisfy begin always permute entries get essentially equivalent matrix case note mutation equivalence shows negative mutate make positive least sign case negate whole matrix thus also assume choice representative exchange graphs consider made two main elements series infinite floor varying complexity attached another edge wish able formally distinguish two components graph following definition helps made chapter definition let quiver corresponding matrix corresponding matrix called fork acyclic vertex called point return qji qir qji qrj acylic resp full subquiver induced vertices incoming arrow resp outgoing arrow property forks relevant mutation corresponding matrix direction excluding point return gives rise another larger fork exchange graphs considering forks found base infinite see lemma make one component graph definition say vertex exchange graph canopy associated matrix fork otherwise say floor given vertex canopy unique path shortest length connecting vertex floor call infinite vertex branch associated seed fork point return know part exchange graph looks following figure dotted dashed lines represent infinite continuation figure depict seed corresponding fork along associated infinite rectangle following figure turns five cases acyclic matrices need consider finite cluster algebras matrices form cases correspond aother mixed case considered section mentioned acyclic matrices special mutation path produces equal degree seeds lemma let one matrices list deg deg deg deg proof noting sign scaling grading vector using lemma follows immediately direct computation case already done equations remark could alternatively used path lemma thing opposite direction fact starting initial seed mutating along traverse precisely vertices exchange graph comprise seeds floor containing acyclic matrices idea traversing exchange graph made precise discuss walks exchange graphs remark make use section difficult show vara vara using partial order definition words variables clusters obtained path lemma grow indefinitely distinct cases dealing infinite beginning fork done way proposition let matrix entries satisfying one let corresponding cluster algebra exchange graph let two cluster variables appear seeds vertices respectively suppose vertices belong different branches write vara vara minimal length mutation paths suppose forks words obtained mutation paths terminating canopy cluster variable proof assume without loss generality occurs branch associated initial cluster need show let mutation path minimal length vara let sda equivalence class representative seed associated seed sdb contain since component containing connected theorem must separate component containing excluding repeated mutation directions one path initial cluster therefore equal turning attention vertices floor graph make note lemma restatement theorem quivers crucial theorem let matrix let starting seed alternating mutations directions produces cycle corresponding exchange graph either aij cycle length first case length second next aim prove following proposition let matrix entries satisfying one let cluster variable vara mutation path suppose vertex corresponds seed sda floor vara cluster variable need prove due manner prove use theorem result higher values automatically implied also make use theorem tells required exchange graphs note since exchange graph repeats every six vertices along mutation path whose corresponding vertices make part floor graph fact actually repeats every three vertices think repetition six vertices need six mutations obtain equal degree seeds therefore may restrict attention one segment graph call segment containing initial seed initial segment label vertices vertex corresponding initial cluster argument proceed follows suppose vara taking fixed representative initial seed denote similarly seed representatives vertices mutation path identifies walk terminal vertex may identify walk since although exchange graph labelled mutation directions still able deduce mutation direction results seed class required vertex every step walk identify fixed seed representative particular vertex next identify corresponding vertex may simply read graph since corresponds mutation path graph repeats end point walk extended six vertices along initial cluster aim show vertex belong showing walk must pass vertex whose corresponding cluster longer contains since connected theorem accomplishes aim shows mentioned also implies holds larger values take larger value consider new corresponding variable vertex clearly still case walk must pass vertex whose cluster longer contains since fixed seed representative superimpose mutation paths onto way show must replaced along walk resulting seed equivalence class denote part walk directed series labelled arrows highlighted blue ending blue square vertex mean vertex longer belongs initial vertex need show three variables cluster representative must replaced point walk words three regions adjacent distinct three regions adjacent vertices need one entry corresponding cluster example cluster representative vertex obtained one mutation initial cluster two entries already dealt new entry needs considered let apply method cases case exhange graph partially drawn figure show different regions proof essentially easy see since representative seed class obtainable initial seed mutation path cla longer contains variables since walk passes done figure case part exchange graph case use figure give detailed explanation deduce mutation directions walks superimposed onto exchange graph show certain vertices separate regions approach also proceed subsequent cases figure corresponds variable var cluster class vertex two variables cluster var already dealt time consider var start case taking representative initial seed vertex need find vertex whose seed class contains sda first note following theorem clear mutating initial seed direction produce seed class either since exchange graph symmetric may simply assume direction corresponds rather however choice fixed need use consistently throughout diagrams case obtain seed class must mutate direction since theorem direction keeps pentagon comprising therefore desired vertex obtained var second entry cluster seed class mark mutation walk leading variable considering red diagram consider diagrams order already dealt two entries cluster wish prove vertex part region var finding variable breaks exchange graph show connected subgraph containing var breaks occur next time mutate direction along walk second entry cluster representative replaced need worry variable appearing different entry point would violate variables cluster form transcendence base clearly walk must pass show var get next mutate direction see direction takes rather direction recall exchange graph must form vertex end initial vertex case direction must indeed take alternatively may argue contradiction case walk corresponding would fact take seed fork matrix fork next get mutate direction finally must mutate direction arrive process replace second entry cluster representative var know contain var mark blue square done check variables diagrams regions corresponding initial variables indicated visual aid although bear mind regions abstract sets rather regions plane geometric sense figures simultaneously superimpose two mutation paths onto exchange graph note representative given var var three variables accounted figures suggested diagram see regions adjacent var alternatively could taken another representative var previous choice makes obvious var var accounted given already considered variables along path similar way variables already accounted elsewhere therefore concern variables figure case two segments exchange graph matrix figure case variable initial cluster figure case variable initial cluster figure case variable initial cluster figure case variable var figure case variable var figure case variable var figure case variable var figure case variable var figure case variable var figure case variable var case see variable question vertex separate region variables case deal variables initial cluster time figure consider var var var var cases similar shown figure case two segments exchange graph figure case variables initial cluster figure case variable var figure case variable var figure case variable var figure case variable var case case show relevant diagrams initial variables well var var var var cases similar figure case two segments exchange graph figure case variables initial cluster figure case variable var figure case variable var figure case variable var figure case variable var case case slightly different property others vertices share common region namely var similarly means prove cluster variables distinct cause problems though case variable replaced move one segment right occur move two segments right showing difficult see figure enough prove infinitely many variables degrees indeed given variable canopy shows distinct variables corresponding cluster two segments right usual degree cluster behaviour propagates inductively corresponding cluster every second segment must contain variable unique one pair adjacent segments degree others show diagrams initial variables well var var var var figure case two segments exchange graph figure case variables initial cluster figure case variable var figure case var figure case variable var figure case var shown want cases establishes proposition putting together proposition proposition following theorem let matrix graded cluster algebra infinitely many variables occurring degree grading also balanced proof given work need justify grading balanced course follows lemma since acyclic able see directly let cluster variable vara minimal length mutation path strictly may need replace path vara satisfies degg degg gives bijection variables degree degree occurring degree singular cyclic case case special mutation direction gives mutation direction gives fork thus unlike cases mutate along path without obtaining strictly increasing sequence degrees thus arising graded cluster algebra forced finitely many variables degree though like cases degrees positive indeed acyclic case obtain floor equal degree seeds different underlying clusters might expect graded cluster algebra behave one generated acyclic matrix however theorem conjectured matrices proved way let consider said case following lemma recall relevant notation regarding denominator vectors definition lemma vara vara proof show even odd base case note assume result true assume without loss generality even current degree cluster current matrix max new denominator cluster required thus since path special mutation path analogous one lemma acyclic matrix corollary graded cluster algebra infinitely many degrees contain infinitely many variables proof let mutation path whose corresponding terminal vertex least three mutations floor exchange graph three mutations vertex equivalence class must cla cla distinct cla cla applying reverse path clusters cla cla implies cla contradiction lemma prepending path infinitely many times obtain infinitely many distinct clusters whose corresponding degree clusters equal pigeonhole principle produces infinitely many variables least one degrees deg say length let mutation directions applying mutation path current cluster replace cluster variable repeating process path yields infinitely many variables different degree previous one since degrees strictly increase increase mutation distance floor exchange graph may continue extending mutation path similar way indefinite number times repeat prepending process time produces infinitely many degrees containing infinitely many variables remark given may longer use theorem obvious approach attempting prove conjecture might inductive argument shown lemma denominator clusters along floor exchange graph grow move along special mutation path would appear suitable base case following argument suppose mutation path resulting degree starting initial cluster denominator vector obtained smaller obtained starting larger denominator cluster may expect able prove path obtained extending one mutation also gives larger denominator vector practice showing true seems difficult might initially hoped difficulty mutate larger denominator cluster must also subtract larger denominator vector computing new variable alternative approach may involve trying find invariant cluster variable appropriate task invariant would need capture notion size cluster variable mutation formula circumvents problem yet another approach may involve considering rational function denominator cluster possibly object capturing relevant information cluster attempting show given two objects inputs larger two must give larger output chapter degree subquivers growth useful make note certain small quivers degree quivers embedded subquiver larger quiver provide way find increasing arrows degrees reduce problem finding infinitely many degrees one finding subquiver mutation class initial exchange quiver reduction particular useful chapter note results concerning quivers automatically also true respective opposite quivers notation given quiver vertices use mean entry corresponding matrix number weight arrows vertex vertex arrow opposite direction counting negative follows refer degree subquivers means subquiver sense definition degree quiver require degrees vertices subquiver must match corresponding vertices quiver important note degree subquiver need degree quiver subquivers growing arrows proposition consider quiver recall label arrow means weight let repetitionfree mutation path let inequality strict least one choice moreover proper rooted subpath pair proof notational simplicity use interchangeably working mutation directions mutation vertex turns quiver mutation turns corresponds matrix note matrix since starts mutation vertex may apply corollary path gives desired result essentially equivalent quiver hence original quiver since applied corollary proof proposition immediately obtain corollary corollary let proposition cyclic growing degrees next give conditions quiver allows obtain increasing sequence degrees remark let positive degree quiver degree subquiver notice carrying degree seed mutation particular vertex subtract degree add multiples degrees adjacent vertices since assumed positive degrees adjacent vertices positive arrows external therefore add degrees would obtained mutation entire quiver mutate vertices inside vertices whose degrees may change thus interested showing mutation paths give increasing degrees may safely ignore arrows adjacent vertex outside proposition let positive degree quiver contains degree subquiver let repetition free mutation path suppose deg max proper rooted subpath deg deg proof prove result induction first mutating vertex obtain degree subquiver mutating next obtain induction step let proper rooted subpath suppose result true isomorphism degree subquiver depending whether last mutation either respectively proposition let rooted subpath length wish show mutation direction results degree larger another degree quiver form satisfying one first write obtained upon mutation direction direction direction direction max max max degree subquivers edge weights least two proposition direction previous mutation direction induction hypothesis either strictly greater two degrees max max since max similarly direction find max max direction max max relax assumptions still obtain infinitely many degrees though possibly without strictly increasing sequence requiring specific mutation path proposition let positive degree quiver contains degree subquiver deg deg proof split proof three cases iii mutate direction obtain next mutate direction obtain result easy see induction notice case need assumptions edge weights directions always mutation path however already know quiver remain cyclic corollary always exactly one incoming one outgoing weighted arrow adjacent vertex mutating means even need assume always use weight arrow get lower bound degree next obtain mutate direction obtain next mutate direction obtain subquiver rop satisfying case done iii alternate mutations vertices weight edge increases one time suppose mutations obtained new degree obtained degree mutation next time mutate vertex obtain lower bound new degree edge weight current degree subquiver continue mutate must either eventually obtain new degree strictly greater always obtain one least great otherwise value eventually grow large enough since case occurs may proceed similar way previous cases corollary suppose graded cluster algebra generated initial quiver contains subquiver satisfying conditions proposition proposition contains cluster variables infinitely many different degrees remark course mutate degree subquiver satisfying hypothesis proposition proposition obtain another degree subquiver whose presence equivalent condition showing cluster algebra variables infinitely many degrees corollary still holds replace quiver mutation equivalent remark may well possible relax assumptions proposition still retain infinitely many degrees property situations may even need quiver positive outside subquiver weaker assumptions would make likely could read given degree quiver associated graded cluster algebra infinitely many degrees line enquiry may good candidate research finally give sufficient condition graded cluster algebra positive degrees based presence degree subquiver proposition applied chapter notation consider three cyclic subquivers given quiver use notation mean vertex shares arrow either triangular subquiver formed three vertices form allowed different triangular subquiver either similarly use mean vertex shares arrow triangular subquiver formed three vertices form refer subquiver mean subquiver containing vertices along vertices share arrow referring subquiver mean corresponding property specified note mutating yields subquiver form increased mutating yields subquiver form increased mutating either change form aside reversing arrows hence mutating subquiver gives subquiver form mutating subquiver gives subquiver form applied subquiver form still form similarly applied quiver form lemma suppose degree quiver subquiver mutation along path yields degree similarly mutation along yields degree proof clear result true suppose result true assume without loss even may comments paragraph notation degree quiver form next mutation carrying mutation obtain degree result follows induction result immediate direction weight arrow matter new degree quivers old sections saw presence degree subquiver gave information cluster algebras arising larger quivers containing subquiver natural question arises whether general understanding properties rank graded cluster algebras reduced understanding properties lower rank cluster algebras precisely might ask following question let degree quiver consider graded cluster algebra gives rise suppose degree quiver contains subquiver properties graded cluster algebra inherited instance infinitely many variables occurring degree must corresponding set degrees associated infinitely many variables rather containing subquiver constructed combination smaller degree quivers way properties smaller quivers inherited progress direction would greatly increase understanding gradings general though establishing answer question appear easy attempt tackle note interesting property degree quivers often broken sums smaller degree quivers definition let degree quivers let sets vertices respectively let subset whose elements called matches match deg deg also match involving either call vertices appear match unmatched denote unmatched vertices respectively define quiver obtained follows match create new vertex words identify vertex vertex define set vertices pair vertices create arrow weight negatively weighted arrow one direction meaning positively weighted arrow opposite direction usual pair unmatched vertices come original quiver set weight arrow original value come different quivers create arrow proposition let degree quivers also degree quiver proof let vertex need check balanced satisfies equation unmatched vertex clearly balanced since original quiver suppose match weight weight since balanced example let degree quiver vertex balanced since similarly definition call degree quiver irreducible degree quivers denote sets vertices respectively example degree quiver example irreducible degree quiver appropriate potential summand must two vertices degree quiver two vertices must assign vertices degree order balanced hand degree quiver example irreducible since thus qop degree quivers irreducible might hoped turns case however see examples restricted settings information obviously obtained information possible progress towards question could involve attempting expand restricted settings attempting find correct relationship similar equation degree quivers summands example let degree quiver vertex deg let quiver markov chapter associated graded cluster algebra infinitely many variables degree easy show mutation path gives infinitely many variables degree let graded cluster algebra must also infinitely many variables degree since mutating along subquiver corresponding change way would need mutate vertex order affect variables obtain essentially obtained mutation degree example another example let gives rise cluster algebra infinitely many degrees corollary let show infinitely many degrees produced alternately mutating two particular directions denote denote entry aij show sequence strictly increasing excluding first term mutating along cthe path gives since negative since negative mutating along obtain satisfy respective relations holds induction thus lemma obtain infinite increasing sequence degrees alternately mutating directions suppose quiver associated label correspond one alternating directions let degree quiver vertex satisfying deg deg let must also infinitely many degrees words attaching another quiver vertex endows arising cluster algebra infinitely many degrees example take two alternating vertices mutate chapter gradings cluster algebra structure coordinate algebras matrices grassmannians chapter consider gradings cluster algebra structures coordinate algebras matrices grassmannians prove infinite type cases contain variables infinitely many different degrees also prove positive degrees occur cluster algebras deal classical cluster algebra structures results gradings quantum analogues also determined classical case follows theorem remark various points use fairly long mutation paths prove certain results given paths difficult prove results hand reader may wonder paths found worth mentioning specific code written help discover paths patterns led noted previously code available electronic version thesis graded cluster algebra structure matrices definition let set matrices coordinate algebra matrices xij coordinate function given aij structure cluster algebra follows row set column set let minor matrix denote corresponding xkl define sets cluster algebra cardinality efficient define initial seed using quiver show initial quiver denote figure generalises replacing grid grid form bottom rightmost vertices frozen see also initial cluster given defining entry cluster corresponding vertex position equal initial grading vector given assigning vertex position degree min easy check definition gives valid degree quiver moreover since algebra corresponding cluster algebra also similarly cluster algebra corresponding introduce section justification initial structure gives rise cluster algebra agrees coordinate ring quantum matrices see corollary infinitely many degrees first consider case without loss generality may consider smallest case prove result considering certain subquiver larger cases always contain subquiver thus possible show also contain infinitely many degrees exactly way initial exchange quiver given recall denote frozen vertices placing square around figure initial quiver lemma graded cluster algebra structure variables infinitely many degrees proof show mutation sequence transforms quiver apply proposition note embedding sequence larger initial quivers embed subquiver figure bottom left larger quiver allows immediately generalise results course using exact mutation sequence would work general since vertices corresponding subquiver relabelled embedded larger initial quiver clear adjust mutation path compensate consider following subquiver figure subquiver perform sequence mutations subquiver shown following diagram mutation often longer interested mutated vertex remove mutated subquiver direct computation shows last subquiver corresponds degree subquiver since apply proposition noting frozen vertex part mutation path stipulated proposition corollary graded cluster algebra structure variables infinitely many degrees infinite type cases covered note need consider start subquiver perform mutation path obtain quiver taining subquiver time write termediate computations corresponding degree quiver satisfies proposition note vertex part stipulated mutation path established following proposition graded cluster algebra associated infinite type cluster variables infinitely many different degrees occurring degrees natural question ask whether variables positive degrees exist graded cluster algebra associated prove indeed case also proves result larger cases therefore infinite type cases lemma cluster variables degree proof show writing certain mutation sequences result degree subquivers form turns case yield sequences degrees increasing mutation initial quiver mutated path subquiver degree subquiver variables degrees form lemma path gives subquiver degree subquiver variables degrees form next gives subquiver degree subquiver leads variables degrees form finally gives leads variables degrees form initial cluster contains variables degrees easy find mutation path yields variable degree example deg covers positive degrees remark lifting result lemma larger cases check subquiver corresponding deal interact rest quiver embedded fact easy see mutation restricted vertices within rectangle arrow created one vertices vertex position since vertices mutate satisfy since time rather embedding pattern bottom left keep vertices corresponding mutation paths larger initial quivers need worry vertices outside leftmost square larger cases lemma cluster variables degree proof case find sequences degrees increasing mutation might expect sequences case increase larger amount vertices mutate means achieving double arrow possible mutations previously turn means degrees longer grow along way proof essentially summarise table leaves degrees consider degrees contained example path quiver degree quiver degrees table mutation paths leading degrees corollary graded cluster algebra associated infinite type cluster variables degree corollaries grassmannians definition grasmmannian set subspaces vector space fixing basis subspace described matrix whose rows linearly independent vectors forming basis subspace let subset coordinate function maps matrix minor indexed coordinate ring isomorphic subalgebra generated coordinates grasmannian coordinate ring structure graded cluster algebra follows initial exchange quiver take quiver add frozen vertex single arrow vertex position assign vertices degree results quiver gives rise cluster algebra explicit expression initial cluster variables given one obtained tracing construction therein references give explicit initial clusters different quivers since need know cluster variables explicitly concern unduly easy check degree quiver valid background associated cluster algebra desired structure see classical case quantum case terms proving existence infinitely many degrees corresponding results immediately follow matrix algebra case since grading since initial exchange quiver added frozen vertex mutation sequences matrix case yield quivers subquivers apply proposition thus proposition graded cluster algebra associated infinite type cluster variables infinitely many different degrees also variables degree prove similar way matrix case lemma cluster variables degree proof case find paths give degree sequences increasing mutation applying gives subquiver degree subquiver lemma variables degrees form path gives subquiver degree subquiver variables degrees form leaves degree easy find example var lemma cluster variables degree proof previous section degree sequences case increase larger amount matrix case summarise three paths work table path quiver degree quiver degrees table mutation paths leading degrees leaves degrees found example corollary graded cluster algebra associated infinite type cluster variables degree chapter gradings finite mutation type quivers arising surface triangulations quivers finite mutation type ones whose associated matrices mutationfinite known fall one two cases adjacency matrices triangulations marked surfaces finite collection quivers correspond triangulations surfaces precisely following theorem theorem rank quiver finite mutation type either corresponds adjacency matrix triangulation bordered surface mutation equivalent one following quivers chapter investigate latter class initial information finite list quivers try determine said gradings cluster algebras associated quivers finite list ultimately detail though give additional information consider table note quivers give rise finite type cluster algebras considered approach use calculation find degree quiver class case yield exhaustive list occurring degrees prove degrees infinitely many variables find mutation paths give rise repeating degree quivers sign produce growing denominator vectors start writing initial grading bases quivers list considered previous chapter lemma initial grading bases given table quiver grading table proof reduces finding bases kernels matrices corresponding quivers see give rise zero grading next wish determine degrees occur corresponding cluster algebras magma algorithm available accompanying file computes mutation class degree quivers well information summarise proposition note mutation degree quiver class sizes computed essential equivalence particular number smaller classes computed quiver isomorphism done java app since quivers isomorphic opposites whereas quiver always essentially equivalent opposite proposition following quivers quiver mutation class degree quivers class occurring degrees table ready investigate question variables per degree quivers although able answer completely every case may still determine part answer grading case whose grading behaviour qualitatively different able determine information associated graded cluster algebra start note lemma proof easy show odd even claim true direct computation assume true suppose even since max recall max taken componentwise equation suppose odd using quiver wrote odd max proves result corollary infinitely many variables degree proof follows since easy show corollary proof assume even proof odd similar lemma denominator cluster mutating direction get max express matrix quiver planar max next max max required omitted second max zero negative entries corollary infinitely many variables degree proof follows since proposition infinitely many variables occurring degree proof immediate previous two corollaries since variables degrees noted table gradings cases certain degrees expect let consider correspond one variable degrees infinitely many variables able show latter true finding mutation path gives infinitely many examples path comparable special path lemma however one minimal path kind cases chapter find one examples always case minimal path always length equal rank paths also usually much difficult find cases facts due higher rank rank acyclic quivers floor exchange graph consists repeating segments comprised twodimensional polytopes along special path traverses higher ranks expect floor consist polytopes whose dimension close rank thus may many routes one segment floor another due size complexity segments finding paths move initial vertex segment difficult general little known structure exchange graph cluster algebras rank greater paths provide show exchange graph example consist repeating segments conjecture similar rank mixed case contrast rank acyclic cases segments bounded hyperplanes rather enclosed canopy infinitely many trees whose vertices forks remark graded cluster algebras arising balanced lemma since corresponding quivers acyclic grading show degrees infinitely many different variables conjecture degrees correspond one variable lemma let deg along results negative mutating initial degree seed initial degree seed proof although straightforward write computation since need refer intermediate quivers involved later result immediate deg deg deg deg deg deg deg required conjecture minimal note equal rank length path desired property soon require use floor ceiling functions note following identities involving functions make frequent use chapter lnm equivalently lnm equation given lemma let proof refer matrices lemma throughout base case assume claim true max max max max max max finally max used identity thus result also true therefore claim proved infinitely many variables deproposition gree proof proof lemma see repeating path gives infinitely many variables degree lemma infinitely many paths must result different variables denominator vectors different one variable degree conjecture one variable degree statement may compared analogous result proposition difficult establish structure exchange graph form simple enough could deduce result using relevant set recurrence relations case higher rank makes much difficult attempt method still expect result hold computer aided calculation gives confidence conjecture computing thousands different mutation paths result degree one variable found degree similar considerations also apply conjecture conjecture make later chapter grading grading let consider show degrees contain infinitely many variables conjecture degrees correspond one variable lemma let deg along results negation mutating initial degree seed initial degree seed proof see proof appendix proof including list matrices obtained use lemma let proof result clearly true provides base case assume true refer matrices proof throughout time write new denominator slice mutation following max max max max max lnm max lnm lnm max lnm lnm max new denominator slice induction step need check iii straightforward write split cases based mod show calculation assert cases easily shown similar way thus obtain induction step gives result proposition finitely many variables degree proof follows combining lemma lemma conjecture one variable degree one variable degree grading expect degrees correspond one variable degrees infinitely many different variables case difficult write simple formula entries third denominator slice entries depend mod lemma let deg proof see proof appendix include list matrices obtained lemma let entries follows lnm proof result clearly true provides base case assume true cases proved inma similar way mrefer matrices proof note since compute denominator entries along mutation path lnm max lnm lnm max lnm lnm max lnm lnm max max lnm max lnm max lnm max lnm lnm lnm max lnm max lnm finally lnm lnm lnm lnm max lnm new denominator slice lnm considering first entry true form similarly result true infinitely many variproposition ables degree proof proof lemma see repeating path gives infinitely many variables degree lemma infinitely many paths must result different variables since denominator vectors obtained grow indefinitely one variable conjecture degree one variable degree chapter gradings quivers arising surface triangulations graded cluster algebras associated surfaces exists class cluster algebras associated oriented bordered surfaces marked points authors describe process cluster algebra arises surface explained cluster algebras may given grading assigning cluster variable degree sum valued marked points though grading agree precisely definition grading assumed thesis chapter review theory cluster algebras arising surfaces following adapt definition setting order define grading sense cluster algebra arising properties associated surface apply theory study graded cluster algebra structure classes examples basic idea associating surface cluster algebra surface triangulated arcs represents cluster variable configuration triangulation also encodes exchange quiver triangulated surface considered seed made precise following definitions definition marked surfaces bordered surface marked points pair connected oriented riemann surface boundary finite set marked points closure connected component boundary least one marked point however technical reasons following particular cases excluded definition sphere one two punctures unpunctured oncepunctured monogon unpunctured digon unpunctured triangle sphere three punctures excludes cases able triangulated define shortly marked point boundary component called puncture however consider gradings arising surfaces later allow surfaces punctures definition arcs ideal triangulations arc curve isotopy intersect except possibly endpoints whose endpoints endpoints marked points arc allowed cut unpunctured monogon unpunctured digon arc contractible marked point boundary use denote set arcs typically proposition two arcs called compatible representatives respective isotopy classes intersect interior ideal triangulation maximal collection distinct pairwise compatible arcs arcs cut ideal triangles thus side ideal triangle arc segment boundary component two marked points deviating slightly use term boundary arc refer segment even though genuine arc three sides ideal triangle required distinct triangle shown figure called although allowed consider gradings furthermore two triangles share one side figure triangle following defines analogue seed mutation triangulated surfaces definition arc complex flips arc complex simplicial complex ground set whose simplices collections distinct mutually compatible arcs whose maximal simplices ideal triangulations denote arc complex dual graph vertices bijection ideal triangulations flip transformation ideal triangulation removes particular arc replaces unique new arc forms new ideal triangulation together remaining arcs thus edges correspond flips clear flip triangulated surface involutive though every arc flipped suitable replacement arc edge interior loop triangle mentioned contrast seed mutation every entry seed may mutated section authors resolve issue introducing tagged arcs flips however class examples gradings arise every arc flippable indeed allow punctures triangle arise examples proposition two ideal triangulations related sequence flips connected proof follows results compare following theorem theorem theorem fundamental group generated cycles length pinned base point still following explain associate exchange matrix triangulated surface definition definition let ideal triangulation define signed adjacency matrix follows label rows columns arcs strictly labelling definition arc labelled map defined follows ideal triangle edge arc remaining edge triangle otherwise ideal triangle define matrix sides labeled followed clockwise order true order otherwise set ranges ideal triangles entries remark alternative using matrix may draw corresponding quiver triangulated surface picking point arc adding arrow arcs time appears clockwise direction triangle may happen given pair arcs leading double arrow cancellation arrows remark suppose two triangulations marked surface true indeed unless gives rise finite type cluster algebra infinitely many different triangulations note also two different marked surfaces may signed adjacency matrix example see example example consider triangulated marked surface ideal triangulation three flipping arc arcs obtain new ideal triangulation corresponding quiver orientation dynkin diagram type indeed cluster algebra associated marked surface type general cluster algebra associated punctures type table notice example matrix corresponding flipping arc one obtained mutating direction happens general see next result relates flips arcs matrix mutation proposition proposition let ideal triangulation suppose ideal triangulation obtained flipping arc labelled following corollary proposition proposition proposition proposition let triangulation mutation equivalence class independent depends definition write unambiguously refer cluster algebra arising particular triangulated surface yet precisely established relationship arcs class ideal triangulations given surface cluster variables corresponding cluster algebra relationship arc complex triangulated surface cluster complex cluster algebra arising surface next theorem main result addresses version presented fact special case theorem allows possibility triangles using tagged triangulations need ultimately excluding surfaces punctures theorem theorem let bordered surface marked points without punctures let cluster algebra generated triangulation arc complex isomorphic cluster complex isomorphic exchange graph apart one two particular cases result extended allow surfaces punctures exclude surfaces study work going introduce allows grading arise triangulation ready introduce mechanism grading arises idea assign values marked points surface set degree arc thus corresponding cluster variable sum endpoints first note following lets read dimension grading theorem theorem let ideal triangulation corank number punctures plus number boundary components even number marked points refer boundary components even number marked points even boundary components ones odd number points odd components thus class examples dimension grading given number even boundary components set definition make alteration definition section set essentially give grading space since changed definition need prove subsequent results altered setting combinatorial way without needing background theory definition let arc ideal triangulation consider marked points common marked point attached end call endpoint call functions endpoint maps given unless otherwise stated tacitly fix two endpoint maps refer definition let marked surface without punctures ideal triangulation boundary arcs call function valuation function along tuple valuation functions valued marked surface although allow take arbitrary rational values genuine vector space practice choose values degree arc define presently integer arc valued marked surface valuation function define function degf degf call degf degree turn proving valuation function gives rise grading cluster algebra associated marked surface seed graded exchange relations mutation direction need homogeneous show corresponding exchange relations homogeneous arc ideal triangulation valued marked surface first need introduce new combinatorial objects let translate notion balanced vertices quiver arcs triangulation definition let arc marked surface without punctures triangulation edge two ideal triangles define configuration subset consisting two triangles standard configuration one two triangles share one edge whose arcs endpoints make four distinct points words standard configuration square diagonal homotopy starred configuration one looks like standard configuration except certain marked points starred symbol identified others starred symbol allow two adjacent boundary arcs endpoints starred symbol valuation function say balanced degf degf runs arcs follow clockwise direction inside one triangles adding two summands occurs triangles similarly direction empty sums defined may extended obvious way tuple valuation functions equivalently quiver superimposed configuration balanced corresponding vertex quiver remark representing diagram use dotted lines mean edge boundary arc may arcs contained region falls inside outer boundary formed two triangles case may write indicate arbitrary set arcs present although part configuration make diagram clear avoid depicting something appears part ideal triangulation note outer boundary formed configuration arc may consist one four arcs example consider valued marked surface shaded area represents region excluded convention adopt throughout annulus class surfaces consider next section six points outer boundary one point inner boundary configuration balanced since degf degf definition setting definition suppose two triangles share exactly one edge standard configuration may use form new starred configuration follows loop encloses region interior one two triangles region marked cut loop open basepoint precisely loop form replaced arcs attached base point loop attached one starred points corresponding side arc originally important maintain distinction loops different base points starred configuration obtain star corresponding pairs points different symbols cutting loops based different points valid starred configuration way could obtain two adjacent boundary arcs endpoints starred symbol original configuration multiple boundary loops basepoint configuration call starred configuration obtained process opened configuration example configuration example standard configuration opened configuration balanced second example let marked surface balanced degf degf obtain opened configuration two steps first open loop whose endpoints upper marked point value loop bounding gives open remaining loop gives opened configuration balanced examples balanced configuration gave rise balanced open configuration coincidence see shortly remark possible obtain configuration starting starred configuration gluing along matching symbols create internal loops example latter opened configuration example reverse two steps carried obtain get back original configuration possible configurations obtained starring standard configuration possible ways including starring two points symbols gluing simple results allow restrict attention small number configurations systematic way lemma let arc ideal triangulation valued marked surface triangles share one edge opened configuration balanced balanced proof clear inspection lemma suppose given standard configuration balanced valuation functions starred configuration obtained replacing marked points starred points starring points proof replacing points standard configuration starred points simply restricts possible values points points starred symbol must share common value configuration already balanced values endowed valuation function particular restricted values lemma let arc ideal triangulation valued marked surface let unique new arc triangulation obtained flipping balanced balanced proof arcs follow clockwise direction exactly follow direction arcs follow direction follow clockwise thus exchanging simply effect swapping left hand side equation right hand side vice versa proposition let arc ideal triangulation valued marked surface balanced proof configuration two triangles share either one two edges thus two cases consider triangles share two edges two possibilities note double arrow interior arc would present superimposed quiver configurations balanced first degf degf second sums automatically zero marked points must zero value valuation function boundaries odd number points suppose two triangles share one edge lemma lemma may assume without loss generality form standard configuration therefore need list configurations check balanced arbitrary valuation function write check list table lhs degf configuration rhs degf lhs lhs rhs rhs lhs lhs rhs rhs lhs lhs rhs rhs table standard configurations proposition let valued marked surface valuation function initial ideal triangulation say arcs graded cluster algebra generated initial degree seed degf degf proof since arcs bijection cluster variables need justify seed degree seed proposition balanced exactly condition required equation corollary given tuple valuation functions proposition extends obvious way shown degf takes triangulation gives valid degree cluster valuation function show given degree cluster valuation function degf gives degree cluster suppose marked surface ideal triangulation arcs assume ker grading theorem know boundary components even number marked points also follows denote even boundary components component denote set marked points miri assume labelled mij adjacent working mod miri adjacent basis given defining marked points otherwise definition triangulation define ker degf degf proposition know image indeed ker easy show linear wish show isomorphism crucial step establishing enough show result single triangulation mutation class lemma mutation direction injective proof suppose injective thus degf fix say degf consider another arc degree clearly still degf assume arc degf consider configuration values corresponding marked points assume marked points distinct apart arcs may boundary arcs since arc degree must configuration must deg configuration means deg thus also injective immediately implies step wished establish corollary let fixed ideal triangulation injective ideal triangulation deal uniformly special class cases lemma suppose marked surface exists triangulation following true valuation function zero every boundary component exactly two marked points proof boundary component odd number marked points must always zero let triangulation satisfying following even boundary component two marked points arc marked points must exist least marked points justify compatible clear since arc endpoints confined unique boundary component also easy see indeed exist triangulation containing arcs ideal triangulation already definition another arc compatible let note since even boundary labelled marked points also degf suppose particular thus implies marked points must also satisfy thus even boundary component containing exactly two marked points given left address boundary components exactly two marked points surfaces theorem let marked surface isomorphism ideal triangulation proof show injective therefore bijective triangulation boundary components exactly two marked points corollary assume triangulation satisfying lemma implies done assume boundary components exactly two marked points suppose valuation function show exists triangulation containing arc goes marked points value implies points zero value triangulation extension indeed corollary may assume even boundary component arc marked points lemma zero even boundary components two points always automatically zero odd boundary components suppose boundary component exactly two marked points must fall one three cases boundary component digon contained inside region bounded another component iii encloses interior boundary component let fixed marked point case trivial nonboundary arcs arc complex empty case use corollary assume contains arc loop encircling triangulation exists since clearly compatible arcs previously assumed thus zero case iii topologically equivalent done shows assume must zero every boundary component thus injective hence bijective therefore isomorphism remark potentially worthwhile line future research investigating whether grading defined extended surfaces punctures possible approach following lemma surface triangulation without self folded triangles choose valuation function occur mutated triangulation produces triangles pass tagged triangulation triangles replaced notched arcs suitable way extend grading may set degree notched arc value point plain end minus value point notched end annulus marked points odd marked points outer boundary annulus component points inner boundary component grading corresponding cluster algebra two different scenarios depending whether one even assume least one even otherwise nontrivial gradings occur theorem former case assume latter even may following remark graded cluster algebra associated annulus see simply note initial triangulation triangulation open cylinder marked points top circle base circle turning cylinder upside gives initial triangulation clearly change corresponding initial quiver grading section consider case even show grading mixed type initial take valued triangulation corresponds initial graded quiver entries tuple degree cluster entries lemma infinitely many variables degrees proof see infinitely many variables degree say fix marked point value outer boundary point inner boundary value let arc starts winds times around inner boundary clockwise fashion let spiralling inwards ends example point top outer boundary point inner boundary arc drawn homotopic corresponding cluster variables distinct theorem degree set infinitely many variables degree degree may dealt similar way lemma finitely many variables degrees proof write exact values upper bounds number variables occur degrees easy see numbers must finite way obtain infinitely many arcs annulus marked points produce sequence arcs loop increasingly many times around inner boundary annulus arc must start one boundary end therefore must degree since points inner boundary value points outer boundary value therefore finitely many arcs degree corollary infinitely many variables degrees finitely many degrees degrees occur thus graded cluster algebra associated triangulation annulus marked points odd mixed type remark possible prove lemma using standard methods involving growing sequences denominator vectors though takes considerably effort needed lemma involves showing certain degrees finitely many variables would difficult prove without theory section often case trying establish finitely many variables given degree remark none relies odd could thus extend results case even provided extend grading appropriately although would alone enough show happens graded cluster algebra even assuming take standard grading recall definition however make use fact following section annulus marked points even let assume even two boundary components even number marked points get grading take initial valued triangulation corresponds initial graded quiver tuple degree cluster entries entries basis grading space given entries entries entries entries note grading vector one case odd first glance addition dimension grading may appear potential effect classification graded cluster algebra however difficult show case fact work done section already almost enough determine classification present case proposition assume even graded cluster algebra cardinality set cluster variables degree determined equal cardinality set cluster variables degree cardinality case odd specifically mixed type infinitely many variables degrees finitely many variables degrees occurring degrees proof write first note case odd mixed type infinitely many variables degrees finitely many degree noted remark may proved exactly way odd justify degrees listed proposition ones occur easy initial valued triangulation may read possible combinations obtained summing two valued marked points degree must correspond priori least one must associated infinitely many variables similarly since noted lemma infinitely many arcs degrees arcs give infinitely many variables degrees easy see fact obtain infinitely many arcs degree degree fixing appropriate points value outer boundary either appropriate inner boundary taking arcs wind around inner boundary component increasing number times similarly infinitely many variables degrees hand finitely many possible arcs result degrees therefore case also degree form since arcs ways obtain corresponding first entries shows happens consider standard grading nonstandard grading however possible get graded structure different given follows behaviour consider valuation corresponds initial graded quiver entries tuple degree cluster thus entries case following easy see proposition even infinitely many variables degrees occurring degrees proof possible degrees clear similar arguments used previous cases considered find infinitely many arcs degrees remark theory developed chapter likely able successfully attack many examples cluster algebras arising marked surfaces examples could include generalisation annulus inner boundaries rather one torus disc multiple discs removed appendix appendix proof lemma write computation result immediate first write matrices obtained along mutation path lemma claimed thus degree seeds deg deg deg deg deg deg deg deg thus deg negation initial cluster claimed proof lemma write matrices obtained along mutation path lemma claimed thus degree seeds deg deg deg deg deg deg deg deg deg deg deg thus deg negation initial cluster claimed references assem blais samson mutation classes comm algebra beineke hille quivers three vertices markov equation algebr represent theory appendix otto kerner berenstein fomin zelevinsky cluster algebras iii upper bounds double bruhat cells duke math berenstein zelevinsky quantum cluster algebras adv buan marsh reineke reiten todorov tilting theory cluster combinatorics adv caldero keller triangulated categories cluster algebras ann sci norm sup carter lie algebras finite affine type volume cambridge studies advanced mathematics cambridge university press cambridge chekhov penner introduction thurston quantum theory uspekhi mat nauk felikson shapiro tumarkin cluster algebras finite mutation type eur math soc jems fomin shapiro thurston cluster algebras triangulated surfaces cluster complexes acta fomin zelevinsky cluster algebras foundations amer math fomin zelevinsky cluster algebras finite type classification invent fomin zelevinsky cluster algebras coefficients compos leclerc cluster structures quantum coordinate rings selecta math gekhtman shapiro vainshtein cluster algebras poisson geometry volume mathematical surveys monographs american mathematical society providence grabowski graded cluster algebras algebraic grabowski launois graded quantum cluster algebras application quantum grassmannians proc lond math soc grabowski pressland graded frobenius cluster categories arxiv graham knuth patashnik concrete mathematics publishing company reading second edition foundation computer science harer virtual cohomological dimension mapping class group orientable surface invent hatcher triangulations surfaces topology keller quiver mutation java quivermutation mosher tiling projective foliation space punctured surface trans amer math muller skein cluster algebras marked surfaces quantum palu cluster characters triangulated categories ann inst fourier grenoble schiffler quiver representations cms books smc springer cham scott grassmannians cluster algebras proceedings london mathematical society warkentin exchange graphs via quiver mutation phd thesis bonn
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aug approximation single hidden layer feedforward neural networks fixed weights namig guliyev vugar ismailov abstract feedforward neural networks wide applicability various disciplines science due universal approximation property authors shown single hidden layer feedforward neural networks slfns fixed weights still possess universal approximation property provided approximated functions univariate phenomenon lay restrictions number neurons hidden layer number probability considered network give precise results note constructively prove slfns fixed weight two neurons hidden layer approximate continuous function compact subset real line applicability result demonstrated various numerical examples finally show slfns fixed weights approximate continuous multivariate functions contents introduction construction sigmoidal function practical computation properties constructed sigmoidal function main results numerical results analysis multivariate case acknowledgements references introduction approximation capabilities single hidden layer feedforward neural networks slfns investigated many works past years typical results show slfns possess universal approximation property approximate continuous function compact set arbitrary precision mathematics subject classification key words phrases feedforward neural network approximation hidden layer sigmoidal function activation function weight namig guliyev vugar ismailov slfn units hidden layer input evaluates function form weights vectors thresholds coefficients real numbers activation function univariate function properties neural network model studied quite well choosing various activation functions many authors proved slfns chosen activation function possess universal approximation property see compact set class functions dense space continuous functions general complete result type obtained leshno lin pinkus schocken proved continuous activation function universal approximation property density property polynomial result shown power slfns within possible choices activation function provided continuous detailed review many results see many applications convenient take activation function sigmoidal function defined lim lim literature neural networks abounds use functions superpositions see possibility approximating continuous function compact subset real line space slfns sigmoidal activation function well studied number papers recent years theory neural networks developed direction example point view practical applications neural networks restricted set weights gained special interest see proved slfns restricted set weights still possess universal approximation property example stinchcombe white showed slfns polygonal polynomial spline analytic activation function bounded set weights universal approximation property ito investigated property networks using monotone sigmoidal functions tending minus infinity infinity weights located unit sphere one coauthors considered slfns weights varying restricted set directions gave several necessary sufficient conditions good approximation networks set weights consisting two directions showed geometrically explicit solution problem hahm hong went direction showed slfns fixed weights approximate arbitrarily well univariate function since fixed weights reduce computational expense training time result particular interest mathematical formulation result reads follows theorem hahm hong assume continuous function finite segment assume bounded measurable sigmoidal function sufficiently small exist constants approximation single hidden layer networks positive integers note theorem depend smaller neurons hidden layer one take approximate required precision phenomenon pointed necessary many papers various activation functions plenty practical examples diagrams tables etc literature showing number neurons increases error approximation gets smaller well known one challenges neural networks process deciding optimal number hidden neurons challenge understanding reduce computational expense training time usual networks fixed weights best fit purpose respect cao xie strengthened result specifying number hidden neurons realize approximation continuous function implementing modulus continuity established upper bound estimations approximation error shown class lipschitz functions lipm lipschitz constant degree approximation bound sup approximation capabilities slfns fixed weight also analyzed lin guo cao taking activation function continuous even function authors showed neural networks form approximate continuous function arbitrary precision note weights fixed equal consequently depend prove first gave integral representation trigonometric polynomials constructed explicitly network formed approximates integral representation finally obtained result trigonometric polynomials used prove upper bound approximation error paper construct special sigmoidal activation function meets mentioned challenges univariate setting mathematical terminology construct sigmoidal function theorem depend error moreover take parameters depend find numbers large class functions especially analytic functions answer question positive give algorithm computer program computing numbers practice results illustrated several examples finally show slfns fixed weights capable approximating multivariate functions arbitrary precision construction sigmoidal function section construct algorithmically sigmoidal function use main result following section besides sigmoidality take care namig guliyev vugar ismailov smoothness monotonicity weak sense weak monotonicity understand behavior function whose difference absolute value monotonic function sufficiently small number regard say real function defined set called respectively exists increasing respectively decreasing function obviously coincides usual concept monotonicity function start construction assume given closed interval sufficiently small real number construct algorithmically based two numbers namely following steps describe algorithm introduce function min log note function strictly increasing real line satisfies following properties want construct satisfying inequalities tend tends obey inequality function proceeding construction need enumerate monic polynomials rational coefficients let sequence see enumerate rational numbers setting note monic polynomial rational coefficients uniquely written positive rational number determines unique finite continued fraction construct bijection set monic polynomials rational coefficients set positive rational numbers follows monic polynomial associate rational number monic polynomial form associate rational number monic approximation single hidden layer networks polynomial form associate rational number monic polynomial degree associate rational number words define rqn example first elements sequence start constructing intervals monic polynomial set note numbers depend avoid complication symbols indicate notation introduce sequence clearly sequence strictly increasing converges define function difficult notice numbers coefficients linear function mapping closed interval onto closed interval besides interval therefore obtain namig guliyev vugar ismailov step construct intervals purpose use smooth transition function obviously set note numbers already defined previous step since numbers lie interval follows first extend smoothly interval take choose one choose min number satisfying example chosen define first half interval function let prove satisfies condition indeed nothing prove since hence follows numbers hand obtain together yields since inclusion valid since belong finally conclude define second half interval similar way approximation single hidden layer networks min sup one easily verify constructed satisfies condition steps construct interval remaining interval define difficult verify strictly increasing smooth function note also tends left final step completes construction whole real line practical computation properties constructed sigmoidal function noted algorithm allows one compute constructed point real axis instantly code algorithm available http practical example give graph see figure numerical table see table containing several computed values function interval figure shows graph function changes interval parameter decreases obeys following properties sigmoidal strictly increasing increasing easily computable practice properties easily seen exposition essential property sigmoidal function ability approximate arbitrary continuous function using fixed number translations scalings precisely two translations scalings sufficient formulate important property theorem next section main results main results paper formulated following two theorems theorem assume continuous function finite segment sigmoidal function constructed section sufficiently small exist constants namig guliyev vugar ismailov figure graph table computed values proof set divide interval segments follows computed respectively follows let continuous function unit interval density polynomials rational coefficients space continuous functions compact subset exists polynomial form approximation single hidden layer networks figure graph denote leading coefficient define otherwise set cases together means namely hand write namig guliyev vugar ismailov hence note valid unit interval using linear transformation difficult interval indeed let constructed arbitrarily small positive number transformed function well defined apply inequality using inverse transformation write last inequality completes proof since compact subset real line contained segment following generalization theorem holds theorem let compact subset real line diameter let positive number one algorithmically construct computable sigmoidal activation function infinitely differentiable strictly increasing increasing satisfies following property exist numbers remark idea using monic polynomials see section proof new numerical analysis neural networks limited number hidden neurons fact one interested theoretical practical result countable dense subset suffices maiorov pinkus used subset prove existence sigmoidal monotonic analytic activation function consequently neural network fixed number hidden neurons approximates arbitrarily well continuous function note result theoretical value authors suggest constructing using sigmoidal function previous work exploited sequence polynomials rational coefficients construct new universal sigmoidal function note problem fixing weights approximation neural networks considered although construction efficient sense computation sigmoidal function difficulties appeared computing approximating neural network parameters relatively simple approximated functions see remark reason avoided giving practical numerical examples usage monic polynomials instance turned advantageous reducing running time algorithm computing mentioned network parameters allows one approximate various functions sufficiently small precision obtain required parameters scaling coefficients thresholds practice give corresponding numerical results next section approximation single hidden layer networks numerical results prove theorem continuous function approximated arbitrarily well slfns fixed weight two neurons hidden layer activation function network constructed section seen proof approach totally constructive one evaluate value point real axis draw graph instantly using programming interface url shown beginning section current section demonstrate result various examples different error bounds find parameters theorem computations done sagemath computations use following algorithm works well analytic functions assume function whose taylor series around point converges uniformly consider function find taylor polynomial satisfies inequality find polynomial rational coefficients denote leading coefficient polynomial find otherwise set evaluate respectively calculate parameters network construct network gives sequel give four practical examples able make comparisons examples considered functions given interval first select polynomial function target function investigate sigmoidal neural network approximation function also considered note authors chose sigmoidal function obtained numerical results see table slfns neurons hidden layer see also additional constructive result concerning error approximation example seen table number neurons hidden layer increases error bound decreases value phenomenon longer true sigmoidal function see section using theorem construct explicitly slfn two neurons hidden layer approximates polynomial arbitrarily given precision explicit construction mean namig guliyev vugar ismailov table heaviside function sigmoidal function number neurons maximum error table several function number neurons parameters network maximum error network parameters computed directly namely calculated values parameters follows turns polynomial exact representation interval identity let consider polynomial function function exact representation nevertheless one easily construct network two neurons hidden layer sufficiently small approximation error table displays numerical computations network parameters six different approximation errors end consider nonpolynomial functions sin cos tables display parameters neural networks six approximation error bounds seen tables bounds alter number hidden neurons figures show graphs constructed networks approximate corresponding target functions analysis multivariate case section want draw reader attention following question slfns fixed weights preserve universal approximation property multivariate setting networks form approximation single hidden layer networks figure graphs approximators table several function number parameters neurons network maximum error table several function sin cos number neurons parameters network maximum error weight fixed units hidden layer may different different networks approximate continuous multivariate function within arbitrarily small tolerance note fixed obvious multivariate function approximated networks form indeed linear functional namig guliyev vugar ismailov figure graphs approximators figure graphs sin cos approximators selected annihilates functions since functional nontrivial set functions denote sequel dense arbitrary compact set containing points hence approximation continuous functions possible compact sets question case different different networks rather complicated positive answer question would mean example theorem admits generalization functions unfortunately answer question negative details follows summand function depending inner product thus whole sum function function form note functions form called ridge functions approximation single hidden layer networks literature abounds use functions linear combinations see great deal references therein see set subset set ridge functions along let also consider sets note vary vectors functions whilst fixed clearly lin pinkus proved exists function compact set inf denotes uniform norm follows result set hence dense topology uniform convergence compacta since obtain set dense either thus always continuous multivariate functions approximated arbitrarily well slfns fixed weights phenomenon justifies researchers see introduction investigate universal approximation property networks univariate case analysis leads following general negative result approximation slfns limited weights theorem continuous function multivariate continuous function approximated arbitrarily well neural networks form vary number pairwise independent vectors weights network uniformly bounded positive integer networks theorem shows particular limitation neural networks one hidden layer refer reader interesting results discussions around limitations networks acknowledgements research second author supported azerbaijan national academy sciences program approximation neural networks problems frames references calkin wilf recounting rationals amer math monthly cao xie construction approximation feedforword neural networks fixed weights proceedings ninth international conference machine learning cybernetics qingdao chen chen approximation continuous functionals neural networks application dynamic systems ieee trans neural networks namig guliyev vugar ismailov chui approximation ridge functions neural networks one hidden layer approx theory chui mhaskar limitations approximation capabilities neural networks one hidden layer adv comput math costarelli spigler constructive approximation superposition sigmoidal functions anal theory appl cotter theorem application neural networks ieee trans neural networks cybenko approximation superpositions sigmoidal function math control signal systems draghici capabilities neural networks using limited precision weights neural networks funahashi approximate realization continuous mapping neural networks neural networks gallant white exists neural network make avoidable mistakes proceedings ieee international conference neural networks vol ieee press new york guliyev ismailov single hidden layer feedforward network one neuron hidden layer approximate univariate function neural computation hahm hong approximation neural networks fixed weight comput math appl hornik approximation capabilities multilayer feedforward networks neural networks iliev kyurkchiev markov approximation step function sigmoid functions math comput simulation ismailov approximation neural networks weights varying finite set directions math anal appl approximation ridge functions neural networks bounded number neurons appl anal approximation sums ridge functions fixed directions russian algebra analiz ismailov savas measure theoretic results approximation neural networks limited weights numer funct anal optim ito approximation continuous functions linear combinations shifted rotations sigmoid function without scaling neural networks jian jinshou neural networks limited precision weights application embedded systems proceedings second international workshop education technology computer science wuhan kolmogorov theorem multilayer neural networks neural networks leshno lin pinkus schocken multilayer feedforward networks activation function approximate function neural networks liao fang nuttle neural network model pattern classification comput oper res lin limitations shallow nets approximation neural networks lin guo cao approximation neural networks scattered data appl math comput lin pinkus fundamentality ridge functions approx theory maiorov pinkus lower bounds approximation mlp neural networks neurocomputing mhaskar micchelli approximation superposition sigmoidal function radial basis functions adv appl math pinkus approximation theory mlp model neural networks acta numerica cambridge univ press cambridge ridge functions cambridge university press cambridge approximation single hidden layer networks sikkema der wert einiger konstanten der theorie der approximation mit numer math stein sage mathematics software version sage developers http stinchcombe white approximating learning unknown mappings using multilayer feedforward networks bounded weights proceedings ieee international joint conference neural networks vol ieee new york institute mathematics mechanics azerbaijan national academy sciences vahabzadeh baku azerbaijan address njguliyev institute mathematics mechanics azerbaijan national academy sciences vahabzadeh baku azerbaijan address vugaris
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robust specification mining demonstrations marcell susmit ashish sanjit feb university california berkeley sseshia sri international menlo park jha tiwari abstract consider problem inferring temporal specifications demonstrations agent interacting uncertain stochastic environment specifications useful control autonomous systems operating uncertain environments demonstrations may errors specification inference method must robust provide novel formulation problem maximum posteriori map probability inference problem give efficient approach solve problem demonstrated case studies inspired robotics introduction formal specifications precise mathematical descriptions desired system properties play central role formal methods specifications equally useful right specification right level abstraction turn intractable problem tractable one infeasible problem solvable one however finding right specification hard practice literature particular many settings robotics natural way specify task demonstrations setting agent human expert gives one demonstrations task must automatically synthesize controller robot execute setting requires way generalize specific demonstrations broader task specification field specification mining attempts address problems positing techniques conjecture mine useful specifications available data one characterize techniques use labeled data many types labels possible iii class specifications considered underlying data generated ability handle inconsistencies data motivate issues example example consider remote controlled omnidirectional robot placed elevated platform white cells robot moves grid falls platform move time step robot capable moving forward backward diagonally staying place action performed small chance performing different action called slip probability operator robot tasked following assignment move southeast cell stay cell one second move northwest cell stay northwest cell one second suppose wish infer specification intended operator providing demonstrations experiments crucially robot slip time step fig illustration running grid world example left grid world cells annotated coordinates white cells indicate platform right example trajectory running example specification demonstrations seen observer include executions violate specification operator mind however slip probability low demonstration succeed high probability thus algorithm information environment may able conclude slip unintentional discount buggy demonstration words algorithm robust erroneous demonstrations long errors infrequent abstractly one imagine specification mining task interaction agent demonstrating task teacher student teacher interacts environment provide series demonstrations may include erroneous demonstrations student observes demonstrations possibly without labels attempts robustly infer intended correct specification problem setting considered paper one application problem temporal logic based planning uncertain stochastic environments see inferred formal specifications used synthesize controllers provable guarantees related work field specification inference vast one going back several decades see lack space point reader elsewhere detailed survey focus closely related work shares following key characteristics problem demonstrations traces intended positive examples otherwise unlabeled underlying system generating demonstration stochastic iii specification set traces inconsistencies data unlabeled negative examples demonstrations viewed positive examples noise thus literature learning positive examples relevant previous works tuned parametric specifications using positive examples used linear temporal logic ltl templates find specification minimizes number times property violate maximizes number times specification satisfied inferred specification minimizes rate within grammar given positive negative examples via optimization based decision trees genetic algorithms similarly recent work learning temporal properties unlabeled data adapting clustering techniques either directly work within parameter space post processing step hierarchical ing dendograms however techniques purely make assumptions model system generating traces reason formal guarantees provable closest work recent work inferring ltl finding specification minimizes expected number violations optimal agent expected number violations agent applying actions uniformly random violation measure defined minimum number time steps must skipped demonstration satisfy temporal logic specification arbitrary specifications violation measure bounded length demonstration however well defined computation optimal agent expected violations done via dynamic programming product deterministic rabin automaton specification state dynamics powerful procedure incurs heavy cost even simple two state two action markov decision processes mdps authors apply genetic algorithms generate series specifications compute costs find minimium contributions work contributes state art specification inference demonstrations given stochastic environment following key ways present novel formulation problem learning specifications unlabeled demonstrations stochastic environment maximum posteriori map probability inference problem formulation covers broad class boundedtime trace properties including specified temporal logics inspired effectiveness maximum entropy inverse reinforcement learning artificial intelligence robotics literature apply principle maximum entropy problem particular introduce new teacher model term maximum entropy teacher find simple analytic formula likelihood demonstrations given satisfaction rate demonstrations satisfaction rate random action policy exploit structure maximum entropy teacher model design algorithm efficiently search candidate set specifications subset relations specifications known demonstrate algorithm case studies additional contributions include providing reduction learning specifications expert demonstrations inverse reinforcement learning recovering cost functions used specification mining algorithms special cases includes characterizing finding likely demonstrations violation minimizing agent coincides maximum entropy teacher outline sec introduce problem specification inference demonstrations sec show problem special case reinforcement learning naive reduction intractable solve sec introduce specialize maximum entropy teacher model show model used motivate derive existing cost functions literature sec exploit special structure induced maximum entropy teacher model efficiently robust specification inference demonstrations general classes bounded specifications finally sec demonstrate performance algorithm larger variant grid world profile much better algorithm performs brute force worst case bounds two families specification classes varying size perform comparison violation minimizing agent model specification inference demonstrations section formulate problem inferring specification demonstrations present formal definition problem illustrated demonstrations provided teacher teacher operates possibly stochastic environment executes sequence actions action probabilistically changes system state simplicity assume set actions states finite system states actions observable provided demonstration system naturally modeled probabilistic automaton formally defined definition probabilistic automaton probabilistic automaton tuple finite set states starting state finite set actions specifies transition probability going given action states remark probabilistic automaton extended markov decision process mdp adding markovian reward function depends current state action paper aim learning temporal specifications consequently reward corresponding specifications depend sequence states actions example grid world robot operating grid world formalized def probablistic automaton follows let platform platform grid def cells fallen platform denote square ring surrounding platform robot enters fallen assumed fallen platform initial state special state action transitions robot occupying def one four cells platform overall set states platform fallen def set actions correspond staying place moving cell cell finally let probability def robot slips uniformly transitioning neighboring cell neighbor finally transition function def platform fallen platform neighbors platform otherwise definition trace sequence pairs called trace trace length element traces state sequences action sequences bolded remark employ order preserving isomorphism sometimes write trace enables pattern matching refer state action sequences separately next develop machinery distinguish desirable undesirable traces simplicity focus trace properties decidable within fixed time steps leave general specifications infinite sequences future work example allow eventually event occur allow event occur step formalize trace properties sets definition bounded specification given set states set actions fixed trace length bounded specification subset def def def define true false true specifications may given formal notation sets logical formulas automata example formalize experiment specification given let grid world let denote southeast northwest corners resp let denote consider specification denotes traces spawn grid world move southeast corner stay put move northwest corner stay put next example illustrates perhaps unsurprisingly exact learning technique relies strictly positive examples incredibly fragile demonstrations imperfect mislabeled given stochastic enviroment example let specification given agent visits southeast northwest corners grid world denote series five demonstrations four satisfy without slipping one slips resulting robot falling grid thus satisfying suppose one told correct specification lies within set true algorithm naively assumes demonstrations positive examples would rule thus must incorrectly select true crucially adding demonstrations salvage previous example contained correct demonstrations incorrect demonstration resulting slip logic holds generally algorithm must take account intent agent addition outcome first step formalize idea strategy policy agent uses satisfy specification formally policy determines action take next general stochastic depend sequence previous states actions called history definition policies policy collection distributions actions indexed history denote collection policies example let grid world transition system possible policy prime prime prime otherwise teacher model attaches specification policy teacher would use demonstrate within particular probabilistic automaton definition teacher model let set transition systems set candidate specifications teacher model map associates transition pair unique policy remark definition makes presumptions quality demonstrations given teacher example model allows teacher occasionally make mistakes given fixed teacher model transition system specification one obtains teaching policy induces stationary distribution set traces definition trace distributions given policy transition system length induced distribution traces length given denotes length prefix obvious context important often write formally learning problem maximum posteriori probability map inference problem definition specification inference demonstrations specification inference demonstrations problem four tuple probabilistic automaton teacher model sequence length demonstrations drawn unknown distribution induced unknown specification family bounded specifications solution arg max denotes probability teacher used specification generate demonstrations given demonstrations observed reduction inverse reinforcement learning def intimately related problem known inverse reinforcement learning irl definition inverse reinforcement learning inverse reinforcement learning problem four tuple set probabilistic automaton states actions probabilistic automaton family state reward functions state reward function maps trace reward function def family trace reward functions induced denoted irl teacher model sequence length demonstrations drawn unknown distribution induced unknown state reward function solution arg next show def special case def theorem given specification inference demonstrations problem def reduction irl problem whose solution reward function maps solution specification proof consider problem learning specifications given demonstrations transition system reduce irl problem sufficient generate probabilistic automaton class state reward functions bijective map state reward function specifications reward function solves irl problem associates specification solves first unroll transition system steps create new transition system state corresponds time step element corresponds history states actions seen step thus tree rooted state attach state reward states reward leaf states reward either finally observe unique path given leaf node reaching leaf nodes requires steps thus satisfaction decided thus reward function naturally maps specification since bijection leaf nodes paths encode bounded specification powerful reduction suffers exponential blow state size thus traditional irl algorithms intractable example specification subset grid world states nine actions thus cardinality greater sequel shall leverage techniques currently exist irl literature exploit structure bijection avoid explicitly unrolling transition system maximum entropy teacher model begin deriving analytic form following assumptions uniform prior set specifications reducing map instance maximum likelihood estimation demonstrations given teacher model adheres principle maximum entropy discussed moment teacher satisfaction rate approximately satisfaction rate demonstrations suggests defining satisfaction indicator serves similar role definition satisfaction indicator let trace bounded specification def else next define satisfaction rate average indicator function definition let bounded specification probalistic automaton policy def definition given series demonstrations demonstrations def satisfied empirical satisfaction rate denoted next manner analogous maximum entropy inverse reinforcement learning apply principle maximum entropy indirectly define teacher model via demonstration distribution many justifications often given using principle merely remark maximizing entropy minimize bias encoded distribution maintaining consistency observations functionally means weighting traces riskiness respect satisfaction weighting enables resilience imperfect demonstrations next theorem provide analytic form likelihood series demonstrations maximum entropy teacher theorem let probabilistic automaton bounded specification demonstration maximum entropy risk averse teacher satisfaction rate probability uniformly random action sequence satisfy thus read probability generating using random action policy proof probability sampling trace given policy observe since transition probabilities fixed degree freedom recalling function mean principle maximum entropy asserts become mean sum traces define probability given namely def next let applying aforementioned constraints yields combining gives implying finally observe substituting factoring yields remark case agent gives positive examples transition system deterministic reduces common heuristic maximize learning positive examples following corollary thm gives likelihood series demonstrations corollary let probabilistic automaton bounded specification finite sequence demonstrations drawn maximum entropy risk averse teacher satisfaction rate likelihood def def remark positive constant fixed one safely take searching specification maximizes remark taking log change maximum comparing specifications negating result rearranging yields def obj recalling satisfaction rate random action policy one dedef fine log log satisfaction rate policy therefore obj violation measure proportional log satisfaction rate minimizing obj finds specification best explains actions violation minimizing agent example recall grid world specification specification pool true demonstrations respectively seek apply select likely specification given observe suppose probability slipping note gtrue independent compute observe probability randomly generating trace slips contains slips recall satisfy robot spawns world must visit fixed sequence four cells grid world one selects action random transitioning particular reachable next state probability accounting slip cases respectively thus summarize gtrue definition take suppressed multinomial coefficient required two demonstrations however since term change varies one simply absorb coefficient recall thus one evaluate denoting natural log evaluating elements yields true therefore arg correctly selects desired example illustrates uses enviroment model provide inferences robust demonstration errors however important observe one still fit demonstrations example consider set add candidate specification pool set demonstrations recall five demonstations single slip slips via thus natural log evaulates would chosen nevertheless assert example paints overly pessimistic view assuming demonstrations really given via teacher model given enough samples one expects close set safe behaviors moreover example number demonstrations sufficiently increased straight forward calculation reveals subset traces expected higher log likelihood thus algorithm tendency fit safe subset specification demonstrated algorithm seek systematically exploit structure imposed find likely specification within fixed potentially large pool specifications remark practice evaluating trace satisfies specification fairly efficent thus assumed easy compute hand often expensive compute analytic solutions often known complex transition pairs thus one resorts probabilistic model checking monte carlo weighted model counting queries comparatively expensive large algorithm seeks queries present series results provide insights demonstration likelihood change specification changes proposition thus take possible values remark via excluded middle every specification thus given maximum entropy teacher fixed transition system fixed demonstration sequence completely characterized suggests piecewise analysis case splitting definition given candidate specifications demonstrations def def split partitions associate partition function agrees function enables commenting likelihood given various theoretical random policy probability values within next three lemmas proofs appendix provide insight systematically find likely specification without enumerating specifications lemma lemma lemma convex global minimum insights combined following theorem theorem let finite sequence specifications ordered subset inclusion max max proof monotonically increasing sequence lemma via lemma convex thus maximum must occur beginning end sequence theorem suggests specializing sets candidate specifications organized finite bounded partial order respects subset inclusion bounded mean always assumed act bottom top partial order respectively remark construction preclude two incomparable specifications related subset inclusion partial orders arise naturally practical constraints require one apply incomplete technique syntax analysis determining subset inclusion sequences specifications ordered subset inclusion generalize naturally ascending chains partial order definition ascending chains given partial order ascending chain chain sequence elements ordered chain said maximal every element incomparable every element chains totally ordered one extend interval notation specifications specifications chain define notation bottom top chain corollary let chain set demonstrations max max max max gbi gbi observe easily computed performing binary search insights combined algorithm algorithm specification inference chains procedure chain inference search bot search top arg maxi return theorem let represent worst case execution time computing chain given chain specifications demonstrations algorithm runs time log proof proof thm first line second lines algorithm perform binary searches compute bottom top partitions input chain binary search log binary searches performed finally algorithm returns maximum likelihood chain according corollary likelihood query requires one operation queries thus running time log implementation uses following proposition prune candidates without computing line def proposition let candidate specification let suppose exists likely let two values theorem called refuted interval course general contain single chain moreover nothing precludes set unrelated chains connected scenario one appears doomed compute maximum chain separately similarly chains shared prefix rooted fork incomparable segments merge incomparable segments must maximums compared separately crucially one need recompute maximum along shared prefix suggests following simple algorithm algorithm specification inference partial orders procedure partialorder inference sample maximal chain chain inference return maximal chain subroutine returns maximal chain analyze running time alg first define height width degree partial order height size largest maximal chain width size largest subset containing incomparable elements finally let degree parital order average number outgoing edges represented hasse diagram see fig example theorem let represent worst case execution time computing bounded partial order height width degree run demonstrations algorithm runs time log proof since height longest possible chain length thus thm call chain inference takes log time let largest set incomparable elements assumption since sampled chains maximal element must sampled round construction specification appears sampled chain appear future chain thus chains sampled hasse diagram represented directed acyclic graph preprocessing step one compute spanning tree rooted bottom partial order sampling chain done walking path unexplored branches tree requires time removing chain tree visited thus alg takes done marking nodes log time remark alg practice running time alg improved exploiting prop namely one passes likelihood best candidate far one compute aggressive refuted intervals avoid unnecessarily computing random satisfaction probabilities experiments conclusion experiment experiment sought test whether maximum entropy teacher model could correctly infer specification unlabeled demonstrations contain errors begin adapted grid world grid world substituting platform fallen platform specification adapted generalizing condition visiting corner remaining one time step visiting within infinity norm corner units time steps allowing agent visit corners order denote generalization def considered candidate pool computed using monte carlo simulations taking average seconds compute required linear pass demonstration count time spent target regions implemented controller generate demonstrations within demonstrations demonstrations satisfied enumeration scored specification using log likelihood log fig illustrates despite many errors demonstrations recovered high confidence confidence increases exponentially demonstrations given scores proportional log likelihood demonstrations demonstrations fig specification scores experiment experiment next experiment tested alg compared brute force enumeration entirely dominated monte carlo queries give performance terms number queries using grid world specification pool controller ran alg demonstrations observing hasse diagram took form grid correct specification inferred using monte carlo queries respectively compared required via enumeration experiment experiment used grid world controller experiments specification hasse diagram added chain nine specifications element chain introduced new random reran alg demonstrations resulting monte carlo queries respectively compared queries required via enumeration combined experiments show performance alg depends intimately class specifications explored case demonstrations correct specification inferred seconds compared estimated enumeration care taken union new random traces along ascending chains experiment final experiment sought compare maximum entropy teacher model expected violation minimizing objective function introduced comparison adapted clean world experiment setting bounded specifications clean world dynamics fig comprises two states dock undocked two state variables dirt battery teacher objective vacuum room dirt problem finite battery means teacher must interleave vacuuming docking charging candidate specifications likely specifications found along new specification captures strategy employed demonstrations name description candidate vacuum room dirt dirt time steps time step dirt dirt point demonstration time step dirt docked state vacuum unless dirt bat less within two time steps specifications organized hasse diagram depicted fig five identical step demonstrations also result action sequence vacuum vacuum dock wait undock vacuum vacuum dock wait undock original experiment violation measure used conclude likely specification followed closely maximum entropy teacher model assigned scores respectively specifications scores near thus violation model maximum entropy teacher model disagree interpretation demonstrations argue maximum entropy teacher model interpretation justifiable three reasons action sequence given never results dirt thus never demonstrated strategy encoded demonstrated exists action sequence bring demonstrates vacuum vacuum dock wait undock vacuum vacuum dock undock vacuum dynamics deterministic one expect teacher trying demonstrate specification provide another positive example thus given positive example given heavily discounted running alg new action sequence results likely demonstrations concluding briefly remark monte carlo query simulations took demonstration sets queries made including true false conclusion work formulated problem learning specifications unlabeled demonstrations stochastic environment map probability inference problem solve problem first observed naive reduction related problem inverse reinforcement learning intractable introduced maximum entropy teacher model derived analytic formula likelihood demonstrations given satisfaction rate demonstrations satisfaction rate random action policy developed algorithm efficiently find mostly likely specification within candidate set specifications subset relations specifications known finally case studies showed maximum entropy teacher model robust demonstration fig cleaning world dynamics hasse diagram candidates tion developed algorithm searches candidate space better brute force future work includes extending infinite horizon specifications infinite state action spaces characterizing optimal set teacher demonstrations student using maximum entropy teacher model references asarin maler nickovic parametric identification temporal properties proc pages beer eisner rodeh efficient detection vacuity actl formulas formal methods system design bombara belta signal clustering using temporal logics pages springer international publishing cham bombara vasile penedo yasuoka belta decision tree approach data classification using signal temporal logic proc hscc pages caplain finding invariant assertions proving programs acm sigplan notices volume pages acm chavira darwiche probabilistic inference weighted model counting artificial intelligence ding smith belta rus ltl control uncertain environments probabilistic satisfaction guarantees ifac proceedings volumes topcu probably approximately correct mdp learning control temporal logic constraints arxiv preprint jaynes information theory statistical mechanics physical review kasenberg scheutz interpretable apprenticship learning temporal logic specifications arxiv preprint kong jones medina ayala aydin gol belta temporal logic inference classification prediction data proc hscc pages kwiatkowska norman parker prism verification probabilistic realtime systems gopalakrishnan qadeer editors proc international conference computer aided verification cav volume lncs pages springer lemieux park beschastnikh general ltl specification mining automated software engineering ase international conference pages specification mining new formalisms algorithms applications phd thesis eecs department university california berkeley mar metropolis ulam monte carlo method journal american statistical association russell algorithms inverse reinforcement learning icml pages robert machine learning probabilistic perspective sadigh sastry seshia dragan planning autonomous cars leverages effects human actions proceedings robotics science systems conference rss june silvetti nenzi bortolussi bartocci robust genetic algorithm learning temporal specifications data corr deshmukh jin seshia logical clustering learning data proc verification cav wegbreit synthesis loop predicates communications acm wolff topcu murray robust control uncertain markov decision processes temporal logic specifications decision control cdc ieee annual conference pages ieee ziebart maas bagnell dey maximum entropy inverse reinforcement learning aaai volume pages chicago usa appendix proof proof lemma follows trivially note thus proof proof lemma note positive thus rearranging proves lemma proof proof lemma global minimum observe sign determined sign lemma therefore moves less larger decreases increases thus local minimum sign expression change thus convex must global minimum
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improved stochastic trace estimation using mutually unbiased bases department engineering science university oxford engineering product development singapore university technology centre design quantum technologies national university singapore jul examine problem estimating trace matrix given access oracle computes input vector make use basis vectors set mutually unbiased bases widely studied field quantum information processing selection probing vectors approach offers new state art single shot sampling variance requiring log random bits generate vector significantly improves traditional methods hutchinson gaussian estimators terms number random bits required worst case sample variance introduction problem stochastic trace estimation relevant range problems physics applied mathematics electronic structure calculations seismic waveform inversion discretized parameter estimation problems pdes constraints approximating log determinant symmetric positive matrices machine learning particular example research domain many uses stochastic trace estimation used efficiently generalised cross validation gcv discretized iterative methods fitting laplacian smoothing splines large datasets computing number triangles graph string pattern matching training gaussian processes using score functions stochastic trace estimation endeavours choose dimensional vectors expectation equal trace implicit symmetrical positive semi definite matrix seen many sampling policies satisfy condition several metrics used order choose sampling policy one sample variance number samples achieve number random bits required create last metric motivated part relatively long timescales hardware number generation concerns parallelising number generators work propose new stochastic trace estimator based mutually unbiased bases mubs quantify single shot sampling variance proposed mubs sampling method corresponding required number random bits refer methods sample fixed set basis functions fixed basis sampling methods example randomly sample diagonal values matrix sampling set columns form identity matrix referred unit vector estimator literature similar methods sample columns discrete fourier transform dft discrete hartley transform dht discrete cosine transform dct hadamard matrix prove sampling set mutually unbiased bases significantly reduces single shot sample variance particular worst case bound paper laid follows section gives brief introduction mutually unbiased basis section iii describes novel approach using mutually unbiased bases trace estimation section iii gives rigorous analysis new estimator section compares proposed mubs estimator established approaches terms analytic expectation sample variance applied synthetic real data task counting number triangles graph considered example application mutually unbiased basis linear algebra found application diverse range fields field drawing common set tools however occasionally techniques developed one field become well known outside community despite potential wider use work make extensive use mutually unbiased bases sets bases arise physical considerations context quantum mechanics extensively exploited within quantum information community quantum mechanics physical states represented vectors complex vector space simplest form measurement projects state onto one vectors fixed orthonormal basis space probability particular outcome given square length projection onto corresponding basis vector setting natural ask existence pairs sets measurements outcome one measurement reveals nothing outcome another measurement effectively erases information outcome alternate measurement instead performed measurement corresponds particular basis requirement implies absolute value overlap pairs vectors drawn bases corresponding different measurements constant leads directly concept mutually unbiased bases mubs set orthonormal bases said mutually unbiased choices every every dimension space real vector spaces number mutually unbiased bases complicated relationship dimensionality complex vector spaces number mutually unbiased bases known exactly either prime integer power prime furthermore number constructions known constructing bases neither prime power prime number mutually unbiased bases remains open even case known least pdi prime numbers iii trace estimators order estimate trace positive semidefinite matrix single call oracle consider four strategies fixed basis estimator fixed orthonormal basis choose uniformly random elements trace estimated mutually unbiased bases mubs estimator fixed choice set mutually unbiased bases choose uniformly random choose uniformly random elements taken maximum number mutually unbiased bases complex vector space dimension fixed basis strategy trace estimated methods paper compare fixed basis estimator mubs estimator table iii first analyse worst case variance fixed base estimator analysis analysis mubs estimator follows make assumption consider worst case variance begin definition variance estimator single query let random variable chosen according fixed basis strategy var first strategy generic formulation approaches sample vectors fixed orthogonal basis efficient sampling method terms number random bits required literature second strategy novel represents main contribution strategies similar randomness requirements first strategy least random bits necessary ensure possibility choosing every element second strategy identical number random bits necessary choose fixed random bits necessary choose note upper bound number mutually unbiased bases one greater dimensionality space bound saturated spaces dimensionality prime integer power prime thus number random bits necessary implement strategies differs factor approximately two third forth strategies significantly outperform fixed basis estimator terms variance cost dramatic increase amount randomness required extensively studied literature conciseness repeat analysis denotes expectation value argument compute term term first dim hence second term equal turning first term hutchinson estimator randomly choose elements independently identically distributed rademacher distribution trace estimated gaussian estimator randomly choose elements independently identically distributed zero mean unit variance gaussian distribution trace estimated analysis fixed basis estimator mii uau fixed unitary matrix vector standard basis mii ith entry main diagonal variance fixed basis estimator given vfixed worst case occurs value maximized fixed trace hence occurs single diagonal entry worst case single shot variance fixed basis estimator fworst ixed analysis mubs estimator turn analysis mubs estimator assume either prime prime raised integer power since case matrix always padded zeros dimension little overhead case established variance remains given except defined terms vectors chosen according mubs strategy analyse individual terms making variance begin worst estimator fixed basis mubs hutchinson gaussian exact fixed precision table comparison single shot variance worst case single shot variance worst number random bits required commonly used trace estimators mubs estimator case mubs estimator quantities provided variances upper bounds rather exact variance hence second term variance fixed basis estimator analysing first term however difficult begin observation expressed terms trace kronecker product two matrices follows notice implies dimensions prime integer powers prime since cases implies eigenvalues nonzero subspace minimize sum squares fixed sum since positive conclude eigenvalue must equal unity returning calculation variance form may appear intimidating prove fact projector eigenvalue either prove indirectly first showing rank using relationship traces conclude remaining eigenvalues equal unity vector form trivially satisfies since vectors form basis subspace dimension conclude rank turning issue trace moving summations inside equation obtain similarly compute trace obtain hence var implies variance estimate bounded fact possible compute variance exactly observing projector onto symmetric subspace integer power prime say vector vector orthogonal vectors eigenspace whereas vector null space compute exact variance mubs estimator using spectral decomposition absolute error quantified worst case performance mubs estimator explore performance practice set numerical experiments example application consider counting number triangles graph important problem number application domains identifying number mutual acquaintences social network efficient method trace triangle algorithm algorithm based relationship dataset absolute error number samples dataset absolute error number samples dataset absolute error numerical results number samples thus mubs estimator better worst case performance hutchinson estimator factor proaches large improvement perhaps unsurprising since symmetric matrices xtr axr xti axi real imaginary parts hence evaluating single complex vector equivalent taking sum two different real vectors leading factor two improvement variance average table compares single shot variance worst case single shot variance randomness requirements trace estimators seen comparison mubs estimator strictly smaller variance either hutchinson gaussian methods requiring significantly less randomness implement given drastic reduction randomness requirements improved worst case performance mubs estimator provides attractive alternative previous methods estimating trace implicit matrices since positive matrices value bounded single shot variance mubs estimator bounded vworst mubs worst case significant improvement bound stemming even restricted positive worst case variance bounded vworst mubs since defined worst case single shot variance mubs estimator least factor better fixed basis estimator furthermore variance widely used hutchinson estimator given worst case hence worst case single shot variance hutchinson estimator vhworst gaussian hutchinson unit mubs vmubs dataset number samples fig comparison performance stochastic trace estimation methods four datasets fixed basis method included competitive experiments performed times solid line indicated empirical mean absolute relative error surrounding transparent region indicates one empirical standard deviation trials dataset vertices edges triangles table datasets used comparison stochastic trace estimation methods counting triangles graphs datasets found triangles graphs presented table results experiment presented figure code experiments efficient python implementation generating mubs sample vectors made publically available mubs estimator outperforms classical method experiments would expected theoretical analysis terms variance addition exponential reduction randomness means implementations making use hardware random number generation generally see significant decrease processing times adjacency matrix number triangles undirected graph acknowledgements trace adjacency matrix cubed sampled per sample opposed explicitly computed compared gaussian hutchinson unit mubs estimators performance predicting number bai fahey golub menon richter tech citeseer van leeuwen aravkin herrmann international journal geophysics haber chung herrmann siam journal optimization boutsidis drineas kambadur zouzias arxiv preprint hutchinson communications computation atallah chyzak dumas algorithmica atallah grigorescu information processing letters avron workshop data mining theory applications vol tsourakakis data mining icdm eighth ieee international conference ieee stein chen anitescu annals applied statistics avron toledo journal acm jacm jff acknowledges support air force office scientific research aoard grant material based research funded part singapore national research foundation nrf award schwinger proceedings national academy sciences durt englert bengtsson international journal quantum information boykin sitharam tarifi wocjan arxiv preprint klappenecker finite fields applications springer butterley hall physics letters hutchinson communications computation silver physical review nielsen chuang quantum computation quantum information cambridge university press comprehensive introduction mathematics quantum mechanics systems refer reader
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scavenger theorem prover based conflict resolution daniyar john bruno woltzenlogel oct itmo university petersburg russia ditegulov australian national university canberra australia abstract paper introduces scavenger first theorem prover pure logic without equality based new conflict resolution calculus conflict resolution restricted resolution inference rule resembles generalization unit propagation well rule assuming decision literals rule deriving new clauses generalization clause learning introduction outstanding efficiency current propositional naturally raises question whether would possible employ similar ideas automating logical reasoning recent conflict resolution regarded crucial initial step answer question perspective generalizes logic two main mechanisms modern based unit propagation clause learning calculus sound refutationally complete derivations isomorphic implication graphs paper goes one step defining proof search algorithms familiarity propositional cdcl procedure assumed even though briefly sketched section main challenge lifting procedure logic unlike propositional logic unit propagation always terminate true clauses necessarily uniformly true literals section solutions challenges discussed section section experimental results presented section related work resolution rule traced back resolution attempts lift dpll cdcl logic include model evolution geometric resolution clause learning goal sensitive procedure brief summary approaches comparison author order alphabetical surname confused homonymous calculus linear rational inequalities found furthermore many architectures theorem proving use black box propositional reasoning without attempting lift semantic resolution yet another related approach uses externally built models guide resolution propositional cdcl search propositional case keeps model trail consisting conjunctive list decision literals propagated literals literals unit clauses automatically added trail whenever clause one literal falsified current model literal added model thereby satisfying clause process known unit propagation reaches conflict situation dual literal already contained model would added backtracks removing model decision literals responsible conflict well propagated literals entailed removed decision literals deriving learning clause duals decision literals responsible conflict empty clause decision literals unit propagation terminates without reaching conflict clauses satisfied model input clause set satisfiable clauses still satisfied chooses assigns another decision literal adding trail satisfying clauses contain conflict resolution inference rules conflict resolution calculus shown figure unit propagating resolution rule chain restricted resolutions unit clauses left premises unit clause final conclusion decision literals denoted square brackets clause learning rule infers new clause consisting negations instances decision literals used reach conflict empty clause clause learning inference said discharge decision literals uses resolution calculus derivations directed acyclic graphs necessarily refutation derivation undischarged decision literals natural deduction point view unit propagating resolution rule regarded chain implication eliminations taking unification account whereas decision literals conflict driven clause learning reminiscent respectively assumptions chains negation introductions also generalized unification therefore considered hybrid resolution natural deduction practice optimizations used sophisticated clauses disjunctions duals decision literals involved conflict derived optimizations inessential focus paper resolution unifier conflict unifier clause learning cli composition substitutions used patha since proof dag necessarily may one path connecting proof fig conflict resolution calculus lifting challenges logic presents many new challenges methods based propagation decisions following singled logic unit propagation may never terminate example clause set clearly unsatisfiable assignment true false would satisfy last four clauses however unit propagation would derive following infinite sequence units successively resolving previously derived units starting consequently proof search strategy would wait unit propagation terminate making decisions would never able conclude given clause set unsatisfiable absence uniformly true literals satisfied clauses propositional case clause true model always least one literal true model logic shared variables create dependencies literals instance clause set satisfiable model uniformly true true instances uniformly true propagation without satisfaction propositional case one literal clause false model literal propagated added model clause necessarily becomes true model need considered propagation anymore least backtracking case hand clause would propagate literal model containing become true model true must remain available propagations instance literal added model clause used propagate without propagation clause model iff one literals false model logic contrast propositional logic even case clause necessarily propagate literal one literals false model instance clause model containing instance propagated first two challenges affect search conceptual level solutions discussed section last two prevent direct generalization data structures watched literals make unit propagation efficient propositional case partial solutions discussed section model construction proof search despite fundamental differences propositional logic described previous section algorithms presented aim adhere much possible propositional procedure sketched section propositional case model construction conjunctive list literals literals may contain universal variables literal model instance said true note checking literal true model expensive logic propositional logic whereas latter suffices check former necessary find literal substitution literal said strongly true model iff straightforward solution second challenge absence uniformly true literals satisfied clauses clause satisfied model iff relevant instances literal true instance said relevant substitutes clause variables terms occur thus instance clause satisfied model relevant instances literals true model however solution costly requires generation many instances fortunately many though cases satisfied clause literal true case clause said uniformly satisfied uniform satisfaction cheaper check satisfaction however drawback uniform satisfaction model construction algorithm may repeatedly attempt satisfy clause uniformly satisfied choosing one literals decision literal example clause uniformly satisfied model without knowing clause already satisfied model procedure would try choose either decision literal choices useless decisions would lead conflicts clauses equal previously derived clause unit clause containing literal part current model clause said weakly satisfied model literals useless decisions first challenge general case crucial make decisions unit propagation example given item section instance deciding moment would allow propagation respectively due clauses triggering conflict learned clause would would trigger conflict propagation time due clauses last conflict depend decision literal empty clause derived thus clause set refuted question interleave decisions propagations one straightforward approach keep track propagation depth implication graph decision literal literal propagated unit clause propagation depth literal propagation depth maximum propagation depth predecessors propagation performed exhaustively propagation depth threshold decision literal chosen threshold incremented eager decisions guarantee decision eventually made even infinite propagation path however eager decisions may also lead spurious conflicts generating useless conflictdriven clauses instance clause set clauses numbered easier reference unsatisfiable conflict decisions obtained propagating repeatedly conflicts former propagation depth propagation depth threshold lower decision literal chosen conflict reached chosen example attempt satisfy sixth clause propagations using clauses depth lower threshold reaching conflict isomorphism implication graphs subderivations conflict resolution propagation depth equal corresponding subderivation height initial axiom clauses learned clauses height height conclusion resolution inference maximum height unit premises generates clause useless showing unsatisfiability whole clause set serious issue useless clauses often generated conflicts decisions well nevertheless example suggests starting threshold strategy increasing threshold chosen wisely since performance may sensitive choice interestingly problem propagation manifest fragments logic infinite unit propagation paths impossible large fragment effectively propositional class consisting sentences prenex forms quantifier prefix function symbols fragment simpler proof search strategy makes decisions unit propagation terminates propositional case suffices infinite unit propagation paths occur effectively propositional fragment function symbols hence term increase arbitrarily whenever term depth bounded infinite unit propagation paths occur finitely many literals bounded term depth given finite set constant function predicate symbols finite arity occurring clause set insight term depth important naturally suggests different approach general case instead limiting propagation depth limit term depth instead allowing arbitrarily long propagations long term depth propagated literals smaller current term depth threshold literal propagated term depth smaller threshold new decisions chosen propagation terminates still clauses uniformly satisfied eager decisions may lead spurious conflicts bounding propagation term depth seems intuitively sensible bounding propagation depth implementation details scavenger implemented scala source code usage instructions available https packrat combinator parsers able parse tptp cnf files although scavenger firstorder prover every logical expression converted simply typed lambda expression implemented abstract class concrete subclasses sym app abs respectively symbols applications abstractions trait var used distinguish variables symbols scala case classes used make behave like algebraic datatype constructors choice simply typed lambda expressions motivated intention generalize scavenger logic logic support tptp tff thf future every clause internally represented immutable sequent consisting set positive literals succedent set negative literals antecedent depth constants variables zero depth complex term maximum depth proper subterms problem unsatisfiable scavenger output refutation internally represented collection proofnode objects instances following immutable classes unitpropagatingresolution conflict conflictdrivenclauselearning axiom decision first three classes correspond directly rules shown figure axiom used leaf nodes containing input clauses decision represents fictive rule holding decision literals class responsible checking typically require statements soundness conditions corresponding inference rule axiom decision conflictdrivenclauselearning classes less lines code conflict unitpropagatingresolution respectively lines code code analyzing conflicts traversing subderivations conflict graphs finding decisions contributed conflict implemented superclass lines long following three variants scavenger implemented aiming effectively propositional fragment propagation bounded decisions made propagation terminates propagation bounded propagation depth threshold starting input clauses assigned depth derived clauses propagated literals obtained depth threshold assigned depth threshold incremented whenever every input clause neither uniformly satisfied weakly satisfied used derive new clause propagate new literal case decision literal chosen assigned depth uniformly satisfy one clauses neither uniformly satisfied weakly satisfied propagation bounded term depth threshold starting propagation terminates stochastic choice either selecting decision literal incrementing threshold made probability option uniform satisfaction clauses checked third fourth challenges discussed section critical performance prevent direct generalization data structures watched literals enables efficient detection clauses ready propagate literals without knowing exactly clauses ready propagate scavenger three variants loops clauses goal using propagation however actually trying use given clause propagation costly order avoid cost scavenger performs two quicker tests firstly checks whether clause uniformly satisfied checking whether one literals belongs model clause dismissed imperfect test however occasionally satisfied clauses dismissed logic satisfied clauses uniformly satisfied secondly every literal every clause scavenger keeps set decision literals propagated literals unifiable clause one literal empty set associated rough analogue watched literals detecting clauses imperfect test logic clauses ready propagate despite imperfections tests reduce number clauses need considered propagation quick simple implement overall three variants scavenger listed implemented concisely main classes lines long respectively attempt made increase efficiency expense code readability pedagogical value premature optimization would inappropriate first scavenger still sophisticated backtracking restarting mechanism propositional scavenger reaches conflict restarts almost completely derived clauses kept model construction reset empty model experiments experiments starexec cluster evaluate scavenger performance tptp benchmarks cnf form without equality comparison provers available starexec tptp community suitable cnf problems without equality evaluated well job pair timeouts cpu seconds wallclock seconds prover problems solved epr prover problems solved epr table number problems solved prover table shows many unsatisfiable cnf problems effectively propositional epr unsatisfiable cnf problems theorem prover solved figures shows performance detail first implementation best variants scavenger show acceptable performance variants scavenger outperformed pepr grande darwinfm paradox raw experimental data available https zenonmodulo additionally outperformed effectively propositional propblems outperformed zenonmodulo solved problem less less although long ceased prover replaced fact scavenger solves almost many problems encouraging mature prover years development implementing language several refinements proof search resolution paramodulation orderings set support splitting demodulation subsumption whereas scavenger yet unrefined concise implementation scala comparatively straightforward search strategy proofs conflict resolution calculus completed slightly months conceptually based geometric resolution darwin based model evolution similar scavenger scavenger already outperforms still far darwin probably due scavenger current eagerness restart every conflict whereas darwin backtracks carefully sections scavenger darwin also treat variables decision literals differently consequently scavenger detects conflicts learning clauses expensive unifiers must collected conflict graph composed fig performance benchmarks provers ordered performance solved problems suggests fig performance epr benchmarks provers ordered performance uncommon issue practice still able solve many problems even though care bound propagation whereas two variants solve fewer problems overhead bounding propagation even necessary nevertheless problems solved problems solved among scavenger variants solve problems tptp difficulty rating syn fld domains problems solved less seconds conclusions future work scavenger first theorem prover based new conflict resolution calculus experiments show promising albeit yet competitive performance comparison performance three variants scavenger shows interleave decisions within possibly unitpropagations research needed determine possibly problem dependent way optimal initial depth thresholds threshold incrementation strategies alternatively entirely different criteria could explored deciding make eager decision propagation instance decisions could made fixed dynamically adjusted amount time elapses performance bottleneck needs urgently addressed future work backtracking restarting currently variants scavenger restart every conflict keeping derived clauses throwing away model construct far must reconstruct models scratch every conflict requires lot repeated therefore significant performance boost could expected sensible backtracking strategy scavenger current naive unification algorithm could improved term indexing might also room improve scavenger rough analogue watched literals data structure even though challenges make unlikely something good propositional watched literals data structure could ever developed experimentation also needed find optimal values parameters used scavenger governing initial thresholds incrementation policies scavenger already acceptable performance despite implementation improvement possibilities discussed indicates automated theorem proving based conflict resolution calculus feasible however much work remains done determine whether approach eventually become competitive today fastest provers acknowledgments thank ezequiel postan implementation tptp parsers skeptik reused scavenger grateful albert giegerich aaron stump geoff sutcliffe help setting experiments starexec research partially funded australian government australian research council google summer code program daniyar itegulov financially supported russian scientific foundation grant references alagi weidenbach nrcl model building approach fragment lutz ranise eds frontiers combining systems international symposium frocos wroclaw poland september proceedings lecture notes computer science vol springer http baumgartner first order procedure proceedings international conference automated deduction cade baumgartner model theorem proving ieee intelligent systems http baumgartner fuchs tinelli lemma learning model evolution calculus hermann voronkov eds logic programming artificial intelligence reasoning international conference lpar phnom penh cambodia november proceedings lecture notes computer science vol springer http baumgartner tinelli model evolution calculus baader automated deduction international conference automated deduction miami beach usa july august proceedings lecture notes computer science vol springer http bonacina plaisted constraint manipulation sggs kutsia ringeissen eds proceedings workshop unification unif seventh international joint conference automated reasoning ijcar sixth federated logic conference floc technical reports research institute symbolic computation johannes kepler linz july http bonacina plaisted sggs theorem proving exposition schulz moura konev eds proceedings fourth workshop practical aspects automated reasoning paar seventh international joint conference automated reasoning ijcar sixth federated logic conference floc july easychair proceedings computing epic vol july bonacina plaisted reasoning model representation journal automated reasoning http bonacina plaisted reasoning inference system completeness journal automated reasoning http boudou fellner woltzenlogel paleo skeptik proof compression system demri kapur weidenbach eds automated reasoning international joint conference ijcar held part vienna summer logic vsl vienna austria july proceedings lecture notes computer science vol springer http brown satallax automatic prover gramlich miller sattler eds automated reasoning international joint conference ijcar manchester june proceedings lecture notes computer science vol springer http claessen anatomy equinox extensible automated reasoning tool logic beyond talk abstract proceedings international conference automated deduction davis putnam computing procedure quantification theory journal acm hodgson slaney system description automated reasoning first international joint conference ijcar siena italy june proceedings http korovin iprover theorem prover logic system description armando baumgartner dowek eds automated reasoning international joint conference ijcar sydney australia august proceedings lecture notes computer science vol springer http korovin modular approach automated reasoning programming logics korovin tsiskaridze voronkov conflict resolution gent principles practice constraint programming international conference lisbon portugal september proceedings lecture notes computer science vol springer http malik clause learning sat solvers handbook satisfiability martin davis loveland machine program theorem proving communications acm mccharen overbeek wos complexity related enhancements automated programs computers mathematics applications mccune otter stickel international conference automated deduction kaiserslautern frg july proceedings lecture notes computer science vol springer http mccune otter reference manual corr http nieuwenhuis hillenbrand riazanov voronkov evaluation indexing techniques theorem proving automated reasoning first international joint conference ijcar siena italy june proceedings http nivelle meng geometric resolution proof procedure based finite model search furbach shankar eds automated reasoning third international joint conference ijcar seattle usa august proceedings lecture notes computer science vol springer http slaney woltzenlogel paleo conflict resolution resolution calculus decision literals clause learning journal automated reasoning http slaney scott theorem prover bajcsy proceedings international joint conference artificial intelligence france august september morgan kaufmann http stump sutcliffe tinelli starexec infrastructure logic solving demri kapur weidenbach eds automated reasoning international joint conference ijcar held part vienna summer logic vsl vienna austria july proceedings springer international publishing cham http sutcliffe tptp problem library associated infrastructure fof cnf parts journal automated reasoning voronkov avatar architecture theorem provers biere bloem eds computer aided verification international conference cav held part vienna summer logic vsl vienna austria july proceedings lecture notes computer science vol springer http
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markov decision processes continuous side information aditya modi admodi computer science engineering univ michigan ann arbor nov nan jiang nanjiang microsoft research new york satinder singh baveja computer science engineering univ michigan ann arbor ambuj tewari tewaria department statistics univ michigan ann arbor abstract consider reinforcement learning setting agent interacts sequence episodic mdps start episode agent access context determines dynamics mdp episode setting motivated applications healthcare baseline measurements patient start treatment episode form context may provide information patient might respond treatment decisions propose algorithms learning contextual markov decision processes cmdps assumption unobserved mdp parameters vary smoothly observed context also give lower upper pac bounds smoothness assumption lower bound exponential dependence dimension consider tractable linear setting context used create linear combinations finite set mdps linear setting give pac learning algorithm based kwik learning techniques keywords reinforcement learning pac bounds kwik learning introduction consider basic sequential decision making problem healthcare namely learning treatment policy patients optimize health outcome interest one could model interaction every patient markov decision process mdp precision personalized medicine want treatment personalized every patient time amount data available given patient may enough personalize well means modeling patient via different mdp result severely suboptimal treatment policies extreme pooling patients data results data perhaps relevant patient currently want treat therefore face large amount shared data learn single policy finding relevant policy patient similar occurs applications involving humans agents environment online tutoring web advertising modi jiang singh tewari key observation many personalized decision making scenarios side information available individuals might help designing personalized policies also help pool interaction data across right subsets individuals examples data include laboratory data medical history patients healthcare user profiles history logs web advertising student profiles historical scores online tutoring access side information let learn better policies even limited amount interaction individual users refer contexts adopt augmented model called contextual markov decision process cmdp proposed hallak assume contexts fully observed available interaction starts new paper study sample complexity learning cmdps worst case consider two concrete settings learning cmdp continuous contexts first setting individual mdps vary arbitrary smooth manner contexts propose algorithm section pac bounds innate hardness learning general case captured lower bound construction section show possible achieve significantly better sample complexity structured cmdps consider another setting contexts used create linear combinations finite set fixed unknown mdps use kwik framework devise kwik algorithm section also provide pac upper bound algorithm contextual markov decision process problem setup notation start basic definitions notations mdps introduce contextual case definition markov decision processes markov decision process mdp defined tuple state space action space defines transition probability function tuple defines initial state distribution mdp consider case fixed horizon denoted episodic mdps use denote policy action state timestep episode initial state observed according distribution afterwards agent chooses action according policy reward next state according reward transition functions policy define value follows optimal policy one achieves largest possible value called optimal value next define contextual model similar definition denoted given hallak hallak assumes latent contexts results significant differences work application scenarios required assumptions results see detailed discussion section markov decision processes continuous side information definition contextual mdp contextual markov decision process cmdp defined tuple context space state space action space function maps context mdp parameters denote mdp context make simplifying assumption initial state distribution independent context assume throughout paper rewards bounded denote respectively also assume context space bounded norm upper bounded constant consider online learning scenario following protocol observe context choose policy based previous episodes experience episode using make distributional assumptions context sequence instead allow sequence chosen arbitrary potentially adversarial manner natural criteria judging efficiency algorithm look number episodes performs main aim pac analysis bound number episodes value algorithm policy dann brunskill although give pac bounds coverrmax algorithm given reader make note made explicit attempts achieve tightest possible result use rmax brafman tennenholtz algorithm base construction handle simplicity approach also combined pac algorithms strehl littman dann brunskill improved dependence section present algorithm provide pac bound smoothness assumption key motivation contextual setting sharing information among different contexts helpful therefore natural assume mdps corresponding similar contexts similar formalized following smoothness assumption definition smoothness given cmdp distance metric context space two contexts following constraints call smooth cmdp smoothness parameters assume distance metric constants known smoothness assumption allows use minimally tweaked version rmax brafman tennenholtz provide analysis smooth cmdps similar existing literature mdps kearns singh strehl strehl littman know transition dynamics expected reward functions pair finite mdp easily compute optimal policy idea rmax distinguish pairs known unknown pair known visited enough number times empirical estimates reward transition probabilities due sufficient data state becomes known actions pairs become known rmax constructs auxiliary mdp encourages optimistic behaviour assigning maximum reward hence name rmax remaining unknown states act according optimal policy auxiliary mdp one following must happen exploit information available achieve value visit unknown states accumulate information efficiently formally set known states define approximate induced mdp following manner let denote number observations pair transitions respectively also let denote total reward obtained pair define values define values use certainty equivalent policy computed induced mdp perform balanced wandering kearns singh unknown states balanced wandering ensures actions tried equally fairly unknown states assigning maximum reward unknown states pushes agent visit states provides necessary exploration impetus generic template rmax given algorithm contextual case infinite number mdps idea behind algorithm one group close enough contexts treat single mdp utilizing boundedness context space create cover finitely many balls radius centered tuning radius control bias introduced ignoring differences among mdps ball allows pool together data mdps ball avoid difficulty infinite mdps instead deal finitely many size cover number balls measured notion covering numbers see defined min resulting algorithm obtained using subroutines algorithm state sample complexity guarantee theorem markov decision processes continuous side information algorithm rmax template cmdp initialize episode receive context set using redict choose else choose unknown pdate end end end algorithm function initialize min create initialize counts balls function predict find return using else return unknown end function update find increment counts rewards end theorem pac bound input values cmdp smoothness parameters probability least coverrmax algorithm produces sequence policies yield episodes min proof sketch first carefully adapt analysis rmax kakade get pac bound episodic mdp let number visits pair model estimate error reward estimate absolute error show lemma let mdp fixed horizon optimal policy computed rmax starting state probability least msa episodes instead learning model contextual mdp separately combine data within ball therefore take care two things choose radius fine enough cover value number visits state becomes known ball satisfying conditions lemma mdps within ball need radius cover min value using lemma obtain upper bound number episodes single ball probability least setting individual failure probability using union bound get stated pac observe pac bound linear dependence covering number context space case euclidean metric space covering number would order however show section dependence would least linear hence indicate difficulty optimally learning cases lower bound prove lower bound number episodes learning algorithm smooth cmdp shows linear dependence covering number context space unavoidable far know existing way constructing pac lower bounds continuous state spaces smoothness simply augment state representation include context information instead prove lower bound theorem builds upon work dann brunskill lower bounds episodic finite mdps slivkins lower bounds contextual bandits theorem lower bound smooth cmdp exists constants every algorithm satisfies pac guarantee computes sequence deterministic policies context hard cmdp smoothness constant number episodes proof overall idea embed multiple mdp learning problems cmdp agent learn optimal policy mdp separately generalize across show maximum number problems embedded scales detailed proofs refer reader appendix markov decision processes continuous side information figure hard instances episodic mdp dann brunskill initial state moves uniform distribution states regardless action states absorbing rewards respectively states reward actions state essentially acts hard bandit instance whose actions move randomly action satisfies one action action satisfies covering number result follows incorporating known pac lower bound episodic mdps start lower bound learning episodic mdps see figure caption details construction due dann brunskill adapt lower bound statement setting theorem theorem lower bound episodic mdps dann brunskill exists constants every algorithm satisfies pac guarantee computes sequence deterministic policies hard instance mhard number episodes constants chosen discuss populate context space hard mdps note figure agent know action rewarding adversary choose element essentially choosing instance family mdps scenario would like allow adversary choose mdp independently individual packing point yield lower bound linear packing number however always possible due smoothness assumption committing mdp one point may restrict adversary choices another point deal difficulty note pair hard mdps differ transition distributions therefore construct packing radius defined set points two points least away lower bound differs original paper value normalized see whereas allow magnitude value grow maximum size known packing number max related covering number radius chosen arbitrary choices hard mdp instances different packing points always satisfy smoothness assumption recall fix mdps specify mdp follows state action max max essentially move away packing point transition become uniform show claim cmdp defined satisfies definition constant choose context sequence given input repetitions arbitrary permutation construction learning different points independent lower bound simply lower bound learning single mdp theorem multiplied cardinality packing number using well known relation desired lower bound refer reader appendix proof claim detailed analysis contextual linear combination mdps previous section clear contextual mdp smoothness assumptions exponential dependence context dimension unavoidable computational requirements algorithm scales covering number context space section focus structured assumption mapping context space mdps show achieve substantially improved sample efficiency specific assumption make section model parameters individual mdp linear combination parameters base mdps use shorthand vectors concatenate parameters different base mdps parameters base mdps unknown need recovered data learning reward function vary context hence reward smoothness satisfied proof claim deferred appendix markov decision processes continuous side information agent combination coefficients directly available context vector assumption motivated application scenario responds according characteristic distribution possible behavioural patterns mathematical difficulty arbitrary context vector always valid transition function may violate normalization constraints therefore require stays probability simplex always kwik first explain estimate model parameters linear setting discuss perform exploration properly model estimation recall section treat mdps whose contexts fall small ball single mdp estimate parameters using data local context ball section however global structure due parametric assumption base mdps shared across contexts implies data obtained context may useful learning mdp parameters another context far away avoid exponential dependence need leverage structure generalize globally across entire context space due linear combination setup use linear regression replace estimation procedure equation episode context observe pair snext drawn therefore indicator whether snext equal forms unbiased estimate esnext snext based observation construct pair snext whenever observe transition tuple snext context relationship governed linear prediction rule coefficients hence estimate data simply collect pairs correspond particular tuple run linear regression recover coefficients case reward function similar hence discussed data abundant observed many times exploratory design matrix consists vectors expect recover accurately guarantee conditions since context chosen adversarially design matrix indeed observe however matrix new contexts lie subspace spanned previously observed contexts make accurate predictions despite inability recover model parameters online linear regression procedure take care issue choose kwik walsh procedure original kwik deals scalar labels used decide whether estimate sufficiently accurate known pair becomes known known approach however generally leads loose analysis use snext denote random variable denote possible realization need predict individual accurately estimate close true distribution error pair already considered known extend kwik analysis handle outputs provide tighter error bounds treating whole introduce extended version kwik explain incorporate knownness information rmax skeleton perform efficient exploration identifying known kwik kwik algorithm propose linear setting still uses rmax template algorithm exploration every episode build induced mdp act greedily according optimal policy balanced wandering major difference lies set known states identified constructed explain see pseudocode algorithm high level algorithm works following way constructing query kwik procedure estimates every pair using redict kwik procedure either returns know returns estimates guaranteed accurate returned consider unknown associate rmax reward exploration optimistic exploration ensures significant probability observing pairs predicted observe pairs episode call pdate pairs formed via equation make progress estimating parameters unknown pairs next walk pseudocode explain redict pdate work detail prove upper bound number updates happen condition holds line forms basis analysis kwik algorithm initialize matrices using initialize update time let design matrix episode row context observed episode matrix inverse rules verify update rule line essentially yields value episode inverse unnormalized regularized empirical covariance matrix plays central role linear regression analysis matrix accumulates outer product feature vector context vector label snext obvious linear regression estimate using data episode new input vector comes check whether predetermined threshold line recall inverse covariance matrix small implies estimate close along direction predict pct otherwise return kwik subroutine rewards similar hence omitted ensure estimated transition probability valid project estimated vector onto done efficiently using existing techniques duchi state kwik bound learning transition function kwik bound learning rewards much smaller hence omitted use kwik bound scalar linear regression walsh property multinomial samples get kwik bound markov decision processes continuous side information theorem kwik bound learning multinomial vectors kwik algorithm executed probability vectors min log suitable constants number updates take place see line bounded max probability least prediction returned pct proof sketch see full proof appendix provide direct reduction kwik bound learning scalar values key idea notice vector sup conceptually view algorithm running scalar linear regression simultaneously projects vector label scalar fixed linear transformation require every scalar regressor kwik guarantee error guarantee vector label follows union bound algorithm kwik learning function initialize function predict return else return end function update snext prediction snext end result ready prove formal pac guarantee kwik theorem pac bound kwik input values linear cmdp model number base mdps probability kwik lrrmax algorithm produces sequence policies yield log max dsa episodes proof kwik subroutine algorithm makes predictions require projection onto update matrices happen unknown state action pair visited kwik subroutine still predicts line kwik bound states fixed number updates unknown pair parameters always known desired accuracy number updates obtained setting desired accuracy transitions failure probability theorem max dsa use lemma instead updating counts number visits look number updates unknown pairs applying union bound state action pairs using lemma easy see episodes msa bounded probability least bound theorem obtained substituting value see contextual mdp linear structure helps avoiding exponential dependence context dimension combined dependence max related work transfer latent contexts general definition cmdps captures problem transfer see taylor stone lazaric surveys empirical results recent papers also advanced theoretical understanding transfer instance brunskill hallak analyzed sample complexity cmdps mdp element finite small set mdps mdp label treated latent unseen context mahmud consider problem transferring optimal policies large set known mdps new mdp commonality papers mdp label context observed hence methods initially explore every new mdp identify label requires episode length substantially longer planning horizon problematic assumption motivating scenarios interact patient user student limited period time data single episode whose length planning horizon enough identifying underlying mdp contrast prior work propose leverage observable context information perform direct transfer previous mdps algorithm works arbitrary episode length markov decision processes continuous side information side information work leverages available mdp inspired use contexts contextual bandits langford zhang use side information also found literature ammar developed policy gradient method context used transferring knowledge tasks killian used parametric forms mdps develop models personalized medicine policies hiv treatment metric space smooth cmdps section pool observations across similar contexts reduce problem learning policies finitely many mdps alternative approach consider infinite mdp whose state representation augmented context apply methods metric state spaces proposed pazis parr however might increase sample computational complexity unnecessarily longer leverage structure particular component augmented state namely context remains episode concretely augmenting approach needs perform planning augmented mdp states contexts makes requirement worse solution perform planning mdps defined whose computational characteristics dependence context space addition allow context sequence chosen adversarial manner corresponds adversarially chosen initial states mdps usually handled methods kwik learning linear hypothesis classes linear combination setting section provides instance parametric assumptions lead substantially improved pac bounds build upon learning framework developed previous work szita use kwik linear regression resulting kwik algorithm sample complexity bound inherently depends kwik bound linear regression well known even linear hypothesis classes kwik bound exponential input dimension agnostic case szita therefore success algorithm relies validity modelling assumption neu studied problem similar linear combination setting proposed algorithm combining jaksch confidence set techniques stochastic linear optimization literature dani filippi work takes independent different approach provide pac guarantee directly comparable regret bound still observe dependence optimal pac whereas optimal bandit regret analysis hand dependence number rounds optimal dependence counterpart pac analysis suboptimal interesting future direction combine algorithmic ideas papers improve guarantees conclusion paper present general setting using side information learning policies large potentially infinite number mdps proposed algorithm algorithm case mdps vary smoothly respect observed side information lower bound construction indicates necessary exponential dependence pac algorithm context dimension smooth cmdp also consider another instance parametric assumption using kwik linear regression procedure present kwik algorithm efficient exploration linear combination mdps pac analysis shows significant improvement structural assumption use context based modelling multiple tasks rich application possibilities personalized recommendations healthcare treatment policies tutoring systems believe setting possibly extended cover large space quite well number mdps contexts mapping context environment hope work spurs research along directions acknowledgments work supported part grant open philanthropy project center part nsf grant iis ambuj tewari acknowledges support nsf grant career sloan research fellowship opinions findings conclusions recommendations expressed authors necessarily reflect views sponsors references yasin gergely neu online learning mdps side information arxiv preprint haitham ammar eric eaton paul ruvolo matthew taylor online learning policy gradient methods proceedings international conference machine learning pages ronen brafman moshe tennenholtz general polynomial time algorithm reinforcement learning journal machine learning research oct emma brunskill lihong sample complexity reinforcement learning arxiv preprint varsha dani thomas hayes sham kakade stochastic linear optimization bandit feedback colt pages christoph dann emma brunskill sample complexity episodic reinforcement learning advances neural information processing systems pages john duchi shai yoram singer tushar chandra efficient projections onto learning high dimensions proceedings international conference machine learning pages acm markov decision processes continuous side information sarah filippi olivier cappe garivier csaba parametric bandits generalized linear case advances neural information processing systems pages assaf hallak dotan castro shie mannor contextual markov decision processes arxiv preprint thomas jaksch ronald ortner peter auer regret bounds reinforcement learning journal machine learning research apr sham machandranath kakade sample complexity reinforcement learning phd thesis michael kearns satinder singh reinforcement learning polynomial time machine learning taylor killian george konidaris finale transfer learning across patient variations hidden parameter markov decision processes arxiv preprint john langford tong zhang algorithm bandits side information advances neural information processing systems pages lazaric transfer reinforcement learning framework survey wiering van otterlo editors reinforcement learning state art springer lihong michael littman thomas walsh knows knows framework learning proceedings international conference machine learning pages acm lihong wei chu john langford robert schapire approach personalized news article recommendation proceedings international conference world wide web pages acm mahmud majd hawasly benjamin rosman subramanian ramamoorthy clustering markov decision processes continual transfer arxiv preprint jason pazis ronald parr pac optimal exploration continuous space markov decision processes aaai aleksandrs slivkins contextual bandits similarity information journal machine learning research alexander strehl michael littman analysis interval estimation markov decision processes journal computer system sciences alexander strehl lihong michael littman reinforcement learning finite mdps pac analysis journal machine learning research nov szita csaba agnostic kwik learning efficient approximate reinforcement learning proceedings annual conference learning theory pages matthew taylor peter stone transfer learning reinforcement learning domains survey journal machine learning research jul thomas walsh szita carlos diuk michael littman exploring compact representations linear regression proceedings conference uncertainty artificial intelligence pages auai press corrected version available technical report department computer science rutgers university december yihong packing covering consequences minimax risk informationtheoretic methods statistics statistics yale university markov decision processes continuous side information appendix proofs section proof lemma adapt analysis kakade episodic case results removal factor since complete episodes counted mistakes count every action episode reproduce detailed analysis completion completing proof lemma firstly look version simulation lemma kearns singh also complete analysis assume rewards lie definition induced mdp let mdp subset states given set define induced mdp following manner define values pmk rmk define pmk rmk lemma simulation lemma episodic mdps let two mdps space transition dynamics reward functions two mdps kpm every policy two mdps satisfy property denote proof consider set trajectories length let probability observing trajectory behaviour policy let expected average reward obtained trajectory mdp bound second term follows proof lemma kakade combining two expressions get desired result lemma induced inequalities let mdp set known states let induced mdp defined respect show policy states escape unknown denotes value policy mdp starting state proof see lemma kakade corollary implicit explore exploit let mdp set optimal policies known states induced mdp respectively states escape unknown proof follows lemma kakade optimal policy also using assumption proof lemma let mdp rmax computes optimal policy denoted lemma vmm combining lemma get escape unknown vmm escape unknown escape unknown escape probability less desired relation true therefore need bound number episodes expected number greater note due balanced wandering msa visits unknown states rmax algorithm execution may encounter extra visits estimates updated termination episode whenever quantity expected number exploration steps episodes least msa hoeffding inequality episodes probability least number successful exploration steps greater therefore msa probability least total number visits unknown state msa using upper bound visits conclude many episodes suffice markov decision processes continuous side information proof theorem need compute required resolution cover number transitions guarantee approximation value functions required previous lemma following result key result lemma cover approximation given cmdp finite cover visit every pair times ball summing servations policy probability least approximate mdp corresponding computed using empirical averages satisfy proof visit state action pair observe transition context tth visit probability pct let encode vector indices except observing transitions create estimate bounding first term use hoeffding bound max exp therefore probability least error becomes one easily verify using similar arguments error rewards context less using simulation lemma get desired result appendix lower bound analysis proof claim instances packing points assigned parameters context state action given max max prove definition smoothness requirements satisfied claim contextual mdp defined valid instance contextual mdp smoothness constants reward function vary context hence reward smoothness satisfied proof need prove defined contextual mdp satisfies constraints definition let assume smoothness assumption violated context pair smoothness constraints rewards satisfied trivially value constant implies exists state action know shows without loss generality assume also leads triangle inequality triangle inequality simplifies definition max max leads contradiction markov decision processes continuous side information lower bound smooth cmdp lower bound construction made key argument defined contextual mapping specific instances requires agent learn models significant number mdps separately get decent generalization construction populates set packing points context space hard mdps argues instances independent algorithm perspective formalize statement let adversary makes choices instances context follows select mdp family hard instances described figure optimal action state chosen randomly independently assignments parameter deciding difference optimality actions figure taken required construction theorem denote instances set individual instance set random choices made adversary construction uniform distribution possible set instances claim assignment optimal actions packing points would define valid smooth contextual mdp independent choice optimal actions makes mdps packing point least difficult learning single mdp formally let sequence transitions rewards observed learning agent packing points due independence individual instances see denotes distribution trajectories thus observing trajectories mdp instances packing points let algorithm deduce anything nature instance chosen one point respect distribution learning contextual mdp equivalent worse simulating single mdp learning algorithm packing points given contextual mdp algorithm alg alg alg alg optimal single mdp learning algorithm expectation respect distribution instances algorithm randomness theorem lower bound expectation right hand side inequality total number mistakes lower bounded setting gives stated lower bound appendix proof theorem section present proof kwik bound learning transition probabilities proof uses reduction technique reduces label setting scalar setting combines kwik bound scalar labels given walsh proof fix state action pair consider sequence contexts transitions observed pair given new context want estimate kwik algorithm estimate described section wish bound error prediction made know kpc sup use representation prove tighter kwik bound learning transition probabilities every fixed formulate new linear regression problem pair recall snext vector label real interest projects scalar value algorithm viewed implicitly running regression thanks linearity since depends input contexts linear simply equal linear regression prediction problem result kwik bound problem establish automatically applies property taking union bound yields desired error guarantee thanks equation establish kwik guarantee new regression problem groundtruth expected label noise label noise constant magnitude conditions invoke kwik bound scalar linear regression walsh theorem kwik bound linear regression walsh suppose observation noise noisy linear regression problem absolute value bounded let upper bound norm true linear coefficients kwik linear regression algorithm executed log min suitable constants number max log probability least sample prediction made prediction markov decision processes continuous side information purpose set theorem kwik linear regression algorithm known status context checked manner done line algorithm therefore kpc sup equation union bound regression implicitly run substituting values theorem get min log number bounded max
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robust visual slam point line features nov xingxing xiaojia yong guoquan paper develop robust efficient visual slam system utilizes heterogeneous point line features leveraging proposed system consists stereo matching frame tracking local mapping loop detection bundle adjustment point line features particular main theoretical contributions paper first time employ orthonormal representation minimal parameterization model line features along point features visual slam analytically derive jacobians errors respect line parameters significantly improves slam solution proposed slam extensively tested synthetic experiments whose results demonstrate proposed system outperforms methods various scenarios ntroduction visual slam one enabling technologies autonomous systems cars unmanned aerial vehicles space robots solutions rely point features due simplicity line features commonly seen environments less sensitive lighting variation position ambiguity used recent work principle combination point line features would provide geometric constraints structure environment either one motivates design robust point line features recently approaches become favorable due superior accuracy per computational unit compared approaches particular slam one popular formulations constructs factor graph whose nodes correspond states estimate edges represent measurement constraints nodes incorporating line features traditional point graph slam framework two challenges arise first one spatial line often parameterized convenience transformation incurs extra computational overhead graph optimization note spatial line four degrees freedom typically represented two spatial endpoints coordinates six degrees freedom secondly known jacobian plays xingxing zuo xiaojia xie institute control zhejiang university zhejiang china yong liu state key laboratory industrial control technology institute control zhejiang university zhejiang china yong liu corresponding author email yongliu guoquan huang department mechanical engineering university delaware newark usa important role using iterative approach solve graph optimization problem part parametrization approaches using line features typically employ numerically computed jacobians incurs approximation contrast analytically compute jacobians graph optimization order improve accuracy well efficiency particular paper introduces robust efficient visual slam system using point line features unified cost function combining reprojection errors points lines spatial line parametrized orthonormal representation minimal decoupled representation based minimal parametrization derive analytical jacobian line error specifically main contributions paper following improved extraction matching method line features introduced robustify data association proposed visual slam employ orthonormal minimal representation parameterize lines analytically compute corresponding jacobians design implement complete visual slam system using point line features includes stereo matching frame tracking local mapping bundle adjustment line feature point feature well based loop detection extensive experimental results presented validate performance elated ork methods proposed parameterize line space efficiently sola summarizes several methods represent line including coordinates anchored lines homogeneouspoints line etc minimizing number parameters bartoli proposed orthonormal representation minimum four parameters represent spatial lines sfm combination point line features utilized slam community recently marzorati proposed slam points lines uses special trifocal cameras detect reconstruct lines rother reconstructed points lines cost requiring reference plane visible views koletschka proposed stereo odometry based points lines computes disparity endpoints line deals partial occlusions fuses point line features form rgbd visual odometry algorithm extracts points lines data also proved fusing two types features point line features detected one image point line map fig proposed visual slam point line features dataset note green lines indicate trajectory camera motion blue frames represent keyframes current frame green local map tracking moment red produced smaller uncertainty motion estimation using either feature type alone work ruben proposed probabilistic approach stereo visual odometry based combination points line segments weighs associated errors points line segments according covariance matrices iii etection epresentation ine eatures extraction description line features line segment detector lsd popular feature detector line segments designed work noisy image various scenes without parameter tuning able provide subpixel accuracy however lsd suffers problem dividing line multiple segments scenarios shown fig causing failures matching tracking line features side line segment darker improvement merge segments according differences direction distance shown fig represents minimum distance endpoints two segments indicates distance midpoint one segment line segment smaller given threshold direction difference also small two segments considered candidates combined improved line detector advantages making robust accurate data association demonstrated experiments fig shows result two different detectors note merged line segments found improved detector represented lbd line descriptor vector orb point descriptor distance two descriptors another criterion fusing two lines line feature matching fig performance lsd left original image right line features detected image lsd therefore paper seek improve lsd algorithm minimizing influence dividing line multiple segments particular merge line segments one straight line divided several parts line segment extracted lsd start point end point distinguished direction encoded fig distances two line segments based lbd line segment descriptor introduce geometric properties line perform effective line matching approach two successfully matched line features need satisfy following conditions angular difference two matched lines smaller given threshold length similar length min max overlapping length two lines greater loverlap certain value min fig comparative results different detectors left original lsd detector right proposed improved detector distance two lbd descriptors less certain value geometric representation line initialized two spatial points assume homogeneous coordinates respectively inhomogeneous coordinates represented coordinates constructed follows fig geometric interpretation four parameters updating orthonormal representation vector consisting direction vector line normal vector plane determined line origin since line four degrees freedom coordinates parameterized backend graph optimization extra degrees freedom increase computational cost cause numerical instability system thus bartoli proposed orthonormal representation minimum four parameters obtain orthonormal representation coordinates knk kvk orthonormal representation knk kvk line consists update orthonormal representation minimum four parameters date vector update specific geometric interpretation updated encapsulates vertical distance origin spatial line shown fig case fixed represented gray vector related rotation line around three axes drawn orange green blue note proposed visual slam system use orthonormal representation optimization minimal decoupled representation however steps coordinates used due convenience camera projection endpoints trimming line initialization raph ptimization oint ine easurements follows present detail line measurements incorporated visual slam system point measurements treated standard way example measurement models point line features use transformation matrix tcw denote transformation world frame camera frame consists rotation matrix rcw translation vector tcw shown first convert line world frame camera frame shown denoted representation coordinates line projected image described image plane according known intrinsic parameters camera noted normal components coordinates provide meaningful information projection error line represented distance two homogeneous endpoints matched line segment line image plane shown rcw tcw tcw rcw tcw rcw hcw rcw denotes matrix vector hcw represents transformation matrix line knc denotes projection matrix line denotes distance function camera pose tkw point position position line denoted vertices graph model two types edge edge edge constructed according data association errors encapsulated edges epk ktkw elk knc hcw stands coordinates point image denotes normal components coordinates simplicity omit conversion homogeneous coordinates inhomogeneous equations assuming observations obey gaussian distribution final cost function obtained inverse covariance matrices points lines robust huber cost functions optimization minimizes cost function ept elk epk elk jacobian line error known jacobian important using iterative approach solve graph optimization problem best knowledge first paper deriving analytical jacobains errors respect line parameters including jacobian respect small pose changes four dimensional vector updates orthonormal representation jacobian line error respect line given two endpoints matched line segment image recall projection line knc assuming orthonormal representation line world frame consists write jacobians directly hcw ith column difficult compute directly divide pose changes two parts translation part rotation part set zeros computing jacobian respect transformation matrix containing translation new line exp tcw tcw rcw tcw rcw tcw rcw rcw rcw tcw rcw rcw exp denotes exponential map lie algebras lie groups hence lie algebra easy deduce partial derivative rcw rcw process deduce similar except shows final result drops coordinate frame subscripts readability readers refer appendix details stacking jacobians obtain final jacobian finally jacobian error respect line parameters found using chain rule analytical jacobians available employ iterative algorithms solve graph optimization problem xperimental esults system implementation proposed visual slam system designed implemented based three main parallel threads see fig tracking local mapping loop closing global thread started finishing loop closing following briefly describe component focusing difference build associations perform guided search bound complexity enhance accuracy matching since line may partially observed projected line handled projected point fig shows simple example dash lines observed camera solid lines order ensure visibility projected line segments image plane propose culling based method described follow fig architecture proposed visual slam system using point line features tracking system uses rectified stereo image sequence input every input frame four threads launched extract point feature keypoints line feature keylines left right image parallel orb features applied point feature detection description line feature detected lsd described lbd descriptor two threads launched stereo matching features classified stereo monocular features according whether feature left image could find stereo match right image shown fig stereo matching lines performs described section monocular feature search match unmatched features keyframes finding matched feature triangulate feature way stereo features transform line world frame current frame according prior tkw compute two endpoints xsk xek discard line xsk xek behind camera one behind camera compute intersection plane line xik xsk xsk xek value depicted fig project two endpoints front camera image plane since projected line maybe lays across even image bound projected lines must dealt algorithm efficient line clipping algorithm retain orientation original line line matching done efficiently thanks restricted searching space binary descriptor last step decide whether current frame new keyframe adopt policy add conditions related line features fig partial observation line dash lines observed camera solid lines red points denotes intersection plane line fig workflow images motion estimation made two types tracking namely tracking last frame tracking local map former one gives initial pose estimation correspondences adjacent frame latter one refines pose much constraints current frame local map data association pose estimated using algorithm use constant velocity motion model predict camera pose prior tracking last frame prior known map points map lines last frame local map projected current frame local mapping new keyframe added connections current keyframe frames updated providing information local mapping triangulates map points lines removes outlier landmarks deletes redundant keyframe camera poses landmarks local map adjusted performing local optimization line parameterized infinite spatial line hence endpoints affect final optimization results however endpoints play important role matching visualizing system need maintain two endpoints line optimization done line current keyframe trimming corresponding line similar slslam loop closing global loop closing thread used reduce drift accumulated exploration loop detection loop correction loop detection try find candidate keyframes based technique bags words visual vocabulary trained offline point line features cluster orb features lbd features build vocabulary dbow respectively every input keyframe converted bag words vector stored online database similarity score two bag vector computed follow empirical weight coefficient related scenes similarity score point feature line feature find correspondences new keyframe candidate keyframe also refine correspondences time consistency test try compute transformation matrix epnp corresponding points ransac scheme failed alternatively compute method proposed using matching lines across two stereo views finally pose graph optimization launched correct loop finished global incorporated achieve optimal solution separate thread pixel added points endpoints lines captured images loop detection disabled display pose error clearly root mean square error relative pose error metric evaluate performance method fig shows estimated trajectory proposed system average result monte carlo experiments runs shown table translation rotation errors obtained scene lots point features result scene containing points scene comparable number points lines odometry based point feature performs better one using lines reason may error infinite long spatial line related normal vector line coordinates shown section matched point features produce constrains number lines table shows method based point features larger error line features scene points method based fusion points lines outperform results various experiments conducted synthetic scene accuracy time efficiency approach analyzed experiments algorithm run computer intel core memory linux operating system fig top oblique views estimated camera trajectory table rpe ethods based ifferent eatures point feature line feature feature rad rad fig synthetic scene lines variable number points synthetic data accurate data association synthetic scene experiment proposed verify correctness advantage introduced line feature optimization derived jacobian line error used optimization synthetic scene fig contains house total lines variable number points construction similar scene virtual stereo camera baseline moves around house collecting images pixels gaussian white noise variance real data scene experiment carried dataset kitti dataset comprehensive assessment approach presented article several open source approaches compared section including slslam plsvo presented paper complete point feature based slam system contains map reuse loop detection relocation slslam based straight line feature constructing scene composed straight lines relatively excellent line based slam system plsvo odometry using two endpoints represent spatial line performing matching fig shows images dataset fig shows results generated dataset fig shows trajectory map camera fig sample images used dataset fig comparison results dataset without loop closure top bottom row show top side views results fig results loop closure left results loop closure right results loop closure loop correction loop closure followed bundle adjustment plsvo poor performance dataset compare slslam proposed dataset provided ground truth degrees drift loop closure compared fair comparison disable loop detection thread use parameters point feature extraction image extract point features scale levels scale factor table rrors efore oop losure method errors loop closure slslam fig shows top side views reconstruction results three systems without loop closures respectively point zero coordinates starting point finishing point table shows drift loop closure translation observed table perform better demonstrate strength including constraints straight line slslam best performance meters error vertical direction reason account dataset contains scenarios reflective white walls windows floor etc time due influence ceiling lights point features prone mismatched bring big errors optimization process proposed approach set different weights error terms points lines consideration versatility component based point feature unstable performance low accuracy proposed system based combination point line features affected coincides synthetic scene experiment terms time efficiency execution time increase much features extracted parallel threads images dimensions feature extraction stereo matching requires system requires additional consideration line features dataset tracking thread achieve performance satisfy requirements fig results kitti dataset left map composed point line features right one frame extracted point line features also evaluate system kitti odometry benchmark sequences scenarios lines selected fig shows result kitti dataset system experiment compare loop detection modules disabled fair comparison experiment absolute trajectory error used evaluation criterion table iii shows results experiment rans rot represent translations rotations respectively smallest sequence marked table shown system acceptable performance several sequences performance improvement achieved compared original plsvo poor performance matching data association accumulated errors onclusions improve accuracy robustness visual slam present approach using point line features spatial line expressed orthonormal representation optimization process compactest decoupled form jacobians reprojection error respect line parameters also derived make good performance proved fusing two types features produce robust estimation synthetic scene robust visual slam also able work future investigate introduce inertial sensors system point line features source code module slslam unavailable include experiments kitti dataset table iii esults plsvo kitti dataset plsvo trans rot rad ate trans rot rad ate trans rot rad sequence sequence sequence ppendix appendix explain jacobian respect detail set zeros computing jacobian respect transformation containing rotation new line denoted exp denotes matrix vector process deducing properties rotation matrix used written directly eferences slam system monocular stereo cameras arxiv preprint rother linear reconstruction points lines planes cameras using reference iccv marzorati matteucci migliore sorrenti integration lines points visual slam uncertain projective emcr citeseer klein murray improving agility slam european conference computer vision springer koletschka puig daniilidis mevo stereo visual odometry intelligent robots systems iros international conference ieee zhang lee lim suh building map using stereo slam ieee transactions robotics vol song robust odometry using point line features proceedings ieee international conference computer vision ate robust stereo visual odometry probabilistic combination points line segments robotics automation icra ieee international conference ieee strasdat montiel davison monocular slam filter robotics automation icra ieee international conference ieee sola civera montiel impact landmark parametrization monocular points lines international journal computer vision vol bartoli sturm using lines representation triangulation bundle adjustment computer vision image understanding vol von gioi jakubowicz morel randall lsd fast line segment detector false detection control ieee transactions pattern analysis machine intelligence vol zhang koch efficient robust line segment matching approach based lbd descriptor pairwise geometric consistency journal visual communication image representation vol rublee rabaud konolige bradski orb efficient alternative sift surf computer vision iccv ieee international conference ieee woo park han beack line matching using geometric intensity data artificial intelligence computational intelligence aici international conference vol ieee bartoli sturm line motion matrix alignment line reconstructions computer vision pattern recognition cvpr proceedings ieee computer society conference vol ieee algorithm implementation theory numerical analysis springer liang barsky new concept method line clipping acm transactions graphics tog vol tardos bags binary words fast place recognition image sequences ieee transactions robotics vol fast relocalisation loop closing slam robotics automation icra ieee international conference ieee lepetit fua epnp accurate solution pnp problem international journal computer vision vol fischler bolles random sample consensus paradigm model fitting applications image analysis automated cartography communications acm vol pradeep lim egomotion estimation using assorted features international journal computer vision vol geiger lenz stiller urtasun vision meets robotics kitti dataset international journal robotics research vol
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nonparametric regression adaptive truncation via convex hierarchical penalty asad ali noah university washington dec january abstract consider problem regression potentially large number covariates propose convex penalized estimation framework particularly highdimensional sparse additive models proposed approach combines appealing features finite basis representation smoothing penalties estimation particular case additive models finite basis representation provides parsimonious representation fitted functions adaptive component functions posses different levels complexity hand smoothing spline type penalty component functions adaptive offer parsimonious representation estimated function proposed approach simultaneously achieves parsimony adaptivity computationally efficient framework demonstrate properties empirical studies real simulated datasets show estimator converges minimax rate functions within hierarchical class establish minimax rates large class sparse additive models proposed method implemented using efficient algorithm scales similarly lasso number covariates samples size introduction motivation consider first univariate function estimation pairs observations assume mean finite variance many proposals estimating local polynomials stone kernels nadaraya watson splines wahba others focus basis expansions estimators also known projection estimators arguably simplest among commonly used let response covariate vectors let kvkn modified referred empirical norm projection estimators solutions linear regression problems based set basis functions along truncation level specifically let matrix entries basis expansion estimate given proj argmin asymptotically balance bias variance allowed vary unfortunately choosing truncation level difficult practice depends properties smoothness choice basis functions usually chosen via split sample validation basis aharis department biostatistics ashojaie departments biostatistics nrsimon department biostatistics expansions hierarchically ordered measure complexity less complex etc projection estimators small would also give parsimonious representation projection estimation approach extends easily additive models hastie xip true underlying model believed form xij components model estimated using basis expansion component solving optimization problem argmin pkj estimate fbj problems often assumed many components popular choice scenario add sparsity inducing penalty basis expansion framework ravikumar solve argmin approach known sparse additive modeling spam practice truncation level used feature keep computation tractable even additive models widely different complexities strategy leads poor estimates scenarios moderate number observations issue often severely limits effectiveness predictive models built using spam addressing limitation one major motivations manuscript propose hierbasis penalized estimation method motivated projection estimator hierbasis truncation level determined rather prespecified hierbasis framework applied fit univariate multivariate models well additive models without sparsity also discuss extension hierbasis multivariate settings applied univariate problems hierbasis performs similarly standard basis estimator little regularization however additive sparse additive models hierbasis automatically chooses truncation level feature truncation levels often differ features based underlying complexity true vastly improve prediction accuracy model additionally allows maintain much parsimony possible estimating fbj illustrate advantages data example section using polynomial basis expansion average find features fbj linear quadratic cubic none selected truncation level larger hierbasis also computationally efficient applied problems thousands observations features addition hierbasis estimates attain minimax optimal rates standard smoothness assumptions univariate multivariate sparse additive models particular univariate hierbasis converges order degree smoothness similarly multivariate hierbasis attains rate sparse additive hierbasis suitable compatibility condition shown converge max log number even compatibility condition additive hierbasis consistent without convergence rate max logn figure examples basis functions natural hierarchical complexity polynomial trigonometric wavelet basis functions shown left center right panels respectively rest paper organized follows section gives formal description hierbasis univariate case well extension additive sparse additive models section contains analysis genomic data comparing hierbasis spam section gives efficient algorithm fitting hierbasis additive extension section present theoretical analysis hierbasis also applies general class sparse additive models section contains simulation study exploring operating characteristics hierbasis comparing performance estimation methods concluding remarks presenting section methodology estimation via basis expansion commonly used technique nonparametric regression basis expansion often truncated achieve parsimonious representations control tradeoff separately tuning truncation level parameter may feasible lowdimensional regressions approach becomes quickly infeasible additive models optimal truncation level requires searching subset proposal motivated need adaptive estimator select truncation level manner achieve goal penalized estimation formulation using novel penalty approach particularly suitable basis functions possess natural hierarchy basis functions become increasingly complex higher values examples basis functions include polynomial trigonometric wavelet basis functions depicted figure emphasize hierarchical nature proposed penalized estimation framework motivation based basis functions natural hierarchy refer hierarchical basis expansion estimator hierbasis hierbasis proposal consider first univariate case projection estimator equation noted section choosing truncation level key small result large bias large variance particular balance necessitates relates smoothness underlying proposal hierbasis addresses challenge consider instead complete basis using penalized regression framework choose truncation level specifically hierbasis estimator defined hier argmin denotes submatrix containing columns subvector containing entries tuning parameters hierarchical group lasso form zhao hierbasis penalty result hier hierarchical sparsity solution sufficiently large many entries given define induced truncation level minimal unlike simple basis expansion estimator truncation level prespecified hierbasis estimator determined two tuning parameters analogous smoothness parameter smoothing splines wahba number bounded derivatives used truncation level simple projection estimator practice using gives good results similar use cubic smoothing splines hand determines tradeoff theoretically optimal value practice suggest using split sample validation choose univariate setting simple basis expansion estimator truncation level chosen split sample validation likely adequate contrast hierbasis adds additional regularization function estimate addition choosing truncation level additional shrinkage may helpful indicated empirical results section however benefit generally relatively small true benefit hierbasis comes application additive multivariate problems described next additive hierbasis noted section projection estimator commonly used fit additive models often using set basis functions features ideally additive projection estimator obtained considering different truncation level feature small achieved using split sample validation searching combinations however number candidate models grows exponentially becomes quickly unwieldy often single used practice difficulty selecting truncation level primary limitation projection estimator additive multivariate models level smoothness component vastly different single truncation level result estimates many degrees freedom giving overly variable function estimates others insufficiently flexible estimates single choice truncation level thus lead poor regression estimates hierbasis proposal designed circumvent limitation projection estimators choosing truncation level models multiple covariate particular additive hierbasis straightforward extension univariate hierbasis defined solution argmin function estimates obtained fbj estimates hierarchically sparse additive hierbasis solution result specifically minimal addition major advantage additive hierbasis induced truncation level adaptive may different feature important characteristic mitigates major disadvantage simple projection estimators result additive hierbasis allows balance parsimony feature individually without exhaustive computational search advantage hierbasis simple projection estimators becomes even significant high dimensions instance popular spam estimator generally obtained using single truncation level noted result poor estimators similar spam sparse additive hierbasis additive models employs additional penalty yuan lin defined argmin defined important feature optimization problem sparse additive hierbasis tuning parameters two penalty terms linked link theoretically justified section briefly oracle choice tuning parameters gives estimates numerical experiments section corroborate finding show choice tuning parameters results strong predictive performance without requiring validation space tuning parameters spam sufficiently large sparse additive hierbasis gives sparse solution two estimators differ however nonzero estimates hierarchically sparse induced truncation level whereas nonzero complexity additional flexibility sparse additive hierbasis proves critical settings achieved without paying price computational sample complexity moreover tuning parameters additional flexibly sparse additive hierbasis achieved number tuning parameters spam relationship existing methods univariate hierbasis section builds upon existing penalized methods estimating regression functions popular penalized estimation method smoothing spline estimator wahba sets basis natural splines knots observed solves following optimization problem minimize derivative order smoothing spline eliminates dependence truncation level closed form solution however estimated functions piecewise polynomial splines degree knots result smoothing spline estimates parsimonious especially multivariate settings achieve parsimonious estimates mammen van geer use approach select knots spline functions locally adaptive regression splines use natural spline basis solve minimize defined proposal mammen van geer closely related recent computationally tractable trend filtering proposal kim tibshirani despite appealing properties univariate setting locally adaptive regression splines trend filtering computationally difficult extend sparse additive models even single feature neither estimator solution spam estimator overcomes difficulty using fixed truncation level components pointed main drawback spam nonzero components additive model level complexity recently proposed sparse partially linear additive model splam lou partly mitigates shortcoming setting thep nonzero components linear functions achieved using hierarchical penalty form coefficient linear term basis expansion depending value tuning parameters first term penalty sets entire vector coefficients jth feature zero whereas second term sets coefficients corresponding terms zero additive sparse additive hierbasis proposals section seen generalizations spam splam wherein complexity nonzero component determined specifically spam becomes special case sparse additive hierbasis weights set similarly orthogonal design matrix splam special case sparse additive hierbasis allows another level hierarchy weights set theoretical analysis section indicates addition improved flexibility choice weights hierbasis result optimal rates convergence analysis colitis data apply hierbasis logistic loss order perform classification using gene expression measurements details hierbasis logistic loss given section consider colitis dataset burczynski gene expression measurements peripheral blood mononuclear cells pbmcs sampled adults ulcerative colitis crohn disease available geo accession number gds aim use gene expression measurements distinguish two diseases given small sample size consider genes largest variance compare performances hierbasis spam lasso tibshirani splits data training test sets standardizing gene mean zero variance one training set choose tuning parameters using training set calculate misclassification rate test set also calculate sparsity model selected defined proportion fitted components identically zero use parametrization hierbasis given maximum number basis functions selected fitted component hierbasis spam fit multiple models basis functions computational reasons use full set basis vectors hierbasis instead used basis functions use smaller basis functions hierbasis discussed section misclassification error rates test set sparsity shown figure clearly show superior performance hierbasis spam hierbasis appears comparable lasso terms mse gains slight advantage sparsity addition fitted function hierbasis monotonic nearly parsimonious linear fits lasso demonstrated figure appendix plot fitted functions one split data also show spam estimates highly irregular would indicate complex relationships computational considerations extensions conservative basis truncation hierbasis proposal uses basis expansion basis functions practice reasonable never nonzero entries generally choice solution misclassification error hierbasis lasso spam basis spam basis spam basis spam basis spam basis spam basis spam basis spam basis spam basis spam basis sparsity hierbasis lasso figure top test errors different splits colitis data method bottom sparsity model selected different splits data method entries instead solve hier argmin long solution identical original proposal even identical long sufficiently large cbn constant theoretical properties maintained bound relies smoothness underlying choosing gives conservative upper bound independent underlying additionally discussed section using rather basis functions computational complexity decreases similar result holds sparse additive hierbasis argmin worth noting easier choose level truncation level simple basis expansion estimator simple basis expansion requires exact truncation level neither large small hand hierbasis requires basis small algorithm solving hierbasis sparse additive hierbasis appealing feature hierbasis efficiently computed fact using results jenatton hierbasis computed via coordinate descent algorithm begin optimization problem consider decomposition optimization problem equivalently written defining minimize equivalent solving minimize formulation directly apply results jenatton detailed algorithm reformulation also used efficiently solve sparse additive extension algorithm coordinate descent hierbasis procedure hierbasis initialize update max algorithm solving end return end procedure via block coordinate descent algorithm specifically given set estimates fix one vectors optimize vector using algorithm iterating convergence yields solution problem described algorithm appendix convergence computational complexity noted section closed form solution hierbasis optimization problem obtained one pass coordinate descent algorithm shown jenatton block coordinate descent algorithm sparse additive hierbasis extensively studied literature guaranteed converge global optimum convex problems solving problem requires decomposition matrix followed multiplication steps require operations respectively however steps needed sequence values additive hierbasis decompositions needed entire sequence proposition jenatton given optimization problem solved operations block update additive hierbasis requires matrix multiplication ujt followed solving proximal problem see appendix requires operations thus sparse additive hierbasis requires npk operations equal computational complexity lasso friedman computational complexity calculations indicate hierbasis sparse additive hierbasis solved efficiently next report timing results implementation hierbasis intel coretm ghz processor solving univariate hierbasis example takes median time seconds solving sparse additive hierbasis simulation setting section grid values takes median time seconds degrees freedom regression fixed design consider definition degrees freedom given stein cov ybi ybi fitted response values apply figure visual representation multivariate hierbasis penalty claim haris derive unbiased estimate solution optimization problem using decomposition section let max let denote first columns furthermore vector define arrive following lemma defined given lemma unbiased estimator degrees freedom trace diag diag diagonal matrix main diagonal multivariate hierbasis begin extending hierbasis penalty multivariate regression define multivariate basis expansion consider let xpp functions univariate basis functions consider following basis representation univariate case let matrix entries multivariate hierbasis estimator simply weights given wqk figure demonstrates multivariate hierbasis penalty identity function clear figure multivariate hierbasis natural extension univariate penalty fitted model multivariate polynomial degree choice basis functions multivariate hierbasis acts procedure selecting level complexity interaction models also follows multivariate hierbasis solved using algorithm single pass basis elements extension classification also extend hierbasis setting binary classification via logistic loss function let response logistic hierbasis obtained following modification arg min log exp given new predicted values given exp extension additive hierbasis binary response also defined similarly log exp arg min logistic hierbasis problem efficiently solved via proximal gradient descent algorithm combettes pesquet see appendix details theoretical results section investigate asymptotic properties hierbasis proving theoretical results hierbasis combine previously developed ideas empirical process theory metric entropy number novel results general convergence rates sparse additive models metric entropy hierarchical class particular new results section allow one establish convergence rates broad class penalized sparse additive model estimators compatibility condition component features rates match minimax lower bound estimation sparse additive models independent component functions established previously raskutti see corollary thus additive sparse additive hierbasis estimators hand assumptions component functions obtain rates additive analog convergence rates lasso chatterjee established theorem knowledge convergence rates previously derived sparse additive models finally also calculate entropy hierarchical class matching upper lower bounds lemma lemma new results allows establish univariate sparse additive estimators minimax within hierarchical univariate hierarchical sparse additive classes respectively rates begin stating two results literature establishing convergence rates present contributions sections firstly theorem yang barron establishes lower bound minimax rate subject certain conditions secondly framework establishing upper bound convergence rates given theorem van geer require slight generalization result van geer state prove appendix first introduce terminology notation entropy set set equipped metric subset smallest respect metric denote rthe function class respect metric measure fixed sample denote empirical measure use notation kqn theorem theorem yang barron consider model assume entropy condition holds function class min max minimum space measurable functions constant depends theorem theorem van geer consider model define arg min function class satisfy entropy condition constant probability least function exp max constant depends state theorems sake completeness results nonparametric literature allow establish convergence rates estimator using entropy bounds relevant function class following section establish entropy bounds hierbasis multivariate hierbasis penalty theoretical results hierbasis section prove minimax rates univariate multivariate hierbasis specializing theorems first introduce nonparametric function classes hierbasis present primary contribution section establishing entropy bounds univariate multivariate hierbasis function class using theorem results immediately establish lower bound minimax rate upper bound use theorem use upper bound truncation error function truncation level proof entropy results presented appendix completeness provide details upper bound appendix define following function class similarly define multivariate function class integers defined probability measure associated allow limiting case abuse notation define notation next subsection dedicated proving main condition theorem entropy appropriate function classes hierbasis entropy results hierbasis specialize theorems analysis hierbasis need characterize hierbasis function class defined establish upper bound next lemma lemma show calculation equivalent entropy calculation subset rkn respectively respect usual norm reduction allows use simple volume arguments existing results establishing entropy conditions lemma considers hierbasis penalty full generality penalty set weights gives similar reduction entropy calculations multivariate case little extra work class hierbasis respeclemma reduction rkn denote fnm tively multivariate hierbasis functions fnm allow limiting case fnm entropy hkn respect norm hkn secondly assume gram matrix finite maximum eigenvalue denoted denoting lemma establishes connections entropy results function classes interest easy see set max hkn proportional hkn proportionality constant depends respectively next lemma establishes univariate multivariate hierbasis weights upper bound upper bound need specialize theorem univariate hierbasis weights lemma upper bound region multivariate hierbasis weights constants lemma sufficient applying theorem invoke theorem need exact value entropy proportionality constant natural way achieve find lower bound entropy matches upper bound following lemma univariate lemma lower bound wkn region hierbasis weights multivariate hierbasis weights constants assume simplicity specializing theorems hierbasis following corollary establishes lower bound minimax rate estimating true function belongs function class consider three different choices hierbasis class multivariate hierbasis class sobolev class prove result use fact upper bound convergence rates found matches lower bound conclude estimator minimax corollary mth order hierbasis function class min max mth order multivariate hierbasis class weights defined min max finally mth order sobolev class fsob min max last step analysis next specialize theorem establish upper bound convergence rate univariate multivariate hierbasis estimators following corollary demonstrates number interesting points firstly note respect empirical norm defined estimators achieve minimax rate classes hierbasis minimax sobolev class fsob sob sobolev class well result also gives insight role corollary consider model mean zero noise define univariate multivariate hierbasis estimators fbuni arg min fbmulti arg min respectively penalty penalty assume maxk gram matrix bounded maximum eigenvalue denoted constant probability least exp buni max constants depend constant probability fsob least exp buni max constants depend assume define integer constant probability least exp bmulti max constants depend result demonstrates achieve usual rates long truncation level satisfies note since appropriate choice truncation level would gives conservative truncation level theoretical results sparse additive models section establish convergence rates sparse additive models terms general entropy condition raskutti proved lower bound minimax rates estimation sparse additive models assuming independent covariates completeness state result theorem first contribution oracle inequality upper bound prediction error additive models inequality establishes consistency estimators slow convergence rates specifically rates minimax lower bound raskutti proceed state compatibility condition leads two corollaries firstly establishes convergence rates order secondly automatically establishes minimax rates univariate regression special case additive model contributions section extend broad class estimators seen additive model analog theorem let true function independent random noise denote sparse additive approximation function call active set subset sample mean ensure identifiability assume consider estimator fbj fbp arg min penalty form kfj think smoothness penalty function theorem theorem raskutti samples sparse additive model iid iid class satisfying entropy condition assume covariates independent constant log min max fbj max minimum set measurable functions next state first key result section establishes oracle inequality additive models well slow rates convergence theorem assume model maxi constants assume entropy condition holds function class constant estimator log max probability least exp exp kfb kfj kfbj positive constants furthermore function class satisfies supf log max depends maxj ready establish fast rates convergence additive models using compatibility condition stated next compatibility condition say compatibility condition met set constant satisfying kfj kfj holds kfj skf corollary assuming conditions condition compatibility met probability least exp exp log kfbj fbj max constant depends maxj constants theorem corollary assuming conditions theorem compatibility condition holds trivially probability least exp exp constant depends simulation studies simulation univariate regression begin simulation compare performance hierbasis smoothing splines wahba trend filtering kim tibshirani smoothing splines trend filtering implemented packages splines core team genlasso arnold tibshirani respectively generate data using different choices function errors generated satisfies snr fixed ratio snr simulation consider fixed design also done facilitate comparison trend filtering become substantially slow random particularly covariates uniformly distributed closed interval consider four different choices denoted snr true functions follows exp sinh sin applied hierbasis values linear log scale applied smoothing splines grid values degrees freedom trend table average relative mse degrees freedom relative hierbasis order value greater indicates lower corresponding value hierbasis results presented averaged datasets standard error shown within parenthesis function method dof relative mse degree polynomial degree polynomial exponential function sine function smoothing splines first order trend filter second order trend filter third order trend filter smoothing splines first order trend filter third order trend filter third order trend filter smoothing splines first order trend filter second order trend filter third order trend filter smoothing splines first order trend filter second order trend filter third order trend filter filtering applied sequence lambda values automatically selected implementation hierbasis smoothing splines fix fit trend filters orders simulation setting plot mse function dof define mse fitted model also generate test set size ntest method find minimizes prediction error test set evaluate mse dof fitted model report relative mse dof hierbasis precise report ratios figure displays mse hierbasis smoothing splines trend filtering orders function degrees freedom also plot results fitting hierbasis hierbasis appears outperform competitors terms mse especially polynomials observe comparable performance exponential sine functions also provides empirical evidence theoretical results proved hierbasis converge rates comparable smoothing splines since functions considered simulation smooth expected see hierbasis converge fast competing methods figure shows examples fitted models fixed value dof see hierbasis seems perform well mostly robust changes value smoothing splines estimates unable well hierbasis number effective degrees freedom bottom panel figure surprising observe first order trend filter perform poorly due model misspecification simulation multivariate additive regression proceed simulation study illustrate performance hierbasis additive setting perform small simulation study compare performance additive hierbasis spam ravikumar spam implemented package sam zhao uses natural spline basis functions facilitate fairer comparison also implement spam using polynomial basis expansion refer due lack packages penalties meier splam lou defer comparison methods future work degree polynomial average mse average mse degree polynomial degrees freedom degrees freedom exponential function sine function average mse average mse degrees freedom degrees freedom figure average mse simulated datasets function degrees freedom true models given colored lines indicate results hierbasis order trend filtering order smoothing splines consider simulation setting meier modifications high dimensional data smaller ratio generate samples features data generated follows normal snr sin sin sin cos covariates uniform implemented parametrization sequence values decreasing linearly fix maximum number basis functions hierbasis implement spam basis functions surprising observe superior performance hierbasis terms mse figure however note figure hierbasis seems even outperform spam small lambda values complex models observe lower mse spam fewer basis functions low sparsity spam able control variance estimator small number basis functions used whereas hierbasis control variance controlling smoothness via penalty large lambda values obtain sparser models hence control variance however bias spam inflated using fewer basis functions figure show fitted functions spam hierbasis using value minimizes test set error spam basis functions mse mse lambda lambda figure average mse simulated datasets function hierbasis spam left hierbasis right colored lines indicate results hierbasis order spam basis functions spam basis functions conclusion paper introduced hierbasis novel approach regression high dimensional models recall original motivation regression especially additive models require estimator adapt function complexity way showed methods like spam splam unable effectively data adaptive proposals sparsity smoothness penalty meier come cost highly complex fitted models even simple underlying surfaces use hierarchical penalty allows adaptively fit simple models simple functions shown sections theoretical analyses section show hierbasis rates faster existing methods also establish fast convergence rates broad class sparse additive estimators sparsity smoothness penalty one special case similar result proved raskutti however considered independent component functions rkhs thus smoothness penalties norm hilbert space covered formulation package hierbasis available https implements methods described paper references arnold tibshirani genlasso path algorithm generalized lasso problems url http package version burczynski peterson twine zuberek brodeur casciotti maganti reddy strahs immermann spinelli schwertschlag slager cotreau dorner molecular classification crohn disease ulcerative colitis patients using transcriptional profiles peripheral blood mononuclear cells journal molecular diagnostics chatterjee assumptionless consistency lasso preprint available combettes pesquet proximal splitting methods signal processing algorithms inverse problems science engineering eds bauschke burachik combettes elser luke wolkowicz springer new york dumer covering ellipsoid equal balls journal combinatorial theory series friedman hastie tibshirani regularization paths generalized linear models via coordinate descent journal statistical software van geer empirical processes vol cambridge university press haris witten simon convex modeling interactions strong heredity preprint available hastie tibshirani friedman elements statistical learning springer new york jenatton mairal bach obozinski proximal methods sparse hierarchical dictionary learning proceedings international conference machine learning kim koh boyd gorinevsky trend filtering siam review lou bien caruana gehrke sparse partially linear additive models preprint available mammen van geer locally adaptive regression splines annals statistics meier van geer additive modeling annals statistics nadaraya estimating regression theory probability applications core team language environment statistical computing foundation statistical computing vienna austria url http raskutti wainwright rates sparse additive models kernel classes via convex programming journal machine learning research raskutti wainwright lower bounds minimax rates nonparametric regression additive sparsity smoothness advances neural information processing systems ravikumar lafferty liu wasserman sparse additive models journal royal statistical society series statistical methodology stein estimation mean multivariate normal distribution annals statistics stone consistent nonparametric regression annals statistics tibshirani regression shrinkage selection via lasso journal royal statistical society series methodological tibshirani adaptive piecewise polynomial estimation via trend filtering annals statistics evaluation unknown distribution density observations doklady gene expression value gene expression value change log odds change log odds change log odds change log odds gene expression value gene expression value figure plots fitted functions single split colitis data training test sets hierbasis lasso spam basis functions wahba spline models observational data siam watson smooth regression analysis indian journal statistics series yang barron determination minimax rates convergence annals statistics yuan lin model selection estimation regression grouped variables journal royal statistical society series statistical methodology zhao rocha composite absolute penalties family grouped hierarchical variable selection annals statistics zhao liu roeder sam sparse additive modelling url http package version additional figures simulation studies data analysis appendix present additional figures referenced section figure shows examples fitted functions one split dataset training test sets figure shows examples fitted models fixed value dof figure show fitted functions spam hierbasis using value minimizes test set error spam basis functions algorithms additive logistic hierbasis give algorithm additive hierbasis well algorithm logistic hierbasis use coordinate descent algorithm solving additive sparse additive hierbasis algorithm cyclically iterates features feature applies univariate solution detailed algorithm exact details given algorithm also give algorithm solving logistic hierbasis based proximal gradient descent begin let log exp denote derivative point algorithm presents steps solving algorithm additive logistic hierbasis similarly derived omitted interest brevity degree polynomial estimated function estimated function estimated function estimated function estimated function estimated function estimated function estimated function estimated function estimated function estimated function estimated function sine function exponential function degree polynomial figure scatterplots simulated data along true estimated functions top row includes plots simulated data along true function used generating data three rows show fitted functions method degrees freedom corresponding fitted model hierbasis smoothing splines trend filtering shown rows respectively trend filtering hierbasis shown green blue red available implementation order shown smoothing splines figure first component functions simulation study section estimates hierbasis shown green whereas spam fitted spam basis functions shown blue red respectively case tuning parameter leading smallest mse used algorithm block coordinate descent additive hierbasis procedure additive hierbasis max iter initialize max iter converged set update arg min end end return end procedure algorithm proximal gradient descent logistic hierbasis procedure logistic hierbasis initialize convergence select step size via line search update algorithm solving arg min end return end procedure proofs section proof lemma firstly final equality follows due orthonormality similarly show thus smallest hkn functions associated form smallest respect norm extended case proves first part secondly note associated functions thus smallest pkn cover respect metric since cover smallest cover since inequality holds select giving result proof lemma ellipsoid ekn show lemma dumer proved upper bound ellipsoids state appendix special case theorem yields desired upper bound shown corollary therefore similarly consider special case multivariate hierbasis weights corollary gives result proof lemma let integer note since define truncated hierbasis region simply viewing subset let hdw radius lemma hdw lower bound entropy ball obtained simple volume argument since hence hdw log log since inequality holds hdw log univariate case hence log log log log multivariate case argument slightly different due presence zero weights hence note assumption hence wqk implies hence finally since therefore hdw log log log log log log log last inequality follows fact details corollary univariate case pkn firstly select secondly note hierbasis estimator brevity drop dependence denote thus term bound hence keep term inequality sob truncation error note last inequality follows proof lemma result follows since multivariate case pkn assume take calculations univariate case truncation error note hence proof theorem recall fbj arbitrary univariate function class denote functions fbj proof theorem functions convenience simply write begin proof theorem basic inequality lemma basic inequality function following basic inequality solution kfbj additive function kfj proof kfb kfb fbin implies fbin kfb fbj kfb kfbj second term note leads kfbj lemma bounding term max log probability least exp constant depends proof lemma van geer exp result follows setting lemma bounding term max logn constant probability least exp fbj fbj positive constants proof firstly assumption cover kfj interested set firstly function min min means set cover set implies equivalently finally since since apply lemma class sufficiently large sup exp kfj union bound max sup kfj exp exp log log exp exp log finally show follows log log holds since log thus probability least exp constant fact fbj last inequality follows bernoulli inequality using active set far shown probability least exp exp following inequality holds kfb fbj kfb fbj notational convenience exclude term following manipulations active set right hand side rhs kfj kfj kfbj inequality holds decomposition fbj fbj left hand side lhs kfb kfbj fbj kfbj fbj fbj kfb kfbj since inequality follows triangle inequality fbj terms obtain inequality kfj implies kfb implies slow rates convergence max completes proof theorem next section prove oracle inequality fast rates via compatibility condition using compatibility condition recall compatibility condition whenever kfj kfj kfj skf assume compatibility condition prove corollary considering following two cases case case kfb holds hence compatibility condition hence function kfb kfb kfb kfb use inequality implies case kfb case max implies max constraining hierbasis penalty region recall following definitions rkn defined respectively lemma regions weights proof sufficient show pkn defined inclusion lemma region rkn akn proof let brevity denote bkn wkn wkn wkn implies figure demonstrate two lemma special case entropy results ellipsoids section establish entropy results ellipsoid circle allow establish entropy rates hierbasis penalty region since potentially arbitrarily large need way handle dimension turns done using simple argument demonstrate following theorem upper bound satisfies theorem dumer ellipsoid following inequality log ekn log figure demonstration lemma lemma special case blue region red black left right plots largest integer largest holds trivially integer corollary sobolev ellipsoids theorem let following upper bound constant depends proof firstly note definition let thus show result follows since hence log log second part use fact obtain log log log log log log log log log log log sterling inequality log log log log log log log log log log log log log log implies log log log corollary multivariate hierbasis theorem let wqk fixed following dimension define upper bound constant depends proof firstly since entropy hence restrict note must integer weights zero wqk definition second line follows inequality implies log similarly integer means term log log log log log log log log log log log log log hence log log log log log induction show log implies log log log log log finally note proof theorem proof definition leads following inequality via simple decomposition kfb obtain kfb kfn max thus basic inequality given max hence basic inequality either kfb implies result note implies thus invoke lemma van geer conclude probability least exp constants kfb define set sup set kfb means either kfb desired form consider case case kfb gives kfb kfb plugging right hand side solving kfb obtain kfb kfb recall definition case case kfb directly get kfb kfb thus shown set max shown probability least exp inequality max complete proof note kfb max max variation lemma van geer lemma assume supf kqn univariate function class given entropy bound constant max constant depending exp sup kqn proof corollary van geer sup kgkqn exp apply peeling device see van geer class let sup kgkqn kgkqn sup kgkqn sup kgk sup kgkqn exp exp exp
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version march symbol grounding living organisms realized artificial agents van hateren johann bernouilli institute mathematics computer science university groningen box groningen netherlands abstract system artificial intelligence usually relies symbol manipulation least partly implicitly however interpretation symbols represent ultimately left humans designers users system symbols acquire meaning system independent external interpretation unsolved problem grounding symbols obtained embodiment causally connecting symbols variables physical environment robot sensors effectors however causal connection produce representation aboutness kind symbols humans present theory explains humans living organisms acquired capability symbols variables represent refer something else theory shows reference physical objects also abstract objects even misguided errors reference objects subsequently abstract primary components theory biological context discuss conditions theory could implemented artificial agents major component theory strong nonlinearity associated potentially unlimited latter likely acceptable artificial systems remains unclear goals inherently serving selfreproduction aboutness goals could stabilized introduction much artificial intelligence relies manipulating symbols even system relies mostly connectionist processing variables brooks must ultimately interpreted symbolically human judges performance terms whether relates inputs outputs meaningful way transforming meaningful inputs meaningful outputs symbolic transformation newell simon conjectured information processing physical symbol system needed producing general intelligent action kind humans produce physical symbols present physical patterns within computing machine regarded forming formal system searle questioned sufficiency formal systems producing major characteristics symbols combinations symbols humans symbols used example human language typically refer something else need resemble symbol example word tree refer actual tree resemblance word symbols therefore refer arbitrary way agreed upon convention addition human symbols typically vague ambiguous meaning shifting depending context word tree mean quite different things forester genealogist computer scientist even within particular field symbols get many different interpretations depending context puzzling aspect symbols capability something else even something else concrete like specific tree quite abstract like tree general abstract like concept danger aboutness referring symbols used humans van hateren apparent lack aboutness symbols artificial systems searle main argument question sufficiency physical symbol systems obtaining general intelligence harnad discussed arbitrariness symbols purely formal system symbol grounding problem problem formal system clear symbols unequivocally connect physical reality proposed solution use connectionist system intermediate formal system sensory data material system sensors possibly effectors robot produce causal connection external reality internal symbolic processing embodying symbols way already discussed robot reply searle deemed sufficient solving problem aboutness also known intentionality philosophical jargon symbol grounding defined harnad focused physical reference specifically aim solve general problem aboutness article use symbol grounding wider sense following sun thus including grounding reference aboutness general sense symbol grounding artificial systems becomes problem produce reference meaning within artificial agent without meaning parasitic human interpretation although many solutions proposed often combining symbolic connectionist processing none appear really solve problem aboutness reviews see ziemke taddeo floridi rodriguez coradeschi bielecka recent theoretical computational work produced plausible explanation aboutness symbols evolved living organisms general humans particular van hateren start reviewing theory explain mechanisms involved abstract mechanisms biological context formulate form independent specific material implementation subsequently discuss prospects copying mechanisms artificial agents identify primary problems would need solved accomplishing biological symbol grounding understanding aboutness likely arisen organic evolution requires appreciating three key notions first basic darwinian theory differential reproduction organisms equipped reproduce effectively others likely become dominant appear naturally selected extension depends organism establishing internal model capability reproduce second notion specific stochastic role internal model produces connection internal external variables indirect represents rather causally connects thus producing primordial form aboutness third notion communication organisms reduce ambiguity results combining internal variables abstract symbols three sections explain notions xtending basi darw inian volution basic darwinian theory evolution natural selection consists reproductive loop loop figure organism called agent produces offspring incorporates slight structural changes rate reproducing given quantity called fitness ftrue figure fitness taken expected rate reproduction number offspring per unit time actual number offspring stochastic random realization rate integrated lifetime symbol grounding living artificial agents figure basic darwinian theory extension agents reproduce reproductive loop depending reproduction rate ftrue estimate fest ftrue stochastically drives structural behavioral changes hereditary learned agent loop fest ftrue agent evolutionary biology fitness often measured empirically retrospectively simple parameter estimated counting offspring however purpose modeling prediction fitness seen result complex dynamical process also denoted term fitness akin function value form argument fitness process timevarying properties environment agent including memory inputs expected reproduction rate output output vary continually instantly example temporarily dropping times resources scarce recovering resources become abundant finally going zero agent dies structure form ftrue extremely complex complex agents natural environment ftrue internal estimate mentioned refers either value structure either clear context specified explicitly agents typically occupy environment continually unpredictably changes compete one another limited resources limited materials limited energy limited space environmental change competition tend drive fitness changing thereby improving fitness agents lineages survive changes take place evolutionary time across generations form mutations also within lifetime single agent learning resulting behavioral change although latter changes usually transferred offspring unless cultural transfer learning still influences fitness defined instantly varying rate reproduction important stress power darwinian process originates fact reproduction multiplicative process fitness replacement level growth numbers fast exponential hand fitness remains replacement level particular line descending agents lineage eventually become extinct addition basic darwinian process directly depending ftrue second process promote evolutionary success utilizing fitness indirectly van hateren illustrated loop figure basic idea structural changes agents either across generations within lifetime controlled internal estimate fitness produced within agents estimate called fest figure subsequently drives variance structural change called variation figure inverse way depending value fest symbolized figure precise form subject evolutionary optimization traces van hateren right illustrate works variable lower trace modulates variance stochastic process upper trace way causing change therefore midway regular deterministic causation usual way things modeled science regular stochastic causation letting random generator drive downstream effects key point noise added deterministic variable multiplied noted fitness best seen process case fest physiological process within agent output rate reproduction case fest estimated rate reproduction however value fest taken present explicitly specific physiological variable implicitly distributed way within agent similarly producing structural change agent also seen distributed process many different structures behaviors could changed partial fitness effects inputs fest outputs structural change would need taken account changes weighted properly required system would hard design first principles readily amenable evolutionary optimization theory shown work evolutionary simulations van hateren structural changes evolutionary well agential timescales reason works understood intuitively follows ftrue high fest likely high well assumed reasonable estimate ftrue fitness high things going well agent reason change much little variation would beneficial facilitate future adaptation environmental change thus fitness high variation low fitness low agent trouble changing thereby likely maintaining low fitness agent descendants likely lead death extinction least average better strategy increase variation hope finding variant higher fitness desperate times call desperate measures variation random considerable chance change worse subsequently increasing variation even loop acts continually change worse decrease fest thereby increase variation however variant higher fitness encountered chance variation strongly reduced agent remain close beneficial state drifting away slowly least environmental change reduces fitness things start loop therefore adaptive optimization process reason works low probability success changing randomly compensated average prospect exponential growth fitness becomes high enough mechanism loop uses random change thus concerns structural change effects fitness foreseen yet many cases effects structural change foreseen least partly foresight possible example knowledge similar situations past available genetic memory established previous natural selection behavioral memory established previous learning perhaps obtained cultural transmission agent obviously make use foreseeable benefits fitness automatically stochastic variation needed loop figure handle parts structural change unknown effects although internally generated behavior agent could seen agency wide sense specific behavior produced loop seen agency narrow sense agency narrow sense involves internal intrinsic overall goal agent goal obtaining high fest addition rather special stochastic process generating behavior process generates new symbol grounding living artificial agents figure expansion relationship fest ftrue factors deriving environment factors deriving agent irrelevant factors fest internal structure behavior derives unpredictable way stochasticity without completely random stochasticity continually driven overall goal high fest elsewhere argue agency narrow sense source intrinsic behavioral freedom living organisms appear van hateren topic article interesting part theory figure establishment reference aboutness least primordial form internal fest somewhat arbitrary factors taken account making reasonable estimate value ftrue fixed indeed likely vary members species moreover fest part agent implies also fest subject structural change evolution learning nevertheless fest still must refer ftrue ftrue agents completely lack connection unlikely exist would survived previous natural selection would outcompeted agents adequate fest even weak inaccurate aboutness low probability agents would best belong small minority heading extinction argue next two sections fact fest stands refers ftrue source aboutness specific variables symbols agent details aboutness ftrue fest complex structure illustrated figure mentioned ftrue arises nature interaction environmental factors arising environment agential factors arising agent together factors history present retained indirectly genetic learned memory agents physically cause ftrue rather complex way also fest likely complex structure although complete model ftrue infeasible agent evolutionary drive make modeled fest good possible given resources agent better value fest mimics value ftrue better mechanism loop figure work increase ftrue way fest faithfully mimic value ftrue across wide range environmental conditions mimicking least structure ftrue structure even approximate van hateren agent direct access factors may relevant modeling fitness external factors derive must sampled environmental sensors accessing internal states processes may also require sensors least specialized subprocesses making implicit estimates relevant factors together estimated environmental factors est agential factors est physical physiological factors producing causing fest internal process agent however factors need relevant ftrue may errors structure fest factors est figure may derive irrelevant parts perhaps may primarily random although evolutionary pressure make fest free errors guarantee fest optimal sense evolution requires good enough flawless optimal errors may inevitable causally coupled factors important simple feasible way agent separate moreover error today may turn asset tomorrow vice versa aboutness fest respect ftrue figure produces aboutness factors est est respect corresponding broken arrows figure however also points several ways reference missing misdirected first may relevant factors ftrue figure completely missed estimated fest perhaps yet discovered factor perhaps would complicated take many resources incorporate one might call incomplete missed reference second estimate est may vary level accuracy vary adequately functional incorporation structure fest reflects true role structure ftrue accurate reference partial best finally factor may false irrelevant case still refers ftrue inherits aboutness overall aboutness fest respect ftrue depicted figure reference wrong factor figure aboutness illustrated figure explains several puzzling aspects aboutness first aboutness different simple physical connection fact somewhat fuzzy although ftrue fest physically connected thereby indirectly arbitrariness connection relative arbitrariness fest complex estimators like fest likely somewhat flexible different variants showing similar performance implies estimates specific factors est also flexible aboutness reference therefore flexible fuzzy well another puzzling aspect aboutness fact error pointing false even factors christiansen chater classical example mental aboutness thinking animal like unicorn seen possibility error built concept right low level factors depicted figure factors combined symbols symbols stabilized discussed section stabilization symbols primary problem agent literal model ftrue infeasible realistic situation complex agent deal natural environment filled complex agents constructing fest sufficiently adequate loop figure therefore requires heuristic strategies using rules thumb one strategy likely useful combining elementary factors abstract compounds tested actual utility example several factors compounded factors may combined compounded factor signifying danger compounded factor may useful approximately corresponds conjectured compounded symbol grounding living artificial agents figure elementary factors constituting fest compounded symbols synchronized aligned communication agents factor ftrue explains ftrue decreases danger increases recall ftrue expected rate reproduction indeed temporarily decreases danger even danger retrospect actually result damage realized rate reproduction compromised heuristic strategies may thus utilize rather abstract somewhat fuzzy factors conventionally factors called symbols symbols corresponding aspects ftrue thereby indirectly reality causally connected ftrue example agent left figure formed symbol signifying danger agent several simpler factors factors may include irrelevant ones factors agent believes signify danger without actually symbols formed single agent likely somewhat arbitrary many ways identify combine factors contributing symbol like danger also many ways include irrelevant marginally relevant factors single agents limited opportunities resources verify accuracy effectiveness symbols species sufficiently social cooperative may evolutionary advantage communicate symbols example symbol signifying impending danger however communicating symbols useful symbols different agents approximately aligned therefore communication drive synchronization stabilization symbols steels example agent right figure symbol signifies agent danger roughly similar identical way left agent communicating danger produces average joint fitness benefits agents incentive synchronize composition symbols reducing differences perhaps getting rid clearly irrelevant factors moreover utility symbol thus defined modeling typical ftrue agents species easily checked many agents sharing symbol implies also drive modify symbols increases utility accurately estimating ftrue symbols become considerably useful incorporated symbolic system human language systems facilitate synchronizing modifying symbols also enable additional efficacy flexibility symbolic expressions moreover environment includes agents partly responsible ftrue implies models agents incorporated fest van hateren thereby enables symbols part fest symbols part ftrue topics obviously form large complex field beyond scope article nevertheless current theory implies human grounding van hateren symbols form meaning words semantics general requires mere embodiment multimodal grounding bruni discussion conclusion abstracting biological context primary functional components theory presented following first strong nonlinearity exponential growth decline agent numbers lying heart reproduction loop figure nonlinearity depends parameter reproduction rate ftrue complex function environment including agents gradually changing agent structure runaway growth ultimately stopped automatic reduction ftrue resources become limiting average mechanism loop results agents sufficiently well matched environment reproduce least fast die process therefore appears optimize ftrue viewed retrospectively external point view actual explicit goals built process agents process naturally arises capability reproduce small changes combined inevitable scarcity resources strong nonlinearity driving process enables extension utilizes stochastic change agent structure driven modulated estimate ftrue estimate fest also depends environment agent structure way fully defined controlled within agent nonlinearity associated reproduction agents loop typically obtain higher ftrue agents without loop obtain extension therefore evolvable sustainable basic process theory requires fest evolved reasonable estimate ftrue values fest ftrue likely similar one another although ftrue fest thus coupled coupling way ftrue produced primarily external processes mostly independent way fest produced processes evolved learned within agent agent induces structural changes within increase ftrue likely though guaranteed increase fest well similarly agent induces structural changes within increase fest likely though guaranteed increase ftrue strength likelihood fest ftrue feature optimized previous evolution maintained continued evolution coupling fest ftrue implies agents must intrinsic drive increase fest increase ftrue average increasing ftrue main feature basic optimization process fest internal process within agent increasing fest actual goal agent van hateren fact agent ultimate goal specific tasks agent subordinate always serving fest guaranteed serve ftrue coupling fest ftrue perfect topic article important consequence theory fest ftrue aboutness arises fact fest ftrue tightly coupled somewhat arbitrary way control agent way constructs fest primary aboutness produced internalized version fitness within agent figure inherited factors agent environment including agents compose fest figure factors may may correspond accurately factors composing ftrue leading varying degrees symbol grounding living artificial agents accuracy aboutness possibility error reference items straightforward extensions basic theory consist combining factors abstract symbols synchronizing stabilizing symbols amongst communicating agents figure clear summary latter parts theory defining stabilizing symbols physical social grounding vogt cangelosi coradeschi organizing symbolic system may complex require anything goes beyond existing robotic technology indeed steels shown letting robots communicate colored samples produce consistent stable framework meaning symbols used technologically challenging part foundation theory requires system inherent exponential growth continually utilize estimate fitness fest modulating stochastic change structure evolutionary robotics floreano bongard normally experimenter decides ftrue defined example robotic designs move faster across plane lipson pollack granted reproduction therefore possibility mutate forms might move even faster theory presented goal would serving ultimate goal fast reproduction high ftrue fest evolving learning biological agents therefore switch whenever switching technically possible given physical physiological constraints constraints pathways open evolution learning whenever increases rate reproduction therefore autonomously abandon like moving fast switch new unpredictable reproduction served way full implementation following organic version theory would require solving several challenging problems first problem realize fully autonomous reproduction including autonomous gathering materials energy needed reproduction second problem designing initial structure sufficiently evolvability serve basic optimization mechanism loop figure continue indefinitely finally problem constructing initial version evolvable learning fest stochastically modulate structural change thereby serve extended optimization mechanism loop figure however successfully solving problems implies evolving artificial agents occasionally give rise exponential growth numbers plague waves extinction also occur occasionally would imply waste resources also competition organic life including humans artificial agents overall goal fast reproduction independent therefore likely unaligned human goals applies composing overall goal thus even autonomous reproduction functioning fest would feasible either materials similar different current biological ones would compelling reasons implement theory would alignment likely conflict human purposes theory also work agential goals kept constrained goals serve human interests clearly basic mechanism loop figure could work like example quickly moving robots given illustrates however agents evolving mechanism symbol grounding proposed lack loop fest goals symbols therefore remain parasitic human goals symbols symbols lack autonomous aboutness loop depends subtle stochastic process modulated stochasticity van hateren change fitness directly mechanism depending statistics variance structural change therefore function driven strong nonlinearity organic life nonlinearity comes exponential growth numbers implied produces strong selection nearly variants die continue also prevents complete extinction quickly multiplying numbers successful variants autonomous goal high fest unacceptable suggested fest must replaced another goal say gest principle possible example simulation arbitrary goal see figure van hateren example gest might defined high robot move quickly across place low gest slow moving would imply increased mutation rate figure however gest high task performed well successful variants must still selected protected extinction letting multiply words gest must coupled gtrue genuine reproductive fitness ftrue either multiplying must controlled purely human autonomous agents gest ftrue generally rather different value structure assigned goal gest little reproduction aboutness figures broken second issue gest much part agent structure anything else evolving learning agent therefore likely change along rest agent unless could somehow isolated organic life fest presumably distributed throughout agent see section isolating change option clear sustained isolation could realized artificial agents least conceivable however isolating fest change may hinder synchronization symbols depicted figure thereby prevent establishment maintenance useful symbolic system moreover isolation turns infeasible practice goal gest eventually align ftrue latter drives basic darwinian optimization process gest change fest gradually inevitably change may happen quite slowly agent little limited intelligence bacterium insect happen much faster agents intelligent high intelligence requires capacity change continually expand space possible unpredictable directions timescale evolution also timescale individual lifetime conclusion considerations clear goals kept constrained serve human goals similar concerns see bostrom even possible example isolating gest agent structures involved learning clear symbol grounding form aboutness could obtained gest would unrelated ftrue primary question another solution organic one found retains special stochastic properties loop would produce aboutness without issues sketched interesting see problem solution human ingenuity find references bielecka symbol grounding problem causal theory reference new ideas psychology bongard evolutionary robotics communications acm alignment human goals could controlled enforced case animal husbandry shows however agents involved would level intelligence requires symbolic reasoning enforcement would presumably unethical well resisted agents goals might actively sabotaged symbol grounding living artificial agents bostrom superintelligent motivation instrumental rationality advanced artificial agents minds machines brooks intelligence without representation artificial intelligence bruni tran baroni multimodal distributional semantics journal artificial intelligence research cangelosi grounding sharing symbols pragmatics cognition christiansen chater symbol grounding emperor new theory meaning proceedings annual conference cognitive science society hillsdale lawrence erlbaum associates coradeschi loutfi wrede short review symbol grounding robotic intelligent systems intelligenz floreano keller evolution adaptive behaviour robots means darwinian selection plos biology harnad symbol grounding problem physica lipson pollack automatic design manufacture robotic lifeforms nature newell simon computer science empirical inquiry symbols search communications acm hermosillo lara meaning artificial agents symbol grounding problem revisited mind machines searle minds brains programs behavioral brain sciences steels symbol grounding problem solved next vega symbols embodiment debates meaning cognition oxford oxford university press sun symbol grounding new look old idea philosophical psychology taddeo floridi solving symbol grounding problem critical review fifteen years research journal experimental theoretical artificial intelligence van hateren new criterion demarcating life origins life evolution biospheres van hateren origin agency consciousness free phenomenology cognitive sciences van hateren active causation origin meaning biological cybernetics vogt physical symbol grounding problem cognitive systems research ziemke rethinking grounding riegler peschl von stein eds understanding representation cognitive sciences new york plenum press
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modularizing specifying protocols among threads jongmans farhad arbab centrum wiskunde informatica cwi amsterdam netherlands jongmans centrum wiskunde informatica cwi amsterdam netherlands identify three problems current techniques implementing protocols among threads complicate impair scalability multicore software development implementing synchronization implementing coordination modularizing protocols mend deficiencies argue use languages dsl based existing models concurrency demonstrate feasibility proposal explain use model concurrency reo protocol dsl offers appropriate abstractions natural separation protocols computations describe compiler illustrate use examples introduction advent multicore processors new era began many software developers general nonnumerical applications harness power multicore processors need writing concurrently executable code instead traditional sequential programs notoriously difficult task currently popular tools technology alleviate burden implementing concurrent applications researchers started developing new techniques multicore programming examples include stream processing transactional memory synchronization however one rather aspect multicore programming received little attention sets rules interacting parties must abide communicate paper investigate new approach implementing protocols among threads many popular programming languages gppl feature threads concurrently executing program fragments sharing address space name gppls multithreading facilities fortran coarrays openmp pthreads openmp pthreads nsthread class visual basic namespace java thread class languages combined share roughly according tiobe indexes january statistics one conclude good portion sixty percent software developers encounters threads regularly consequently many developers benefit improvements existing techniques implementing protocols among threads consider improvements merely relevant sheer necessity current models languages apis libraries fail scale http http gay kelly eds programming language approaches software places eptcs jongmans arbab comes implementing protocols importantly refer scalability terms performance also terms aspects software development correctness maintainability reusability approach paper takes aspects account organization section identify three problems current techniques implementing protocols among threads section sketch abstract solution based general notion domainspecific languages section concretize approach one particular language namely reo section concludes paper problems implementing protocols threads prevail implementing concurrency programming languages gppl also provoke controversy programming threads would inflict unreasonable demands reasoning capabilities software developers partly due unpredictable ways threads interact typically one analyze ways threads may interleave consequently potentially paths may exist propose discard present notion threads unless improve ways programming many existing gppls support since seems unlikely change near gear efforts toward getting improvements particular interest lies solving problems implementing protocols among threads section identify three problems first sight writing computation code protocol code program using single language may seem natural indeed many popular gppls sufficient expressive power nevertheless consider inappropriate approach many cases typically language designers gear gppls toward implementing computations implementing protocols suitable level abstraction seems secondary concern consequently languages work well writing computation code developing protocol code concurrency constructs provide coincide concepts needed express protocols directly results two problems complicate writing code protocols problem implementing synchronization threads communicating using shared memory directly reading writing common address space must synchronize actions however implementing synchronization using primitives locks mutexes sempahores comprises tedious activity problem implementing coordination threads interacting structured ways according protocol require coordination ensure respect protocol however implementing coordination using constructs assignments produces code indirectly conveys protocol make tedious activity interestingly two problems common cause lack appropriate abstractions implementing communication interaction gppls example software developers specify thread sends two integers receives array rationals thread allocates shared memory performs pointer arithmetic developers specify communication two threads inhibits interaction among threads acquire release locks developers specify threads exchange data elements synchronously atomically threads wait monitor get notified believe programming languages enable developers implementing communication interaction focus logic protocols modularizing specifying protocols among threads import import public class main private linkedlist object buffer private semaphore notempty private semaphore notfull public main buffer new linkedlist object notempty new semaphore notfull new semaphore new producer new producer new consumer private class producer extends thread public run true object produce private class consumer extends thread public run true object consume figure java implementation example algorithm realization necessary synchronization coordination compilers addition two problems identified lack appropriate abstractions gppls causes third problem absence proper structures enforce least encourage good protocol programming practices programmers frequently succumb temptation isolating protocol code conceptually problem differs problems complicate writing code directly however perplex essentially everything else involved software development process although notions modularization separation concerns long histories computer science linguistic support application programming concurrency protocols scarcely received due attention modularization separation concerns driven development modern programming languages software development practices decades fact already early parnas attributed three advantages abiding principles time shortened separate groups would work module little need communication product possible make drastic changes one module without need change others possible study system one module nevertheless popular gppls enforce modularization protocols consequently dispersing protocol code among computation code comprises common practice implementing protocols among threads illustrate discuss java code figure based algorithm two producer threads produce data elements append queue buffer size concurrently consumer thread takes elements queue consumes queue buffer contains element producers append data consumer takes element queue skip methods produce consume easily one get gist protocol involved example producers reliably arbitrary elements consumer contrast one easily point coherent segments source code actually implement protocol indeed combination lines example thus isolated protocol distinct module separated concerns therefore advantages modularization identified parnas apply fact monolithic program figure suffers disadvantages dispersion groups work independently computation code protocol code monolithic jongmans arbab public interface protocol public void offer object public object poll public class main private protocol protocol public main protocol new new producer new producer new consumer private class producer extends thread public run true object produce private class consumer extends thread public run true object consume figure reimplementation program figure grams moreover one straightforwardly reuse computation code protocol code monolithic programs programs would require dissecting disentangling former small changes protocol require nontrivial changes throughout monolithic program instance suppose allow producers example send data elements alternating order implementing requires significant changes one study computation code protocol code isolation entangled reason protocol correctness one must analyze monolithic programs entirety impact shortcomings increases programs grow larger interaction among threads intensifies protocol complexity likely situation current multicore era problem modularizing protocol code lack appropriate abstractions gppls tempts developers disperse protocol code among code computational tasks case developers isolate protocols modules classes packages namespaces intermix computations practice makes independently developing maintaining reusing modifying testing verifying protocol code problematic impossible avoid disadvantages dispersion propose isolate protocol code separate modules languages instance one achieve encapsulating protocol logic separate class illustrate approach rewrote monolithic program figure modular program figure moved protocol code class see section implementation implements interface programs advantages modularization apply first groups develop protocol code implementations methods offer poll independently computation code moreover one easily reuse protocol implementations second changing protocol requires changing class implementing protocol class computation code however remains unaffected third analyze protocol isolation studying class implementing protocol class solution protocol dsls previous section explained lack appropriate abstractions complicates three aspects implementing protocols implementing synchronization problem implementing definition interface protocol figure serves present discussion every protocol methods offer poll general interface protocol provide methods computation code invoke executing protocol context present discussion offer poll seemed appropriate names modularizing specifying protocols among threads tion problem modularizing protocol code problem believe languages dsl offer solution problems definition language language programming language offers appropriate notations abstractions expressive power focused usually restricted particular problem domain languages dedicated implementation protocols solve problems definition moreover protocol dsls naturally force developers isolate protocols modules specifying protocols different language leads clear syntactic separation computation code protocol code consequently using protocol dsls following workflow secures advantages modularization developers write computation code application gppl developers specify protocols among threads protocol dsl dsl compiler compiles protocol specifications gppl code seemlessly integrating protocols computations benefits seem clear one question remains get protocol dsls must invent scratch kinds appropriate abstractions provide fortunately need design everything ground interaction concurrency received plenty attention theoretical computer science community past decades researchers investigated models concurrency many years petri nets process algebras led various formalisms synchronizing coordinating parties running concurrently actors agents components services processes believe models concurrency provide appropriate abstractions specifying reasoning protocols albeit equally well words protocol dsls look already exist however many formalisms lack sophisticated tool support particular kind compiler mentioned therefore consider implementing code generation tools among main goals efforts toward alleviating burden programming threads existing concurrency formalism focus reo protocol dsl one model concurrency particular interest reo model concurrency graphical syntax originally introduced coordinating components systems models concurrency reo solid foundation exist various compositional semantics describing behavior reo programs called connectors along tools analyzing includes functional analysis detecting deadlock reasoning nonfunctional properties computing guarantees declarative nature however distinguishes reo models concurrency using reo software developers specify interaction takes place indeed reo feature primitive actions sending receiving data elements rather reo considers interaction protocols constraints actions stark contrast traditional models concurrency reo notion interaction advantage formulate specify verify etc protocols one need even consider alternative sequences actions give rise using reo computational threads remain completely oblivious protocols compose coordinate interactions within concurrent application code contains concurrency jongmans arbab syntax semantics figure syntax semantics port automaton mergerwithbuffer syntax semantics figure syntax semantics port automaton alternatorwithbuffer primitive semaphore operations signals mutex even direct communication sole means communication computational thread consists actions performs ports construct application one composes set threads together protocol identifying ports computational threads appropriate nodes reo connector implements protocol reo compiler generates proper multithreaded application code proceed follows section explain main concepts reo three example connectors represents protocol one use example section section discuss compiler reo example protocols figures show three example connectors reo programs graphs nodes arcs refer channels refer nodes admit operations boundary nodes draw open circles figures intuitively one interpret graph representing reo connector follows data elements dispatched input boundary nodes output operations move along arcs nodes replicate multiple outgoing channels along output boundary nodes input operations fetch groups input output transport activities may take place atomically importantly communicating parties perform operations boundary nodes connector remain oblivious connector routes data parties fetch dispatch data elements know elements come connector figure specifies protocol one embedded java code figure explain behavior connector named mergerwithbuffer best discussing port automaton describes semantics figure shows automaton derived automatically figure every state corresponds internal configuration mergerwithbuffer every transition describes step protocol specified mergerwithbuffer transitions carry synchronization constraint set containing nodes data element passes atomic protocol step thus initial state mergerwithbuffer data element passes either nodes nodes every element passes subsequently arrives buffer capacity represent buffer rectangle figure buffer remains full data elements pass case admissible step results element stored buffer leave buffer pass figure shows another connector named alternatorwithbuffer one use consumer example contrast mergerwithbuffer alternatorwithbuffer forces producers modularizing specifying protocols among threads foo syntax foo semantics figure syntax semantics constraint automaton synchronize represented edge nodes dispatch data elements simultaneously connector allows case data element dispatched passes node enters horizontal buffer concurrently data element dispatched enters diagonal buffer next protocol step data element horizontal buffer leaves buffer passes node subsequently data element diagonal buffer leaves buffer passes enters horizontal buffer finally data element horizontal buffer originally dispatched leaves buffer passes thus alternatorwithbuffer first synchronizes producers second offers data elements alternating order consumer figure shows third connector named one use consumer example protocol specified connector differs two significant ways mergerwithbuffer alternatorwithbuffer first difference relates lack synchronization unlike alternatorwithbuffer force producers synchronize dispatch data elements similar alternatorwithbuffer however orders sequence data elements arrive consumer second difference relates exhibits zigzagged edge figure represents filter channel call filter constraint data elements satisfying filter constraint propagate filter channel example assume simple filter constraint namely foo means data element passing equals string foo words producer dispatches foo buffer becomes filled foo otherwise filter loses dispatched data element means essentially producer wasted turn general filter channels facilitate specification protocols whose execution depends content exchanged data port automata express semantics connectors filter channels need stronger formalism constraint automata support richer transition structures port automata instead synchronization constraint transitions carry also data constraint expression data passing particular nodes look like protocol step figure shows constraint automaton describes semantics omitted nonboundary nodes symbols refer content buffer transition fires means buffer contains value exchanged transition means content buffer passes transition alternatively one formulate filter constraints regular expressions patterns see jongmans arbab compiling reo java next discuss use reo dsl implementing protocols present early version compiler simpler explain although focus java emphasize generality approach nothing reo prevents compiling reo fortran visual basic compile anything need means implement drawings connectors use reo ide purpose called extensible coordination tools consists collection eclipse including editor drawing connector diagrams hood stores manipulates diagrams xml documents xml documents serve input compiler detailed next previously introduced reo terms data elements move connector unlike dataflow programming compiling connectors kind distributed application therefore may seem obvious choice nodes naturally map processing elements cores connections processing elements serve channels however approach several shortcomings first network topology hardware may correspond topology connector want deploy second emulating reo channels hardware connections requires additional computations processing elements connected destroys original idea mapping reo channels hardware connections third achieving global atomicity synchronization exclusion emerging connector requires complex distributed algorithms algorithms inflict communication processing overhead deteriorating performance short construing connectors topology literally seems bad idea context compilation instead compiler compiles connectors based constraint automaton semantics ships many common channels including figures combining primitive act composition automatically compute larger connectors use open source library compiler input xml document specifying connector compiler first computes corresponding subsequently annotates java identifiers finally produces java class using ntlr stringtemplate technology one use resulting class java class using compiling connectors conveniently abstract away nonboundary nodes illustrate compilation process suppose want compile mergerwithbuffer use example section drawing mergerwithbuffer using feed corresponding xml document compiler tool automatically generates java class mergerwithbuffer based semantics mergerwithbuffer precisely mergerwithbuffer objects run state machines representing protocol specified mergerwithbuffer figure shows parts java class generated compiling mergerwithbuffer discuss salient aspects class mergerwithbuffer extends class thread line running connectors thread enable proactively sense environment operations concurrent hardware ample cores run connectors always exist instances mergerwithbuffer listen three ports line grant computation threads access boundary nodes mergerwithbuffer interaction computation threads mergerwithbuffer object occurs latter ports computation threads perform ports turn suspend threads http modularizing specifying protocols among threads public class mergerwithbuffer extends thread current state private state current private enum state full empty boundary nodes connector private port private port private port check synchronization data constraints process writes takes constructs mergerwithbuffer public mergerwithbuffer port port port update memory cells initialize data constraints update state current lock get pending takes set take takesond abort return random number generator selecting transitions random random new random lock get pending writes set write writesona abort return data constraints connector checks memory cells connector access makes transition state empty state full firing private void tfromemptytofulla runs state machine modeling mergerwithbuffer public void run true switch current case switch case tfromemptytofulla break case tfromemptytofullb break break case break abort makes transition state empty state full firing private void tfromemptytofullb aborts transition unlocking may locked private abort figure parts java class generated compiling mergerwithbuffer see also figure erations succeed technically ports extend concurrent data structures called synchronization pairs containing pending writes another containing pending supporting admissible locking schemes see class syncpoint exposes following methods lockandgetwrites locks returns set pending write operations line lockandgettakes locks returns set pending take operations line unlockwrites unlocktakes unlock sets writes takes lines performandunlock write performs specified write operation first parameter unlocks set pending write operations line performandunlock take object performs specified take operation first parameter passing data element take second parameter unlocks set pending take operations line overridden method run implements state machine corresponding input compilation process lines main loop never terminates line iteration synchronization points resemble channels jongmans arbab public class implements protocol private port new port private port new port private port new port private map thread port threads new concurrenthashmap thread port public new mergerwithbuffer public object poll return public void offer object thread thread thread synchronized thread thread figure class randomly selects transition line going current state line collect code responsible making transitions separate methods lines important step process making transition consists checking synchronization data constraints line manner mergerwithbuffer object employs locking scheme growing phase acquires locks set pending writes port providing access input node line set pending takes port providing access output node line mergerwithbuffer object locks sets boundary nodes actually occur constraints investigation later shrinking phase releases locks lines phases mergerwithbuffer object checks constraints investigation hold fires corresponding transition transporting data elements accordingly removing operations involved sets locked lines use class mergerwithbuffer example section incorporate implementation class encountered line figure figure shows implementation line specifies producer performs blocking write operation port assigned line specifies consumer performs blocking take operation rest consists initialization code latter characterizes use reo protocol dsl implementations protocol interface serve wrappers compiled connectors encapsulating protocol logic change figure respects protocol specified alternatorwithbuffer requires nontrivial modifications across source code contrast straightforwardly implement class implements protocol replace line figure fact would differ line figure would construct alternatorwithbuffer object instead mergerwithbuffer object similarly use protocol specified shows using reo easily change protocols without affecting computation subsection demonstrates feasibility modularizing protocols implementing protocol dsl remark approach preclude use dedicated implementations certain parts protocol instance consider buffer mergerwithbuffer compiler implements buffer using shared memory explicit locks transparent software developers using reo though suppose architecture deploy program features also hardware transactional memory htm approach allows one write dedicated precisely handwritten computation code protocol code generated reo compiler communicate shared ports ports change replacing one connector another thus unless number ports changes syntactically valid program remains syntactically valid modularizing specifying protocols among threads implementation buffers exploits htm subsequently replace standard buffer implementation dedicated thus besides constructs default approach offers developers flexibility applying languages necessary concluding remarks current work focuses improving compiler instance classes currently generated compiler execute sequentially parallelize rather straightforwardly checking ports appropriate operations concurrently different transitions however speculate get even better performance instead optimize semantic level wish decompose automata smaller automata execute concurrently without synchronization preserving original semantics see preliminary results another potential optimization involves scheduling formal semantics connectors provide tangible information scheduling threads exploiting information yield substantial performance gains hopefully improvements make approach competitive alternative lower level approaches also terms performance recent years session types entered realm programming recent work includes although session types comprise valuable new technique programming threads wonder abstractions provided suffice still consider interesting development especially since reo feature types extending reo session types would comprise significant improvement finally although focused implementing protocols among threads paper compiler presented proven useful also domain web service orchestration references farhad arbab reo coordination model component composition mscs farhad arbab puff magic protocol gul agha olivier danvy meseguer editors formal modeling actors open systems biological systems lncs springer christel baier marjan sirjani farhad arbab jan rutten modeling component connectors reo constraint automata scico mordechai semaphores principles concurrent distributed programming edition philip bernstein vassos hadzilacos nathan goodman two phase locking concurrency control recovery database systems roughly first write htm implementation buffers second remove connection nodes figure third generate code subconnector containing nodes subconnector containing node latter consists node however semantics reo replace equivalent connector two synchronizing nodes thus two compiled connectors finally place implementation buffers two compiled connectors active entity dedicated buffer implementation performs writes takes runs operates fourth party involved protocol interestingly real communicating parties remain oblivious introduction fourth party optimization eliminate active entity performs writes takes merging functionality behavior nodes jongmans arbab sara capecchi mario coppo mariangiola sophia drossopoulou elena giachino amalgamating sessions methods languages generics tcs choi christopher lewis study common pitfalls simple programs proceedings sigcse arie van deursen paul klint joost visser languages annotated bibliography acm sigplan notices mariangiola sophia drossopoulou dimitris mostrous nobuko yoshida objects session types edsger dijkstra role scientific thought selected writings computing personal perspective texts monographs computer science springer simon gay vasco vasconcelos ravara nils gesbert alexandre caldeira modular session types distributed programming proceedings popl maurice herlihy eliot moss transactional memory architectural support data structures acm sigarch computer architecture news kohei honda aybek mukhamedov gary brown chen nobuko yoshida scribbling interactions formal foundation raja natarajan adegboyega ojo editors distributed computing internet technology lncs springer kohei honda vasco vasconcelos makoto kubo language primitives type discipline structured programming chris hankin editor programming languages systems lncs springer kohei honda nobuko yoshida marco carbone multiparty asynchronous session types proceedings popl raymond dimitrios kouzapas olivier pernet nobuko yoshida kohei honda eventful sessions java chris hankin editor ecoop programming lncs springer jongmans farhad arbab overview thirty semantic formalisms reo sacs jongmans dave clarke procedure splitting processes application coordination eptcs jongmans francesco santini mahdi sargolzaei farhad arbab hamideh afsarmanesh automatic code generation orchestration web services reo flavio paoli ernesto pimentel gianluigi zavattaro editors cloud computing lncs springer christian koehler dave clarke decomposing port automata proceedings sac edward lee problem threads computer david parnas criteria used decomposing systems modules cacm terence parr definitive antlr reference building languages pragmatic bookshelf
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greedy blind calibration method compressed sensing unknown sensor gains valerio cambareri amirafshar moshtaghpour laurent jacques feb image signal processing group catholique louvain belgium realisation sensing modalities based principles compressed sensing often hindered discrepancies mathematical model sensing operator necessary signal recovery actual physical implementation amply differ assumed model paper tackle bilinear inverse problem recovering sparse input signal unknown unstructured multiplicative factors affecting sensors capture compressive measurement methodology relies collecting snapshots new draws sensing operator applying greedy algorithm based projected gradient descent principles iterative hard thresholding explore empirically sample complexity requirements algorithm testing phase transition show practically relevant instance problem compressive imaging exact solution obtained snapshots index sensing blind calibration iterative hard thresholding optimisation bilinear inverse problems ntroduction implementation practical sensing schemes based compressed sensing often encounters physical nonidealities realising mathematical model sensing operator whose accuracy paramount attaining highquality recovery observed signal among nonidealities focus case compressive measurement affected unknown multiplicative factor sensor gain focus sensing model diag input signal independent identically distributed random sensing matrices respective snapshots measurements obtained applying sensing matrix reason acquisition partitioned snapshots cleared uncalibrated sensing model unknown set gains remains identical throughout snapshots whose value unknown hence sensing model bilinear retrieving quantities given measurements bilinear inverse problem bip note practically realised compressive imaging schemes using snapshot parallel acquisition convolving input signal one authors funded belgian part study funded project lter ense random masks detailed sensor gains calibrated presence noise strong taking snapshots allows blind calibration without missing instance signal due calibration process showed previous contributions proved instances sensing matrices entries rigorous definition see either unstructured endowed subspace models solved simple suitably initialised projected gradient descent objective number measurements ensuring recovery exact solution shown linear sample complexity dimensions unknowns log factors referring findings paper focus case single input signal representation known basis leverage involved model simply resort hard thresholding operator iterate former algorithm turning greedy scheme proposed greedy approach allows blind calibration actual schemes additional requirement methodology set snapshots collects sufficient amount information emphasis assessing least empirically sample complexity reduced function sparsity log factors hence provided sufficiently sparse show empirically total amount measurements lower still recovering related work blind calibration sensor gains tackled recent literature starting initial approaches uncalibrated sensor networks recently radiointerferometry context algorithms proposed cope model errors interestingly algorithms use sparse known subspace models several input signals rather random draws sensing operator typically feasible optical schemes moreover works attain sample complexity results grant exact recovery first hereafter given two functions indicates constant work proposing provable guarantees using single sparse input signal introduced ling strohmer based lifting approach problem improved guarantees outlined main drawback approach computational complexity given corresponds semidefinite programming former contributions showed nonconvex approach could provide exact recovery guarantees computational advantages respect lifting approaches inspired methodologies sanghavi sun used closely related problem phase retrieval concerns related task blind deconvolution recent approaches bip adopt similar schemes alternating minimisation yet targeting general context blind calibration therefore subject different requirements conditions encountered independently ling strohmer proposed linear least squares settings including covered study practical solver blind calibration sensing model sparse signal priors identifiability bip extent solution uniquely unambiguously determined given completeness refer reader recent contributions aspect contributions outline paper extends algorithm devised account sparse model signal domain fundamental prior whereas sparse models gains could inapplicable drawn random sensor manufactured thus adopt greedy algorithm enforce sparsity detail empirical performances function bip dimensions findings presented follows sec introduce problem propose greedy algorithm based hard thresholding algorithm studied numerically sec iii focus empirical phase transition problem dimensions vary simulate practical case blind calibration compressive imaging sec conclusion drawn afterwards reedy onvex pproach lind alibration solving argmin minimisers scaling anyway unrecoverable adopted constraint also serves fixes one admitted solution kgk control iterates algorithm assume orthonormal basis thus enforce sparsity aim solve scaled probability simplex vector ones begin scaling points argmin problem would due bilinear nature union kdimensional canonical subspaces differs solver devised assumed support supp given known subspace model proceed devise algorithm solving accounts two constraints firstly consider always verified scaling exists value quantifies deviation gains ideal case equal note also avoids component null would correspond losing measurements corresponding sensor hence gains subset closed convex set search start compute projected gradient initial approach blind calibration problem involved defining simple euclidean loss kdiag initialise exact sparsity level stop criteria met line searches signal estimate zhk gain estimate end algorithm blind calibration iterative hard thresholding diag diag ensures steps taken theory would use projection operator ensure gradient step still belongs convex set however start initialisation observed algorithm remain inside convergent conversely diverge independently presence thus practically use projector necessary devising guarantees denotes denotes orthogonal complement vectors projection matrix figure empirical phase transition alg increases top bottom increases left right function fixing report estimated contour levels probability successful recovery exceeds value indicated curve secondly typically done greedy algorithms instead adopting proxy sparsity iteratively enforce evaluating gradient diag diag applying gradient step hard thresholding operator sets entries argument operator heart iterative hard thresholding iht allows enforce signaldomain sparsity finally initialisation ppchoose backprojection unbiased estimate previous considerations approach version blind calibration iterative hard thresholding summarised alg line searches reported step computed closed form crucial accelerate algorithm albeit fashion could optimised cost may yield faster convergence see subject future improvement iii mpirical hase ransition propose extensive experimental assessment phase transition explore effect problem dimensions successful recovery signal gains varying generating random instances problem configurations detail drawn standard gaussian random drawn uniformly random drawn gaussian random matrices let algorithm run given relative change signal gain updates measure probability successful recovery max trials corresponds early termination algorithm convergent reach exact solution provided let run sufficient number convention indicates generation matrix vector gaussian entries mean variance true signal recovery provided iht rsnrx recovery provided bciht rsnrx true gains recovery provided rsnrg figure numerical example blind calibration compressive imaging test image detail tous les jours magritte charly herscovici kind authorization artwork retrieved intended fair use comparison original retrieved signal gains reported respectively iterations results reported fig terms contour levels function theoretical sample complexity result grants provable convergence still study already appreciate effect increasing fixed sparsity levels mild effect region sharpen transition region typically observed standard moreover appreciate impact increasing transition region keeping ratios fixed larger values region algorithm fails converge almost surely rapidly reduced moreover reported red curve matches roughly follows contours shape experiments highlight empirical evidence collected context correctly suggests agrees previous finding log see proposition structure leveraged gains one snapshot always needed algorithm collect sufficient amount information thus interpreting results expect widely used rule thumb blind calibration method converge instances let able recover furthermore sufficiently low total undersampling factor lind alibration ompressive maging proceed apply practical case process rgb image dimension made sparse orthonormal wavelet basis coefficients acquired means gaussian random sensing matrices experiment could carried matrix ensembles bernoulli sensing matrices results substantially unaltered since sparsity level chosen test image high simulate acquisition sensor array use snapshots meet requirements method thus gains retrieved scheme could revert benefiting improved model accuracy provided blind calibration gains set draw uniformly random run rgb channels separately relative change signal gain estimates falls quality data reported worst case among colour channels causes algorithm run iterations achieving estimate rsnrx rsnrg quality estimates observed fig see beneficial effect blind calibration use accelerated version iht given exact sparsity level snapshots corresponding sensing matrices form standard model concatenated vertically hence accelerated iht attempts recover estimate neglecting model error algorithm converges iterations local minimiser whose rsnrx modest performances seen directly fig comparison blind calibration algorithms explored since choice using single sparse input multiple snapshots specific framework nevertheless note computational complexity algorithm competitively low amounts iht plus additional projected gradient step gain domain per iteration proof convergence iht local minimiser devised expect provable convergence results fashion lead bound sample complexity ensures retrieval exact solution onclusion proposed novel approach blind calibration based use snapshots multiple draws random sensing operator greedy algorithm enforces sparsity steps resulting gradient descent nonconvex objective approach capable achieving within snapshots perfect recovery signal gains computationally efficient fashion hence conclude sensor calibration cause concern sensing scheme introducing modality follows using method could viable option cope model errors envision method may used blind calibration imaging sensors well distributed sensor arrays networks suitably modified allow compressive sensing presented empirical evidence phase transition algorithm rigorous convergence guarantee subject current study presented future communication eferences donoho compressed sensing ieee trans inf theory vol herman strohmer general deviants analysis perturbations compressed sensing ieee sel top signal vol romberg compressive sensing random convolution siam imag vol bjorklund magli parallel compressive imaging architecture acquisition ieee picture coding symposium pcs liutkus martina popoff chardon katz lerosey gigan daudet carron imaging nature compressive imaging using multiply scattering medium sci vol bahmani romberg compressive deconvolution random mask imaging ieee trans comput imaging vol dumas lodhi bajwa pierce computational imaging highly parallel architecture challenges solutions opt express vol mar hayat torres armstrong cain yasuda statistical algorithm nonuniformity correction arrays appl vol cambareri jacques blind calibration method randomised sensing strategies international workshop compressed sensing theory applications radar sonar remote sensing cosera haze approach blind calibration linear random sensing models arxiv preprint submitted information inference journal ima vershynin introduction analysis random matrices compressed sensing theory applications cambridge university press balzano nowak blind calibration networks sensors theory algorithms networked sensing information control springer lipor balzano robust blind calibration via total least squares ieee international conference acoustics speech signal processing icassp repetti birdi dabbech wiaux optimization algorithm joint dde calibration imaging radio interferometry arxiv preprint zhu leus giannakis total leastsquares perturbed compressive sampling ieee trans signal vol parker cevher schniter compressive sensing matrix uncertainties approximate message passing approach asilomar conference signals systems computers bilen puy gribonval daudet convex optimization approaches blind sensor calibration using sparsity ieee trans signal vol friedlander strohmer bilinear compressed sensing array asilomar conference signals systems computers caltagirone blind sensor calibration using approximate message passing stat mech theory vol ling strohmer biconvex compressive sensing inverse vol ahmed recht romberg blind deconvolution using convex programming ieee trans inf theory vol jung krahmer blind deconvolution compressed sensing international workshop compressed sensing theory applications radar sonar remote sensing cosera soltanolkotabi phase retrieval via wirtinger flow theory algorithms ieee trans inf theory vol apr sanghavi ward white local convexity solving systems quadratic equations results sun wright geometric analysis phase retrieval ieee international symposium information theory isit jul ling strohmer wei rapid robust reliable blind deconvolution via nonconvex optimization arxiv preprint mansour kamilov multipath removal online blind deconvolution ieee international conference acoustics speech signal processing icassp mar lee tian romberg fast guaranteed blind multichannel deconvolution bilinear system model arxiv preprint lee junge bresler blind recovery sparse signals subsampled convolution ieee trans inf theory vol ling strohmer via linear least squares arxiv preprint choudhary mitra identifiability bilinear inverse problems ieee international conference acoustics speech signal processing icassp sparse blind deconvolution done ieee international symposium information theory isit jun lee bresler identifiability blind deconvolution subspace sparsity constraints ieee trans inf theory vol jul identifiability bilinear inverse problems applications subspace blind gain phase calibration ieee trans inf theory vol blumensath davies iterative hard thresholding compressed sensing appl comput harmon vol blumensath accelerated iterative hard thresholding signal vol
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reinforcement learning probabilistic model predictive control feb sanket kamthe department computing imperial college london abstract based reinforcement learning seen rapid advancements recent times especially advent deep neural networks however majority autonomous algorithms require large number interactions environment large number interactions may impractical many applications robotics many practical systems obey limitations form state space control constraints reduce number system interactions simultaneously handling constraints propose modelbased framework based probabilistic model predictive control mpc particular propose learn probabilistic transition model using gaussian processes gps incorporate model uncertainty longterm predictions thereby reducing impact model errors use mpc find control sequence minimises expected cost provide theoretical guarantees optimality transition models deterministic approximate inference planning demonstrate approach achieve data efficiency also principled way constrained environments introduction reinforcement learning principled mathematical framework autonomous learnproceedings international conference artificial intelligence statistics aistats lanzarote spain pmlr volume copyright author marc peter deisenroth department computing imperial college london ing control policies learning process one distinguishing features despite many recent advances main limitation current algorithms remains data inefficiency required number interactions environment impractically high example many approaches problems state spaces fairly benign dynamics require thousands trials learn data inefficiency makes learning real systems without taskspecific priors impractical prohibits approaches challenging scenarios promising way increase data efficiency without inserting prior knowledge learn models underlying system dynamics good model available used faithful proxy real environment good policies obtained model without additional interactions real system however modelling underlying transition dynamics accurately challenging inevitably leads model errors account model errors proposed use probabilistic models explicitly taking model uncertainty account number interactions real system substantially reduced example authors use gaussian processes gps model dynamics underlying system pilco algorithm propagates uncertainty time planning learns parameters feedback policy means policy search achieves unprecedented data efficiency learning control policies scratch pilco algorithm data efficient shortcomings learning feedback policies needs full planning horizon stabilise system results significant computational burden requires specify parametrised policy priori often hundreds parameters handle state constraints control constraints enforced using differentiable squashing function applied sanket kamthe marc peter deisenroth rbf policy allows pilco explicitly take control constraints account planning however kind constraint handling produce unreliable predictions near constraint boundaries paper develop algorithm data efficient require look full planning horizon handles constraints naturally require parametrised policy theoretically justified key idea reformulate optimal control problem learned models equivalent deterministic problem idea similar reformulation allows exploit pontryagin maximum principle find optimal control signals handling constraints principled way propose probabilistic model predictive control mpc learned models propagating uncertainty time mpc formulation allows plan ahead relatively short horizons limits computational burden allows control applications approach find optimal trajectories constrained settings offers increased robustness model errors unprecedented data efficiency compared state art tions aico model uses approximate inference known locally linear models probabilistic trajectories obtained reformulating stochastic optimal control problem divergence minimisation implicitly linearise transition dynamic via moment matching approximation contribution contributions paper following propose new deterministic formulation probabilistic mpc learned models uncertainty propagation planning reformulation allows apply pontryagin maximum principle pmp planning stage probabilistic mpc gps using pmp handle control constraints principled fashion still maintaining necessary conditions optimality proposed algorithm theoretically justified optimal control theory also achieves data efficiency maintaining probabilistic formulation method handle state control constraints preserving data efficiency optimality properties related work recent survey model based robotics highlights importance models building adaptable robots instead dynamics model zero prior mean used paper rbf network linear mean functions proposed accelerates learning facilitates transferring learned model simulation real robot even implicit model learning beneficial unreal learner proposed learns predictive model environment auxiliary task helps learning mpc transition models predictive control used boiler building control model uncertainty discarded predictive variances used within scheme actively reject periodic disturbances although setting similarly authors used prior model additive noise model improved episodically authors considered mpc problems models posterior mean used ignoring variance planning mpc methods deterministic models useful model errors system noise neglected problem optimal control application optimal control theory models based dynamics employs structure transition model explicit assumption control affinity linearisation via locally quadratic controller learning via probabilistic mpc consider stochastic dynamical system states admissible controls actions state follows markovian dynamics unknown transition function tem noise diag paper consider setting seek control signals minimise expected cost terminal cost stage cost associated applying control state assume initial state gaussian distributed data efficiency follow strategy learn model unknown transition function use find optimal controls minimise every application control sequence update learned model newly acquired experience refers fact control signals independent state state feedback incorporated sanket kamthe marc peter deisenroth section summarises model learning step section details obtain desired trajectory probabilistic transition model learn probabilistic model unknown underlying dynamics robust model errors particular use gaussian process prior plausible transition functions probabilistic model regression finite number function values jointly gaussian distributed fully specified mean function covariance function kernel inputs dynamics given tuples corresponding targets denote collections training inputs targets respectively furthermore assume gausby sian rbf squared exponential covariance function exp signal variance diag diagonal matrix trained via standard procedure evidence maximisation make standard assumption gps target dimension transition function independent given hyperf training targets parameters training inputs yields predictive distribunew test input tion diag predictive dimensions control find desired control sequence follow procedure proposed use learned model predict evolution state given control sequence compute corresponding expected cost find control sequence minimises expected cost following detail steps predictions obtain state distributions given control sequence iteratively predict dxt making deterministic gaussian approximation using moment matching approximation shown work well practice contexts computed closed form using gaussian kernel key property exploit moment matching allows formulate uncertainty propagation deterministic system function mean covariance deterministic control signal define moments distribution blkdiag zet equivalently written deterministic system equation optimal control sequence find optimal sequence first compute expected cost using gaussian approximations obtained via given control sequence second find control sequence minimises expected cost following detail steps computing expected cost compute expected cost sum expected immediate costs choose expectation partial derivatives computed choices include standard quadratic polynomial cost also costs expressed fourier series expansions radial basis function networks gaussian basis function sanket kamthe marc peter deisenroth similar allows define deterministic mappings onto correthat map mean covariance sponding expected costs remark optimisation turns sparse however optimisation via value function dynamic programming valid unconstrained controls address practical shortcoming define pontryagin maximum principle allows formulate constrained problem maintaining sparsity detail sparse structure constrained dynamics problem section feedback control mpc thus far presented way efficiently determining controller however controller stabilise system therefore essential obtain feedback controller mpc practical framework interacting system mpc determines control trajectory starting current state first control signal applied system system transitions update model newly available information mpc procedure turns controller implicit closedloop feedback controller repeated steps ahead current state typically mpc even allows section provided algorithmic framework probabilistic mpc learned models underlying system dynamics explicitly use uncertainty predictions following section justify using optimal control theory additionally discuss account constrained control signals principled way without necessity control signals depend variables neighbouring time index depends variables index furthermore allows explicitly deal constraints states controls following detail solve optimal control problem ocp pmp learned dynamics deterministic uncertainty propagation additionally provide computationally efficient way compute derivatives based maximum principle facilitate discussion first define notation practical control signals often constrained formally define class admissible controls piecewise continuous functions defined compact space definition fairly general commonly used signals satisfy requirement applying admissible controls deterministic system dynamics defined yields set admissible controlled trajectories define tuple control system single admissible control trajectory unique trajectory pair called admissible controlled trajectory define control system formulation centre piece pontryagin approach ocp vector viewed lagrange multiplier dynamics constraints associated ocp successfully apply pmp need system dynamics unique solution given control sequence traditionally interpreted system deterministic interpretation considered limitation pmp paper however exploit fact approximation deterministic operator similar projection used yields deterministic system equations map moments state distribution time moments state distribution time theoretical justification bellman optimality principle yields recursive formulation calculating total expected cost gives sufficient optimality condition pmp provides corresponding necessary optimality condition pmp allows compute gradients expected cost variables state distribution would work equivalently framework local solution apply pmp need extend important characteristics odes system particular need show existence uniqueness local solution difference equation existence solution need satisfy difference equation entire horizon uniqueness need system one sanket kamthe marc peter deisenroth singularity system equation via approximation following lemma moment matching mapping lipschitz continuous controls defined compact set proof based bounding gradient detailed supplementary material existence uniqueness trajectories moment matching difference equation given lemma solution exists unique proof sketch difference equations always yield answer given input therefore solution trivially exists uniqueness directly follows theorem apply due lemma theorem requires system function deterministic see appendix due system dynamics follows directly given control sequence unique pontryagin maximum principle dynamics lemmas definition controlhamiltonian state pmp control system follows theorem let admissible controlled trajectory defined horizon optimal exists vector satisfying following conditions equation vector solution discrete difference equation transversality condition endpoint adjoint vector satisfies minimum condition every find optimal controls arg minimisation problem possesses additional variables variables interpreted lagrange multipliers optimisation capture impact control whole trajectory hence variables make optimisation problem sparse dynamics compute multipliers closed form thereby significantly reducing computational burden minimise expected cost detail calculation section remark optimal control problem aim find admissible control trajectory minimizes cost subject possibly additional constraints pmp gives optimality conditions admissible controlled trajectories generalised handle additional state control constraints remark hamiltonian constant unconstrained controls dynamics equals everywhere final time fixed remark linear dynamics proposed method generalisation ilqg moment matching transition implicitly linearises transition dynamics time step whereas ilqg explicit local linear approximation made linear quadratic cost write lqg case shown theorem iterate successive corrections linear approximations obtain ilqg efficient gradient computation definition hamiltonian efficiently calculate gradient expected total cost time horizon write accumulated cost bellman recursion since control impacts future costs via derivative total cost given comparing expression definition hamiltonian see make tion obtain remark minimum condition used find optimal control hamiltonian minimised admissible control set min sanket kamthe marc peter deisenroth implies gradient expected longterm cost efficiently computed using hamiltonian next show tution valid entire horizon terminal cost valid transversality condition time steps differentiate yields identical equation hence setting pmp implies gradient descent hamiltonian equivalent gradient descent total cost algorithmically setting find optimal control sequence follows given initial random control sequence follow steps described section determine corresponding trajectory ally compute lagrange multipliers forward propagation note traditionally equations propagated backward find multipliers given cost function determine hamiltonians find new control sequence via gradient descent method using return exit converged use sequential quadratic programming sqp bfgs hessian updates lagrangian sqp partially separable function pmp separation explicit via hamiltonians function variables index leads hessian sqp lagrangian structure exploited approximate hessian via within bfgs experimental results evaluate quality algorithm two ways first assess whether probabilistic mpc leads faster learning compared pilco current state art terms data efficiency second assess impact state constraints performing task consider two benchmark problems underactuated fully actuated tasks pilco algorithm date constraint double pendulum constraint figure state constraints benchmarks position cart constrained left side wall angle inner pendulum enter grey region cart pole system system freely swinging pendulum mounted cart balancing task solved using linear model system state space consists position cart cart velocity angle pendulum angular velocity horizontal force applied cart starting position pendulum hangs downwards objective automatically learn controller swings pendulum balances inverted position middle track constrained statespace constraint experiment place wall track near target see fig wall along force limitations requires system swing right side double pendulum system robot arm links lengths two actuators joint state space consists angles angular velocities torques limited starting position links downwards position objective learn control strategy swings balances inverted position constrained doublependulum constraint angle inner pendulum motion range spin see fig constraint blocks clockwise system underpowered swing clockwise first without violating constraints baselines compare approach following baselines pilco algorithm algorithm flavour predictive variances discarded due lack exploration approach within pilco policy search method learn anything useful already demonstrated include baseline average independent experiments every algorithm initialised first random trajectory performance differences algorithms therefore due different approaches controller learning induced exploration data efficiency benchmark experiments double pendulum use exact saturating cost penalises euclidean distance tip outer pendulum target position setting pilco performs well fig shows controller blue approach complete task fewer trials pilco method red repeated trials see learns faster reliably pilco particular zerovariance approach solve task high probability three trials seconds first trial random pilco needs two additional trials reason approach model uncertainties works mpc context within policy search setting include every observed state transition immediately dynamics model makes mpc fairly robust model errors even allows deterministic models simple settings fig highlights proposed approach yellow requires average six trials define success pendulum tip closer target position ten consecutive time steps pilco trial sec per trial success trials general setting follows algorithms start single short random trajectory used learning dynamics model used predict state differences learned dynamics model used determine controller based iterated moment matching applied system starting model learning controller learning application controller system constitute trial trial model updated newly acquired experience learning continues success sanket kamthe marc peter deisenroth pilco trial sec per trial figure performance algorithms error bars represent standard error double pendulum blue consistently outperforms pilco red mpc approach yellow terms data efficiency mpc approach works well task fails task attribute inability explore state space sufficiently well experience achieve success including first random trial pilco requires four additional trials whereas mpc approach completely fails setting reason failure deterministic predictions poor model complicated state space allow sufficient exploration also observe robust variations amongst trials experiments proposed requires pilco experience report unprecedented learning speed benchmarks even settings pilco performs well identify two key ingredients responsible success learning speed approach ability immediately react observed states adjusting plan augment training set model fly soon new state transition observed updated every time step properties turn crucial early stages learning tip outer pendulum closer target sanket kamthe marc peter deisenroth success experiment pilco pilco double pendulum table state constraint violations number trials resulted state constraint violation corresponding trial data shown fig trial sec less per trial constraint success pilco trial sec less per trial double pendulum constraint figure performance constraints error bars represent standard error double chance constraints blue method able consistently solve problem expected violations constraint yellow fails pilco red violates state constraints struggles complete task little information available ignored updates dynamics model approach would still successfully learn although learning efficiency would slightly decreased state constraints scenario pilco struggles setting state space constraints modify tasks setting tasks symmetric impose state constraints way one direction feasible system place wall near target position cart see fig double pendulum constraint angle inner pendulum motion range spin see fig constraints constitute linear constraints state use quadratic cost penalises euclidean distance tip pendulum target along implicit linearisation makes optimal control problem implicit rollout violates state constraint immediately abort trial move next trial use experimental data efficiency experiments state constraints implemented expected violations xlimit chance constraints xlimit fig shows mpcbased controller chance constraint blue successfully completes task small acceptable number violations see table expected violation approach considers predicted mean yellow fails complete task due repeated constraint violations pilco uses saturating cost little hope learning quadratic cost partial success completing task struggles especially initial trials due repeated state violations one key points observe table incorporation uncertainty planning crucial successful learning use predicted means determine whether constraint violated learning reliably safe incorporation predictive variance however results significantly fewer constraint violations conclusion discussion proposed algorithm based probabilistic mpc learned transition models using gaussian processes exploiting pontryagin maximum principle algorithm naturally deal state control constraints key theoretical underpinning practical algorithm optimal control problem uncertainty propagation via moment matching deterministic optimal control problem mpc allows learned model updated immediately leads increased robustness respect model inaccuracies provided empirical evidence framework theoretically sound also extremely data efficient able learn settings hard state constraints one critical components approach incorporation model uncertainty modelling planning complex environments model sanket kamthe marc peter deisenroth uncertainty drives targeted exploration additionally allows account constraints way important early stages learning references bellman dynamic programming princeton university press princeton usa bischoff koller markert knoll learning throttle valve control using policy search proceedings european conference machine learning knowledge discovery databases bischoff van hoof mchutchon rasmussen knoll peters deisenroth policy search learning robot control using sparse data proceedings international conference robotics automation bock plitt multiple shooting algorithm direct solution optimal control problems proceedings ifac world congress budapest pergamon press boedecker springenberg wulfing riedmiller approximate optimal control based sparse gaussian process models symposium adaptive dynamic programming reinforcement learning clarke optimization analysis society industrial applied mathematics cutler efficient reinforcement learning robots using informative simulated priors proceedings international conference robotics automation deisenroth efficient reinforcement learning using gaussian processes kit scientific publishing deisenroth fox rasmussen gaussian processes learning robotics control transactions pattern analysis machine intelligence deisenroth rasmussen pilco approach policy search proceedings international conference machine learning diehl lecture notes optimal control estimation girard rasmussen quinonerocandela gaussian process priors uncertain ahead time series forecasting advances neural information processing systems grancharova kocijan johansen explicit stochastic predictive control combustion plants based gaussian process models automatica griewank toint partitioned variable metric updates large structured optimization problems numerische mathematik pannek stability suboptimality using stabilizing constraints nonlinear model predictive control theory algorithms pages springer hennig optimal reinforcement learning gaussian systems advances neural information processing systems jaderberg mnih czarnecki schaul leibo silver kavukcuoglu reinforcement learning unsupervised auxiliary tasks international conference learning representations klenske zeilinger hennig gaussian predictive control periodic error correction transactions control systems technology kocijan modelling control dynamic systems using gaussian process models advances industrial control springer international publishing lee srinivasa mason robust optimal control uncertain nonlinear dynamical systems mackay introduction gaussian processes neural networks machine learning volume pages springer berlin germany mayne rawlings rao scokaert constrained model predictive control stability optimality automatica minka family algorithms approximate bayesian inference phd thesis massachusetts institute technology cambridge usa minka expectation propagation approximate bayesian inference proceedings conference uncertainty artificial intelligence sanket kamthe marc peter deisenroth mnih kavukcuoglu silver rusu veness bellemare graves riedmiller fidjeland ostrovski petersen beattie sadik antonoglou king kumaran wierstra legg hassabis control deep reinforcement learning nature naidu naidu optimal control systems crc press jordan pegasus policy search method large mdps pomdps proceedings conference uncertainty artificial intelligence nghiem jones demand response modeling control buildings gaussian processes proceedings american control conference nocedal wright numerical optimization springer opper winther tractable approximations probabilistic models adaptive tap mean field approach physical review letters ostafew schoellig barfoot collier nonlinear model predictive control improve mobile robot path tracking journal field robotics ostafew schoellig barfoot robust constrained nmpc enabling reliable mobile robot path tracking international journal robotics research pan theodorou probabilistic differential dynamic programming advances neural information processing systems process regression journal machine learning research raiko tornio variational bayesian learning nonlinear hidden models model predictive control neurocomputing rasmussen williams gaussian processes machine learning mit press cambridge usa rawlik toussaint vijayakumar stochastic optimal control reinforcement learning approximate inference proceedings robotics science systems ledzewicz geometric optimal control theory methods examples volume schneider exploiting model uncertainty estimates safe dynamic control learning advances neural information processing systems silver huang maddison guez sifre van den driessche schrittwieser antonoglou panneershelvam lanctot dieleman grewe nham kalchbrenner sutskever lillicrap leach kavukcuoglu graepel hassabis mastering game deep neural networks tree search nature sutton barto reinforcement learning introduction mit press cambridge usa todorov efficient computation optimal actions proceedings national academy sciences united states america pan theodorou kontitsis sample efficient path integral control uncertainty advances neural information processing systems todorov weiwei generalized iterative lqg method feedback control constrained nonlinear stochastic systems proceedings american control polydoros nalpantidis survey reinforcement learning applications robotics journal intelligent robotic systems toussaint robot trajectory optimization using approximate inference proceedings international conference machine learning pontryagin mishchenko boltyanskii gamkrelidze mathematical theory optimal processes wiley yahya kalakrishnan chebotar levine collective robot reinforcement learning distributed asynchronous guided policy search rasmussen unifying view sparse approximate gaussian sanket kamthe marc peter deisenroth appendix sequential quadratic programming lipschitz continuity use sqp solving optimization problems nlp form lemma moment matching mapping lipschitz continuous controls defined compact set proof lipschitz continuity requires gradient bounded gradient derivatives computed lytically first show derivative bounded defining obtain state dimensions qdi exp qdi size training set dynamics ith training input corresponding gradient given last elements min lagrangian associated nlp lagrange multipliers sequential quadratic programming sqp forms quadratic taylor approximation objective linear approximation constraints iteration min lagrange multipliers associated equality constraint ones defined control hamiltonian hessian matrix computed exploiting block diagonal structure introduced hamiltonian moment matching approximation following law iterated expectations target dimensions obtain predictive mean qdi efa mfa mfa let examine individual terms sum rhs given trained constant definition qdi contains ponentiated negative quadratic term bounded since positive definite qan entries inverse determinant defined bounded finally computed using standard results multiplying makes qdi remaining term integrating gaussians given product matrix reguz lar inverse exists bounded constant qai function since compact also conclude vector difference finite overall proves locally exp lipschitz continuous lemma define difference training input mean test input distribution sanket kamthe marc peter deisenroth computing predictive covariance matrix requires distinguish diagonal elements elements using law total variance obtain target dimensions varf xat xat xat xbt respectively known offdiagonal terms contain additional term covf xat xbt conditional independence assumption models different target dimensions covary given start computation covariance matrix terms common diagonal entries law iterated expectations obtain xat xbt xat xbt maf mbf conditional independence xat xbt given using definition mean function obtain xat xbt using standard results gaussian multiplications integration obtain entries qij qij exp zij zij define zij taken hence entries fully determined references deisenroth fox rasmussen gaussian processes learning robotics control transactions pattern analysis machine intelligence
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macgyver test framework evaluating machine resourcefulness creative problem solving apr vasanth sarathy matthias scheutz tufts university medford usa abstract current measures machine intelligence either difficult evaluate lack ability test robot capacity open worlds propose novel evaluation framework based formal notion macgyver test provides practical way assessing resilience resourcefulness artificial agents introduction consider situation suit covered lint lint remover resourceful reason roll duct tape might good substitute solve problem lint removal peeling full turn worth tape backwards onto roll expose sticky side around roll rolling suit pick lint type everyday creativity resourcefulness hallmark human intelligence best embodied television series macgyver featured clever secret service agent used common objects around like paper clips rubber bands inventive ways escape difficult yet current proposals tests machine intelligence measure abilities like resourcefulness creativity even though exactly needed artificial agents robots agents even home helpers robust resilient ultimately autonomous paper thus propose evaluation framework machine intelligence capability consisting practical tests inventiveness resourcefulness resilience specifically introduce notion macgyver test practical alternative turing test intended advance research society place high value human ability solve novel problems remain resilient beyond media patent system publication systems additional examples rewarding creative problem solving elegance solution background turing test progeny alan turing asked whether machines could produce observable behavior natural language humans would say required thought people turing suggested interrogator unable tell long conversation machine whether dealing machine person conclude machine thinking turing intend test rather prediction sorts cooper van leeuwen nevertheless since turing others developed tests machine intelligence variations turing test address common criticism easy deceive interrogator levesque designed reading comprehension test entitled winograd schema challenge agent presented question ambiguity referent pronoun possessive adjective question asks determine referent ambiguous pronoun possessive adjective selecting one two choices levesque feigenbaum proposed variation turing test machine tested team subject matter specialists natural language conversation feigenbaum tests attempted study machine ability produce creative artifacts solve novel problems boden bringsjord bringsjord sen riedl extending capabilities beyond linguistic creative harnad total turing test suggested range capabilities must expanded full set robotic capacities found embodied systems harnad schweizer extended incorporate species evolution development time proposed truly total turing test test individual cognitive systems whether species candidate cognitive architecture question capable evolutionary achievement schweizer finding turing test variants helping guide research development many proposed approach specific goals designed couched toy problems representative task cohen research communities benefited greatly approach focused efforts towards specific machine capabilities like object recognition automatic scheduling planning scene standing localization mapping even many public competitions challenges emerged tested machine performance applying capabilities image recognition contests machine learning contests competitions even tested embodiment robotic capacities combining multiple tasks example darpa robotics challenge tested robot ability conduct tasks relevant remote operation including turning valves using tool break concrete panel opening doors remove debris blocking entryways unfortunately turing test variants well taskbased challenges sufficient true measures autonomy autonomy requires ability integrated embodied system interact environment achieve goals solving problems limited resources available none tests interested measuring sort intelligence capability sort relevant practical standpoint macgyver evaluation framework proposed evaluation framework based idea creativity intended answer question whether embodied machines generate execute learn strategies identifying solving seeminglyunsolvable problems idea present agent problem unsolvable agent initial knowledge observing agent problem solving processes estimate probability agent creative agent think outside current context take exploratory actions incorporate relevant environmental cues learned knowledge make problem tractable least computable agent general ability solve problems type problem solving framework typically used area automated planning describing various sorts problems solution plans naturally suited defining problem creative solution strategy ready formalize various notions macgyver evaluation framework preliminaries classical planning define first order language predicates negations represents terms variables constants predicate grounded terms constants use classical planning notions planning domain represented represents set states set actions transition functions classical planning problem triple initial state goal state plan sequence actions plan solution planning problem also note proposed subset harnad instead requiring robots everything real people focused requiring robots exhibit resourcefulness resilience also subset schweizer expands notion intelligence consider notion state reachability set successor states defines set states reachable macgyver problem formalize macgyver problem mgp define universe world within universe world describes full set abilities agent includes abilities agent knows unaware define agent subdomain representing proper subset world within awareness agent mgp becomes planning problem defined world outside agent current subdomain definition universe first define universe classical planning domain representing aspects physical world perceivable actionable agents regardless capabilities includes allowable states actions transitions physical universe definition world define world portion universe corresponding aspects perceivable actionable particular species agent agent species particular set sensors actuators allowing agents species perceive proper subset states actions transition functions thus world defined follows definition agent subdomain next define agent sit ati type planning subdomain corresponding agent perception action within world words agent fully aware capabilities times agent domain corresponds portion world agent perceiving acting time sit ati sit ati sit ati definition macgyver problem define macgyver problem mgp respect agent planning problem agent world goal state currently unreachable agent formally mgp sit initial state agent set ground predicates naturally follows context world mgp classical planning problem agent current perspective unsolvable reformulate mgp language recognition problem able brief complexity analysis definition given set statements planning problems let set statements represents macgyver problem without syntactical restrictions theorem decidable proof proof simple number possible states agent subdomain agent world finite possible search see whether solution exists agent world agent initial domain theorem proof membership mgp amounts looking see problem solvable problem upon concluding solvable problem becomes one determining solvable problem world corresponding agent species problems problems expspace unrestricted case ghallab thus mgpexistence expspace hardness reduce classical planning problem mgp mgpexistence defining new world define new world extend classical domain one state defining new state goal state adding actions transitions every state new goal state also set agent domain classical planning domain iff agent domain thus mgpexistence solving macgyver problem theorems know possible intractable agent know whether given problem mgp agent perspective solving mgp like solving planning problem additional requirement sense learn previously unknown state transition function action specifically solving mgp involve performing actions environment making observations extending contracting agent subdomain exploring different contexts solution strategies definition agent domain modification domain modification involves either domain extension domain domain extension agent time agent world agent subdomain previous time agent extends subdomain sensing perceiving environment self agent extend domain making observation receiving advice instruction performing introspection formally interest brevity consider domain extensions agent subdomain results domain extension domain modification set set domain modifications subdomain let subdomain resulting applying definition strategy strategy strategy tuple plan set domain modifications strategy involves least one domain modification definition context context tuple representing agent subdomain state time ready define insightful strategy set actions domain modifications agent needs perform allow goal state problem reachable agent definition insightful strategy let agent current context let mgp agent context insightful strategy strategy applied results context formalizing insightful strategy way somewhat analogous moment insight reached problem becomes tractable definition computable solution plan becomes feasible specifically solving problem involves amount creative exploration domain extensions contractions point agent information needs within subdomain solve problem classical planning problem need domain extensions alternatively define insightful strategy terms goal state reachable solution discovered polynomial time next review simple toy examples illustrate concepts discussed thus far operationalizing macgyver problem consider two examples help operationalize formalism presented thus far first modification popular blocks world planning problem second practical task tightening screws however caveat certain common tools unavailable problem solver must improvise specifically discuss various capabilities agent must possess order overcome challenges posed examples toy example world consider agent tasked moving block one location another agent able execute without first discovering new domain information let agent subdomain consist set locations two objects towel block function locationof representing location object suppose agent aware following predicates negations object location near agent near location touching agent touching object holding agent holding object define set actions agent domain follows reach move robot arm near object precond effect near grasp grasp object precond near effect touching lift lift object precond touching effect holding carryt carry object precond holding effect release release object precond touching effect given agent domain start state defined define agent context tuple agent domain start state locationof locationof consider simple planning problem world agent must move block location agent could execute simple plan follows solve problem reach locationof grasp locationof lif locationof carryt release course plan agent touches holds block moves location using similar plan agent could move towel location well consider difficult planning problem agent asked move block without touching given constraints imposed problem goal state reachable agent must discover alternative way move block agent must uncover states world previously subdomain example agent may learn moving towel location towel covers block might discover new predicate covered would prevent touching block agent may also uncover new action push would allow push object along surface uncover new predicates actions agent may execute insightful strategy agent domain extended problem becomes standard planning problem agent discover solution plan covering block towel pushing towel block location order autonomously resolve problem agent must able recognize stuck discover new insights build new plans additionally agent must able actually execute operation agent must suitable robotic sensory action mechanisms locate grasp manipulate objects practical example makeshift screwdriver consider practical example attaching fastening things together critical task many domains depending situation require resilience unexpected events resourcefulness finding solutions suppose agent must affix two blocks set blocks order agent tool box containing set tools screwdriver plier hammer set fasteners screw nail addition objects agent environment towel coin mug ducttape assume agent sense following relations predicates negations respect objects isavailable tool available use fastenwith tool designed fastener grabw ith tool designed grab fastener isholding agent holding tool isreachable tool reach fastener iscoupled tool coupled fastener isattachedt fastener attached inserted blocks issecured fastener tightly secured blocks also define set actions agent subdomain follows select tool use fastener precond isavailable astenw ith effect isholding grab grab fastener tool precond isholding grabw ith effect iscoupled placeandalign place align fastener blocks effect isattachedt reachandengage reach engage tool fastener precond isholding astenw ith isreachable effect iscoupled install install fastener tool precond iscoupled isattachedt effect issecured suppose screw loosely inserted two blocks isattachedt screw needs given space limitations presented entire domain represented example nevertheless analysis properties still hold tightened screw tightening screw would quite straightforward performing actions select reachandengage install reason screwdriver gone missing screwdriver macgyver problem way agent given current subdomain knowledge tighten screw goal state issecured screw unreachable agent current context hence agent must extend domain one approach consider one objects coin could used screwdriver well mug towel might agent must able switch around variables existing knowledge expose previously unknown capabilities tools example switching grab grab agent explore possibility grabbing coin plier similarly relaxing constraints variables relations agent perform reachandengage action whereby couple makeshift tool namely coin screw screw recessed location therefore difficult access without elongate arm coin might fit head screw necessary elongation would able reach screw approach might grab coin plier use assembly tighten screw maybe even additionally duct tape extra support noted earlier generally agent must able relax preexisting constraints generate new actions exploring hypothesizing testing variation agent expand domain example still relatively simple humans helps highlight complexity resources needed agent perform task successfully identifying building makeshift screwdriver standard screwdriver available shows degree resilience events autonomy resourcefulness believe important component everyday creativity intelligence formulating notion resourcefulness manner better study complexity cognitive processes also computationalize abilities even formally measure agent requirements intelligence physical embodiment humans solve problems particularly creative insight problems tend use various heuristics simplify search space identify invariants environment may may relevant knoblich agent solving mgp must possess ability execute types heuristics cognitive strategies moreover mgps merely problems classical planning sense require ability discover problem unsolvable planning standpoint discover environmental exploration relevant aspects surroundings order extend domain knowledge discoveries turn likely require additional cognitive resources heuristics allow agent make discoveries ficiently finally agent must also able remember knowledge able efficiently solve future instances similar problems capabilities standpoint agent must possess sensory action capabilities able execute exploration discovery process including grasping manipulating unfamiliar objects practical capabilities trivial combination intelligent reasoning provide clear demonstration agent autonomy solving practical problems examples provide sense types planning problems might qualify mgp certain mgps challenging others next present theoretical measure difficulty mgp optimal solution generally assume solvable mgp best solution involves agent taking effective actions making required observations needed uncovering solution using elegant strategy formalize notions first defining optimal solutions measure complexity insightful strategy optimal solution definition optimal solutions let mgp agent let optimal solution plan set optimal domain modifications set minimum set domain modimain modifications fications needed inclusion actions optimal solution plan optimal solution strategy solution set optimal domain strategy modifications definition let mgp agent let set optimal solution strategies exists insightful strategy let set optimal insightful strategies set represented program prefix universal turing machine capable listing elements halting use kolmogorov complexity set insightful strategies computes define intrinsic difficulty mgp kolmogorov complexity set optimal insightful strategies shown intractable measuring intrinsic difficulty mgp computable use kolmogorov complexity even instead choose use alternative computable approximation kolmogorov complexity normalized compression distance determining difficult must consult oracle determine optimal solution reality agent know problem facing mgp even know agent would tough time determining well measuring progress agent success challenge creative problems often know getting closer warmer solution formalize idea using solomonoff induction first designate judge based strategy currently executed agent guesses probability finite number steps agent likely completed insightful strategy consider agent performing strategy attempt solve mgp judge evaluating performance agent judge must first attempt understand agent trying thus judge must first hypothesize agent model capable generating let agent defined probability measure measure represents probability agent generates strategy given mgp particular context judge know advance measure could change depending type agent example random agent could whereas macgyver agent could represented different probability measure knowing type agent want judge able evaluate many different types agents possible infinitely many different types agents accordingly infinitely many different hypotheses agent thus simply take expected value respect uniform distribution hypotheses must weighed heavily others solomonoff devised universal distribution set computable hypotheses perspective computability theory solomonoff universal prior hypothesis defined judge applies principle occam razor given many explanations case hypotheses simplest likely approximate kolmogorov complexity measure able measure progress agent solving mgp must able define performance metric paper develop particular performance metric suggest performance metric proportional level resourcefulness creativity agent generally measuring progress may depend problem scope control variables length elegance solution factors nevertheless simple measure sort serve placeholder develop theory ready define performance progress agent solving mgp definition expected progress consider agent context solving mgp agent executed strategy comprising actions domain modifications let space programs compute measure agent resourcefulness consider judge observing agent fully aware agent context knowledge mgp let judge prefix universal turing machine let kolmogorov complexity function let performance metric interpretation cumulative state agent resourcefulness solving mgp expected progress agent adjudicated judge also interested seeing whether agent given strategy likely improve performance next actions judge need predict continuation agent strategy taking possible hypotheses agent behavior account let possible continuation let represent concatenation judge solomonoff predictor predicted finite continuation likely one less complex kolmogorov sense judge measures state agent attempts solving mgp also predict agent likely perform future conclusion future work apollo space mission astronauts together ground control overcome several challenges bring team safely back earth lovell kluger one challenges controlling carbon dioxide levels onboard space craft two days straight worked odysseys canisters aquariuss life support system using materials known available onboard spacecraft sock plastic bag cover flight manual lots duct tape crew assembled strange contraption taped place carbon dioxide levels immediately began fall safe cass team proposed macgyver test practical alternative turing test formal alternative robotic machine learning challenges require specific internal mechanism agent instead focuses observed behavior akin apollo team flexible dynamic allowing measuring wide range agents across various types problems based fundamental notions set theory automated planning turing computation complexity theory allow formal measure task difficulty although kolmogorov complexity solomonoff induction measures computable formally rigorous substituted computable approximations practical applications future work plan develop examples mgps also begin unpack interesting aspects problem structure study complexity draw comparisons problems believe formally captures concept practical intelligence everyday creativity quintessentially human practically helpful designing autonomous agents importantly intent accompanying mgp formalism help guide research providing set mathematically formal specifications measuring progress based agent ability solve increasingly difficult mgps thus invite researchers develop mgps varying difficulty design agents solve references boden margaret boden turing test artistic creativity kybernetes bringsjord sen selmer bringsjord atriya creative cars hire computational logicians fast page forthcoming bringsjord selmer bringsjord paul bello david ferrucci creativity turing test better lovelace test minds machines cass stephen cass apollo solution ieee spectrum cohen paul cohen turing test magazine cooper van leeuwen barry cooper jan van leeuwen alan turing work impact feigenbaum edward feigenbaum challenges grand challenges computational intelligence journal acm ghallab malik ghallab dana nau paolo traverso automated planning theory practice harnad stevan harnad bodies minds machine incarnation old philosophical problem minds machines knoblich knoblich psychological research insight problem solving recasting reality pages springer levesque hector levesque ernest davis leora morgenstern winograd schema challenge thirteenth international conference principles knowledge representation reasoning paul introduction kolmogorov complexity applications second edition computers mathematics applications lovell kluger jim lovell jeffrey kluger apollo houghton mifflin harcourt riedl mark riedl lovelace test artificial creativity intelligence arxiv preprint page schweizer paul schweizer externalist foundations truly total turing test minds machines solomonoff solomonoff preliminary report general theory inductive inference zator technical bulletin team mission evaluation team mission operations report apollo turing alan turing computing machine intelligence mind lix
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synthesizing dynamic patterns generative convnet jianwen xie zhu ying nian university california los angeles ucla usa may jianwen sczhu ywu abstract models used unsupervised learning terms synthesis various approaches based convnet proposed synthesize realistic static images however much work literature synthesizing dynamic patterns based convnet focus present paper specifically propose synthesize dynamic patterns generalizing generative convnet model recently proposed generative convnet derived discriminative convnet random field model model form exponential tilting reference distribution gaussian white noise distribution uniform distribution exponential tilting parametrized convnet involves multiple layers linear filters rectified linear units relu seek capture features patterns different scales generative convnet sampled langevin dynamics model learned stochastic gradient algorithm analysis synthesis scheme seeks match synthesized signals generated langevin dynamics observed training signals specifically learning algorithm iterates following two steps initializing parameters synthesized signals step updates synthesized signals langevin dynamics samples currently learned model step updates parameters based difference synthesized data observed data order shift density model synthesized data towards observed data shown learning algorithm synthesize realistic spatial image patterns textures objects article generalize spatial generative convnet adding temporal dimension resulting convnet consists multiple layers filters seek capture patterns various scales show learning algorithm training generative convnet synthesize realistic dynamic patterns also show possible learn model incomplete video sequences either occluded pixels missing frames model learning video sequences contain rich dynamic patterns dynamic texture patterns exhibit stationarity temporal domain action patterns either spatial temporal domain show spatialtemporal generative convnet used model synthesize dynamic patterns model defines probability distribution video sequence log probability defined convnet consists multiple layers filters capture spatialtemporal patterns different scales model learned training video sequences analysis synthesis learning algorithm iterates following two steps step synthesizes video sequences currently learned model step updates model parameters based difference synthesized video sequences observed training sequences show learning algorithm synthesize realistic dynamic patterns introduction wide variety dynamic patterns video sequences including dynamic textures textured motions exhibit statistical stationarity stochastic repetitiveness temporal dimension action patterns either spatial temporal domain synthesizing analyzing dynamic patterns interesting problem paper focus task synthesizing dynamic patterns using generative version convolutional neural network convnet cnn convnet proven immensely successful discriminative learning machine convolution operation convnet particularly suited signals images videos sounds exhibit translation invariance either spatial domain temporal domain recently researchers become increasingly interested generative aspects convnet purpose visualizing knowledge learned convnet synthesizing realistic signals developing generative pattern completion accomplished simultaneously related work work generalization generative convnet model adding temporal dimension work dynamic patterns video sequences discriminative convnet used analyzing video data connection discriminative convnet generative convnet studied dynamic textures textured motions studied instance proposed vector model coupled dimension reduction single value decomposition linear model gaussian innovations proposed dynamic model based sparse linear representation frames see recent review dynamic textures generative convnet model expected flexible capturing complex patterns dynamic textures multiple layers filters recently generalized generative adversarial networks model dynamic patterns model model also adversarial interpretation see section details temporal data popular model recurrent neural network causal model requires starting frame contrast model require starting frame compared recurrent network model convenient direct capturing temporal patterns multiple time scales filters generative convnet generative convnet energybased model random field model defined image sequence form exponential tilting reference distribution fix notation let image sequence video defined square rectangular image domain time domain indexes coordinates pixels indexes frames video sequence treat three dimensional function defined filter let denote filtered image sequence feature map let denote filter response feature pixel time convnet composition multiple layers linear filtering relu expressed following recursive formula operates linear filtering operation followed relu max bottom layer indexes three color channels sub sampling may implemented filters multiple layers expected capture patterns multiple scales possible filters fully connected spatial domain well temporal domain feature maps spatial domain dynamic pattern exhibit spatial temporal stationarity exp scoring function generative convnet indexes layers filters layer filters layer used index filters layers respectively numbers filters layers respectively filters locally supported range within local support box image sequence weight parameters define linear filter consists weight bias terms define filters layer gaussian white noise model exp counts number pixels domain without loss generality shall assume scoring function tilts gaussian reference distribution model fact purpose identify spatialtemporal features patterns definition sum filter responses top layer filters positions times spatial temporal pooling reflects fact assume model stationary spatial temporal domains dynamic texture spatial temporal domain top layer filters fully connected spatial temporal domain simple consequential property relu nonlinearity max indicator function otherwise result scoring function piecewise linear linear piece defined multiple layers vari ables tells whether local pattern represented filter detected position time let activation pattern divides image space large number pieces according value piece image space fixed scoring function linear defined fact computed backpropagation process defines deconvolution process filters multiple layers become basis functions layers activation variables different layers become coefficients basis functions deconvolution model whose energy function combination norm comes reference distribution piecewise linear scoring function kik const const kbw constant piece image space fixed since piecewise quadratic function piecewise gaussian piece image space fixed value truncated use denote identity matrix mean gaussian piece within also local mode local mode satisfies hierarchical autoencoder encoding process decoding process general image sequence considered reconstruction reconstruction exact local mode sampling learning algorithm one sample model langevin dynamics indexes time steps step size dynamics driven reconstruction error finiteness step size corrected step langevin dynamics extended hamiltonian monte carlo sophisticated versions learning training image sequences accomplished maximum likelihood let log defined expectation approximated monte carlo samples produced langevin dynamics see algorithm description learning sampling algorithm algorithm keeps synthesizing image sequences current model updating model parameters order match synthesized image sequences observed image sequences learning algorithm keeps shifting probability density low energy regions model synthesized data towards observed data learning algorithm langevin sampling step involves computation parameter updating step involves computation convnet structure gradients computed efficiently two gradients share chain rule computations term mcmc sampling langevin dynamics samples evolving distribution keeps changing thus learning sampling algorithm runs chains adversarial interpretation model model exp update based approximated synthesized image sequences generated langevin dynamics algorithm learning sampling algorithm input training image sequences number synthesized image sequences number langevin steps number learning iterations output estimated parameters synthesized image sequences river let initialize initialize repeat run steps langevin dynamics update starting current step follows equationp calculate obs syn update obs syn step size let ocean figure synthesizing dynamic textures spatial temporal stationarity category first row displays frames observed sequence second third rows display corresponding frames two synthesized sequences generated learning algorithm river ocean zero temperature limit langevin dynamics becomes gradient descent consider value function updating increase shifting low energy regions synthesized image sequences observed image sequences whereas updating decrease moving synthesized image sequences towards low energy regions adversarial interpretation learning sampling algorithm also considered generalization herding method exponential family models general models work let also let assuming uniform reference distribution experiments show model uniform also synthesize realistic dynamic patterns generative adversarial learning generator network unlike model based convnet generator network generates convnet latent vector follows known prior distribution collects parameters convnet recently developed alternating algorithm train generator network without involving extra network recently developed cooperative training method recruits generator network reconstruct regenerate synthesized image sequences speed mcmc sampling experiments learn generative convnet video clips collected dataset internet code experiments based matconvnet show synthesis results displaying frames video sequences posted synthesis results project page http reader watch videos experiment generating dynamic textures spatial temporal stationarity first learn model dynamic textures stationary spatial temporal domains use filters convolutional spatial temporal domains first layer filters size pixels frames flashing lights figure comparison synthesizing dynamic texture waterfall top bottom segments observed sequence synthesized sequence method synthesized sequence method fountain burning fire heating pot category one observed video prepared size range intensities mean subtraction used use chain langevin sampling number langevin iterations every two consecutive updates parameters number learning iterations add one layer every iterations use learning rates learning rate higher layer less lower layer order obtain stable convergence experiment generating dynamic textures temporal stationarity spring water figure synthesizing dynamic textures temporal stationarity category first row displays frames observed sequence second row displays corresponding frames synthesized sequence generated learning algorithm flashing lights fountain burning fire heating pot spring water second layer filters size third layer filters size figure displays results category first row displays frames observed sequence second third rows show corresponding frames two synthesized sequences generated learning algorithm use learning scheme starting first layer sequentially add layers one one time learn model generate synthesized image sequence using algorithm learning new layer filters refine lower layers filters learn generative convnet many dynamic textures structured background objects stationary spatial domain case network used experiment may fail however modify network experiment using filters fully connected spatial domain second layer specifically first layer filters size pixels frames second layer spatially fully connected layer contains filters fully connected spatial domain convolutional temporal domain temporal size filters frames size frames temporal dimension due spatial full connectivity second layer spatial domain feature maps third layer reduced third layer filters size temporal dimension use learning scheme learn generative convnet dynamic textures iteration layers filters updated different learning rates learning rate higher layer much less lower layer avoid issue large gradients learn generative convnet category one training video synthesize observed sequences frame observed sequences synthesized sequences running cows frame synthesized sequences examples synthesized sequences figure learning observed fire videos implementation videos using langevin dynamics figure displays results category first row shows frames observed sequence second row shows corresponding frames synthesized sequence generated learning algorithm use set parameters categories without tuning figure compares method linear dynamic system model image sequence generated model appears blurred sequence generated method learning model scaled learn fire pattern training videos implementation size videos video contains frames pixels parallel chains langevin sampling used experiment slightly modify network using filters size pixels frames first layer spatially fully connected filters temporal size frames size second layer keeping setting third layer unchanged number learning iterations figure shows one frame observed sequences corresponding frame synthesized sequences two examples synthesized sequences also displayed experiment generating action patterns without spatial temporal stationarity experiments show generative spatialtemporal convnet learn sequences without observed sequences synthesized sequences running tigers figure synthesizing action patterns action video sequence continuous frames shown running cows frames training sequences displayed corresponding frames synthesized sequences generated learning algorithm displayed running tigers frames observed training sequences displayed corresponding frames synthesized sequences displayed ment also specialize learning roughly aligned video sequences action patterns either spatial temporal domain using single toplayer filter covers whole video sequence learn generative convnet video sequences aligned actions first layer filters size pixels frames second layer fully connected layer single filter covers whole sequence observed sequences size figure displays two results modeling synthesizing actions roughly aligned video sequences learn model category number training sequences running cow example running tiger example videos collected internet frames example figure displays segments observed sequences segments synthesized action sequences generated learning algorithm run paralleled chains experiment running cows paralleled chains experiment running tigers experiments show model capture action patterns one limitation model involve explicit tracking objects parts experiment learning incomplete data model learn video sequences occluded pixels task inspired fact videos contain occluded objects learning method adapted task minimal modification modification involves iteration running steps langevin dynamics recover occluded regions observed sequences iteration use completed observed sequences synthesized sequences compute gradient update model parameters method simultaneously accomplishes following tasks recover occluded pixels training video sequences synthesize new video sequences learned model learn model updating model parameters using recovered sequences synthesized sequences see algorithm description learning sampling recovery algorithm table recovery errors occlusion experiments salt pepper masks flag fountain ocean playing sea world traffic windmill avg single region masks algorithm learning sampling recovery algorithm input training image sequences occluded pixels binary masks indicating locations occluded pixels training image sequences number synthesized image sequences number langevin steps synthesizing image sequences number langevin steps recovering occluded pixels number learning iterations output estimated parameters synthesized image sequences recovered image sequences flag fountain ocean playing sea world traffic windmill avg missing frames flag fountain ocean playing sea world traffic windmill avg design types occlusions type salt pepper occlusion randomly place masks image domain cover pixels videos type single region mask occlusion randomly place mask image let initialize initialize initialize repeat run steps langevin dynamics recover occluded region starting current step follows equation occluded pixels updated step run steps langevin dynamics update starting current step follows equationp calculate obs syn update obs syn step size let domain type missing frames randomly block image frames video figure displays one example recovery result type occlusion video frames quantitatively evaluate qualities recovered videos test method video sequences collected dataset types occlusions use model structure one used experiment number langevin steps recovering set equal number langevin steps synthesizing experiment report recovery errors measured average per pixel difference original image sequence recovered image sequence occluded pixels range pixel intensities compare results salt pepper masks removing moving boat lake single region masks removing walking person front fountain figure background inpainting videos experiment first column displays frames original video second column shows corresponding frames black masks occluding target removed third column shows inpainting result algorithm moving boat walking person missing frames figure learning occluded video sequences experiment first row shows segment occluded sequence black masks second row shows corresponding segment recovered sequence type salt pepper mask type single region mask type missing frames results obtained generic markov random field model defined video sequence model markov random field whose potentials pairwise differences nearest neighbor pixels nearest neighbors defined spatial temporal domains image sequences recovered sampling intensities occluded pixels conditional observed pixels using gibbs sampler table shows comparison results types occlusions see model recover incomplete data learning experiment background inpainting moving object video occluded frame turns recovery algorithm become algorithm background inpainting videos goal remove undesired moving object video use model one experiment figure figure shows two examples removals moving boat walking person respectively videos collected example first column displays frames original video second column shows corresponding frames masks ing target removed third column presents inpainting result algorithm video size example example experiment different video inpainting interpolation synthesize image patches fill empty regions video running langevin dynamics experiments run single langevin chain synthesis conclusion paper propose generative convnet model synthesizing dynamic patterns dynamic textures action patterns experiments show model synthesize realistic dynamic patterns moreover possible learn model video sequences occluded pixels missing frames experiments included paper show method also generate sound patterns mcmc sampling model sped learning sampling models multiple scales recruiting generator network reconstruct regenerate synthesized examples cooperative training acknowledgments work supported nsf dms darpa simplex onr muri darpa aro references denton chintala fergus deep generative image models using laplacian pyramid adversarial networks nips pages doretto chiuso soatto dynamic textures international journal computer vision dosovitskiy tobias springenberg brox learning generate chairs convolutional neural networks cvpr pages ghanem ahuja maximum margin distance learning dynamic texture recognition eccv pages springer girolami calderhead riemann manifold langevin hamiltonian monte carlo methods journal royal statistical society series statistical methodology goodfellow mirza wardefarley ozair courville bengio generative adversarial nets nips pages gregor danihelka graves rezende wierstra draw recurrent neural network image generation icml pages han zhu alternating backpropagation generator network aaai han zhu video primal sketch unified representation video journal mathematical imaging vision hochreiter schmidhuber long memory neural computation yang convolutional neural networks human action recognition ieee transactions pattern analysis machine intelligence krizhevsky sutskever hinton imagenet classification deep convolutional neural networks nips pages kulkarni whitney kohli tenenbaum deep convolutional inverse graphics network arxiv eprints lecun bottou bengio haffner gradientbased learning applied document recognition proceedings ieee lecun chopra hadsell ranzato huang tutorial learning predicting structured data zhu learning frame models using cnn filters aaai montufar pascanu cho bengio number linear regions deep neural networks nips pages neal mcmc using hamiltonian dynamics handbook markov chain monte carlo newson almansa fradet gousseau http ngiam chen koh learning deep energy models icml pages sun https vedaldi lenc matconvnet convolutional neural networks matlab corr vondrick pirsiavash torralba generating videos scene dynamics nips pages wang zhu generative method textured motion analysis synthesis eccv pages springer wang zhu analysis synthesis textured motion particles waves ieee transactions pattern analysis machine intelligence welling herding dynamical weights learn icml pages acm williams zipser learning algorithm continually running fully recurrent neural networks neural computation xie gao zhu cooperative training descriptor generator networks arxiv preprint xie zhu theory generative convnet icml guo tao kernel learning dynamic texture synthesis ieee transactions image processing younes convergence markovian stochastic algorithms rapidly decreasing ergodicity rates stochastics international journal probability stochastic processes zeiler taylor fergus adaptive deconvolutional networks mid high level feature learning iccv pages
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canonical embedded resolution singularities excellent schemes feb vincent cossart uwe jannsen shuji saito february introduction basic invariants singularities permissible embedded case case main theorems strategy proofs bases characteristic polyhedra transformation standard bases termination fundamental sequences blowups case additional invariants case proof case key lemmas proof case separable residue extensions proof case iii inseparable residue extensions maximal contact dimension versailles laboratoire avenue des versailles cedex france mathematik regensburg regensburg germany interactive research center science graduate school science engineering tokyo institute technology ookayama meguro tokyo japan sshuji alternative proof theorem functoriality locally noetherian schemes algebraic spaces stacks introduction principal aim paper show following three theorems resolution singularities arbitrary reduced excellent noetherian scheme dimension two following schemes assumed noetherian see end introduction section locally noetherian schemes theorem canonical controlled resolution exists canonical finite sequence morphisms regular permissible center contained sing singular locus sequence functorial sense compatible automorphisms zariski localizations note implies isomorphism xreg xsing recall subscheme called permissible regular normally flat along see compatibility automorphisms means every automorphism extends sequence unique way compatibility localizations means via localization canonical resolution sequence suppressing morphisms become isomorphisms theorem implies theorem canonical embedded resolution let closed immersion regular excellent scheme canonical commutative diagram regular closed immersion proper surjective morphisms inducing isomorphisms xsing xsing xsing xsing moreover morphisms compatible automorphisms zariski localizations fact starting theorem one gets canonical sequence closed immersions sing identified strict transform proper fact projective surjective regular since several applications following refinement useful theorem canonical embedded resolution boundary let closed immersion regular scheme let simple normal crossings divisor irreducible component contained canonical commutative diagram closed immersion projective surjective isomorphisms outside xsing exceptional locus closed subscheme isomorphism moreover regular simple normal crossings divisor intersects transversally furthermore compatible automorphisms zariski localizations precisely prove existence commutative diagram vertical morphisms closed immersions permissible center sing regular identified strict transform complete transform union strict transform exceptional divisor furthermore permissible normal crossing see definition implies simple normal crossings divisor holds fact second main theorem paper theorem states somewhat general version contain irreducible components one assume contained strongly locus xbsreg see definition one gets normal crossing definition implies isomorphism xbsreg xreg particular xreg addition theorem also treats schemes case red regular normal crossing normally flat along red moreover obtain variant consider strict transforms normal crossings divisor strict transform get normal crossing xred case strict transform theorem case assume embedding also proved general version first main theorem theorem allows scheme well boundary notion newly introduced paper see section theorem comes two versions one complete transforms one strict transforms approach implies theorem implies theorem particular canonical resolution sequence theorem strict transforms theorem variant coincides intrinsic sequence theorem thus readers interested theorems skip sections ignore mentioning crossings divisors assuming empty note following corollary corollary let regular excellent scheme dimension let reduced closed subscheme dimension two exists projective surjective morphism isomorphism reduced subscheme structure simple normal crossings divisor fact applying theorem get projective surjective morphism regular regular closed subscheme simple normal crossings divisor isomorphism fact xsing moreover intersect transversally particular normal crossing sense definition hence obtain wanted situation composing regular subscheme letting simple normal crossings divisor see lemma moreover mention theorem applied paper second third author prove conjecture kato finiteness certain motivic cohomology groups varieties finite fields main motivation authors work subject knowledge none three theorems known least stated generality even dim know reference results although may integral dimension theorem found section proof theorem written zariski proved theorem without discussing canonicity functoriality irreducible surfaces algebraically closed fields characteristic zero five years later proved corollary without canonicity functoriality surfaces fields characteristic zero embedded threefold book abhyankar extended last result algebraically closed fields making heavy use papers around time hironaka sketched shorter proof result algebraically closed ground fields based different method recently shorter account abhyankar results given cutkosky excellent schemes characteristic zero whose residue fields characteristic zero arbitrary dimension theorems proved hironaka famous paper main theorem corollary theorem holds well except approach constructive give canonicity functoriality issues addressed solved later literature especially papers villamayor see particular see related different approaches references scheme fixed embedding regular scheme considered process depends embedding last issue remedied different approach positive characteristics canonicity addressed abhyankar results weaker type resolution surfaces replacing blowups regular centers different techniques zariski showed resolve surface necessarily algebraically closed field characteristic zero local uniformization based methods abhyankar extended algebraically closed fields positive characteristics later extended several results general schemes whose closed points perfect residue fields lipman gave simple procedure obtain resolution singularities arbitrary excellent schemes following way finite sequence proper surjective morphisms regular sequence obtained alternating normalization blowing finitely many isolated singular points processes uniformization normalization controlled sense theorem obtained permissible known extend ambient regular scheme like theorem neither clear get theorem procedure particular weaker results sufficient mentioned applications even case weak resolution singularities proved jong remains mention results weak resolution singularities threefolds field zariski char abhyankar algebraically closed characteristic see also cossart piltant arbitrary topic present paper approach roughly based strategy used precisely follows approach still surfaces given hironaka paper cited general strategy common one develops certain invariants measure singularities aims constructing sequence invariants finally decreasing end one concludes one reached regular situation choices centers made considering strata invariants fact one blows worst locus strata invariants maximal possibly desingularising strata main point show invariants finally decrease characteristic zero done technique introduced hironaka called method maximal contact see ahv giraud papers theoretic background induction dimension known theory maximal contact work positive characteristic theoretic counterexamples explicit counterexamples threefolds characteristic two narasimhan see also interpretation sense clear counterexamples threefolds positive characteristic used way section paper show maximal contact even exists surfaces characteristic even maximal contact considered weakest sense therefore strategy proof different follow one outlined based certain polyhedra see paper considers case hypersurface another paper hironaka develops theory polyhedra ideals several generators terms certain standard bases also appear introduction expresses hope theory polyhedra useful resolution singularities least surfaces paper seen fulfilment program fundamental paper hironaka uses two important invariants measuring singularity point arbitrary scheme primary secondary one dimension dim directrix dirx depend cone hironaka proves permissible point image equality holds one says near suitably normalized main problem show infinite sequence near points means control hironaka introduces tertiary complex invariant polyhedron associated singularity lies depends local ring also various choices regular local ring quotient system regular parameters parameters directrix dirx equations quotient precisely base ker situation theorem naturally given case always exists completion question ruling infinite sequence near points depends completion well case considered section single strictly decreasing invariant comes polyhedra rather behavior shape tells end infinite sequence near points exits sufficient purpose might interesting find strictly decreasing invariant also case particular situation considered hypersurface algebraically closed field done hironaka see also variant counterpart local question one consider global strategy global behavior invariants understand choice permissible centers global improvement regularity since nice local computations geometric behavior nice use invariant hox alternative primary invariant extensively studied bennett proved similar results permissible somewhat improved singh bennett also defined global functions however work well give nice strata case weakly biequidimensional excellent schemes introduce variant definition works arbitrary finite dimensional excellent schemes solves question raised bennett associated strata locally closed closures contained particular closed maximal although main results schemes major part paper written schemes arbitrary dimension hope might useful investigations part section sections exploit specific features situation according understanding mainly two obstructions extension schemes fact theorem gives crucial information locus near points one assume char char dim lack good invariants polyhedra suitable tertiary invariants case tried write paper way well readable experts resolution singularities like two want understand results techniques apply arithmetic algebraic geometry also reason use notion idealistic exponents would given extra burden recall theory define characteristic polyhedra idealistic exponents rephrase statements equipped theory treatment functions defining scheme functions defining boundary would looked symmetric hand global algorithm clearly distinguishes two briefly discuss contents sections section discuss primary secondary invariants local global singularities mentioned section discuss permissible behavior introduced invariants based fundamental results hironaka bennett singh section study similar questions setting theorem logsituation one boundary normal crossings divisor define class functions depending choice history function subdivisors characterizing old components number old components gives associated strata relate two functions strata study transversality properties section extend theory situation excellent scheme embedding regular scheme turns one also define notion boundary tuple rather multiset forgetting ordering locally principal closed subschemes embedded situation normal crossings divisor associated boundary given traces components show carry information needed moreover approach makes evident constructions strategies defined later intrinsic depend embedding results section carried section perfect matching see lemma could started section derived embedded situation section special case felt illuminating start familiar classical setting moreover results section later paper reduced embedded situation passing local ring completing see remark lemma applications thereafter section state main theorems corresponding somewhat general versions theorem respectively explain strategy prove based important theorem hironaka see theorem following remark suffices find succession permissible blowups invariants decrease although principle seems might obvious surfaces could find suitable reference provided precise statement short proof fact dimension see corollaries case without boundary corollaries case boundary problem arising set hilbert functions ordered total product order iff order infinite decreasing sequences set functions overcome fact subset hilbert functions quotients fixed polynomial ring noetherian ordered set see theorem preparations define canonical resolution sequence see remark definition corollaries definition whole resolution sequence point remark define canonical resolution sequences explicit strategy resolution singularities dimension would interesting see strategy always works proof finiteness resolution sequences dimension two reduced two key theorems theorem exclude possibility certain infinite chains near points key theorems concern isolated singularities hence local ring closed point hold arbitrary dimension condition geometric dimension directrix holds dim mentioned local situation may assume embedded situation basic tool various considerations study situation mentioned local ring arbitrary dimension quotient regular local ring section discuss suitable systems regular parameters suitable families generators good choice obtained admissible definition means affine parameters directrix recalled study valuations associated initial forms respect valuations elements behavior change system parameters special case generated one element case hypersurface singularities choice good general choices better favorable choice standard basis definition introduced hironaka introduced general notion base flexible work plays important role paper section recall slightly different way hironaka definition polyhedron associated system parameters basis polyhedron intersection choices fixed recall hironaka crucial result admissible hironaka calls namely normalized definition solvable definition vertices also certain process making given normalized changing solvable changing finitely many vertices vertices complete one significance result provides natural way transforming base standard base assumption admissible explained important study permissible near points situation system associate certain new systems key result proved section standard base base next key result chosen admissible hence hironaka crucial result mentioned transform system standard base key theorems concern certain sequences permissible blowups arise naturally canonical resolution sequence call fundamental sequences blowups definition fundamental units permissible blowups definition use principal tool sequences bpermissible blowups first blowup closed point initial point later blowups certain maximal centers map isomorphically onto lie consist points near fundamental sequence still center properties fundamental unit none chosen closed point terminal point near section study first properties fundamental sequences particular show certain bound associated polyhedra suffices show first key theorem dealing case also used section second key theorem dealing case one needs information polyhedra particular additional invariants introduced section theorem proved next three sections states infinite sequence fundamental units closed initial points terminal points match isolated strata preparations section section treats case residue field extension trivial separable much inspired however treats special situation hypersurface regular threefold algebraically closed field contain proofs claims section treats case occur inseparable residue field extensions case basically treated give detailed account fill gaps original proof aid results section giraud notion ridge french notion generalizes directrix section show maximal contact exist surfaces positive characteristic counterexample given works field characteristic section give proof fact suffices show eliminate maximal stratum finally section give functoriality obtain resolution arbitrary flat morphisms regular fibers apply show resolution excellent schemes locally noetherian excellent algebraic stacks atlas dimension two clear much owe earlier work resolution singularities particular work hironaka gave general strategy also important tools used paper conventions concluding remarks schemes assumed finite dimensional regular schemes always assumed locally noetherian recall also excellent schemes definition locally noetherian introduction sections readers best assume schemes noetherian places write locally noetherian indicate certain definitions make sense certain results still hold schemes locally noetherian resolution schemes treated section basic invariants singularities section introduce basic invariants singularities invariants graded rings homogeneous ideals polynomial rings let field polynomial ring variables let homogeneous polynomials degree including fix homogeneous ideal definition integers define supremum satisfying condition exist homogeneous definition write call following result iii lemma lemma let homogeneous elements degree definition let system homogeneous elements ideal generates weakly normalized satisfies condition lemma standard base satisfies conditions lemma following easy consequence corollary let homogeneous ideal let system homogeneous elements weakly normalized following conditions equivalent deg minimal degree conditions satisfied extended standard base lemma standard base obtained follows put min pick put min pick proceed get standard base remark let homogeneous generators deg considerations show standard base obtained possibly omitting follows space field extension write lemma following clear lemma ideal second invariant directrix lemma lemma let field extension smallest space kxi symk symk words minimal generated elements simply write recall spec called cone graded ring definition field extension closed subscheme dirk dir defined surjection called directrix definition dirk spec symk define dim dirk dim dimk dimk dirk simply write remark definition dirk determined pair indeed intrinsic definition depending let symk polynomial ring surjection factors canonical surjection directrix identified directrix defined via ker way dir defined graded generated elements degree similarly standard graded finitely generated graded generated may define intrinsic ker symk canonical epimorphism situation definition entries dimk variable obviously situation hand dirk dirk lemma let assumptions dim field extensions equality holds one following conditions holds separable necessarily algebraic dim particular holds perfect field proof inequality trivial follows since turn implies claim condition claim condition proved lemma arbitrary degree transcendence case finite separable extension easy obviously sufficient consider case finite galois galois group hilbert theorem space acts way canonical map isomorphism implies claim follows finally recall hilbert function confused hilbert polynomial graded algebra let set natural numbers including let set functions endow product order defined definition finitely generated graded hilbert function element defined dimk integer define inductively note remark obviously field extension variable define inductively certain sense hilbert function measures far away polynomial ring definition define function define inductively one lemma let finitely generated graded algebra dimension field generated elements degree one standard graded algebra equality holds suitable integer one proof may take base change extension field therefore may assume infinite case noether normalization elements mapped degree one part see chap theorem means monomorphism graded equality holds isomorphism since finitely generated also finitely many homogenous generators degrees gives surjective map graded smodules grading hence follows different asymptotics shall need following property theorem let field let hfn set hilbert functions standard graded hfn independent hfn noetherian ordered set hfn every strictly descending sequence hfn finite every infinite subset hfn elements proof standard graded holds quotient homogeneous ideal hand known monomial ideal ideal generated monomials variables see clo attributed macaulay wit one may take ideal leading terms respect lexicographic order monomials loc cit proposition proves moreover may assume considered hilbert functions form monomial ideal thus led consideration infinite set monomial ideals main theorem macl theorem says may assume form monomial ideals macl one finds infinite chain among ideals necessarily strictly increasing since sequence contradiction noetherian remark functions instead functions property shown theorem proof modeled corollary formulated monomial ideals see also corollary another argument set monomial ideals respect reverse inclusion noetherian ordered set finally shown sophisticated methods even set hilbert functions noetherian invariants local rings ring prime ideal set grp graded algebra follows assume noetherian regular local ring maximal ideal residue field moreover assume regular grp denotes symmetric algebra free concretely let system regular parameters grp identified polynomial ring grp mod fix ideal case set grp define ideal inp grp exact sequence inp grp grp note inp prime ideal put max called order prime ideals following semicontinuity result define initial form inp mod pvp pvp grp case inp inp definition system elements standard base inm inm standard base inm polynomial ring grm define definition inm grm absolute noetherian local ring maximal ideal defined absolute remark grn shown standard base generates next define directrix dir noetherian local ring first introduce basic notations let maximal ideal let corresponding closed point spec let residue field define spec symk zariski tangent space spec spec grn grn tangent cone spec note dim dim map symk grn gives rise closed immersion definition let field extension directrix dirk defined directrix dirk grn grn grn remark set dim dirk simply write remark definition regular ideal dirk spec grm grm spec grm inm smallest vector space inm grm inm inm generated elements moreover dirk symk simply write lemma let assumptions dim field extensions equality holds one following conditions holds separable necessarily algebraic dim proof follows lemma dim dim definition define algebraic closure lemma dim extension later use note following immediate consequence construction standard base corollary lemma let lemma exists standard base inm finally define hilbert functions noetherian local ring maximal ideal residue field associated graded ring grm explicitly hilbert function element defined dimf integer define inductively particular length called hilbert samuel function sometimes call functions form functions hilbert function measures far away regular ring lemma let noetherian local ring dimension define definition equality holds regular proof see also theorem property since dim dim grm maximal ideal since regular grm follows lemma later purpose note following facts lemma let assumptions let quotient ring equality holds proof let maximal ideal inequality holds since natural maps surjective assume definition implies particular implies natural maps isomorphisms noting ker get hence lemma let noetherian local rings integers one dim integers one dim dim proof let dim get lemma conversely assume form lemma get integer contradiction asymptotic behavior hence let dim note statement empty apply invariants schemes let locally noetherian scheme definition point define dirx dir dimk dirx field extension closed subscheme fixed regular excellent scheme define also define idirx grmx ideal defining dirx spec grmx maximal ideal note always dirx called tangent cone called zariski tangent space similarly lemma let locally noetherian scheme let morphism locally noetherian let point lying assume sense bennett flat respective maximal ideals particular holds canonical isomorphism separable dirx hence consider addition cartesian diagram closed immersions regular excellent schemes let morphism let assume locally noetherian flat fibre regular assume smooth around isomorphisms adk dim hence dirx adk let excellent scheme let morphism locally noetherian let spec completion lying isomorphisms adk fibre morphism dim hence dirx adk assume commutative diagram regular excellent scheme regular locally noetherian scheme closed immersions denote spec completion regarded closed subscheme regular proof suffices show let flat local morphism local noetherian rings isomorphisms mna mnb fact holds induction flatness injection mna induces injection mna whose image mna mnb deduce isomorphisms mna hence claim consider local rings maximal ideal residue field maximal ideal residue field assumption flat regular fibers hence true local morphism since obtained localization hence lemma closed subscheme spec spec permissible regular flat proposition get isomorphism hand flatness get canonical isomorphisms hence get isomorphism grm isomorphism becomes grm exactly first isomorphism since regular dimension ismorphism gives second isomorphism statements directrix follow remark claims involving follow applying morphism spec spec fact since excellent morphism flat geometrically regular fibers true base change consider diagram applied gives isomorphisms adk dim regular lemmas implies regular final equality follows isomorphism remark next introduce functions excellent schemes recall excellent scheme catenary irreducible closed subschemes maximal chains irreducible closed subschemes finite length denoted codimz irreducible closed subschemes codimw codimw codimz definition let locally noetherian catenary scheme excellent scheme fix integer dim recall schemes assumed finite dimensional let set irreducible components define function min codimz define follows let hox define set called stratum sending generic point set identified set generic points hence set irreducible components local ring therefore depends depend remark choice matter sake definiteness could taken dim readers invited prefer dealing two different schemes difference dim dim would always appear note even dim dim open subscheme applications usually common bound dimensions schemes considered take sophisticated way would consider whole array function possibly negative defined inductively formulae second function one product order converse hold general definitions could use ordering confused ordering applications choice appears note rest paper choice assumed often suppressed notations shall need following property lemma one codim open subset codim proof inclusion clear one codimz codimz codim thus codim result follows let irreducible components contain may take fact get codim study function analogue lemma lemma following lemma one equality holds regular point proof hox first inequality follows lemma second inequality holds codimx inequalities equalities hence lemma regular point conversely regular one irreducible component lies hence codimx second inequality equality moreover lemma first inequality equality remark particular regular except reg reg regular point independent choice regular point equal following important upper function theorem let locally noetherian catenary scheme specialization dense open subset function upper closed proof hox hox hox first inequality holds results bennett theorem improved singh see remark theorem second holds lemma first open set permissible see codim hox hox see hand lemma open subset codim thus hox hox hox following lemma equivalent conjunction let locally noetherian topological space zariski every closed irreducible subset admits generic point example let locally noetherian scheme recall map ordered abelian group called upper set closed note property compatible restriction topological subspace particular scheme also upper restricting subscheme arbitrary localization lemma map upper following holds dense open subset assume upper set locally closed closure contained particular closed maximal element moreover set xmax maximal closed noetherian takes finitely many values proof may restrict case noetherian taking open covering noetherian subspaces fact subset closed resp locally closed holds intersection moreover may take noetherian iii lemma closed following conditions hold every open obviously equivalent assuming get taking following set conversely assume let noetherian induction show closed assume minimal closed subset wrong since finitely many irreducible components must irreducible let generic point let dense open subset closed hence closed minimality hence contradiction similarly show takes finitely many values noetherian shows assume minimal closed subset wrong necessarily irreducible let generic point let minimality takes finitely many values closed set contradiction first claim follows equality claims follow easily recall may assume definition locally noetherian catenary let let set maximal elements set xmax called locus note disjoint union definition lemma lemma let locally noetherian catenary scheme set locally closed closure contained closed set particular closed xmax closed noetherian finite following regard sets xmax locally closed subschemes endowed reduced subscheme structure lemma let locally noetherian scheme let morphism locally noetherian let assume flat fibre regular assume smooth around sense lemma one hox hox dim hence catenary particular flat regular fibers assume smooth moreover regular regular let excellent scheme let morphism locally finite type let spec completion lying hox particular proof shall use following two lemmas lemma let flat morphism locally noetherian schemes let codimw codimz dim owz particular generic point generic point codimwz proof see ega lemma morphism excellent schemes let assume irreducible codimw codimz proof see ega proof lemma first equality follows lemma show second equality may assume reduced since flat lemma implies generic point generic point moreover generic point exists generic point indeed one take generic point lemma applied flat morphism implies codimension hence generic point shows may consider case integral suffices show following claim assume integral let irreducible component containing codimw codimx since question local may assume spec local noetherian ring normal ega local ring normal well fibers regular hence normal therefore integral dim dim claim follows lemma general let normalization let spec normalization consider cartesian diagram vertical morphisms flat since excellent morphism finite surjective base change claim point generic generic point indeed lemma generic maps generic point codimension zero fibre since fibres generic point case satisfies corresponding properties generic point let irreducible component containing last claim together surjectivity irreducible component dominating since finite closed map even thus point conclude codimw codimw codimx first resp third equality follows lemma applied finite morphism resp second equality follows first case proof noting normal shows claim completes proof next claims including first equality obvious see isomorphism schemes note following morphism schemes subscheme identified subscheme whose underlying topological space homeomorphic show reduced follows ega flatness last statement follows remark see also ega another proof finally follows applying morphism spec compare proof lemma deduction remark bennett defined global functions hox dim showed functions good properties socalled weakly biequidimensional excellent schemes looking generic points closed points one easily sees function coincides function dim biequidimensional permissible discuss fundamental results behavior functions permissible let locally noetherian scheme let closed reduced subscheme let ideal sheaf put grid definition normally flat along stalk grid grid flat normally flat along normally flat along points grid flat permissible regular normally flat along contains irreducible component containing permissible permissible points blowup permissible center called permissible blowup closed subscheme normal cone defined spec grid theorem dense open subset normally flat along following conditions equivalent normally flat along dirx natural map induces isomorphism acts addition assume addition closed subscheme regular locally noetherian scheme let set maximal ideal let resp ideal defining resp following conditions equivalent conditions iii let grp grm natural map inm generated grm inp exists standard base let morphism locally noetherian let regarded closed subscheme let assume flat fiber regular normally flat along normally flat along moreover regular resp permissible regular resp permissible proof follows theorem page theorem page since flat one grd grd thus first claim follows following general fact flat morphism local rings hence faithfully flat flat flat since flat fiber hence regular next claim regularity follows ega last claim nowhere density follows flatness arguments used beginning proof lemma generic point generic point conversely start latter situation generic point mapping numerical criterion normal flatness due bennett carry setting let assumption beginning section assume addition catenary let excellent let definition theorem assume regular let let generic point component contains following conditions equivalent normally flat along codimy hox hox closure proof equivalence proved bennett theorem rest special case following lemma lemma let following equivalent codim hox hox conditions hold codimy proof definition considered functions equivalence suffices show either implies codimy assume lemma dim dim hand dim codimx codimy codimx codimy dim thus get dim dim codimy implies lemma next assume let let prime ideal corresponding suffices show contains minimal prime ideals fact means contained irreducible components contain since irreducible codimz codimz codimy deduce equality codimy hence time proved last claims lemma claim let reduced primary decomposition zero ideal let rad prime ideal associated set contains minimal prime ideals suffices show contains ideals note ideal contained rad contained assume contrary may assume put let resp localization resp resp hop codimy dim dim first inequality follows lemma second inequality follows theorem hence implies lemma implies contradicting assumption primary decomposition reduced criterion complemented following observation lemma let connected irreducible component proof show irreducible assume exists contained two different irreducible components let local ring let reduced primary decomposition zero ideal let rad prime ideal associated assumption trace given let spec spec generic point corresponding assumption hox definition hand dim dim codim codimz first inequality holds results recalled proof theorem second inequality follows lemma last inequality holds since since minimal prime ideal reduced isomorphism opi therefore equalities direction lemma conclude contradiction prove property dimension directrix theorem let excellent scheme scheme embeddable regular scheme let permissible dim question depends local ring lemma may assume consider spectrum complete local ring closed point moreover cohen structure theory complete local noetherian rings see ega every ring quotient complete regular local ring therefore theorem implied following result theorem let regular local ring maximal ideal let ideal let prime ideal spec spec permissible let localization jrp dim proof set let quotient field set definition exists idir dimension dimk dim inp inp grp lemma assume exist free let grp grm natural map idir note assumption lemma satisfied dim theory elementary divisors theorem consequence conclusion lemma noting dimk dimk dimk dim lemma finishes proof theorem case dim show lemma assumption lemma implies grp syma syma resp syma sub grp generated resp claim inp inp grp theorem iii claim implies inm generated inm implies lemma thus suffices show claim note inp grp inp grp inp indeed implies grp flatness grp grp implies inp grp inp take inp exists inp grp choosing basis allows identify grp polynomial ring expanding get since inp implies inp inp grp completes proof claim finish proof theorem suffices reduce case dim remarked lemma assume dim take prime ideal regular dimension let localization jrq noting grprq grp assumption implies spec spec permissible induction dim dim dim note regular local ring catenary ega hence reduced show dim completes proof theorem bennett hironaka proved results behavior functions permissible fundamental resolution singularities recall results well improvements singh carry setting theorem let excellent scheme scheme embeddable regular scheme let permissible closed subscheme let blowup center take points let trdegk hox hox hox hox equalities hold morphism injective denotes strict transform equalities hold field extension one assume addition closed immersion regular scheme let blowup center lexicographic order hox hox proof slightly weaker form hox hox first inequality proved bennett theorem hironaka gave simplified proof theorem stronger form proved singh remark theorem second inequality since dim dim suffices show note universally catenary assumption excellent schemes closed subschemes regular schemes universally catenary definition ega respectively let irreducible components let strict transform irreducible components ega implies note oyi universally catenary since codimyi follows immediately last inequality follows hox claims proved hironaka theorems iii hence remains show assume equality holds everywhere show hox hox suffices show let dim dim lemma assumption implies hand lemma inequality hox hox implies deduce implies view conversely assume hox hox show suffices show show view notations suffices show third claim lemma suffices show injectivity second claim lemma let homomorphism noetherian rings assume minimal prime ideal lie image spec spec injective proof let primary decomposition zero ideal let rad prime ideal assume injective primary decomposition therefore rad set associated prime ideals hence contains minimal prime ideals see theorems contradicts assumption note following consequence lemmas claims theorem may via base change assume spec quotient regular local ring fact last property holds either assumption excellent may base change completion quotient regular ring cohen structure theorem also suffices check injectivity base change faithfully flat use results hironaka first consider case let prime ideal corresponding let grp let scheme theoretic fibre proj inequalities hox hof hcx hox dim spec assumption equalities moreover regular local ring maximal ideal system regular parameters ideal generated initial forms respect fiber blowup spec center isomorphic proj polynomial ring homogeneous ideal identifies canonical immersion proj proj may assume lies standard open subset proj furthermore assume equality middle proof lemma graded isomorphism graded hand let maximal ideal let elements whose images form basis one isomorphism graded algebras grn induced canonical map grn permissible proposition shows image zero zero divisor grn therefore image zero zero divisor claim every element kernel annihilated power gives contradiction kernel let monoidal transform center corresponding let strict transform since xxr follows iii lemma generators natural numbers generated evidently implies every element kernel annihilated power consider case residue field extension arbitrary reduce residually rational case technique one may replace specox consider cartesian diagram faithfully flat monogenic map either finite projection permissible moreover generic point monogenic field extension point maps satisfies furthermore one inequalities hox hox algebraic otherwise assumption inequalities become fact equalities induction number generators starting residually rational case proved may assume injective since injective obtained injectivity completes proof also proved claims finally prove still assumption embedded regular scheme hironaka proved theorem equality implies inequality together implies finishes proof theorem corollary definition either definition let assumption theorem put near near near recall another result hironaka improved mizutani plays crucial role paper let excellent scheme scheme embeddable regular scheme let permissible closed subscheme let blowup center take points theorem assume near assume char char dim residue field dirx projective space associated vector space proof first note inclusion induced inclusion cones spectra graded algebras dirx isomorphism cones theorem precisely induced applying surjection graded grnx sym dirx identify affine spaces associated vector spaces note proj since claim local may pass local ring proof theorem may assume embedded regular scheme denotes blowup inclusion therefore claim dirx follows theorem theorem miz fact second reference certain canonical subgroup scheme depending following properties defined homogeneous equations coordinates hence subcone associated subspace contains moreover first reference vector subspace char char dim miz improved sharp bound dim hand first reference action respects near since conclude contained hence dimension dim dim dim therefore assumption dim vector subspace thus contained biggest subspace dirx therefore dirx lemma consider let blowup permissible center contained let irreducible closed subset contains proper subset proper transform assume char char dim proof let resp generic point resp take points first inequality follows theorem second theorem last equality follow fact since inequalities equalities implies proves next show point near theorem thus implies show let closure assumption proper closed subset since local invariant may localize assume regular theorem permissible assertion follows previous assertion applied embedded case let regular scheme let simple normal crossing divisor let subdivisor union irreducible components containing definition let regular subscheme say normal crossing exists system regular parameters satisfying following conditions spec spec spec spec say transversal addition parameters chosen say resp transversal resp transversal point note transversal intersection set irreducible components contain let consider let strict transform let exceptional divisor let complete transform easily see following simple normal crossing divisors lemma following comparison next section convenient consider set irreducible components definition simple normal crossings boundary set regular divisors associated divisor div simple normal crossings divisor let often elements also called components equivalent condition intersect transversally subset intersection bir regular pure codimension associations div give mutual inverse bijections set simple normal crossing divisors set simple normal crossing boundaries use second language via definitions correspond following setting boundaries definition regular subscheme transversal boundary intersects multiple intersections bis transversally normal crossing transversal transversal resp transversal resp points strict complete transform defined respectively strict transform exceptional divisor note boundaries lemma following consider regular scheme simple normal crossing boundary moreover let closed subscheme definition let regular closed subscheme say permissible say permissible definition history function function subsets satisfies following conditions exists open subset function put component called old resp new component resp basic example history function given following lemma function history function fact satisfies without condition proof left readers define function cardinality endow lexicographic order theorem conditions immediately imply following theorem let assumption specialization dense open subset words see lemma function upper let let max set maximal elements definition define call xmax max locus define diro dirx dimk dirx lemma locally closed closure contained moreover closed subset closed max union disjoint xmax finite noetherian theorems imply following theorem let regular closed subscheme following conditions equivalent permissible open neighborhood every specialization conditions diro definition closed subscheme called satisfies equivalent conditions theorem called remark note theorem let assumption theorem assume irreducible assume dim theorem generic point dim proof first claim hold replacing point regular indeed claim follows inequalities dim dim dim dim follows theorems claim reduced case dim argument proof theorem let let ideals defining respectively let localization jrp lemma implies exists part system regular parameters idir hinp inp grp idir hinm inm grm take part system regular parameters exists irreducible component spec spec idiro hinm inm inm inm idiro grm ideal defining diro implies hence part system regular parameters idiro hinp inp inp inp implies conclusion theorem let closed subscheme consider diagram let complete strict transform respectively subsets given history function define functions follows let define otherwise strict transform otherwise near note history functions lemma functions proof lemma given later complete strict transform definition call respectively theorem take points particular place holds proof follows immediately theorem following mostly use complete transform ease notation max often write instead similarly everything depends definition say following equivalent conditions hold near contained strict transforms call near following result immediate consequence theorem definition theorem assume assume char char dim residue field diro proof lemma take want show put theorems first assume implies get next assume lemma get next show exists open subset put lemma exists open subset theorem lemma exists open subset show satisfies desired property take put assumption first assume implies get next assume implies get similar completes proof lemma proof let set new components near strict transform near near similarly otherwise study transversality certain regular subscheme definition subspace say transversal notation dimk dimk lemma let assume closed subscheme proof let maximal ideal take spec spec put inm view lemma follows following facts ideal hhb grm defines subschemes spec grm proj grm strict transform lemma let assume dimk consider let closed subscheme induces isomorphism proof suffices consider assumptions lemma imply argument last part proof lemma implies thus reduced showing follows lemma let regular scheme regular closed subschemes intersect transversally let let strict transform suppose closed subscheme intersect transversally induces isomorphism transversality implies proof definition thus hence get last isomorphism follows assumption completes proof lemma theorem let take assume char char dim near assume diro proof first show assume near sake later proof also assume diro lose generality proof since may take put resp maximal ideal resp resp assumptions theorem exists system regular parameters satisfying following conditions fixing identification grm inm inm inm inm spec spec idirx grm definition iii exists spec spec diro dirx spec spec spec let resp resp strict transform resp resp spec let theorem exists part system regular parameters polynomial ring subring also implies spec spec note spec defined moreover defined defined defined hvj hand theorem assumption near implies exist idiro clearly implies assertion inclusion equality implies diro thus proof theorem complete corollary let take closed points assume char char dim assume integer following hold either separable diro diro proof claim first show first equality implies near indeed assumption implies lemma theorem hence remains show separable follows lemma assume theorem implies contradiction since near theorem implies shows second equality claim proved claim must hence suffices show assuming dirx claim second assumption implies therefore assertion follows theorem definition call admissible call admissible admissible note admissibility implies bin bin defined follows definition call inessential contains irreducible components contain let bin set inessential boundary components set irreducible components containing definition call reg reg remark call lemma admissible regular normal crossing proof first claim follows lemma since regular assumption means transversal hand unique connected component lies bin last equality holds assumption thus following transition property lemma let admissible admissible proof proof somewhat similar theorem near empty definition therefore may consider case near look surjection kernel let corresponding surjection local rings blowups kernel system regular parameters satisfying following conditions standard basis maximal ideal initial forms define inside spec spec iii spec given div div div div let div exists part system regular parameters since near hox hox theorem trdegk evaluating get dim dim similarly get dim dim hence dim dim dim dim follows initial forms already define inside shows defined finally give following functoriality objects introduced lemma consider cartesian diagram regular scheme hence closed immersion hence flat morphism regular fibers let simple normal crossing divisor let history function associated boundary let regular simple normal crossing divisor let boundary associated let map maps unique component containing bijection inverse maps reduced subscheme structure function subsets history function one particular let closed subscheme let regarded closed subscheme let transversal resp normal crossing resp resp transversal resp normal crossing resp resp unique morphisms making diagrams commutative moreover diagrams cartesian morphism induces morphism diagrams diagrams identify well proof view remarks definition suffices show regular closed subscheme pure codimension regular pure codimension well first property clear lemma second one follows lemma generic point image generic point codimension fiber zero latter fiber thus lemma implies codimz codimz particular preimage regular since flat follows well generic point maps generic point thus disjoint union irreducible components whose generic points lie exactly one component contains condition definition holds construction let let lemma property hence shows show exists open subset maps let preimage let already know hand disjoint union irreducible components unique component containing since hence unique component containing since must equal hence shown hence equation follows lemma bijection equality follows isomorphism follows proof follows via arguments used first claim follows universal property fact ideal sheaf ideal sheaf coincides image ideal sheaf flatness hand since ideal sheaf follows ega left diagram cartesian letting get finally universal property closed immersions give last claim first claim follows applying taking base change first diagram treat exceptional divisor remaining claim let images respectively induce bijections second claim follows first claim also gives bijection case let locally noetherian scheme start following definition definition boundary multiset locally principal closed subschemes recall multisets sets multiplicities precisely multiset elements one forgets ordering one also think sets element appear several times makes clear one define elements cardinalities inclusions intersections unions multisets note also locally principal subschemes need divisors could use multisets convenient questions functoriality see following let locally noetherian scheme let boundary sometimes also call elements components although neither irreducible connected general let submultiset given components containing note definition compatible arbitrary localization morphism also write note even start true set locally principal divisors multiset also might divisor let spec boundary spec definition let regular subscheme let say transversal submultiset bir intersection bir regular codimension hold true set say normal crossing transversal say transversal resp normal crossing transversal resp every remark obviously transversal resp normal crossing resp true set simple normal crossings boundary regular scheme see definition defines divisor normal crossings definition let regular closed subscheme say permissible say let locally integral closed subscheme let locally principal closed subscheme define canonical locally principal subscheme locally spec ring given prime ideal given principal ideal situation proj graded define homogenous element graded principal ideal depends equation change multiplied unit thus proj proj gives locally principal subscheme divisor show definition glues general gives locally principal subscheme show compatible localization let consider nothing show holds consider case hence since prime ideal consider associated chart proj spectrum subring union sets definition assumption trace chart defined ideal given union sets defined union sets hand also spectrum ring union sets png definition assumption ideal ring union sets union sets definition locally principal subscheme defined called principal strict transform following always use principal strict transforms call simply transforms remark always commutative diagram natural proper morphisms strict transform morphisms isomorphisms closed immersion however general isomorphism need locally principal subscheme fact notations locally given graded ideal indicated inclusion need equality give isomorphism taking proj regular regular divisor fact notations regular prime ideal locally discrete valuation associated moreover assumption closed immersion regular scheme simple normal crossings divisor set irreducible components simple normal crossings boundary particular boundary sense definition boundary may multiset construction connects present section previous one see also lemma let boundary let principal strict transform let exceptional divisor definition call strict transform complete transform note always locally principal divisor boundaries moreover note following useful functorialities lemma let closed immersion assume nowhere dense locally integral closed subscheme closed immersion one let flat morphism regular fibers let closed subscheme let regarded closed subscheme regular regular case one canonical morphism proof question local may assume spec affine take notations spec ideal spec given thus case distinction claim follows equality first case equality second case exceptional divisor one exceptional divisor since flat regular fibers first claim follows following may assume spec flat spec equals spec defined ideal integral map flat hence map also injective thus case distinction one hand hence hence claimed definition history function boundary function submultisets satisfies following conditions exists open subset function put divisor called old resp new component resp boundary history pair boundary history function basic example history function given following lemma function history function fact satisfies without condition proof left readers define function cardinality endow lexicographic order conditions theorem immediately imply following theorem let assumption specialization dense open subset words see lemma function upper noetherian induction theorem implies finite define max set maximal elements definition define call xmax max locus define diro dir dimk dirx theorem lemma locally closed closure contained particular closed max union definition disjoint xmax closed subset theorems imply following theorem let regular closed subscheme following conditions equivalent permissible open neighborhood every specialization conditions diro definition closed subscheme called satisfies equivalent conditions theorem called every remark note let closed subscheme consider let complete transform given history function subsets follows let define functions define otherwise strict transform otherwise near note proof following lemma identical lemma lemma function subsets history function complete strict transform definition call respectively results previous section embedded case companions nonembedded situation start following theorem analogous theorem theorem take points particular place holds proof follows immediately theorem following mostly use complete transform ease notation max often write instead similarly everything depends definition say following equivalent conditions hold near contained strict transforms call near following result analogue theorem immediate consequence theorem definition theorem assume assume char char dim residue field diro next show compatibility notions case present section corresponding notions embedded situation previous section lemma let embedding regular excellent scheme let simple normal crossings boundary let closed regular subscheme transversal resp normal crossing sense definition transversal resp normal crossing sense definition intrinsic condition let history function sense definition define function submultisets history function sense definition one diro dirx hence also subspace two notions transversality definition definition equivalent regular closed subscheme sense definition sense definition moreover permissible sense definition sense definition let let respective blowups closed immersion moreover let history function equalities complete transforms strict transforms respectively proof claims easily follow definitions claim directrix note claim follows lemma results case depend local ring point base scheme often reduced embedded case relies lemma following two observations remark let excellent scheme let boundary let assume property concerning shown passing local ring completion following construction useful ring quotient regular excellent ring let let local functions defining get surjection mapping functions define simple normal crossings boundary spec spec may thus assume embedded regular excellent scheme simple normal crossings boundary apply remark lemma theorem let irreducible subscheme assume dim theorem generic point dim proof question local around may pass completion since excellent may assume embedded situation thus claim follows corresponding result embedded case theorem let let set new components near strict transform near near similarly otherwise study transversality certain regular subscheme definition subspace say transversal notation dimk dimk lemma assume dirx closed subscheme hence also proof way claim follows corresponding result embedded case lemma lemma let assume dirx dimk consider let closed subscheme induces isomorphism proof way follows corresponding result embedded case lemma theorem let take assume char char dim near assume diro proof reduction embedded case theorem corollary let take closed points assume char char dim assume integer following hold either separable diro diro follows theorem like corollary follows theorem definition call admissible call admissible admissible note admissibility implies bin bin defined follows definition call inessential contains irreducible components contain let bin set inessential boundary components set irreducible components containing definition call reg call lemma admissible regular normal crossing proof proof identical lemma lemma let admissible admissible proof way proof lemma reduced embedded case theorem later shall need following comparison closed immersion lemma let closed immersion excellent schemes let boundary let closed regular subscheme transversal resp normal crossing transversal resp normal crossing let history function define function submultisets history function admissible admissible converse always hold let regular closed subscheme permissible moreover let let respective blowups closed immersion moreover let history function equality complete strict transforms respectively proofs along lines lemma note subspaces one remark since regular subscheme regular scheme always permissible lemma seen special case lemma following analogue lemma lemma let flat morphism regular fibers locally noetherian schemes let boundary let history function let let map maps bijection function subsets history function one particular let closed subscheme let regarded closed subscheme let transversal resp normal crossing resp resp transversal resp normal crossing resp resp unique morphism making following diagram commutative structural morphisms moreover diagram cartesian identifies well proof trivial conditions history function easily checked since bijection equation follows lemma remark first three cases follow lemma last claim follows first two claims shown like proof lemma last four claims follow applying first two claims every main theorems strategy proofs treat following two situations parallel way embedded case excellent noetherian scheme closed immersion excellent regular noetherian scheme simple normal crossings boundary definition case excellent noetherian scheme boundary definition section schemes assumed noetherian definition point called regular regular normal crossing call every regular normal crossing call strongly regular every contains unique irreducible component lies amounts equation bin bin multi set inessential boundary components see definitions case case denote xreg resp xbreg resp xbsreg set regular resp resp strongly points open subsets dense reduced call xbsing xbreg locus introduce following definition case schemes definition call xred regular normally flat along xred call xred regular normally flat along xred compare definition reserved name permissible subschemes containing irreducible component call xred normally flat along xred call xred normally flat along xred similar remark comparison call strongly xred strongly normally flat along xred note regular reduced similarly reduced finally strongly strongly reduced denote xqreg xbqreg xbsqreg sets strongly points respectively theorem dense open subsets moreover inclusions xqreg xbqreg xbsqreg xqreg xqreg xreg xbreg xbsreg xreg xreg last inclusions rows equalities contains irreducible component vertical inclusions equalities reduced lemma let connected excellent scheme one xqreg thus constant xqreg let admissible boundary history max one xbsqreg thus constant max xbsqreg proof let xqreg let irreducible component containing let generic point since xqreg open dense quasiregular contained xqreg constant xqreg theorem therefore since closed conclude lemma conclude proves first claim direction trivial second claim obvious consequence direction first claim let consequently irreducible generic point conclude hence since latter set closed second claim follows immediately study lemma implies lemma subscheme complete transform xbsreg sreg xbsqreg sqreg first consider case definition case sequence complete resp strict blowups diagram boundary center complete strict transform transform resp call sequence contracted none morphisms isomorphism given sequence blowups define associated contracted sequence suppressing isomorphisms sequence renumbering abbreviate short prove theorem following general form theorem let dimension two canonical finite contracted sequence complete isomorphism sqreg particular morphism isomorphism xbsqreg moreover following functoriality holds equivariance action automorphism group extends sequence unique way localization sequence compatible passing open subschemes arbitrary localizations morphisms following sense denotes pullback associated contracted sequence contr coincides also canonical finite contracted sequence strict blowups properties except strict transform reduced every reduced regular normal crossing moreover xbsqreg xbsreg particular theorem obtained case xbsreg xreg sequence apply reduced xbsreg xreg well sequence empty obtain extra information collection strict transforms created exceptional divisors simple normal crossing divisor definition let category schemes closed localization say canonical functorial resolution boundaries holds statements theorem hold schemes boundaries say canonical functorial resolution holds statements theorem theorem hold schemes consider case definition case sequence complete resp strict blowups sequence blowups center permissible complete transform resp bei strict transform call sequence contracted none morphisms isomorphism given sequence blowups define associated contracted sequence suppressing isomorphisms sequence renumbering abbreviate short prove theorem following form theorem let dimension two canonical finite contracted sequence complete isomorphism sqreg regular particular morphism isomorphism xbsqreg moreover following functoriality holds equivariance action automorphism group automorphisms respect extends sequence unique way localization sequence compatible passing open subschemes arbitrary localizations morphisms following sense denotes pullback associated contracted sequence contr coincides also canonical finite contracted sequence strict blowups properties except strict transform reduced reduced xbsqreg xbsreg regular normal crossing simple normal crossings divisor definition let category schemes closed localization say canonical functorial embedded resolution boundaries holds statements theorem hold triples regular excellent scheme simple normal crossing divisor closed subscheme remark follows lemma theorem implies theorem following way constructed one obtains consecutively blowing center identifying strict transform conversely restriction generally approach canonical functorial embedded resolution boundaries holds category schemes definition canonical functorial resolution boundaries holds set strategy proof theorems general setting let excellent scheme recall consider noetherian schemes definition call excellent noetherian scheme equisingular constant call locally equisingular connected components equisingular see strategy make locally equisingular remark open one hence equisingular lemma two possibilities connected component either equisingular irreducible umax nowhere dense first case follows theorem theorem permissible equisingular well definition permissible centers nowhere dense reduced connected component equisingular regular remark hence locally equisingular iff equisingular iff regular way example following situation occur schemes disjoint union three irreducible components blowing xmax make locally equisingular motivated remarks define definition let connected equisingular morphism composite sequence morphisms blowup permissible center let elements assume given noting isomorphism glue composition permissible get morphism define definition let connected equisingular morphism called following conditions hold composition permissible blowups isomorphism xmax note theorem imply exists definition excellent scheme morphism called elimination restriction connected component equisingular isomorphism connected components theorem let excellent noetherian scheme let sequence morphisms locally equisingular isomorphism proof see also theorem alternative proof theorem suppose exists infinite sequence locally equisingular let union connected components equisingular let choose element follows finiteness theorem functions occurring contained set hfm hilbert functions standard graded algebras thus follows theorem noetherianess hfm see proposition several equivalent conditions characterizing noetherian ordered sets choose hxi hxj let image remark morphism maps theorem remark hxk since conclude inequalities equalities therefore contradicting assumption elimination corollary prove canonical functorial resolution singularities excellent reduced schemes dimension suffices prove every connected excellent reduced scheme dimension exists canonical functorial equivalently suffices show every scheme every canonical functorial functoriality means analogues properties theorem hold respectively analogue property following either pullback sequence canonical passing associated reduced sequence fact assumptions one gets canonical functorial sequence theorem locally equisingular means regular remark consider case corollary prove canonical functorial resolution singularities excellent schemes dimension suffices prove exists canonical functorial every connected excellent scheme dimension equivalently suffices show every scheme every canonical functorial functoriality defined corollary fact first get canonical functorial sequence eliminations locally equisingular corollary get similar sequence red regular blowing centers get sequence identified red strict transform since red reduced homeomorphic equisingular see remark theorem permissible follows red regular normally flat along red assume first sequence sequence red functorial immediate sequence functorial localizations well automorphisms follows inductively via localization automorphisms respect center therefore extend unique way remark point choice strategy might tempting start desingularizing xred xred permissible xred general permissible however made first series arguments show get permissible xred achieve xred becomes regular time normally flat along xred consider variant schemes boundary let excellent scheme let boundary boundary case case following consider complete transforms boundaries sequences complete simply speak sequences easy see analogous results also hold case strict transforms sequences strict definition call constant locally every connected component remark follows lemma connected component either umax nowhere dense first case every one reduced locally even regular scheme obviously happen union three irrereg reg reg ducible components reg made blowing xmax definition let history function admissible let sequence written boundaries neither regular schemes case let complete transform let center connected max called connected called max max max arbitrary called max max restriction connected component isomorphism restriction connected components call sequence contracted none morphisms isomorphism general define associated contracted sequence omitting isomorphisms renumbering final index may decrease remark glueing one gets canonical functorial max connected one canonical functorial max canonical functorial max one connected components functoriality defined theorem similar way theorem one proves theorem infinite sequence max eliminations following result obtained embedded case corollary case show canonical functorial resolution singularities boundaries noetherian reduced excellent schemes dimension suffices show existence canonical functorial max connected reduced excellent schemes dimension admissible boundaries history functorial last statement means obvious analogues conditions theorem hold sequences considered case obvious analogous statement holds fact given start history function strongly holds done assumption canonical functorial max let strict transform obtained successive transforms sequence blowups whose composition admissible lemma done lemma repeat process time iterate necessary theorem finitely many steps process obtains hence achieves resolution lemma case lemma case case obtain corollary case show canonical functorial resolution singularities boundaries excellent noetherian schemes dimension suffices show existence canonical functorial max connected excellent schemes dimension admissible boundaries history constant functorial last statement means obvious analogues conditions theorem hold sequences considered case obvious analogous statement holds follows corollary similar way corollary follows first get canonical functorial sequence locally look canonical functorial resolution sequence red corollary comes via complete transforms blowing centers obtain sequence red identifies thus moreover normally flat along constant connected components red prove theorem remark theorem follows well corollary suffices produce canonical functorial max connected excellent noetherian schemes dimension two admissio ble boundaries history constant remarks definition suffices produce canonical functorial max slightly modify procedure deduce theorem following partly weaker partly general result theorem case let excellent connected noetherian scheme let admissible boundary history constant let max assume following char char dim dim integer closed point either diro exists canonical reduced satisfies analogues properties theorem analogue following either else max reduced sequence associated pullback sequence canonical theorem follows conditions theorem hold blowup center conditions hold well fact holds since dim dim condition holds theorem condition holds corollary moreover condition holds see proof theorem assume dimension condition holds condition holds moreover constant condition holds lemma holds well hand presence condition suffices consider admissible boundaries history satisfy condition fact procedure outlined proof corollary property trivially fulfilled beginning remarked lines also fulfilled sequence blowups therefore keep assumptions theorem sequence blowups arguments corollaries apply modified setting remark start proof theorem outline strategy works conditions theorem stated generally would interesting see also works schemes let excellent connected noetherian scheme let admissible boundary history constant let max construct canonical contracted sequence blowups follows introduce notations constructed let give labels years irreducible components inductive way follows irreducible components label irreducible component dominates irreducible component inherits label otherwise gets label write union irreducible components label start actual definition dim dim induction dimension canonical resolution sequence sequence blowups successively blowing centers also permissible lemma obtain sequence blowups strict transform moreover blow get call obtained sequence first resolution cycle proceed canonical resolution sequence get sequence blowing centers blow get call sequence second cycle repeating process finitely many times producing cycles get empty situation fact proceed several cycles reach empty proceed etc procedure ends ynr empty situation fact eliminated situation theorem see already see proposition sequence constructed functorial automorphisms morphisms geometrically regular fibers remark corollaries procedure always finite gives max canonical functorial resolution boundaries formulated proof theorem let blowup center seen conditions hold complete transform well show condition may assume irreducible closed point regular irreducible curve step let closed point consider note trivial reasons theorem diro convention hence condition also satisfied moreover dim diro condition implies diro lemma hence union closed points projective line cases step let regular irreducible dimension theorem let generic point consider let closed point theorem diro hence one point dim condition satisfied similarly diro theorem hence empty consists unique point one implies induces isomorphism thus regular theorem moreover case condition implies dirx lemma hence hence collection closed points regular irreducible curve cases step consider special case dim dim every point isolated moreover dim canonical sequence defined remark consists blowing points repeating process long theorem process stops finitely many steps theorem holds noticed theorem shows theorem holds dim exists canonical functorial resolution sequence step consider general case construct canonical reduced sequence blowups follows let constructed let give labels irreducible components inductive way follows irreducible components label irreducible component dominates irreducible component inherits label otherwise gets label write union irreducible components label step assumption dim let byo let canonical resolution sequence theorem exists finite step regular normal crossing write boundary obtained let sequence obtained inductively blowing center blowup identifying strict transform collection closed points hence write boundary obtained therefore sequence could also use lemma remark lemma since nowhere dense subscheme contained fact equal label part defined first stage claim subschemes regular dimension fact holds construction moreover statements step conclude schemes disjoint unions closed points projective lines hence regular moreover let union components since closed points closed point dominate curve morphism isomorphism hence regular normal crossing direct check application lemma since regular contained conclude step next blow subscheme regular obtain claim subschemes regular dimension moreover intersection empty first part follows similar arguments fact arguments exactly careful since consists irreducible components strict transform blowup scheme possibly structure since regular dimension isomorphism second part consists finitely many closed points definition contained step next blow otherwise obtain proceed way blowing smallest number obtain always claim subschemes regular dimension intersection empty first part follows second part follows claim may assume induction definition implies proves desired assertion step thus defined wanted canonical sequence reduced construction show finiteness sequence lemma let scheme satisfying assumptions theorem let irreducible regular curve let let step dim dim regular case put repeat procedure get sequence morphism process stops finitely many steps dim proof let generic point remarked step get longer sequence however must finite theorem applied localization spec point max note definition result set components note schemes satisfy conditions theorem therefore shown claim regular clear resolution sequence step following property centers blowups always lie subscheme let connected components let open subscheme containing meeting resolution sequence obtained glueing resolution sequences subsets show finiteness resolution sequence may thus assume regular irreducible applying lemma may assume collection finitely many closed points isolated stratum moreover clear case remaining part resolution sequence canonical resolution sequence step thus reduced case isolated point fact may assume consists first step canonical sequence form blowup done empty consists unique point lying latter case therefore theorem otherwise must assumption theorem thus obtain sequence points sequence stops either exn sequence exi sequence must finite theorem remains case follows theorem step last step show functoriality properties theorem property course easier special case written reference show generally proposition canonical sequence defined remark functorial arbitrary flat morphisms geometrically regular fibers following sense let max let sequence defined remark max natural history function induced see lemma let analogous sequence let sequence consisting one term obtained notations lemma closed subscheme let base change note canonical way particular let base change etc canonically isomorphic associated contracted sequence contr obtained omitting isomorphisms precisely claim following canonical isomorphism define function inductively following property otherwise following holds canonical isomorphisms let canonical isomorphism induced unique morphisms diagrams commutative moreover isomorphisms left hand morphism identity morphism occurring right hand morphism base change morphism induces isomorphisms analogous statement holds sequence remark replaced max sequence deduced gluing canonical resolution sequences outlined proof corollary corollary using mentioned proof prove claims induction dimension quite clear deduce dimension fixed claims trivial dimension zero therefore suffices prove assumed smaller dimension since flat regular fibers morphism isomorphism lemma property follows since flat compare proof lemma claims use induction empty show isomorphism follows immediately since definition moreover follows trivially show strategy remark following holds first resolution cycle already completed blow since flat geometrically regular fibers conclude regular well follows determined first center canonical resolution sequence ymax similarly flat max note regular fibers hence max ymax lemma shows suppose already constructed morphisms proved define morphisms composition isomorphism canonical projection flat regular fibers holds isomorphism unique morphism necessarily isomorphism making diagram commutative trivially cartesian define composition projection induces isomorphism induction flat unique morphism making diagram commutative diagram cartesian see lemma base change right column obtain diagram wanted flatness regularity fibers lemma isomorphism cases empty note cartesian diagram fact show isomorphism induces isomorphisms factorization condition means map generic points equivalently generic points show corresponding property place assume since flat generic points points lie generic points generic points fiber analogous statement holds place let let generic point turn maps maps generic point generic point let image generic point lies since maps note fibers obtained base change field extension still generic point generic point fiber flat fiber obtained base change residue field extension therefore generic point fiber implies generic point hence lies finally show induces isomorphism provided suppose prescription remark means either first case second case also get hence also therefore cases strategy regular strategy regular flat regular fibers also claim follows regular lies singular locus strategy one resolution cycles suppose cycle started obtain centers canonical resolution sequence first assume start resolution cycle well induction dimension apply canonical commutative diagram cartesian squares upper line part canonical resolution sequence lower line come canonical resolution sequence function defined analogous way using instead note definition resolution cycle center center induction vertical morphisms induce isomorphisms last scheme claim coincides morphism induced proves claim since claim true assume true hence first scheme empty last one shows since moreover existence uniqueness morphisms induced diagram implies remaining claim regular induction regular well note flat regular fibers hence strategy start resolution cycle well proof cycle starting renumbering corollary situation proposition sequence resp max sequence resp resolution sequence finite also finite turn two key theorems used proof let excellent scheme let boundary history let assume char char dim diro consider diro dirx let generic point note theorem point lies lemma point point proof take point theorems follows theorems dim dim used second inequality fact cases apply hence assumption implies near theorems get dim dim implies conclusion assume implies dim lemma remark respect complete transform consider complete transform theorem one point exists let closure induces isomorphism lemma theorem implies diry point lemma unique point consider proceed way construction occurred proof theorem leads following definition assume let integer fundamental sequence blowups length canonical possibly infinite sequence permissible blowups satisfies following conditions diro hxq let generic point iii isomorphism generic point point near sequence infinite complete transform proof following first key theorem given theorem assume regular closed subscheme spec dimension sequence stops finitely many steps remark note assumption theorem holds particular dim spec thus special case theorem following corollary isolated locus fundamental sequence consists sequence blowups closed points finite particular dim obtain canonical sequences constructed steps proof theorem hence obtain finiteness needed proof consider fundamental sequence blowups definition second case needed proof theorem namely isolated theorem remark deduce fundamental sequence finite exists point let nonempty points lie maximality image consists finitely many closed points since contain generic point argument used step proof theorem points one point latter set consist finitely many closed points well pick one point since already treated case isolated always led following definition definition sequence blowups called fundamental unit blowups length following conditions satisfied closed point diro iii natural morphism induces isomorphism surjective closed point exm exm considered successive complete transform convention fundamental unit blowups length sequence blowups call resp initial resp terminal part remark definition assumed resp isolated xmax resp max definition chain fundamental units blowups sequence blowups fundamental unit blowups terminal part coincides initial part finiteness canonical case needed proof theorem consequence following second key theorem whose proof given theorem let chain fundamental units bpermissible blowups let initial part assume regular closed subscheme max dimension holds isolated chain must stop max finitely many steps fact show finiteness considered case show infinite sequence closed points lies happen exn construction canonical sequence would give rise infinite chain fundamental units remark claims fundamental sequences fundamental units chains fundamental units depend localization spec moreover results lemma lemma may assume spec complete local ring thus assume embedding regular excellent scheme moreover lemma simple normal crossings boundary whose thus may consider embedded version constructions blowup embedded diagram proofs theorems situation assumed bases let regular noetherian local ring maximal ideal residue field let ideal turns directrix dir important invariant singularity spec useful consider system regular parameters idir grm inm inm form coordinates affine space dir aek consequently useful distinguish grm observation leads following definition system regular parameters called strictly admissible satisfies condition sequence elements called admissible extended strictly admissible system let strictly admissible let system elements called admissible inm let tuple new variables note admissible isomorphism grm inm mod idir map induces following isomorphism use later dir proj admissibility play essential role next section moment shall work following general setup setup let let system elements extended system regular parameters follows fix work various choices choice induces identification grm inm inm let hui put mod hui note hui let write expansion completion yrbr uae define form mod define easy see depends presentation need make expansion uniquely determined ega choose ring coefficients subring complete local ring maximal ideal char choose set representatives note char char quotient field back general situation expanded unique way use following map sets representative introduce notion base see definition generalizes standard base definition following facts crucial permissible blowup standard base necessarily transformed standard base base see theorem hand standard procedure transform base standard base see theorem linear form given called positive resp resp definition let setup let linear form define respect min come presentation set fix representative initial form respect defined inl inl range satisfying set inl ideal define hinl case positive define inl inl hinl grm easy see one following inl element polynomial ring coefficients formal power series ring positive inl grm independent choice remark note inm proofs following lemmas easy left readers lemma let assumptions definition independent choice min assume char automatic positive inl inl sum ranges let another system parameters system regular parameters assume definition let assumption definition let system elements linear form called effective inl lemma let assumption definition following conditions equivalent effective inl iii write exist positive linear form effective contained hui effective linear form effective precisely one following inl definition let setup let system elements called base tuple setup positive form effective inl grm called base addition standard base inl cases called reference datum base lemma let setup let standard base inm base reference datum remark assume admissible definition standard base base proof immediate consequence definition remark show first note inm inm standard base inm choose strictly admissible identify grm exist form standard base inm note homogeneous degree hinm inm inm writing inm implies standard base inm hence suffices show exists positive linear form inl may write sum ranges easy see exists positive linear form satisfying following inl proof lemma complete crucial fact bases following theorem let resp standard base setup positive linear form effective reference datum going proof theorem deduce following corollary let resp standard base let resp standard base proof assumption implies contained hui lemma exists positive linear form effective assumption theorem implies inl inl generate resp form standard base inl lemma get inl inl implies corollary proof theorem let reference datum exists assumption definition system regular parameters positive linear form generate resp form standard base first assume theorem follows proposition view lemma note condition proposition always satisfied positive consider general case since systems regular parameters exists glr hui take positive linear form easy computation shows implies inm mod view lemma proof theorem reduced case proposition let setup let let linear forms assume char lemma assume following conditions exist positive one take assume exist positive hinl inl inl hinl inl proof let expand put bmax bmax max maximum taken respect lexicographic order lemma assumption proposition exist bmax bmax positive one take proof assumptions proposition write degree homogeneous writing implies take bmax looking homogeneous part degree get homogenous part degree therefore bmax bmax put bmax lift bmax note positive sum finite shows lemma view proposition last inequality holds min min bmax therefore lemma get min lemma implies bmax hence get bmax bmax proves lemma finally min proves lemma proof lemma complete proceed proof proposition construct lemma applying lemma repeatedly get sequence bmax bmax min min note bmax drop forever lexicographic order must infinitely many noting discrete subset implies given taking sufficiently large shows first assertion proposition view implies exist implies lemma inl inl inl sum ranges hinl inl shows also implies last assertion proposition faithful flatness completes proof proposition lemma let setup assume admissible definition choose strictly admissible base inm equality makes sense via isomorphism grm induced choice base standard base inm proof follows second assertion follows first view show first assertion take positive linear form effective theorem reference datum definition lemma implies suffices show inm strict admissibility lemma implies exists standard base admissible definition lemma implies inm take positive linear form proposition lemma imply inl inm completes proof characteristic polyhedra section always setup beginning introduce polyhedron plays crucial role paper provide useful invariants singularities spec see also give natural way transform base standard base see corollary definition closed convex subset implies essential boundary subset consisting unless write positive linear form put min called face slope one easily sees bounded positive consists unique point call vertex remark call min definition let setup let contained hui write polyhedron defined smallest containing points fact polyhedron depends depend presentation defined inv inv inv writing inv inv sum ranges linear form write definition min face slope namely definition write simply let define inel inel sum ranges note inel different inl definition write inel one easily sees following lemma let notation definition inv independent presentation bounded inel independent presentation otherwise may depend abuse notation bounded inel inv inv inv vertex lemma inm inm proof definition equivalent condition equivalent lemma follows easily follow similar argument details omitted definition let system elements hui define polyhedron smallest containing defined inv inv inv noting inel face defined similarly linear form put min let denote set vertices put inv call set essential points definition smallest contains lemma following fact easily seen lemma max particular finite set theorem let assumption definition following conditions equivalent base standard base inm admissible conditions imply strictly admissible proof implication follows lemma view remark show effective lemma remark thus desired assertion follows theorem assume admissible lemma conditions imply inm generated polynomials thus must idir assumption admissible proves last assertion definition let assumption setup polyhedron intersection basis reference datum remark original definition given formulated intrinsically follows definitions equivalent polyhedron defined equations different linear forms another important result hironaka provides certain condition see theorem first introduce notion normalizedness definition let polynomial ring field homogeneous ideal define homogeneous homogeneous polynomial leading exponent biggest exponent lexicographic order occurring max generated homogeneous elements also write note definition assume given homogeneous degree normalized writing normalized normalized easy see normalized weakly normalized sense definition way transform weakly normalized standard base homogeneous ideal normalized standard base lemma theorem definition let assumption definition weakly normalized weakly normalized normalized sense definition normalized inv inv sense definition normalized along face inel inel sense definition introduce notions solvability preparedness definition let assumption definition called solvable inv case tuple called solution remark possible inv definition hence solvable definition let assumption definition call prepared normalized solvable call prepared along face normalized along solvable call prepared along call well prepared prepared call totally prepared well prepared normalized along bounded faces state hironaka crucial result theorem let assumption definition assume standard base following condition holds proper let vertex prepared vertex particular note condition satisfied admissible definition corollary let assumption theorem assume admissible following conditions equivalent prepared lying strictly admissible standard base admissible conditions hold proof clearly implies equivalence follows theorem remains show implies lemma find strictly admissible standard base admissible lemma hence theorem implies since vertex finally theorem implies argument shows completes proof following refinement theorem theorem let assumption definition assume admissible base let spec spec assume permissible exists vertex face prepared particular proof theorem theorem last assumption implies let lemma thus suffices show implies let standard base usual write note lemma hence get following equivalences base noting standard base base lemma used assumption theorem implies exists base satisfies conditions since implies implies desired assertion finally last assertion follows lemma expect get desirable situation theorem right away need procedures attain situation given following results first discuss normalizations theorem let assumption definition assume weakly normalized exist xij hui following hold xij normalized iii remark situation passage called normalization easy see need following slight generalization theorem theorem let assumption definition let bounded face assume normalized exist xij putting xij normalized along proof write positive linear form write ine homogeneous degree put nothing done assume contrary let min bmax maximal element respect lexicographic order note bmax assumption normalized construction exist homogeneous degree bmax exponents smaller bmax note homogeneous degree take inm take bmax bmax bmax mod bmax put bmax ine ine bmax bmax inv inv hence elements smaller bmax lexicographical order proves theorem induction discuss dissolutions theorem let assumption definition let solution form particular exists solution always let huv image grm let iii inm remark situation passage called dissolution easy see come preparation let assumption definition apply alternately repeatedly normalizations dissolutions vertices polyhedra precise endow order defined lexicographical order let smallest point apply normalization theorem dissolution theorem solvable get prepared repeating process arrive following conclusion theorem let assumption definition assume weakly normalized integer exist xij hui hui putting xij prepared along bounded faces contained complete obtain stronger conclusion remark theorem make theorem satisfy additional condition normalized along bounded faces polyhedron contained complete make totally prepared corollary let assumption notation theorem base base admissible strictly admissible standard base admissible proof follows corollary follows theorem end section prepare key result relates certain localizations ring certain projections polyhedra let definition let let localization let jrs maximal ideal want relate characteristic polyhedron assume given presentation rewritten uae introduce conditions naturally verified case spec exceptional divisor blowup closed point see lemma assume subfield mod get following mod uae mod mod hence following equivalences assume fixed finitely many condition implies theorem let linear form holds hold initial form along face respect presentation definition lies polynomial ring inel inels considered equation assume assume proper prepared along face els prepared along face proof get min min follows show use compute inel uae first resp second sum ranges resp resp follows easily show assumption implies inel normalized suffices show following claim let vertex solvable prove claim descending induction case obvious easily see exists vertex vertex induction hypothesis may assume solvable assume solvable exist elements fraction field inv claim lies localization admit claim moment claim lift take positive linear form localization set elv hence theorem apply instead note proof used carries replacement get assumption theorem implies thus get inequalities follow contradicts proof complete show claim note discrete valuation ring prime element mod thus suffices show assume contrary may assume set recall consider mod claim indeed structure theorem complete local rings implies contained separable extension lemma smallest thus claim follows fact expand get since unit noting implies hand imply absurd completes proof claim transformation standard bases section study transformation standard base permissible particular respect near points begin setting local description situation theorem setup let excellent regular scheme closed subscheme take closed point put maximal ideal put write spec spec ideal define integers let closed subscheme permissible let prime ideal defining theorem dirx find system regular parameters strictly admissible definition gives identification grm inm inm inm grm consider diagram note proj fix point proj proj theorem char char dim point near lies proj moreover near theorem implies without loss generality assume lies chart let maximal ideal let ideal defining put part system regular parameters choose system regular parameters note trdegk assume given standard base admissible definition inm inm lemma finally assume assumption satisfied trivial reasons conditions theorem example assumptions imply uau sum ranges lemma implies mod later use choose resp ring coefficients resp also choose set representatives resp set representatives note choices independent demand end setup want compare properties downstairs upstairs especially properties polyhedra initial forms let linear form downstairs resp upstairs inl denote resp corresponding initial form resp respect resp theorem setup basis standard base basis precisely exists positive linear form upstairs first need show following lemma let assumption setup choose consider linear forms downstairs upstairs respectively downstairs upstairs following holds assuming assuming inl proof let write uau notation note implies hence min min proves lemma next assume note therefore see inl note last implication independent choice representative moreover conditions hold inl completes proof lemma proof theorem lemma choose consider linear forms downstairs upstairs lemma desired positive linear form theorem take upstairs lemma proposition inl inm inl clearly proposition suffices show satisfies condition char proposition since char char first part follows lemma show second part choose take integer proposition implies exist used fact vlp downstairs let discrete valuation respect ideal implies therefore calculate lemma lemma implies therefore may assume lemma imply last sum ranges proves second part proof theorem complete keep assumptions notations setup assume standard base theorem base corollary assures form standard base preparation admissible hence following result important theorem let near definition exist linear forms idir particular admissible let field extension consider following map first isomorphism maps recall idir grm idir idir theorem immediate consequence following general result theorem assume near idir idir mod idir linear forms proof need following proposition near exist hij setting hij following mod particular standard base proof first note since near last assertion proposition follows view iii lemma put map notations setup implies mod prove suffices show hij letting proposition indeed follows replacing hij elements sufficiently close implies corollary imply one take let maximal following holds exist hij holds hij since otherwise want show suppose holds contradicts maximality implies assumption mod since normalized definition normalized therefore corollary imply exist homogeneous degree setting claim completes proof indeed implies holds contradicts maximality find apply argument instead setting repeating process get holds contradicts maximality show claim noting suffices show write similarly write homogeneous degree imply let noting char property implies shows desired assertion proof theorem second assertion follows first one show first proposition exist homogeneous mod let idir exist homogeneous hand implies homogeneous regarding everything mod get second equality implies mod first equality implies since idir minimal subspace implies desired assertion conclude section following useful criteria nearness nearness keep assumptions notations setup theorem near converse holds prepared vertex lying assume near converse holds assumption proof write assume lemma write put first inequality follows theorem last lemma lemma assumption weakly normalized definition weakly normalized implies therefore get near also get implies assume near let proposition claim basis indeed basis theorem claim follows fact using corollary claim exists vertex prepared vertex obtain theorem assume lemma assumed equality remark since inm implies idir near finally assume near theorem basis theorem admissible thus corollary implies completes proof theorem remark theorem important compute also important see implies issues discussed later paper various situations see lemma termination fundamental sequences bpermissible blowups case section prove key theorem deducing stronger result theorem moreover give explicit bound length fundamental sequence polyhedron beginning first introduce basic setup setup let excellent regular scheme let closed subscheme take point let maximal ideal residue field write spec spec ideal define integers also assume given simple normal crossing boundary history function subsets definition note may empty introduce notations definition prelabel system regular parameters admissible definition base lemma prelabel label corollary means strictly admissible standard base admissible lemma inm idir grm inm grm inm grm label resp totally prepared sense definition resp remark theorem label choose element spec spec positive linear form let definition writing min easy see depends choice define min irreducible component theorem assume char char dim assume fundamental sequence length starting definition let label case assume satisfies following condition exist spec spec diro dirx assume prepared along faces elq first show deduce theorem theorem suffices show assumption infinite fundamental sequence blowups assume contrary write write definition hence assumption last statement implies setting spec implies theorems conclude image spec similar argument second part implies contradicts assumption theorem since dim completes proof theorem prepare proof theorem consider theorem point near contained dirx take label definition identification determined dirx proj let dirx near theorems imply let maximal ideal let ideal defining note dirx spec spec denote seen setup system regular parameters localization use usual identifications moreover consider maps linear called monic lemma minimal containing monic linear form inel inele inele inel prepared along ele prepared along assume prepared along near resp near resp totally prepared assume near prelabel assume prepared along resp totally prepared resp totally prepared label proof compute follows linear form min min compute inel sum ranges follows easily follows theorem first assertion follows theorem theorem assertion follows corollary consider diro dirx proj let generic point let assume making suitable choice coordinate may assume exists spec spec diro dirx spec spec note trdegk theorems near also implies permissible theorems theorem near write let maximal ideal note localization system regular parameters lemma let characteristic polyhedron minimal containing lemma monic linear form particular prepared along face totally prepared assume prepared along face near holds assume prepared along face near holds label proof follows lemma theorem applied place need check conditions well theorem replacement holds since view presentation holds fact fixed finitely many theorem consequence assumption label definition finally theorem follows lemma consequences theorems lemma completes proof lemma proof theorem write assume extends sequence let label write ozq maximal ideal let ideal defining write claim let label prepared along faces elq localization label proof claim follows lemma induction follows applied spec spec place note condition satisfied spec lemma recalling near convention get theorem claim remains show rewrite follows map easily see noting factors strict transform spec defined valuation defined ideal see hence get implies min since assumption shows desired assertion proof theorem complete corollary let label assume example assumptions theorem hold length fundamental unit definition greatest integer proof claim case together theorem assumption near near additional invariants case order show key theorem recall invariants singularities defined hironaka definition works dimension long directrix definition polyhedron define inf inf inf sup inf inf inf picture follows slope three vertices play crucial roles let consider situation setup assume let prelabel recall write write see inf inf inf sup inf inf definition prelabel definition prepared say exists prelabel theorem complete extend definition situation old components taken account assume spec definition let system regular parameters admissible choose spec spec implies hui definition define minimal containg easy see independent choice prelabel let minimal containg note assume define min prelabel called prelabel exists lemma lemma lemma let prelable preparation vertex assume integer exists label moreover complete one make totally prepared proof consequence view corollary prove setting vertex prepared lies range theorem implies normalization thus suffices show dissolution write remark dissolution given coordinate transformation write simplicity choose spec may write get case implies second equality follows third follows case implies second equality follows third equality holds since completes proof lemma lemma let prelabel assume regular closed subscheme dimension particular holds isolated proof corollary prepare vertices get label since thus may replace assume standard base assume letting since implies implies theorems point corresponding aro gument prove thus spec contradicts assumption lemma assertion shown way lemma let system regular parameters strictly admissible assume dimk dirx assume addition label proof assumption idirx hinm inm hence implies easily deduce first assertion lemma second assertion obvious consequence first proof case key lemmas section prepare key lemmas proof theorem let assumption setup assume char char dim fix label adopt notations definition recall inm grm inm lemma assumption implies also implies always satisfied choose spec spec study two cases case point blowup consider note trivial reasons let complete transform definition case strict transform theorem point near contained dirx let pair new variables let dir proj isomorphism determined take closed point near theorems put maximal ideal let ideals spec spec spec spec lemma assume proj let system regular parameters near prelabel minimal containg vertices move horizontally prepared prepared along face totally prepared assume putting assertions hold replacing proof definition strictly admissible standard base admissible hence seen setup follows theorem theorem follow lemma assumption implies strict transform spec spec spec implies first assertion second assertion follows first following lemma shown way previous lemma except last assertion follows lemma lemma assume proj put system regular parameters near prelabel minimal containg prepared prepared along face totally prepared assume regular closed subscheme dimension let generic point theorem theorem near write maximal ideal note system regular parameters lemma let characteristic polyhedron assume well prepared near near near near label proof lemma special case lemma case curve blowup let assumption beginning section assume given regular curve containing definition spec spec theorem assumption implies generic point consider let complete transform definition case strict transform theorem unique point possibly near given dirx proj follows assume near theorems let maximal ideal let ideal spec spec seen setup system regular parameters lemma near prelabel resp resp totally prepared assertions hold replacing proof first assertion follows theorem second first setup follows theorems show write compute immediately implies first assertion shown way lemma finally last assertion shown way lemma completes proof lemma lemma assume label note implies let spec let generic point unique point possibly near near label proof first assertion direct consequence theorem second assertion follows lemma applied base change via diagram proof case separable residue extensions section prove theorem implies key theorem assumption residue fields initial points separably algebraic proof divided two steps step one fundamental unit let assumptions notations beginning previous section assume given fundamental unit blowups definition denoted let generic point let image definition following conditions hold near hxq exq let ozq maximal ideal let ideals spec spec spec spec let rpq localization jpq rpq let pair new variables consider isomorphism dirx proj definition prelabel resp label prelabel resp label point homogeneous coordinates proj called coordinates say sense definition prelabel prelabel definition remark coordinates depend lemma let label prepared along face assume coordinates let following holds label minimal containg vpq well prepared label proof lemma label vprepared lemma well prepared label near lemma follows lemmas applied place definition call regular closed subscheme dimension note holds particular isolated especially lies locus isolated lemma let label prepared along face assume coordinates set following hold label minimal containg proof shown way lemma using lemma instead proposition assume proof assumption take prelabel coordinates either assume first case lemma preparation may assume prepared along face lemma lemmas applied place get label deduce first inequality comes case coordinates shown way using lemma instead lemma assume coordinates let mod prelabel proj coordinate hence proof reduced first case view following lemma lemma let inv inv grm inm grm hence also hence proof compute implies vertices line together initial forms affected transformation thus first assertion follows last assertion shown argument applied instead step chain fundamental units step consider following situation assume given chain fundamental units blowups definition let initial part length let maximal ideal spec spec ideal theorem assume separably algebraic see definition sequence stops finitely many steps proof lemmas suffices show claim replacing base changes via spec maximal unramified extension completion use excellent thus may assume complete hence vadmissible proposition implies note lemma lemma theorem hence strict inequality may occur finitely many hence may assume follows prelabel write completeness choose totally prepared label assumption implies lemma coordinate must put lift prepare get totally prepared label lemma coordinate lemma implies totally prepared label assumption implies lemma coordinate put lift prepare get totally prepared label lemma totally prepared label lemma moreover coordinate lemma implies totally prepared label argument repeats imply following claim assume sequence proceeds infinitely many steps exists sequence elements following holds recalling complete set lim prepare get totally prepared label totally prepared label coordinate write lemma implies totally prepared label moreover minimal containing length implies see definition lemma assumption theorem implies sequence must stop finitely many steps claimed proof case iii inseparable residue extensions section complete proof key theorem see theorem let assumptions notations beginning case let irreducible homogeneous polynomial corresponding proj set grm assume divide let deg choosing lift set following two lemmas shown way lemma lemma let system regular parameters near prelabel lemma assume setting state main result section proposition let assume following conditions regular closed subscheme xmax dimension prepared normalized exists part system regular parameters following holds prelabel unless latter case particular proving proposition complete proof theorem view proposition theorem proof theorem obvious consequence following theorem assume given fundamental unit blowups aso sume regular closed subscheme xmax dimension holds isolated xmax assume definition vadmissible proof assumption take prelabel lemma preparation may assume let lemma proposition applied place claim proposition since corollary implies prepare vertices faces lying get label note lemma let imply lemma applied place implies label get first inequality resp equality comes resp lemma completes proof theorem start proof proposition may write either homogeneous degree write either homogeneous degree divisible get compute sup sup sup denotes degree polynomial set deg lift letting lift imply lemma implies noting get inf argument applied shows let identification get residue class lemma proof first inequality holds general picture definition second follows proves first assertion second assertion follows applying argument view lemma completes proof corollary solvable proposition holds proof indeed suffices take case proposition follows lemma proposition implies view normalized hence assumption implies prepared remains show lemma suffices show assumption lemma hence proof proposition remains treat case solvable assume case implies indeed since assumption also implies exist let homogeneous polynomials deg deg divisible implies mod choose lift set mod define note since noted consider following condition lemma assume holds following true system regular parameters strictly admissible conditions proposition satisfied place lemma holds may replace show proposition solvable done corollary solvable holds apply procedure get new system regular parameters process must stop finitely many steps last inequality lemma thus proposition follows corollary lemma following lemma lemma assume hold solvable dissolve let proof lemma conditions imply last inequality holds general picture definition implies second assertion obvious let valuation respect together implies coordinate transformation affects vertices lying shows condition proposition holds first assertion follows equality implied first inequality second assertion consequence first assertion since lemma implies implies completes proof remains show condition proposition holds introduce tuple independent variables write substitute get write condition normalizedness lemma implies normalizedness lemma assume given subset let tuple independent variables write proof thus completes proof lemma proof lemma let lift tuple independent variables write hence implies implies set write mod either homogeneous divisible choose lift set deg implies gives lemma implies noting need following lemma lemma exist equivalent proof lemma given later using lemma see vertex line since implies moreover exist defined equation lemma condition hold proof assumption claim indeed recalling assertion equivalent follows hand max implies implies since lemma imply hence solvable show normalized let setting implies first sum ranges mod mod let proposition implies implies normalized sense definition lemma implies hence lemma implies proves desired assertion finally remains show proof divided following two cases case case note always since assume case first inequality holds assumption second holds general assumption implies since coordinate transformation affects vertices lying inequality holds since thus get hand lemma get shows desired inequality thanks lemma implies since noted hence implies hand first inequality holds general second inequality holds since lemma assumption last inequality follows proves desired inequality case assume case write assumption implies definition also implies since coordinate transformation affects vertices lying implies hence recall initial form along face written linear form either homogeneous degree divisible note compute sup choose lifts respectively set note implies lemma implies substituting get lemma implies since hence get inf implies implies desired hand seen last inequality noted proves desired assertion proof lemma complete proof lemma proof lemma assume contrary write may negative multiplying get fact left hand side note either homogeneous degree right hand side recall view equations imply claim admitting claim implies one dissolve vertices line contradicts assumption completes proof lemma remains show claim show recall case denote variables define derivations new variables apply lemmas assumptions lemmas satisfied lemma normalizedness implied proposition according results possibly changing ordering find integer homogeneous polynomials variables indexed coefficients numbers powers exponential characteristic char xnj additive polynomials homogeneous degree moreover equations right hand side define ridge french tangent cone spec grm biggest group subscheme spec grm spec respects respect additive structure since dir definition assumption schemes dimension hence must since variables occur equations thus get hence permutation variables equations become implies yrqr gives hence noting easily conclude inductively completes proof claim maximal contact dimension let excellent regular scheme let closed subscheme definition closed subscheme said maximal contact following conditions satisfied take sequence permissible blowups permissible assume exists sequence points near strict transform remark definition much weaker hironaka original definition see ahv section prove following theorem let field characteristic let three variables consider spec let hypersurface defined equation yun integers satisfying condition let origin let open neighborhood smooth hypersurface maximal contact observe following consequence maximal contact derive contradiction let definition let maximal ideal residue field let ideal defining spec lemma assume spec set dim assume let part system regular parameters assume spec maximal contact let label see definition assume let integer following holds particular system regular parameters proof consider fundamental sequence permissible blowups definition corollary integer coincides length sequence let strict transform definition let generic point let ozq maximal ideal let ideal defining spec write claim know label claim system regular parameters proof condition implies induction strict transform spec defined also implies transversal exception divisor defined spec thus claim follows claim lemma follows easily using fact definition implies inj definition implies hand indeed noting system regular parameters lemma write compute noting factors see claim see min give theorem implies thus get lemma shown start proof theorem let origin let maximal ideal write claim label proof one easily checks idirx inm polyhedron two vertices since vertices integral points polyhedron claim follows definition use three sequences permissible blowups sequence first consider blx strict transform look point parameters let spec defined equation upa using one check unique vertex hence near theorem theorems base admissible hand prepared vertex dissolve coordinate change setting compute using see polyhedron three vertices first vertex fact since first vertex integral point polyhedron hence label definition extend following sequence permissible blowups unique point lying strict transform spec strict transform write ozq convention write claim near label vertex proof induction one easily shows system regular parameters one computes implies spec spec also implies last assertion claim noting fact vertex coming second term see worst case case last term even simpler since vertex claim integral point noting get claim follows theorems corollary sequence consider following sequence permissible blowups unique point lying strict transform spec strict transform write ozq claim near prepared label polyhedron unique vertex proof induction one easily shows system regular parameters one computes shows equation spec last assertion easily follows noting polyhedron get noting claim follows theorems corollary sequence iii sequence permissible blowups looks sequence except unique point lying strict transform spec assume exists spec spec maximal contact spec want deduce contradiction let completion easy see assumption claims hold even replacing thus may work spec spec spec maximal contact closed point spec claim exists hsi hti proof lemma claim write noting write setting implies hsi hti get tun definition implies max since claim lemma implies hence completes proof claim ease notation write follows write homogeneous degree let max two cases case case implies tun min homogeneous degree divisible consider sequence let argument proof claim show resp system regular parameters resp ozq compute upa note since divisible compute definition implies faster true polyhedron slower phantom polyhedron setting noting get claim hence contradicts lemma since label strict transform spec maximal contact spec definition case exist get tun assume consider sequence let argument proof claim show system regular parameters ozq compute used definition implies set claim implies hence contradicts lemma reason previous case remains treat case symmetry proof given argument using sequence iii instead completes proof theorem alternative proof theorem give another proof theorem uses classical tools algebraic geometry recall hilbert polynomial graded field unique polynomial known degree dim definition zero polynomial degree following property see corollary following remarks proof lemma polynomial hilbert polynomial standard graded certain integers moreover one deg family uniquely determined equality dim follows fact degree case obtained case empty family lemma also shows set hilbert polynomials standard graded depend recall totally ordered respect ordering shall need following description ordering lemma two hilbert polynomials one lexicographic ordering formally fill shorter family entries right length longer family proof since since orderings total orderings suffices show implies let assume proceed induction min min must deg deg hence polynomials hilbert polynomials lemma associated invariants respectively implies lexicographic ordering induction together implies claim lemma immediately implies theorem ordered set every strictly descending sequence finite let set hilbert functions standard graded write associated hilbert polynomial trivially let excellent scheme let hilbert polynomial associated let pxmax pxmax max theorem implies lemma closed fact union finitely many closed sets theorem imply pxmax lemma permissible pxmax another proof theorem suppose exists infinite sequence locally equisingular let union connected components equisingular let remark lemma max pxmax pxmax pxmax theorem may assume theorem implies get infinite sequence proper morphisms lemma one choose sequence points theorem theorem may assume hxn exists claim contradicts let want show let elements put thus claim follows following lemma finite proof first claim follows assumption exists second third claim follow respectively lemma let infinite sequence proper morphisms noetherian schemes exists sequence points proof let composite put properness closed clearly put noetherian condition exists implies thus get infinite sequence proper surjective morphisms inductively gives desired claim functoriality locally noetherian schemes algebraic spaces stacks section reformulate obtained functoriality resolution apply obtain resolution singularities locally noetherian excellent schemes algebraic spaces algebraic stacks definition let noetherian excellent scheme dimension two resolution morphism defined composition morphisms canonical resolution sequence theorem amounts morphisms canonical resolution sequence theorem state functoriality mentioned theorems follows theorem morphism unique morphism making diagram commutative moreover diagram cartesian functoriality holds boundary analogous resolution morphism obtained theorem composing morphisms embedded situation theorem resolution morphism obtained composing morphisms sequence theorem generally let arbitrary dimension let boundary let flat morphism geometrically regular fibers smooth morphism let assume canonical resolution sequence constructed remark corollary finite corresponding sequence finite well unique morphism making diagram commutative vertical morphisms compositions morphisms respective resolution sequences moreover diagrams cartesian boundary boundary proof finiteness resolution sequence existence fact diagram cartesian follow proposition since composition sequence uniqueness follows following lemma lemma consider cartesian diagram schemes flat composition unique morphism making diagram commutative proof let another morphism making diagram commutative write composition get commutative diagram cartesian squares since flat flat well hence morphism identifies write composition right column composition left column show inductively gives claim assumption holds since equation holds show follows ideal sheaf invertible sheaf universal property since flat invertible oyn invertible fact morphism obvious moz invertible invertible ideal sheaf consider morphism since shown oyn invertible follows universal property unique morphism since get wanted first apply obtain resolution singularities excellent schemes dimension locally noetherian necessarily noetherian theorem let excellent scheme dimension exists canonical resolution morphisms boundary embedded situation regular excellent normal crossings divisor satisfy properties corresponding morphisms theorem proof follows theorem looking open cover noetherian excellent schemes gluing corresponding morphisms uniqueness intersections argument uniqueness functorial properties follow finally apply get resolution algebraic stacks dimension theorem let excellent algebraic stack grothendieck topology whose coverings flat morphisms geometrically regular fibers stack artin stack assume representable covering morphism chosen topology excellent scheme dim exists proper surjective morphism stacks regular proof follows theorem functoriality obtained way section assumption one may represent groupoid scheme usual structural morphisms excellent schemes morphisms flat regular fibers functorial resolution gives finiteness canonical resolution strategy well using say one defines structural morphism cartesian diagram vertical resolution morphisms way one defines similarly one defines composition structural morphism cartesian diagram vertical resolution morphisms since flat regular fibers well structural morphisms obtained via base change rules follow functoriality equalities checked flat morphisms regular fibers special case algebraic space one sees one gets resolution algebraic space references abhyankar local uniformization algebraic surfaces ground fields characteristic ann math abhyankhar resolution singularities arithmetical surfaces arithmetical algebraic geometry proc conf purdue harper row new york abhyankar algorithm polynomials one indeterminate coefficients two dimensional regular local domain ann mat pura appl abhyankar resolution singularities embedded algebraic surfaces pure applied mathematics vol academic press new abhyankar nonsplitting valuations extensions two dimensional regular local domains math ann abhyankar good points hypersurface adv math ahv aroca hironaka vicente theory maximal contact memorias del instituto jorge juan mathematical memoirs jorge juan institute instituto jorge juan consejo superior investigaciones cientificas madrid aschenbrenner hemmecke finitenness theorems stochastic integer programming found comp math aschenbrenner pong orderings monomial ideals fund math bennett characteristic functions local ring ann math bierstone milman canonical desingularization characteristic zero blowing maximum strata local invariant english summary invent math bierstone milman uniformization analytic spaces amer math soc cossart desingularization embedded excellent surfaces tohoku math cossart desingularization bad examples dim characteristic english summary topics algebraic noncommutative geometry contemp amer math providence cgo cossart giraud orbanz resolution surface singularities appendix hironaka lecture notes mathematics springerverlag berlin cossart piltant resolution singularities threefolds positive characteristic reduction local uniformization purely inseparable coverings english summary algebra cossart piltant resolution singularities threefolds positive characteristic algebra clo cox little shea ideals varieties algorithms introduction computational algebraic geometry commutative algebra second edition undergraduate texts mathematics new york cutkosky resolution singularities graduate studies mathematics american mathematical society providence cutkosky resolution singularities positive charactersitic preprint jong smoothness alterations inst hautes tudes sci publ math encinas hauser strong resolution singularities characteristic zero english summary comment math helv giraud etude locale des cours cycle orsay publication http giraud sur contact maximal french math giraud contact maximal positive french ann sci cole norm sup hauser excellent surfaces taut resolution resolution singularities obergurgl progr birkhuser basel hironaka resolution singularities algebraic variety field characteristic zero ann math hironaka characters singularities math kyoto univ hironaka characteristic polyhedra singularities math kyoto univ hironaka certain numerical characters singularities math kyoto univ hironaka additive groups associated ponts projective space ann math hironaka desingularization excellent surfaces advanced science seminar algebraic geometry summer bowdoin college mimeographed notes bennet appendix cgo hironaka idealistic exponents singularity algebraic geometry sylvester johns hopkins baltimore johns hopkins univ press baltimore jannsen resolution singularities embedded curves appendix jannsen saito kato conjecture motivic cohomology finite fields preprint kunz introduction commutative algebra algebraic geometry boston boston lipman desingularization schemes ann math macl maclagan antichains monomial ideals finite proc amer math soc electronic miz mizutani hironaka additive group schemes nagoya math nag nagata generalization imbedding problem abstract variety complete variety math kyoto univ narasimhan hyperplanarity equimultiple locus proc amer math soc narasimhan monomial equimultiple curves positive characteristic proc amer math soc saito sato finiteness theorem fields ann math appear singh effect permissible local hilbert functions invent math singh formal invariance local characteristic functions seminar vogel vol zur teubner leipzig temkin functorial desingularization schemes characteristic zero case preprint pages villamayor patching local uniformizations english summary ann sci norm sup zariski reduction singularities algebraic surface ann math zariski simplified proof resolution singularities algebraic surface ann math zariski reduction singularities algebraic three dimensional varieties ann math ega grothendieck publ math ega grothendieck publ math
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mar neuromorphic hardware loop training deep spiking network brainscales system sebastian johann guillaume andreas maurice andreas stephan dan kai sebastian vitali mitja christoph alexander christian eric paul johannes mihai stefan stefan vasilis bernhard robert wolfgang christian johannes karlheinz sschmitt kljohann agruebl gguettle ahartel husmann khusmann sjeltsch vkarasen mkleider koke akononov cmauch mueller pmueller mpedro schemmel meierk guillaume maass heidelberg technische university physics neuenheimer feld heidelberg dresden chair neuromorphic circuits dresden graz university technology institute theoretical computer science graz university bern department physiology bern spiking neural networks analog neuromorphic hardware offers several advantages simulating conventional computers particularly terms speed energy consumption however usually comes cost reduced control dynamics emulated networks paper demonstrate iterative training network compensate anomalies induced analog substrate first convert deep neural network trained software spiking network brainscales neuromorphic system thereby enabling acceleration factor compared biological time domain mapping followed training training step network activity first recorded hardware used compute parameter updates software via backpropagation essential finding parameter updates precise need approximately follow correct gradient simplifies computation updates using approach several tens iterations spiking network shows accuracy close ideal prototype presented techniques show deep spiking networks emulated analog neuromorphic devices attain good computational performance despite inherent variations analog substrate ntroduction recently artificial neural networks anns emerged dominant machine learning paradigm many pattern recognition problems although anns extent inspired architecture biological neuronal networks differ significantly biological counterpart many respects first computation biological neurons performed analog voltages continuous time anns typically implemented digital hardware fig brainscales system currently installed consisting five cabinets containing four neuromorphic systems upstream connectivity control cluster provided prominent red cables communicating gigabit speed enables fast system configuration spike output additional rack hosts support infrastructure comprising power supplies servers control cluster network equipment thus operate discretized time second communication neurons ann based arithmetic computed discrete time steps communication biological neuronal networks largely based stereotypically shaped voltage events continuous time events called action potentials spikes recent years several analog neuromorphic computing platforms developed better match features biological neural networks due low power consumption speedup compared simulations run conventional architectures systems promising precursors computing devices rival computational capabilities energy efficiency human brain spiking neural networks principle able emulate ann unclear whether neuromorphic hardware efficiently used implement contemporary deep anns one obstacle lack adequate training procedures anns typically trained fig brainscales wafer module dimensions hosting wafer fpgas positioning backpropagation learning algorithm propagates mask used align elastomeric connectors link wafer precision errors layers network recently large main pcb support pcbs provide power supply successful training neural networks demonstrated circuits well access analog dynamic variables neuron membrane voltages connectors usb slots truenorth chip fully digital neuromorphic connectivity slots distributed design specifically performance four edges main pcb mechanical stability provided learning benchmarks impaired hardware aluminum frame photograph fully assembled wafer module quantization constraints training step errors computed quantized parameters binarized activations section describe mapping neural backpropagating full precision advance network hardware section subsequently however left question open whether similar strategy could describe training detail demonstrate used analog neuromorphic systems since truenorth application procedure handwritten digit recognition fully digital exact software model available therefore task section section parameter neuron activations corresponding gradients available appropriately approximated rain cale wafer cale ystem point time training contrast neural brainscales system follows principle circuits analog hardware precisely controllable physical modeling wherein dynamics vlsi circuits making exact mapping hardware software designed emulate dynamics biological domains challenging archetypes instead numerically computing work demonstrate successful training conventional simulation approach von neumann archianalog neuromorphic system configured implement deep tectures neurons synapses implemented analog neural architecture system used brainscales circuits operate continuous time governed time system neuromorphic architecture constants arise properties transistors features analog neuromorphic circuits digital capacitors microelectronic substrate contrast based communication implemented training procedure neuromorphic devices see analog circuits similar used coarse software model system designed operate regime approximate behavior show nevertheless characteristic time constants syn much smaller backpropagation algorithm capable adapt synaptic typical corresponding biological values defines parameters neuromorphic network quite effectively intrinsic hardware acceleration factor respect running training hardware loop biological system based ideas similar approaches already used context described meantime advanced various network architectures smaller analog lab prototype larger installation comprising wafer phic platforms hagen spikey modules see fig chips parameter updates used recorded activity wafer module neuromorphic system computed corresponding heart brainscales wafer module see fig gradients using parameters ann adaptation silicon wafer hicann high input count analog possible spite fact algorithm neural network chips produced cmos technology explicit knowledge exact parameter values comprises reticles containing hicanns neurons synapses brainscales system connected step chip hosts remainder article structured follows neurons emulating adaptive exponential section describe brainscales neuromorphic adex dynamics able reproduce platform discuss extent parameter variability firing regimes discussed forming logical system starting simple approximate software model neurons combining neuron circuits maximum table ardware utilization power ratings different neural network architectures maxhw ghz thz input synapses reached circuit contributes synapses synapse neuron dynamics emulated analog circuits continuous time action potentials transported digital data packets action potentials spikes injected asynchronously routing structures chip statically routed target synapses transported digital events via network xilinx fpgas one per reticle provide interface configuration spike data connection fpgas control cluster network established using standard gigabit auxiliary pcbs provide brainscales system power control analog readout specified maximum design power single module operating point maxhw assumes average spike rate applied hardware synapses currently power management techniques use numbers reported table based maximum design power table also provides data regarding hardware utilization previously published neural network architectures running neuronal network experiments brainscales software stack transforms abstract neural network description network topology model parameters input stimuli corresponding experiment configuration descriptions spiking neural networks often formulated using dedicated languages based either declarative syntax nineml neuroml use procedural syntax api called pynn current brainscales system uses pynn describe neural network experiments based experiences previous implementations design choice enables use versatile software packages developed pynn ecosystem connection set algebra elephant neo starting experiment description pynn transformation process maps model neurons hardware circuits routes connections neurons create synapses translates model parameters hardware settings translation neuron synapse ghz synaptic time constant hicanns neurons synapses average rate bio speedup bio total rate event network dac synaptic time constant fig example calibration synaptic time constant left measured synaptic time constants different neurons function digital parameter dac controlling responsible analog parameter right measured synaptic time constant blue without white calibration neurons hicann right membrane potential model model brainscales simulation time fig comparison recorded membrane trace neuron simulated nest neuron receives excitatory poisson stimulus followed inhibitory simultaneous excitatory inhibitory poisson stimuli frequency calibrations applied hardware response converted emulated biological domains model parameters requires calibration data see section well rules conversion biological hardware time voltage domains result whole transformation process abstract experiment description converted configuration data acquiring hardware access using fair resource scheduling queuing system based slurm hardware configured experiment ready run system although brainscales software stack provides userfriendly modeling interface hides hardware specifics settings available expert user particular experiments presented make use feature enabling fast iterative modification synaptic weights input stimuli calibration neuron calibration provides translation rules target parameters membrane time constant set corresponding hardware control parameters thereby accounts variations caused transistor mismatch inherent wafer manufacturing process data stored hardware domains two scaling rules used conversion biological time voltage domains time constants scaled acceleration factor hardware time corresponds emulated biological time voltages scaled according vhardware vbio input layer hidden layer label layer scaling factor offset units given biological domain stated otherwise fig exemplifies calibration technique particular case synaptic time constant every neuron analog parameter controlling synaptic time constant varied resulting synaptic time constant determined recorded potential fit data provides mapping desired synaptic time constant value control parameter calibration reduces variation significantly perfectly remaining variability mostly caused variation analog parameter storage fig shows two membrane time courses comparing calibrated silicon neuron numerical simulation nest cases model parameters input spike trains used despite overall match seen calibration perfect neuron used fig inhibitory stimulus weaker compared expectation simulation due analog nature system variations always occur certain extent rendering training essential networks sensitive parameter noise discuss following hidden layer fig topology neural network one input layer two hidden layers one label layer dimension input layer equal number pixels input image number label units equal number image classes network trained recognize iii raining eep piking etwork following describe network model training setup since using abstract network fig examples input data used training original mnist rectified linear units relus equivalent spiking image upper left image lower left middle right column images four classes network leaky lif neurons parallel first describe networks structure abstract terms network modeled directed graph shown fig input layer consisting units resulting weights converted synaptic weights used represent input patterns network later appropriately parametrized lif network learns classify classes represented one brainscales hardware label unit input label layers two synaptic weights trained hardwarehidden layers learn particular features input space software training loop weights directed edges learned several phases training described farther software model network trained modified subset mnist training software model performed similarly dataset handwritten digits first decreased using tensorflow software properties resolution pixels pixels bicubic detailed following interpolation account lower dissimilarity input grayscale value input image pixels reduced resolution images restricted dataset transformed number set five digit classes results activation units input layer training set test set images units output relu unit given spiking neural network trained three phases software model network rectified linear wkl max units relus trained classical backpropagation table euron parameters typical calibration variations parameter value inhibitory reversal potential reset potential resting potential spike threshold excitatory reversal potential synaptic time constant membrane time constant relative variation wkl weight connection unit unit activation function relu sum runs indices units previous layer weights initial weights layer containing units drawn normal distribution mean zero standard deviation weight magnitudes dropped training network trained gradientdescent momentum minimizing cost function matrix containing network weights vector true digit scaled activity label layer samples current batch samples first term euclidean distance predicted labels true labels rewarding correct penalizing wrong activity second term regularizes weights leading suppression large weights prevent overfitting per training step weights updated according wkl wkl change weight learning rate momentum parameter foresight hardware implementation wkl clipped neuromorphic implementation input input image converted poisson spike trains following firing rate input corresponding pth pixel grayscale value pth pixel targeted total firing rate input layer receives case set pattern presented followed silence allow activity decay hardware configuration network mapped brainscales hardware using software stack detailed section neurons layers including input layer randomly placed hicanns input routing additional chips used hicanns connected different fpgas artificial neuron four hardware neuron circuits connected form one logical neuron increase number possible inputs except stimulus input layer pair neurons consecutive layers connected inhibitory excitatory synapse allows weights change sign learning without change configured topology therefore network total synapses neuron parameters despite different input output domain activation functions relus lif neurons share features threshold output zero positive gradient suprathreshold input neuron features required mimic relu behavior therefore disable adaptation exponential features adex model parameters variation calibration see section listed table allow balanced representation positive negative weights reversal potentials chosen symmetric around resting potential refractory period set small possible close linear relation input output activity equation used convert hardware units target value resting potential hardware equals weights trained weights wkl artificial network converted bit hardware weights wkl wkl round positive negative weights assigned excitatory inhibitory synapses corresponding inhibitory excitatory synapse turned hardware loop section laid necessary steps convert artificial network network neurons analog hardware conversion found classification accuracy significantly reduced compared initially trained ann compensate reduced classification accuracy training continued hardware loop see fig training consists series training steps performed follows first neuron activity recorded batch training samples firing rates equated relu unit response used following heuristic label layer resulting vector used compute cost function defined weight updates computed using using relu activation function approximation activation functions hardwareemulated neurons experiments described used parameters batch size samples neuromorphic substrates however two problems exist first relationship abstract backpropagation units used typical deep networks needs mapped spiking neuron dynamics second case analog hardware distortions dynamics need take account latter especially problematic performance weight updates relu activity network usually relies precise parameter training addressed problems context brainscales system accelerated analog neuromorphic platform emulates biologically inspired bit weight discretization spikes neuron models mapping activities abstract domain spikes used scheme translation network topology including connectivity structure parameters described section prediction mnist brainscales following mapping pretrained network forward pass hardware substrate resulting distortions dynamics parameters compensated training fig illustration training procedure antecedent described section step shown relu network see fig mapped approach evaluated small network equivalent lif network brainscales hardware iteration training consists two passes forward pass output trained handwritten digits exemplary scenario firing rates lif network measured hardware backward possible almost completely restore performance pass rates used update synaptic weights lif network abstract model hardware computing corresponding weight updates relu network mapping back hardware implicit essential component methodology fact backpropagation errors needs precise computing cost function gradients using relu activation esults function sufficient adapting weights spiking example activity neuromorphic hardware network circumvents difficulty otherwise classification training one choice determine exact derivative cost function hardware neurons initial software parameters respect lif activation function would found fig figure shows spike times exacerbated diversity neuronal activation functions neurons network five presented samples every analog substrate digit image considered classified correctly distortions network dynamics neuron associated input digit shows highest configuration parameters compensated activity label neurons training images additional training complementary approach would correctly classified except first example digit modify network way makes robust mistaken comparing weights distortions discussed training see fig shows slight approaches inherently robust adjustments needed compensate hardware effects jitter timing spikes robust architectures evolution accuracy per training batch become particularly important coding schemes software model training hardware discussed shown fig different sets hardware neurons experiments presented initial weights software model total classification part commissioning phase brainscales system accuracy computed sum correctly classified lay groundwork extensive studies patterns divided total number patterns test interesting question addressed next whether set training steps accuracy software results achieved also hold larger networks model negligible uncertainty arising deal complex datasets fully functional choice initial weights directly converting artificial system able accomodate large networks network network spiking neurons accuracy without reduction processing time due reduced increases end inherently parallel nature training close performance long run potentially rewarding challenge software model uncertainty given interquartile fully port training hardware well range iqr end integrated plasticity processor designed allow emulation different learning iscussion rules runtime learning also profit problems involving spatial pattern recognition deep acceleration benefits operation neural networks become state art almost fully trained network use analog spiking hardware definition lend implementation might allow accelerated data processing backward pass fig correlation hardware weights training projections first left second hidden layer center label layer right weights zero training omitted relative frequency encoded grayscale area corresponding square accuracy training batch software model brainscales median iqr training step median iqr iteration fig classification accuracy per batch function training step software model left iteration hardware implementation right runs uncertainty given interquartile range iqr expresses variations repeating software model different initial weights training using different initial weights relu training different sets hardware neurons networks also facilitate fast training biologically inspired architectures certain contexts even outperform classical machine learning algorithms acknowledgment work received funding european union sixth framework programme grant agreement facets european union seventh framework programme grant agreement hbp brainscales horizon framework programme grant agreement hbp well manfred foundation authors wish thank simon friedmann matthias hock ioannis kokkinos tobias nonnenmacher lukas pilz moritz schilling dominik schmidt sven schrader simon ziegler holger zoglauer contributions development commissioning system elektronik gmbh schopfheim development manufacturing special pcb used brainscales system und mikrointegration izm berlin germany developing technique required communication external connectivity wafer first two authors contributed equally work eferences lecun bengio hinton deep learning nature vol may furber neuromorphic computing systems neural eng vol maass fast sigmoidal networks via spiking neurons neural comput vol esser merolla arthur convolutional networks fast neuromorphic computing proc natl acad sci hohmann fieres meier training fast neural networks data classification proc int conf neural netw vol ieee press jul fieres schemmel meier convolutional neural network tolerant synaptic faults analog hardware proceedings iapr international workshop artificial neural networks pattern recognition ser springer lecture notes artificial intelligence vol ulm germany springer international publishing pfeil jeltsch six networks universal neuromorphic computing substrate frontiers neuroscience vol time input layer hidden layer hidden layer label layer fig spike raster plot neural activity layers neuromorphic hardware training horizontal dash denotes time certain neuron spiked five examples per digit presented plot digits denoted background color correctly classified images marked green circle liu neuromorphic systems john wiley sons schemmel neuromorphic hardware system neural modeling ieee int symp circuits syst proc may brette gerstner adaptive exponential model effective description neuronal activity vol millner meier vlsi implementation adaptive exponential neuron model adv neur lafferty williams vol naud marcille clopath firing patterns adaptive exponential model biological cybernetics vol nov online available http schemmel fieres meier integration analog neural networks proc int conf neural netw hong kong jul thanasoulis vogginger partzsch pulse communication flow ready accelerated neuromorphic experiments ieee int symp circuits syst proc jun scholze schiefer partzsch vlsi implementation aer interface routing event sorting functionality front neurosci vol petrovici vogginger characterization compensation anomalies neuromorphic modeling platforms plos one vol raikov cannon clewley nineml network interchange neuroscience modeling language bmc neuroscience vol gleeson crook cannon neuroml language describing data driven models neurons networks high degree biological detail plos comput biol vol jun davison eppler pynn common interface neuronal network simulators front neuroinform vol davison establishing novel modeling tool interface neuromorphic hardware system front neuroinform vol djurfeldt novel formalism representation connectivity structure neuronal network models neuroinformatics vol denker yegenoglu holstein elephant tool analysis electrophysiological proceedings meeting german neuroscience society neuroforum german neuroscience society mar garcia guarino jaillet neo object model handling electrophysiology data multiple formats front neuroinform vol february jette grondona slurm simple linux utility resource management proceedings clusterworld conference expo san jose california gewaltig diesmann nest neural simulation tool scholarpedia vol lecun cortes mnist database handwritten digits online available http cao chen khosla spiking deep convolutional neural networks object recognition int comput vis vol abadi agarwal barham tensorflow machine learning heterogeneous systems google research whitepaper software available online available http qian momentum term gradient descent learning algorithms neural vol connor neil liu classification sensor fusion spiking deep belief network front neurosci vol petrovici schroeder breitwieser robustness structure fast inference neuromorphic device hierarchical lif networks submitted ijcnn friedmann nux processor electronic vision group physics heidelberg university user guide online available https friedmann schemmel demonstrating hybrid learning flexible neuromorphic hardware system ieee trans biomed circuits vol leng petrovici martel spiking neural networks superior generative discriminative models cosyne abstracts salt lake city usa february
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aug zeta functions finite schreier graphs zig zag products asif shaikh hemant bhate abstract investigate ihara zeta functions finite schreier graphs basilica group show sheeted unramified normal covering fact galois group graph sheeted unramified non normal covering order give definition generalized replacement product schreier graphs also show corresponding results zig zag product schreier graphs cycle mathematics subject classification keywords ihara zeta functions schreier graphs zig zag product graphs basilica groups introduction main aim paper study ihara zeta functions finite schreier graphs basilica group well ihara zeta functions zig zag products schreier graphs cycle ihara zeta function graph theoretic analogue riemann zeta function defined connected graph sufficiently small prime cycle prime equivalence class tailless backtrackless primitive cycles length see connection adjacency matrix given theorem says det rank fundamental group diagonal matrix whose diagonal entry qjj degree vertex proof theorem found unramified covering finite graph say finite graph covering map onto map every vertex mapped every set points adjacent onto vertices adjacent covering normal galois covering iff graph automorphisms automorphisms form galois group gal see details motivation comes question raised terras zeta functions behave respect graph products particular zig zag products able prove following non normal covering graph fact sheeted normal covering graph thus divides order prove result new generalized replacement product defined two schreier graphs gives resultant graph also schreier graph also non normal covering graph fact sheeted normal covering graph divides paper organized follows section define schreier graphs basilica group give several examples section also contains definition generalized replacement product computations ihara zeta functions schreier graphs using artin functions section ends results needed next section section starts definition zig zag product two graphs see computations ihara zeta functions zig zag product schreier graphs cycle also presented also prove corresponding results zig zag product graphs schreier graphs basilica group let binary alphabet denote set consisting empty word set words length alphabet basilica group acting group generated three state automaton states automaton generators group action given let schreier graph associated action two vertices connected edge labeled near near action transitive graph regular connected graph vertices labels near given every denote information finite infinite schreier graphs basilica group figure graph schreier graphs basilica group respectively found complete classification isomorphism limiting case infinite schreier graphs associated basilica group acting binary tree terms infinite binary sequence given angeli donno matter nognibeda angeli donno savahuss used schreier graphs basilica groups computations zig zag product graphs basilica group belongs important class selfsimilar groups introduced grigorchuk example first four finite schreier graphs basilica group figure substitution rules used construct schreier graph graph next sub section give new definition product two schreier graphs call generalized replacement product schreier graphs generalized replacement product replacement product two graphs well known literature two regular graphs regularity respectively replacement product regular graph regularity details product found generalization product along resultant graph regularity motivation generalized version following using two schreier graphs basilica group able produce resultant graph schreier graph basilica group let two schreier graphs basilica group define generalized replacement product first choose spanning subgraph graph edges vertex labels near respectively note vertex mean word length alphabets let vertices adjacent edges respectively thus every vertex set degree note equation sum lhs regularity graph notice also schreier graphs basilica group edge color say near near rotation map rot defined definition generalized replacement product regular graph vertex set whose edges described following rotation map even odd even odd call edges given lifts edge lifts edge one imagine vertex set graph partitioned sheets indexed vertices definition sheet consists vertices within construction idea put copy around vertex keeping edges thought using given rotations equations connectedness guarantees connectedness graph edge colors near respectively call edge hence say graph two types edges remark figure one observe number edges graph number edges lifts imagine whole sheet single vertex along lift edges interestingly get graph hence say following even edges lifts edges respectively odd edges lifts edges respectively edges lifts edges call sheet edges figure choose graph still normal sheeted cyclic covering graph isomorphic sheets copies spanning subgraph dashed lines black continuous lifts note definition depends spanning subgraph second graph different choice spanning subgraph taken graph need schreier graph basilica group example choose vertex corresponding spanning subgraph edge set edges vertex labels respectively near graph need schreier graph basilica group see figure following result show graph actually schreier graph action basilica group set proposition let following holds graph schreier graph action basilica group set unramified sheeted graph covering non normal covering graph sheeted normal covering graph proof let let define map show adjacency preserving map notice two types edges first type contains edges lifts edges described equation second type contains edges lifts edges described equation edges first type note lift connect lift connect even odd even odd let given edge lift words let adjacent action basilica group even odd even odd even odd adjacent shows adjacency preserving map argument similar given edge lift edges second type edges lifts connect suppose given edge type means least one alphabet therefore definition basilica group elements hence adjacent gives adjacency preserving map definition difficult bijection map therefore exists reverse implication shows also adjacency preserving map bijection thus isomorphisms definition contains sheets graph show unramified cover graph define map show covering map sending neighborhoods onto neighborhoods suppose adjacent definition means adjacent adjacent let suppose adjacent definition gives adjacent let ubv ubv covering map thus unramified covering proof contradiction assume normal covering galois group gal gal generated permutations corresponding frobenius automorphism see permutations arises action basilica group elements ordered set respectively consider unramified covering sheets indexed sheets set two sheets say ith connected lift say write similarly write permutation corresponds permutations written directly using action basilica group elements set respectively suppose write remark therefore product comes action using induction shown action order shows galois group element different generators order hence galois group order bigger contradiction suppose therefore sheeted covering first show denote two sheets recall galois group gal see definition following even odd therefore show sheeted normal covering remains show non identity frobenius automorphism vertex let vertex first sheet vertex second sheet therefore corollary spectrum contained spectrum moreover zeta function divides proof direct consequence proof proposition given let normal covering artin function associated representation gal defined product prime cycles det rob prime denotes lift rob denotes frobenius automorwhere phism defined rob starts labeled ends labeled proposition adjacency matrix block diagonalized blocks form irreducible representation taken degree times defined following formula suppose vertices define matrix defining entry number edges det denotes identity setting deg following analogue formula see thus zeta functions factors follows see corollary matrices called artinized adjacency matrices label vertices using table figure graph normal sheeted cyclic covering graph sheets copies spanning subgraph dashed lines map graph isomorphism table vertex labeling label vertex example covering given figure normal covering obtain spanning subgraph cutting edges gives dashed line spanning subgraph draw covering graph placing sheets spanning subgraph labeling sheet given table connections sheets cover graph given table case representations cyclic galois group trivial representation representation defined two cases table notations sheets vertex set vertex set group element table connections sheet vertex adjacent vertices case trivial representation adjacency matrix case representation reciprocals functions equation example coverings given figure non normal covering follow labeling vertices given table obtain spanning subgraph cutting edges gives dashed line spanning subgraph draw covering graph placing sheets spanning subgraph labeling notations given table connections first sheet sheets given table assume normal covering hence gal element order thus contradiction figure graph non normal sheeted covering graph sheets copies spanning subgraph dashed lines graph isomorphic graph table notations sheets vertex set sheet index table connections sheet sheets vertex adjacent vertices non normal covering proposition let unramified non normal covering following holds even alternate action produces spanning path sheet visits every vertex sheet twice odd alternate action produces spanning path sheet visits every vertex sheet twice let positive number alternate actions type goes outside sheet proof even definition generalized replacement product sheet contains lift edge connects sheet actually copy graph write therefore simultaneous actions stated produces path visits following vertices one even prove using induction shown action order thus times gives right hand side equation shows vertices sheet graph counted twice hence get spanning path sheet visits every vertex sheet twice odd number definition generalized replacement product sheet contains lift edge connects sheet actually copy graph write therefore simultaneous actions stated produces path visits following vertices right hand side equation shows vertices sheet graph counted twice hence get spanning path sheet visits every vertex sheet twice let positive integer lifts sheet another sheets lifts connect sheets hence sheet actually copy graph therefore even action gives vertex therefore action remain within sheet similarly odd show action remain within sheet hence alternate actions goes outside sheet proposition last alphabets words distinct word proof even odd therefore last alphabets word distinct word similarly using action prove last alphabets word also distinct word zig zag product graphs zig zag product two graphs introduced reingold vadhan wigderson taking product large graph small graph resulting graph inherits roughly size large one degree small one expansion properties graphs important property product good expander large small graphs expander angeli donno provided sufficient condition product two graphs connected definition let connected regular graph let connected regular graph let resp rotation map resp zig zag product regular graph degree vertex set identify set whose edges described rotation map angeli donno shown proposition graph isomorphic double cycle graph construction schreier graph basilica group see figure get natural labeling label graph follows proposition let schreier graph action basilica group length cycle graph let graph unramified graph covering proof proposition shown map covering map define map show adjacency preserving map consider adjacent hence adjacent using similar argument see choices also gives adjacency preserving map given vertex form definition zig zag product graphs see vertices adjacent form vertices adjacent form notice length word action word containing first alphabets actually word remaining alphabets form word say thus hence sending neighborhoods one one onto neighborhoods similar argument one make case notice let spanning subgraph even odd corollary two fold normal covering graph galois group gal proof adjacency matrices graphs circulant matrices see therefore exists following order vertices even words applying alternate actions word alternation second coordinate vertices inner cycle graph given equation also applying action given alternate actions word alternation second coordinate vertices outer cycle graph given equation proposition also seen isomorphism defined isomorphism defined hence able write thus vertex written first coordinate vertex vertex sheet graph even case proposition get produces spanning path sheet proposition onto map gives first coordinate first vertices lists given equations contains last position vertices sheet therefore definition zigzag product graphs equation proposition vertices forms connected copy spanning subgraph first coordinate vertices ends denote copy using similar argument show first coordinate next vertices given ends hence get another copy denote following vertices define non identity frobenius automorphism map since thus galois group contains two elements one make similar argument odd corollary non normal covering graph proof corollary given end paper show galois covering need following definition zigzag product graphs let vertices graph proposition sheeted cover following partition uvj call svj uvj therefore svj restriction graph definition let graph define restriction graph subgraph suppose find restriction subgraphs yej taking yej svj one see yej next result connectedness restriction subgraphs proposition following holds restriction subgraph yej connected vjth sheet graph restriction subgraph yej vjth sheet graph sheet disconnected proof recall spanning subgraph even write write set show yej connected enough show yej svj yej action order yej sets yej contain elements yej therefore yej proposition equation elements forms copy spanning subgraph therefore yej connected vjth sheet graph proposition nye therefore nye yej means one components yej vjth sheet graph sheet disconnected prove corollary proof contradiction let sheeted normal covering therefore exists frobenius automorphism sends vjth sheet say vkth sheet proposition vjth sheet connected frobenius automorphism mapping disconnected vkth sheet contradiction table vertex labeling label vertex table notations sheets vertex set group element example graph normal sheeted cyclic covering graph shown figure label vertices using table obtain spanning subgraph cutting edges gives dashed line graph shown figure graph vertices ends alphabet remaining ends proposition vertices forms connected figure graph normal sheeted cyclic covering graph sheets copies spanning subgraph dashed lines continuous lines shown graph lifts edges table connections vertex adjacent vertices sheets label table representations cyclic galois group trivial representation representation defined therefore two cases case trivial representation use information given table write hence aye adjacency matrix case representation find eigenvalues given table table eigenvalues matrix eigenvalues write spectrum see table table spectrum eigenvalue multiplicity reciprocals functions equation table notations sheets vertex set sheet index example figure contains example non normal covering obtain spanning subgraph cutting edges gives dashed line graph shown figure graph vertices ends alphabet remaining ends labeling vertices given table proposition vertices forms connected sheets labeling sheets given table table connections sheets vertex adjacent vertices table vertex labeling label vertex table easily observe following nye figure graph non normal sheeted covering graph sheets copies spanning subgraph dashed lines corollary graph non normal sheeted covering graph propose following conjecture basis study work presented paper may special case conjecture normal covering galois group gal conjecture graph also normal covering galois gal group gal references abdollahi loghman replacement products filomat angeli donno matter nagnibeda schreier graphs basilica group arxiv preprint journal modern dynamics angeli donno connectedness isomorphism properties product graphs arxiv preprint dedeo medrano minei stark terras michelle zeta functions heisenberg graphs finite rings theory applications special functions mourad ismail erik koelink editors springer grigorchuk weakly branch group defined three state automaton international journal algebra computation grigorchuk topics dynamics group actions rooted trees proceedings steklov institute mathematics reingold vadhan wigderson entropy waves graph product new expanders annals mathematics terras zeta functions graphs stroll garden vol cambridge university press
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sound relative error bounds arithmetic anastasiia izycheva eva darulova aug max planck institute software systems saarland informatics campus germany email izycheva eva static analysis tools verifying code compute absolute error bounds numerical errors however often good estimate accuracy take account magnitude computed values relative errors compute errors relative value magnitude thus preferable today tools report relative error bounds merely computed via absolute errors thus necessarily tight informative furthermore whenever computed value close zero part domain tools report relative error estimate surprisingly quality relative error bounds computed today tools systematically studied reported date paper investigate static techniques computing sound absolute error bounds used extended combined computation relative errors experiments standard benchmark set show computing relative errors directly opposed via absolute errors often beneficial provide error estimates six orders magnitude tighter accurate also show interval subdivision another commonly used technique reduce less benefit computing relative errors directly help alleviate effects inherent issue relative error estimates close zero ntroduction numerical software common embedded systems scientific computing usually designed arithmetic implemented finite precision digital computers finite precision however introduces unavoidable roundoff errors individually usually small accumulate affect validity computed results thus important compute sound roundoff error bounds ensure results accurate enough especially applications due unintuitive nature arithmetic discrepancy continuous real arithmetic automated tool support essential one way measure roundoff absolute error errabs max denote possibly multivariate function variable respectively finiteprecision note absolute roundoff errors meaningful restricted domain unrestricted error would unbounded general paper consider interval constraints input variables input variable furthermore focus arithmetic common choice many programs several tools exist compute absolute errors fully automatically nonlinear code absolute errors however always adequate measure result quality consider instance absolute error whether error small thus acceptable computation depends application well magnitude result value error may acceptable probably revise computation increase precision relative error captures relationship errrel max note input domain needs restricted also relative errors today static analysis tools usually report absolute well relative errors latter however computed via absolute errors tools first compute absolute error divide largest function value clearly equation equation compute sound relative error bounds due loss correlation nominator denominator whether loss correlation leads coarse error bounds practice perhaps surprisingly studied yet context automated sound error estimation beyond curiosity interested automated computation relative errors several reasons first relative errors informative often also natural user specifications secondly computing sound error bounds necessarily compute absolute errors become bigger larger input ranges error bounds less tight since relative errors consider range expression expect overapproximations smaller thus making relative errors suitable modular verification one may think computing relative errors challenging computing absolute errors case two reasons first complexity computing relative errors higher compare equation equation due division expression nonlinear even linear complexity nonlinearity already challenging absolute errors computed automated tools usually leading coarser error bounds furthermore whenever range includes zero face inherent division zero indeed today static analysis tools report relative error standard benchmarks reason today static analysis tools employ variety different strategies orthogonal dealing overapproximation absolute roundoff errors due nonlinear arithmetic tool rosa uses forward dataflow analysis linear abstract domain combined nonlinear decision procedure fluctuat augments similar linear analysis interval subdivision fptaylor chooses approach backed algorithm paper investigate today methods used extended combined computation relative errors best knowledge first systematic study fully automated techniques computation relative errors mainly focus issue computing tight relative error bounds extend optimization based approach computing absolute errors computing relative errors directly show experimentally often results tighter error bounds sometimes six orders magnitude furthermore combine interval subdivision aware interval subdivision applied approach however note perhaps surprisingly benefits rather modest compare direct error computation forward analysis computes relative errors via absolute errors standard benchmark set note latter outperforms direct relative error computation single univariate benchmark hand approach clearly benefits interval subdivision also observe interval subdivision beneficial dealing inherent division zero issue relative error computations propose practical preliminary solution reduces impact potential small subdomains allowing tool compute relative errors remainder domain demonstrate benchmarks approach allows tool provide useful information tools contributions implemented techniques within tool daisy available https background first give brief overview arithmetic well techniques automated sound absolute roundoff error estimation arithmetic error definitions section include function highly irregular discontinuous thus unsuitable automated analysis abstract following ieee standard replacing every variable constant operation floatingpoint real arithmetic operations respectively relative error introduced rounding operation bounded machine epsilon denormals subnormals values special representation provide gradual underflow roundoff error expressed absolute error bounded addition subtraction abstraction valid absence overflow invalid operations resulting number nan values values detected separately reported errors paper consider double precision arithmetic approach parametric precision thus applicable types well using abstraction replace function input variables roundoff errors introduced operation general ease reading write without vector notation note abstraction abstraction analysis tools approximate absolute errors errabs max absolute error estimation reviewing existing automated tools absolute roundoff error estimation focus techniques reducing due nonlinear arithmetic order compute tight error bounds rosa computes absolute error bounds using forward analysis combination abstract domains note magnitude absolute roundoff error arithmetic operation depends magnitude operation value easily seen equation turn determined input parameter ranges thus rosa tracks two values intermediate abstract syntax tree node sound approximation range absolute error transfer function errors extend approach bounding absolute errors relative errors thus provide first feasible fully automated approach computing relative errors directly perform first experimental comparison different techniques automated computation sound relative error bounds show interval subdivision beneficial reducing absolute error computations less relative errors computed directly demonstrate interval subdivision provides practical solution division zero challenge relative error computations certain benchmarks remainder term bounds higher order terms ensures soundness computed error bounds approach symbolic sense taylor approximation taken wrt symbolic argument thus represents function point inputs remain symbolic roundoff errors present choosing point perform taylor approximation replacing upper bound reduces initial optimization problem uses ranges propagate errors subexpressions compute new roundoff error committed arithmetic operation question one may think evaluating expression interval arithmetic interpreting width resulting interval error bound would sufficient sound computes pessimistic error bounds especially consider relatively large ranges inputs distinguish part interval width due input interval due accumulated roundoff errors hence need compute ranges errors separately rosa implements different range arithmetics different tradeoffs bounding ranges errors compute ranges rosa offers choice interval arithmetic affine arithmetic tracks linear correlations variables combination interval arithmetic nonlinear arithmetic decision procedure latter procedure first computes range expression standard interval arithmetic refines tightens using calls nlsat decision procedure within smt solver tracking errors rosa uses affine arithmetic since roundoff errors general small tracking linear correlations general sufficient fluctuat abstract interpreter separates errors similarly rosa uses affine arithmetic computing ranges variables error bounds order reduce introduced affine arithmetic nonlinear operations fluctuat uses interval subdivision user designate two variables program whose input ranges subdivided intervals equal width analysis performed separately overall error maximum error subintervals interval subdivision increases runtime analysis significantly especially multivariate functions choice variables subdivide much usually fptaylor unlike daisy fluctuat formulates roundoff error bounds computation optimization problem absolute error expression equation maximized subject interval constraints parameters due discrete nature arithmetic fptaylor optimizes continuous abstraction equation however expression still complex features many variables optimization procedures practice resulting bad performance well bounds coarse useful see subsection experiments fptaylor introduces symbolic taylor approach objective function equation simplified using first order taylor approximation respect errabs max upper bound error term details found fptaylor uses interval arithmetic estimate value term second order thus small general solve optimization problem equation fptaylor uses rigorous optimization also used compute resulting real function range needed instance compute relative errors another tool takes optimizationbased approach uses programming optimization iii ounding elative rrors main goal paper investigate today sound approaches computing absolute errors fare bounding relative errors whether possible advantageous compute relative errors directly via absolute errors section first concentrate obtaining tight bounds presence nonlinear arithmetic postpone discussion orthogonal issue division zero next section thus assume range function want bound relative errors include zero within given input domain furthermore consider arithmetic expression conditionals loops shown challenging even absolute errors thus leave proper treatment future work consider function calls orthogonal issue handled inlining thus reducing code require suitable summaries postconditions also one motivations work forward dataflow analysis approach rosa fluctuat easily generalize relative errors requires intertwining range error computation instead extend fptaylor approach computing relative errors directly subsection furthermore implement interval subdivision subsection orthogonal measure reduce experimentally evaluate different combinations techniques set standard benchmarks subsection based algorithm thus principle similar gelpia approach note queries send solvers satisfiability queries optimization queries nonlinear decision procedure necessary suitable direct optimization algorithm dreal performing optimization internally reason roundabout approach dreal tool optimization interface uses custom input format difficult integrate expect approach affect mostly performance however accuracy bounding relative errors directly first strategy explore compute relative errors directly without taking intermediate step absolute errors extend fptaylor optimization based approach maximize relative error expression using abstraction equation max max hope preserve correlations variables nominator denominator thus obtain tighter informative relative error bounds call optimization equation without simplifications naive approach mentioned previously approach scale well include experiments subsection nonetheless aware concrete results actually reported expected naive approach returns error bounds large essentially useless thus simplify applying symbolic taylor approach introduced fptaylor take taylor approximation around point first term approximation zero obtain following optimization problem interval subdivision committed static analysis techniques grows general width input intervals thus width intermediate ranges intuitively error consider usually achieved small part domain error remaining inputs additionally domain errors obtained may different arithmetic operation thus subdividing input domain usually obtain tighter error bounds note interval subdivision applied error estimation approach fluctuat applied interval subdivision absolute error estimation aware combination approach study effectiveness relative errors apply subdivision input variables thus compute max max max upper bound remainder term unlike equation pull factor term absolute value plan compute tight bounds future upper bounds change affect accuracy uniform precision computations computing upper bounds second order remainder still expected small use interval arithmetic compute sound bound experiments indeed small benchmarks except doppler need sound optimization bound first order terms procedure maintain overall soundness limits available choices significantly fptaylor uses global optimization tool gelpia internally uses based algorithm unfortunately found difficult integrate custom interface furthermore observed unpredictable behavior experiments nondeterministic crashes substantially varying running times repeated runs expression instead use rosa approach combines interval arithmetic refinement approach parametric solver experiment dreal support interface provide rigorous results based fundamentally different techniques implements complete decision procedure nonlinear arithmetic whereas dreal implements framework decision procedures internally number subdivisions input interval multivariate functions subdivide input interval variable maximize error cartesian product clearly analysis running time exponential number variables fluctuat limits subdivisions two variables userdefined number subdivisions choose limit total number analysis runs parameter given desired number subdivisions variable number input variables first variables domains subdivided times larger input domains subdivided first remaining variable ranges remain undivided implementation implement described techniques tool daisy daisy successor rosa compiler generates implementations specifications given following format def real real require daisy parametric approach naive forward dataflow analysis solver used dreal number subdivisions including none combination three orthogonal choices run simply changing daisy input parameters furthermore daisy simplifies derivative expressions approach unsimplified expressions may affect running time solvers thus also accuracy error bounds observed solvers necessarily perform otherwise simplifications finally maintain soundness need make sure introduce internal roundoff errors computation error bounds purpose implement internal computations daisy using rationals andling ivision error computations however running time increases correspondingly ideally could exclude relative error computation small domain around zeros function univariate functions approach zeros instance obtained nonlinear decision procedure multivariate functions challenging zeros simple points curves secondly subdivision could used determining exhibit potential division zero actual relative error bound computation performed remainder input domain without subdividing would lead performance improvements even though domain still consist several disconnected parts univariate functions extension multivariate ones finally could compute ero important challenge arising computing relative errors handle potential divisions zero tools today simply report error whenever function range contains zero unfortunately rare occurrence standard benchmark set verification many functions exhibit division zero see table experiments note divisions zero inherent expression consider usually due overapproximations analysis thus reduce effect division zero eliminate entirely aim reduce domain compute relative errors similar relative absolute errors combined ieee model equation want identify parts input domain relative error computation possible due division zero compute absolute errors remainder input domain compute relative errors use interval subdivision subsection attempting compute relative errors one methods described every subdomain computation fails due potential division zero compute absolute error report user max small standard approach scientific computing sound however consider xperimental valuation compare different strategies relative error computation set standard benchmarks fptaylor forward dataflow analysis approach rosa implemented daisy representatives tools compare fluctuat directly underlying error estimation technique based forward analysis affine arithmetic similar daisy indeed experiments performed previously show small differences computed error bounds rather choose implement interval subdivision within daisy experiments performed double precision precision fptaylor supports although techniques daisy parametric precision experiments performed desktop computer running debian cpu ram benchmarks bsplines doppler jetengine rigidbody sine sqrt turbine nonlinear functions invertedpendulum traincar benchmarks linear embedded examples himmilbeau kepler nonlinear examples project relerror several relative error computed computing absolute error intervals abserror reported relative error bound sound parts domain believe information nonetheless informative either result absolute error bound today tools report may suffer unnecessary realize approach provides practical solution still preliminary improved several ways first smarter subdivision strategy would beneficial currently divide domain intervals vary number subdivision bigger part domain covered relative comparing techniques relative error bounds evaluate accuracy performance different approaches case division zero occurs modify standard input domains benchmarks whenever necessary function ranges contain zero tools thus compute relative error bound table shows relative error bounds computed different techniques tools table corresponding multivariate univariate table elative error bounds computed different techniques daisyopt benchmark underapprox daisy fptaylor naive approach daisy subdiv dreal sine sqroot doppler himmilbeau invpendulum jet table nalysis time different techniques relative errors benchmarks without division zero daisyopt subdiv benchmark daisy fptaylor naive approach daisy subdiv dreal bsplines sines sqroot doppler himmilbeau invpendulum jet kepler rigidbody traincar turbine total analysis times bold marks best result tightest computed error bound benchmark column underapprox gives unsound relative error bound obtained dynamic evaluation inputs values provide idea true relative errors naive approach column confirms simplifications relative error expression indeed necessary note exponents computed bounds last four columns show error bounds relative errors computed directly using optimization based approach denoted daisyopt subsection two solvers without subdivisions subdivisions use univariate benchmarks multivariate experiments parameters showed good performance accuracy benchmarks find direct evaluation relative errors computes tightest error bounds dreal subdiv acceptable analysis times furthermore benchmarks resp nonlinear decision procedure able compute slightly tighter error bounds three benchmarks dreal performs significantly better running times comparable somewhat surprisingly note interval subdivision limited effect accuracy combined direct relative error computation also increasing running time significantly comparing techniques columns daisy fptaylor compute relative errors via absolute errors notice results sometimes several orders magnitude less accurate direct relative error computation six orders magnitude doppler benchmarks column shows relative errors computed table iii elative error scalability respect size input domain benchmark small via absolute errors large ratio small directly large ratio sine sqroot doppler himmilbeau invpendulum jet comparing numbers ratio columns notice direct computation ratio significantly smaller computation via absolute errors means committed direct computation smaller committed relative error computation via absolute errors prominent example doppler benchmark directly computed error relative grew larger domain two orders magnitude relative error computed via absolute grew seven orders magnitude based results conclude relative errors computed directly scale better relative errors computed via absolute respect size input domain via absolute errors using forward analysis subdivision parameters observe unlike directly computed relative errors interval subdivision mostly beneficial scalability relative errors also evaluate scalability direct computation relative errors respect size input domain compare scalability computation relative errors via absolute since magnitude roundoff errors absolute relative depends magnitude input values larger input domains cause larger roundoff errors also approximation errors grows together size input domain ideally want overapproximation grow slowly possible experiments table use large input domains possible without introducing result ranges include zero scalability comparison also compute relative errors small input domains modify standard input domains benchmarks width input intervals reduced function ranges still contain zero experiments performed solver timeout set second relative errors computed without interval subdivision since noticed limited effect accuracy increasing running times table iii presents relative error bounds computed smaller larger input domains columns small large show relative errors computed smaller larger input domains respectively column ratio presents relation values large small domains relation characterizes scalability approach smaller better handling division zero evaluate whether interval subdivision helpful dealing inherent division zero challenge consider standard benchmark set standard input domains table summarizes results first note division zero indeed occurs quite often missing results daisy fptaylor columns show last three columns show results using interval subdivision note obtain results many benchmarks possible change parameters subdivision univariate multivariate benchmarks result consists three values first value maximum relative error computed relative error possible compute brackets report maximum absolute error subdomains relative error computation possible integer amount absolute errors computed report result number division zero less total amount subdomains larger numbers would probably impractical used within modular verification techniques whenever report table means division zero occurred many subdomains observe interval subdivision provide result benchmarks nonetheless computes information benchmarks techniques another common theme run program alongside original one obtain error bounds testing testing also used verification method optimizing computations approaches based testing however consider limited number program executions thus prove sound error bounds elated work presented first experimental investigation suitability different static analysis techniques sound accurate relative error estimation provided function range include zero computing relative errors directly usually yields error bounds orders magnitude accurate relative errors computed via absolute errors current surprising interval subdivision beneficial absolute error estimation applied direct relative error computation often significant effect accuracy furthermore note today rigorous optimization tools could improved terms reliability well scalability finally interval subdivision help alleviate effect inherent division zero issue relative error computation still remains open challenge vii onclusion goal work automated sound static analysis technique computing tight relative error bounds arithmetic related current static analysis tools computing absolute roundoff error bounds another closely related tool gappa computes absolute relative error bounds coq appears relative errors computed directly via absolute errors automated error computation gappa uses intervals thus computation via absolute errors less accurate daisy performs direct computation amounts naive approach shown work poorly direct relative error computation also used context verifying computations mix arithmetic operations roundoff errors computed using optimization based approach similar fptaylor approach targeted specific operations including polynomials authors use taylor theorem however tight error estimates focus paper authors report use whichever bound absolute relative better aware systematic evaluation different approaches sound relative error bounds broadly related abstract static analyses sound wrt arithmetic formalized coq domains however quantify difference semantics show absence runtime errors overflow arithmetic also formalized theory solvers exist include floatingpoint decision procedures however suitable roundoff error quantification combination theory reals would propositional level thus lead useful results arithmetic also formalized theorem provers coq hol light automation support exists form verification condition generation reasoning ranges entire numerical programs proven correct accurate within tight error bounds proven specific computations verification efforts large part manual require substantial user expertise arithmetic well theorem proving work focuses different accuracy automation generality eferences goubault putot static analysis finite precision computations vmcai solovyev jacobsen gopalakrishnan rigorous estimation errors symbolic taylor expansions darulova kuncak sound compilation reals popl magron constantinides donaldson certified roundoff error bounds using semidefinite programming corr vol daisy framework accuracy analysis synthesis numerical programs https ieee standard arithmetic ieee std aug moore interval analysis figueiredo stolfi affine arithmetic concepts applications numerical algorithms vol moura solving arithmetic ijcar moura efficient smt solver tacas darulova kuncak towards compiler reals acm toplas vol baranowski briggs gelpia global extrema locator parallelization interval arithmetic https gao kong clarke dreal smt solver nonlinear theories reals cade darulova kuncak majumdar saha synthesis fixedpoint programs emsoft daumas melquiond certification bounds expressions involving rounded operators acm trans math vol lee aiken verifying pldi chen cousot sound polyhedra abstract domain aplas jeannet apron library numerical abstract domains static analysis cav jourdan laporte blazy leroy pichardie static analyzer popl table elative error bounds computed different techniques standard benchmarks potential division zero daisy subdiv benchmark daisy fptaylor sine sqroot doppler himmilbeau invpendulum jet subdiv daisyopt dreal subdiv ppendix xperimental omparison ubdivision parameters martin brain cesare tinelli wahl automatable formal semantics arithmetic technical report http brain silva griggio haller kroening deciding logic abstract conflict driven clause learning formal methods system design vol boldo melquiond flocq unified library proving algorithms coq arith jacobsen solovyev gopalakrishnan parameterized formalizaton hol light electronic notes theoretical computer science vol linderman dill meng nolan towards program optimization automated analysis numerical precision cgo ayad verification programs ijcar boldo mayero melquiond weis wave equation numerical resolution comprehensive mechanized proof program journal automated reasoning vol ramananandro mountcastle meister lethin unified coq framework verifying programs computations cpp graillat muller maximum relative error computing arithmetic pierre marie curie paris cnrs research report benz hildebrandt hack dynamic program analysis find accuracy problems pldi chiang gopalakrishnan rakamaric solovyev efficient search inputs causing high errors ppopp lam hollingsworth stewart dynamic floatingpoint cancellation detection parallel computing vol panchekha wilcox tatlock automatically improving accuracy floating point expressions pldi nguyen nguyen demmel kahan sen bailey iancu hough precimonious tuning assistant precision lam hollingsworth supinski legendre automatically adapting programs computation ics subdivision approach parametric amount subdivisions interval total amount optimizations appendix present experimental findings good default configuration interval subdivision parameters expect presence zeros function range might affect choice good default configuration thus perform two separate experimental evaluations input domains function range include zero subsection standard input domains function range potentially contains zero subsection potential division zero define good defaults subdivision parameters input domains function range include zero executed multiple tests experiments compute relative errors directly several subdivision values combination two different values limit total analysis runs number used solver timeout second testing configurations table shows relative error bounds different subdivision values table gives running times configurations column vars shows amount input variables benchmark columns show bounds computed subdivisions input intervals respectively noticed perhaps surprisingly subdivision gain accuracy benchmark vars univariate sine sqroot multivariate table omparison different configurations subdivision division zero doppler himmilbeau invpendulum jet table omparison running times different configurations subdivision sine sqroot doppler himmilbeau invpendulum jet total benchmark larger running times indeed case input intervals subdivided recall limit effect introduced total amount optimizations univariate benchmarks influence noticeable since total amount optimizations equal amount subdivisions case limit multivariate benchmarks observe limits amount variables intervals subdivided hence marks benchmarks observed different error bounds three five tightest bounds himmilbeau jet computed configuration second best choice terms accuracy bounds himmilbeau jet kepler differ tightest computed bounds insignificantly one may think greater number subdivisions table vii omparison different configurations subdivision benchmark vars sine sqroot doppler himmilbeau invpendulum jet limits runtime thus configuration greatest tested subdivision value smallest total running time keeping mind select following default configuration univariate benchmarks multivariate benchmarks computation relative errors failed second total amount means relative error successfully computed underline marks results relative error computation failed due division zero domains consider results impractical thus report error bounds cases whenever report table means relative error computations reported division zero domains univariate benchmarks see almost benchmarks configurations provided relative error bounds sineorder subdivision sufficient obtain result interestingly computations reported division zero subdivision division zero occurred possible compute relative error bound since configuration allows compute relative error univariate benchmarks set choose default univariate benchmarks note univariate benchmarks play role whether take amount subdomains tested values lower noticed multivariate benchmarks one configuration allowed compute relative error benchmarks means set multivariate benchmarks beneficial subdivide individual interval less intervals variables subdivided choose configuration default multivariate benchmarks benchmarks however independent subdivision parameters still possible compute estimate relative error computed error bound valid small jet potential division zero find good default subdivision configuration relative error computation potential division zero recall semantic parameter parameter regulates subdivision input interval parameter intended limit running time bounds total amount optimization runs also regulates many variables input intervals subdivided balance subdivisions one interval input intervals subdivided may change deal potential division zero thus reuse configuration found subsection perform comparison several values compare results focus comparison find configuration computes relative errors many benchmarks possible big part input domain possible therefore different configurations compare amount subdomains computations failed division zero table vii summarizes results columns show amount division zero occurred subdivisions input interval upper limit total amount optimizations respectively result consists two values first value amount
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mar implementation practical distributed calculation system browsers javascript application distributed deep learning ken miura miura tokyo tatsuya harada harada tokyo machine intelligence laboratory department university tokyo abstract tional complexity often require use graphics processing units gpus numerous computers practical distributed computation difficult construct distributed computation environment frequently requires preparation certain operating systems installation specific software hadoop shvachko practical application deep learning algorithms new instant distributed calculation environment eagerly anticipated deep learning achieve outstanding results various fields however requires significant computational power graphics processing units gpus numerous computers often required practical application developed new distributed calculation framework called sashimi allows computer used distribution node accessing website also developed new javascript neural network framework called sukiyaki uses general purpose gpus web browsers sukiyaki performs times faster conventional javascript library deep convolutional neural networks deep cnns learning combination sashimi sukiyaki well new distribution algorithms demonstrates distributed deep learning deep cnns web browsers various devices libraries comprise proposed methods available mit license http sashimi distributed calculation framework developed new distributed calculation framework called general distributed calculation framework difficult increase number node computers users must install client software node computer sashimi computer become node computer accessing certain website via web browser without installing client software proposed system execute code written javascript distributed manner implementation introduction sashimi consists two servers calculationframework distributor servers figure user runs project includes distributed processing using calculationframework accesses distributor via web browsers processes distributed executed multiple web browsers results processed distributed machines used processed local machine without conscious differences using calculationframework utilization big data recently come play increasingly important role various business fields big data often handled deep learning algorithms achieve outstanding results various fields example almost teams participated ilsvrc image recognition competition russakovsky used deep learning algorithms algorithms also used speech recognition dahl hinton molecular activity prediction kaggle calculationframework calculationframework describes calculations include distributed processing user writes code according certain interface processes distributed computed multiple web browsers via distributor results computed web browsers automatically however deep learning algorithms huge miljs brand new javascript libraries matrix calculation machine learning sukiyaki browsers note distributor consists two servers httpserver ticketdistributor distributed calculation framework calculation framework execute heavy project distributor httpserver ticketdistributor http httpserver web server implemented provides static files include basic program discloses apis offer datasets used distributed calculation user wants make browser function node user needs access basic program provided httpserver via web browser basic program consists static html file javascript file basic program works follows websocket access via browsers process small tasks user mysql figure sukiyaki architecture connection ticketdistributor established using websocket lected allows results used processed local machine subsection introduce primelistmakerproject finds prime numbers example ticket request sent ticketdistributor project programming unit calculationframework endpoint process starts project user execute distributed processing creating task instance note processes require distributed processing also supported request required external datasets files sent httpserver task process distributed executed web browsers user writes task according certain interface arguments automatically divided distributed web browsers processed results automatically collected primelistmakerproject task determines whether input integer prime number called isprimetask task distributed among web browsers task given arguments generated project must return calculated results using callback function note user use external libraries datasets example task calls prime function external library determines whether input integer prime number calculated result returned ticketdistributor task request sent ticketdistributor downloaded task described ticket task arguments described ticket executed return step task external data cached browser program runs long time memory usage increases due cache therefore implemented garbage collection basis least recently used algorithm error occurs task running error report includes stack trace sent ticketdistributor browser reloads thus task described tickets generated calculationframework continuously executed without special maintenance user accesses program project generates task instance arguments framework generates tickets divided argument framework sends codes arguments tickets external libraries datasets distributor via mysql tickets distributed distributor collected used calculationframework via mysql since project implemented javascript task implemented javascript used without considering whether code executed server browsers users check progress task tickets via httpserver control console console users see project name number tasks number tickets waiting processed number executed tickets number error reports client information project note console used execute code web browsers also provided console user make browsers reload redirect another distributed system use responsive web design rwd techniques user interface basic program control console rwd techniques adapt screen size tablet smartphone makes easy use devices distributed calculation check progress distributor distributor distributes tasks tickets sent calculationframework via mysql browsers distributor also collects results calculated miljs brand new javascript libraries matrix calculation machine learning tasks tickets generated calculationframework distributed browsers via websocket ticketdistributor processed results also collected ticketdistributor unlike httpserver ticketdistributor runs single process communicates web browser unitarily efficiently table specifications devices distributed mnist benchmark model cpu ram ticketdistributor receives ticket request browser obtains tickets ascending order virtual created time mysql server virtual created time determined follows dell optiplex dell optiplex windows professional intel core nexus nexus android krait remarkable proposed system used tablet tablet lower computational power desktop computer overhead time required distribution becomes relatively shorter believe proposed distributed computing method become effective feature extraction methods high computational costs sift deep learning virtual created time ticket creation time undistributed tickets virtual created time five minutes ticket distribution tickets distributed virtual created time five minutes last ticket distribution ticket redistributed sukiyaki deep neural network framework thus results returned within five minutes tickets treated way note tickets redistributed ascending order distribution time tickets distributed thus web browser terminated receives ticket clients low computational capability another client execute task therefore throughput enhanced tickets redistributed intervals least seconds prevents last ticket distributed many clients prevents next calculation delayed implemented algorithm using sql quickly select tickets distributed implemented learning algorithm deep neural networks dnns browsers based sashimi explain proposed framework implementation dnns also discuss advantages proposed method existing library environment distributed computation dnns explained next section primarily implemented deep convolutional neural networks deep cnns obtain high classification accuracy image recognition tasks convnetjs karpathy implemented library using javascript however computational speed limited runs single thread therefore developed deep neural network framework called sukiyaki utilizes fast matrix library called sushi miura sushi matrix library fast implemented webcl utilize general purpose gpus gpgpus efficiently benchmark experiment condition using sashimi demonstrate task high computational cost computed parallel efficiently compare time required classify mnist dataset nearest neighbor method changing number clients experiment images mnist test images classified comparing training images used one four clients desktop computer tablet pcs described table accessed distributor using google chrome web browser desktop tablet environments implementation sukiyaki dnn framework consists sukiyaki object handles procedures learning testing neural network layer objects version deep cnns implemented convolutional layer max pooling layer layer activation layer note add layers implement certain methods forward backward update forward backward update methods layer implemented using sushi matrix library thus executed parallel gpgpus results results shown table environments calculation time reduced distributed computation effect distributed computation use adagrad duchi online parameter leaning method learn parameters miljs brand new javascript libraries matrix calculation machine learning table results distributed mnist benchmark environment dell optiplex nexus clients elapsed time sec quickly original update rule adagrad follows elapsed time ratio table specifications device neural network libraries benchmark model cpu gpu ram scalar learning rate element parameter time step element gradient time step however update rule learning usually becomes unstable sum squared gradients minuscule early learning process therefore modified update rule using constant follows macbook pro retina late mac yosemite intel core intel iris table numbers batches learned per min convnetjs firefox sukiyaki firefox converts input elements output elements estimating class probabilities via softmax function designed sukiyaki dnn framework used browsers dnns trained distributed manner using sashimi distributed calculation framework example model file wherein parameters encoded formatted json note although model file platform independent string format exchanged among machines without rounding errors trained network using firefox web browser macbook pro specs described table compared learning speeds results results shown table figure observed sukiyaki learned network faster convnetjs firefox convergence speed sukiyaki also faster convnetjs note sukiyaki learned network times faster convnetjs benchmark experiment condition compared learning speed sukiyaki dnn framework convnetjs existing javascript library distributed deep learning experiment used deep cnn model shown figure fifty images per learned training images krizhevsky note consists color images classes model convolves input images kernels convolutional layer generates three feature maps size convolutional layer followed activation layer max pooling layer size output halved fourth layer layer ran sukiyaki dnn framework sashimi distributed calculation framework realized distributed learning deep cnns distribution algorithms previous studies proposed methods distributed learning dnns deep cnns dean dean proposed distbelief miljs brand new javascript libraries matrix calculation machine learning input image convolutional layer layer figure deep cnn benchmark deep learning allelize training deep cnns using model parallelism data parallelism efficiently generally deep cnns consist many convolutional layers fullyconnected layers due weight sharing convolutional layers incur significant computational cost relative small number parameters however layers many parameters convolutional layers less computational complexity krizhevsky krizhevsky developed efficient method parallelize training deep cnns applying data parallelism convolutional layers model parallelism layers however focus distributed computation via internet thus must reduce communication costs proposed framework error rate convnetjs convnetjs firefox sukiyaki sukiyaki firefox elapsed time min figure error rate implemented another effective method distributed deep learning parallelize training convolutional layers using data parallelism gpus synchronized layers trained single gpu computational complexity training layers relatively small model parallelism fullyconnected layers necessarily contribute fast learning method combining parallelized standalone learning works efficiently easy implement however method computational resources stay idle layers learned single gpu still room improvement efficient distributed computing method dnns distbelief network partitioned subnetworks different machines responsible computation different subnetworks nodes edges cross partition boundaries must share state information machines however since consider machines connected via internet slow throughput difficult share nodes state different machines proposed framework distbelief focuses network thus also difficult directly apply approach convolutional networks share weights among different nodes singa wang distributed deep learning platform supports model partition data partition manage automatically distributed array data structure without much awareness array partition singa designed accelerate deep learning using mpi unclear whether approach appropriate distributed calculation via internet meeds meeds developed mlitb wherein different training data batches assigned different clients clients compute gradients send master computes weighted average gradients clients updates network new network weights sent clients clients restart compute gradients basis new weights approach simple easy implement however must communicate network weights gradients master clients thus communication overhead becomes excessively large large network study developed new method parallelizes training convolutional layers using data parallelism apply model parallelism fullyconnected layers proposed method trains fullyconnected layers server clients train convolutional layers unlike method convolutional layers krizhevsky krizhevsky proposed method learning speed miljs brand new javascript libraries matrix calculation machine learning tion number clients furthermore developed sukiyaki dnn framework utilizes gpgpus train deep cnns times faster conventional javascript dnn library building sukiyaki sashimi also developed parallel computing method deep cnns suitable distributed computing via internet shown deep cnns trained parallel using web browsers client clients clients clients layer libraries available mit license http future plan improve efficiency distribution algorithm considering clients computational capabilities supporting another network sukiyaki note welcome suggestions improvements code documentation hope many programmers develop sukiyaki sashimi become distributed computing platform anyone use easily convolutional layer figure learning speed distributed deep learning ers learned concurrently method reduces communication cost among machines utilizes computational capability server awaits responses clients benchmark references experiment conditions dahl deng acero contextdependent deep neural networks large vocabulary speech recognition ieee transactions audio speech language processing experiment parallelized training deep cnn shown figure using sukiyaki dnn framework sashimi distributed calculation framework used computer shown table server clients client machine four gpu cores thus ran one four firefox web browsers client machine browsers accessed server running began training network parallel compared training speeds convolution layers varying number clients dean corrado monga chen devin mao ranzato senior tucker yang large scale distributed deep networks advances neural information processing systems duchi hazan singer adaptive subgradient methods online learning stochastic optimization journal machine learning research results results shown figure proposed distributed computation method train layers times faster computation method independently number clients server devoted training layers training speed convolutional layers becomes faster proportion number clients four clients train convolutional layers proposed method two times fast method zhang ren sun delving deep rectifiers surpassing performance imagenet classification hinton deng dahl mohamed jaitly senior vanhoucke nguyen sainath kingsbury deep neural networks acoustic modeling speech recognition shared views four research groups ieee signal processing magazine conclusion future plans paper proposed distributed calculation framework using javascript address increasing need computational resources required process big data developed sashimi distributed calculation framework allow web browser function computation node applying sashimi image classification problem task executed parallel calculation speed becomes faster kaggle merck molecular activity chalhttps lenge merckactivity karpathy convnetjs http miljs brand new javascript libraries matrix calculation machine learning input image convolutional layers figure deep cnn distributed deep learning benchmark table specifications devices distributed deep learning model cpu gpu ram server mac pro mac pro late mac yosemite intel xeon amd firepro client dell alienware dell alienware windows intel core nvidia geforce gtx titan krizhevsky learning multiple layers features tiny images master thesis department computer science university toronto url http krizhevsky one weird trick parallelizing convolutional neural networks meeds hendriks faraby bruntink welling mlitb machine learning browser miura mano kanehira tsuchiya harada miljs brand new javascript libraries matrix calculation machine learning russakovsky deng krause satheesh huang karpathy khosla bernstein berg imagenet large scale visual recognition challenge shvachko kuang radia chansler hadoop distributed file system ieee symposium mass storage systems technologies wang chen dinh gao ooi tan singa distributed system deep learning technical report http layer miljs brand new javascript libraries matrix calculation machine learning appendix sashimi sample program source code prime list maker var projectbase require var isprimetask require var primelistmakerproject function function var task isprimetask var inputs var candidate inputs function results var results prime number new projectbase source code prime var taskbase require var isprimetask function function input output output true else output false new taskbase source code function candidate var candidate candidate return false return true
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automatic program repair roopsha oswaldo allen mar university texas austin ist austria roopsha university texas austin olivo emerson abstract present formal framework repairing imperative sequential programs possibly recursive procedures multiple assertions framework generate repaired programs modifying original erroneous program multiple program locations ensure readability repaired program using expression templates framework also generates set inductive assertions serve proof correctness repaired program step toward integrating programmer intent intuition automated program repair present formulation given cost function associated permissible statement modifications goal ensure total program modification cost exceed given repair budget part predicate abstractionbased solution framework present sound complete algorithm repair boolean programs developed prototype tool based smt solving used successfully repair diverse errors benchmark programs introduction program debugging process fault localization error elimination integral part ensuring correctness existing evolving software essentially manual program debugging often lengthy expensive part program development cycle evident need improved formalization mechanization process however program debugging hard formalize multiple types programming mistakes diverse manifestations multiple ways eliminating detected error moreover particularly challenging assimilate mechanize expert human intuition involved choices made manual program debugging paper present formulation automated program debugging problem addresses concerns formulation obviates need separate fault localization phase directly focusing error elimination program repair fix set update schemas may applied program statements modifying update schema compact description class updates may applied program statement order repair instance update schema assign assign permits replacement assignment statement assignment statements assign skip permits deletion assignment statement etc paper includes deletion statements replacement assignment statements assignment statements replacement guards conditional loop statements guards assume given cost function assigns cost application update schema program statement given erroneous program cost function repair budget goal automatic program repair compute correct obtained modifying using program set update schemas total modification cost exceed postulate quantitative formulation flexible convenient way incorporating user intent intuition automatic program debugging instance user define appropriate differs statements cost functions search penalize modification within trusted program fragment favor application particular update schema another approach repair imperative sequential programs based predicate abstraction routinely used verification tools slam satabs etc analyzing programs tools generate boolean programs equivalent expressive power pushdown systems enjoy desirable computational properties decidability reachability inevitably boolean programs also explored use automatic repair sequential programs partial correctness total correctness papers however accommodate quantitative formulation repair problem compute repaired programs differ original erroneous program exactly one expression moreover papers attempt improve readb obtained concretizing repaired ability concrete program boolean program predicate approach automatic program repair relaxes limitations besides erroneous framework requires boolean program obtained iterative predicate exhibits path error present algorithm casts question repairability given smt query query satisfiable algorithm extracts correct boolean program also extract set inductive witness satisfiability along assertions witness constitute proof correctness algorithm boolean program repair sound complete relab along proof tive repaired boolean program along proof concretized obtain repaired concrete program correctness however concretized repairs may succinct readable hence framework also accept templates constrain specifying desired syntax modified expressions concretization alternate approaches automatic repair synthesis sequential programs rely abstract interpretations concrete programs also often encode problem problem whose solution extracted using sat smt solvers except approaches due bounded semantics imprecise handle total authors use smt reasoning search repairs satisfying templates templates needed ensuring readability generated repairs also ensuring tractability inherently undecidable repair generation query also include notion minimal diagnoses subsumed general formulation given constraints specifying space desired programs associated proof objects program synthesis approach attempts synthesizes program along proof total correctness consisting program invariants ranking functions loops contrast framework interacts user improving readability generated repairs cost function predicates involved generation repaired boolean program proof discovered automatically besides proposals program repair based computing repairs winning strategies games abstraction interpretation mutations genetic algorithms using contracts focusing data structure manipulations also customized program repair engines grading feedback generation programming assignments finally multitude algorithms proposed fault localization based analyzing error traces framework extended handle total correctness synthesizing ranking functions along inductive assertions techniques used preprocessing step improve efficiency algorithm cost giving completeness boolean program repair module summary contributions define new formulation automatic program repair incorporate programmer intuition intent sec present formal solution framework sec sec repair imperative sequential programs possibly recursive procedures multiple assertions method modify original erroneous program multiple program locations ensure readability repaired program using expression templates method succeeds generating repaired program generates proof correctness consisting inductive asserp tions guarantees satisfaction assertions original program part predicate solution present sound complete algorithm repair boolean programs finally present experimental results repairing diverse errors benchmark programs using prototype implementation sec background review predicate abstraction predicate abstraction effective approach model checking imperative programs respect safety properties technique computes conservative abstraction concrete program partitioning state space based valuation finite set red predicates resulting abstract program termed boolean program see fig fig set variables boolean variables boolean variable represents predicate given concrete program overall abstraction refinement method proceeds follows step one initial boolean program computed step two respect specification found correct method concludes correct otherwise abstract counterexample path leading violated assertion computed examined feasibility found feasible method terminates reporting error found infeasible step three refined new boolean program eliminates spurious counterexample thereafter steps two three repeated needed note overall method incomplete may always able possible compute suitable refinement eliminates spurious counterexample check abstract counterexample indeed spurious main int skip else assert main bool goto goto goto goto goto assert assume assume true ass ume assume assume err exit fig example concrete program corresponding boolean program transition graph work interesting case method terminates reporting error henceforth fix concrete program corresponding boolean program exhibits counterexample path let denote set predicates used abstraction predicate first order expression variables let denote corresponding boolean variables let denote mapping boolean variables respective predicates mapping extended standard way expressions boolean variables program syntax technical presentation fix common simplified syntax sequential concrete abstract programs partial definition syntax shown fig syntax denotes variable htypei denotes type variable denotes procedure denotes statement label location hexpri denotes expression hbexpri denotes expression hpgmi hvardecli hproclisti hproci hstmtseqi hlabstmti hstmti hvardecli hproclisti decl htypei hvardecli hvardecli hproci hproclisti hproci begin hvardecli hstmtseqi end hlabstmti hstmtseqi hstmti hstmti skip hexprm hbexpri hstmtseqi else hstmtseqi hexpri hstmti assume hbexpri call hexprk return goto assert hbexpri fig programming language syntax thus concrete abstract boolean program consists declaration global variables followed list procedure definitions procedure definition consists declarations local variables followed sequence labeled statements statement skip parallel assignment conditional loop assume procedure call return goto assert statement make following assumptions distinguished initial procedure main called procedure variable formal parameter names globally unique number actual parameters procedure call matches number formal parameters procedure definition goto statements used arbitrarily used simulate flow control structured programs last statement loop body every statement skip statement htypei includes integers booleans addition boolean programs assume variables formal parameters htypei boolean expressions hexpri hbexpri boolean expressions defined follows hbexpri hdetbexpri hdetbexpri true false hdetbexpri hdetbexpri hdetbexpri hdetbexpri hdetbexpri hdetbexpri hdetbexpri hdetbexpri hdetbexpri hdetbexpri boolean variable thus boolean expression either deterministic boolean expression expression nondeterministically evaluates true assume expresses fair nondeterministic choice permanently evaluate value assume boolean expressions assume hbexpri assert hbexpri statements always deterministic thus concrete program contains nondeterministic expressions boolean program contains nondeterministic expressions rhs assignment statements note syntax permit return values procedures however return values easily modeled using extra global variables hence syntax simplification affect expressivity programming language indeed syntax quite general notation let fix notation proceed program let set procedures main procedure let denote set global variables procedure let denote sets statements locations respectively let lvi denote sets formal parameterssand local variables respectively lvi let lvi denote set variables denote set locations location within procedure let inscope lvi denote set variables whose scope includes denote stmt ormal local statement sets formal parameters local variables procedure containing respectively denote entryi location first statement practice nondeterministic boolean expression boolean expression containing expression choose deterministic boolean expressions true choose evaluates true else true choose evaluates false else choose evaluates handle arbitrary nondeterministic boolean expressions prototype tool see sec consider expressions exposition simplicity context clear simply use instead etc transition graphs addition textual representation often find convenient use transition graph representation programs transition graph representation denoted comprises set labeled rooted directed graphs exactly one node err common informally ith graph captures flow control procedure nodes edges labeled locations corresponding statements respectively precise labi set nodes given exiti err includes unique entry node entryi unique exit node exiti error node err set labeled edges labi defined follows iff stmt assignment assume call statement next sequential stmt stmt skip statement either stmt last statement loop body statement empty label next sequential location empty label stmt either denoted succ location first statement branch assume denoted succ location first statement else branch assume stmt either denoted succ location first statement loop body assume denoted succ next sequential location end loop body assume stmt assert either denoted succ next sequential location empty label denoted succ node err empty label stmt goto statement includes label empty label stmt return statement exiti return let succ denote set path sequence labeled connected edges overloading notation denote sequence statements labeling edges next sequential location last statement else branch conditional statement location following conditional statement next sequential location last statement main procedure stmt every node path entryi exiti transition graph boolean programs defined similarly see fig main modification follows defining set labeled edges graph labi transition graph representation stmt given succ succ defined labels succ succ set assume true program semantics correctness given set variables valuation function maps variable appropriate value type naturally extended map expressions variables values operational semantics defined programs formalizing effect type program statement program configuration configuration program tuple form valuation variables inscope stack elements element form valuation variables local program state pair form defined thus program state excludes stack contents configuration called initial configuration entry node main procedure empty stack use denote transition configuration configuration transitions rules type program statement exit nodes procedures presented fig let take closer look last two transition rules fig transition rules affect stack contents upon execution statement call program configuration control moves entry node called procedure new valuation program variables constrained agree values global variables maps formal parameters values actual arguments according finally element succ pushed onto stack succ location control returns completes execution valuation local variables calling procedure recorded last transition rule fig captures return control calling procedure say completion execution called procedure say top stack element ret removed used retrieve location exiti inscope lvi err inscope undefined cases skip return goto stmt assume assert call exitj succ succ true succ either true succ false succ either true succ false succ err succ entryj ormal succ local ret local ret fig transition rules ret control must return well valuation local variables new valuation program variables constrained agree values global variables agree values local variables execution path program sequence configurations obtained repeated application transition rules fig starting initial configuration note execution path may finite infinite last configuration finite execution path may either terminating configuration error configuration err stuck configuration execution path ends stuck configuration none transition rules fig applicable particular notice notice transition configuration stmt assume defined true operational semantics boolean programs defined similarly main modifications follows stmt given say succ stmt given assignment statement say succ either true false transition rule extended handle scenarios assignment statements multiple expressions rhs call statements expressions actual arguments assertion program statement form assert first order expression representing expected values program variables inscope use term assertion denote statement assert well expression say program configuration satisfies assertion embedded variable valuation satisfies given program annotated set assertions partially correct iff every finite execution path ends terminating configuration say totally correct iff every execution path finite ends terminating configuration follows assume programs annotated set assertions specifying correctness boolean programs interpret nondeterminism dijkstra demonic nondeterminism given program annotated set assertions partially correct iff every finite execution path ends terminating configuration nondeterministic choices might make totally correct iff every execution path finite ends terminating configuration nondeterministic choices might make unless otherwise specified incorrect program one partially correct remark found convenient define boolean programs worth noting formalisms pushdown systems recursive state machines equivalent boolean programs program repair problem let denote set statement types program seen fig suffices consider set statement types given skip assign assume assert call return goto given statement let element denoting statement type let set permissible update schemas identity update schema maps every statement function assert maps statement type statement type update schema given say applied statement get statement given example given assign assign applied assignment statement get assignment statements etc notice update schemas affect label statement permit modification assert statement paper fix following set permissible update schemas programs assign assign assign skip assume assume call call call skip extend notion update let respective sets locations update follows programs let stmt tmt denote respective statements location let function maps location iff update schema say tmt obtained applying stmt let cost function maps tuple consisting update schema location certain cost thus cost applying update schema stmt impose obvious restriction since already fixed set set locations program equivalently boolean program henceforth use instead respectively total cost costc performing rupdate given given incorrect concrete program annotated assertions cost function repair budget goal program repair compute totally correct exists costc say exists addition problem propose another problem follows let set templates grammars representing syntactical restriction modified expressions syntax example template say defining linear arithmetic expressions program variables denoted hblaexpri shown hblaexpri atom hblaexpri hblaexpri hblaexpri hatomi hlatermi hcmpi hlatermi hlatermi const var const var hlatermi hlatermi hcmpi const var denote constants program variables respectively expressions satisfy syntactical requirements template said belong language template denoted let function maps location template let stmt denote set includes expressions certain statement types defined follows stmt stmt else stmt call stmt else stmt assume stmt else stmt empty set given along incorrect goal templateb based program repair compute correct exists costc location tmt conjecture insightful choice cost function help prune search space repaired programs help incorporate expert user intuition intent automatic program repair exploration suitable beyond scope dissertation would like emphasize quite flexible diverse ways used constrain computation example user choose search differs statements defining modify stateor user choose search ment within trusted procedure defining prohibitively large number user choose favor application particular update schema say others defining prohibitively large number similarly insightful templates choices help guide search repairs based user input solution overview present predicate framework program repair recall fixed boolean program sec obtained via iterative predicate exhibits counterexample path addition framework requires boolean program corresponding function maps boolean variables respective involves predicates computation suitable repaired program two main steps repair obtain concretization obtain problem repair boolean program defined manner identical repair concrete program concretization involves mapping statement modified corresponding statement using function repair concretization needs ensure modified expressions meet syntactic requirements corresponding templates following sections describe two steps detail repair boolean programs solution repair boolean program relies autob matically computing inductive assertions along suitable follows explain together certify partial correctness adaptation method inductive assertions program repair let set nodes transition graph representation define set nodes called every entryi exiti every edge procedure call every edge assert statement every cycle contains least one node pair said adjacent every path contains verification path path adjacent note one verification path two adjacent example set exit valid boolean program fig verification paths corresponding follows assume assume assume assume assume assume assume assert inductive assertions denote inductive assertion associated informally inductive assertion property whenever control reaches program execution must true current values variables scope thus boolean program inductive assertion general boolean formula variables whose scope includes precise boolean formula denotes copy subset program variables ormal exitt inscope otherwise thus except main procedure inductive assertions exit nodes procedures exclude local variables declared procedure let denote set inductive assertions associated labeling edge assert slight abuse semantics assert statement justification constraints formulated later section require assertion true whenever control reaches location execution path verification conditions popular approach verification sequential imperative programs compute satisfies set constraints called verification conditions let verification path adjacent verification condition corresponding denoted essentially hoare triple stmt stmt sequence statements labeling unknown seen constraint encoding possible solutions every program execution along path starting set variable valuations satisfying terminates set variable valuations satisfying note definitions adjacent ensure worry along verification paths hoare triple stmt defined using weakest preconditions strongest postconditions paper see shortly find convenient use strongest postconditions program verification using inductive assertions method given program annotated assertions set partially correct one compute set inductive assertions every verification path every pair adjacent cutpoints valid repairability conditions partial correctness let function mapping locations costs find convenient use denote value location set true entryt informally seen recording cumulative cost applying sequence update schemas statements procedure location entryi thus specific update function cost function records total cost costc performing program given verification path adjacent extend definition define repairability condition corresponding denoted crc crc seen constraint encoding possible solutions inductive assertions update functions along associated functions every program execution proceeds along path via statements modified applying update schemas starting set variable valuations satisfying terminates set variable valuations satisfying nondeterministic choices program might make along proceed recall boolean formula ormal exitt inscope otherwise thus locations verification path inscope follows notation juk stmt represents class statements may obtained applying update schema stmt defined permissible update schemas fig etc denote unknown boolean variables inscope note update schema assign assign modifies assignment statement one assigns unknown boolean expressions variables assign skip assume skip call skip assign assign assume assume call call juk stmt stmt skip skip skip assume call stmt call fig definition juk stmt define crc three cases consider stmt contain procedure call assert statement let denote assertion associated location crc given conjunction following set constraints juk stmt denotes natural ordering sequence locations consecutive locations succ keep exposition simple assume unknown boolean expressions deterministic however prototype tool see sec also ability compute modified statements nondeterministic expressions choose notation ustmt denotes set update schemas may applied stmt notation juk stmt denotes strongest postcondition assertion class statements juk stmt define strongest postcondition using multiple variable copies copy location let assume boolean formula consecutive locations succ boolean expression copies representing path condition imposed program boolean expression representing copy variable terms copy program variables note form juk stmt skip goto juk stmt assume assume vbi fig definition juk stmt given form juk stmt defined fig observe juk stmt boolean formula form variable copies entries assume expressions general disjunction boolean formulas form juk stmt obtained computing disjunction strongest postconditions obtained propagating boolean formula juk stmt using rules fig known beforehand entries correspond entries assume expressions unknown entries correspond assume assume assign assign respectively notation denotes unknown boolean expression nondeterministic expressions rhs assignment statement strongest postcondition computed disjunction strongest postconditions possible assignment statements obtained substituting expression either false true thus summarize set constraints encodes applied sequence statements stmt get modified sequence statements say tmt program execution proceeds along tmt strongest postcondition tmt equals cumulative modification cost counting stmt contains procedure call say call path given call verification path length suppose formal parameters crc given following set constraints cexitj ientryj entryj iexitj exitj call skip call skip call call cexitj call call ientryj entryj iexitj exitj constraints involve replacing entryjth exitth copies formal parameters ientryj iexitj corresponding actual parameters expressed copies program variables respectively call call similar substitution performed except actual parameters unknown expressions finally call skip inductive assertion essentially stays variable copies appropriately adjusted general sum cumulative modification cost cexitj procedure cost applying update schema question stmt contains assert statement say assert given assert verification path length crc given following set constraints tmp tmp tmp tmp uniformly convert expressions expressions temporary copy program variables enable checking implications informally implications boolean program repair given let set verification paths every pair adjacent given incorrect program annotated assertions set cost function repair budget say repairable within budget given one compute set inductive assertions update function along models unknown expressions associated applications update schemas valuations function every verification path crc valid constraints met mathematically repairable within budget following formula true nknown crc assumeconstraints nknown set unknowns set boolean program variables copies used encoding constraint crc set unknowns includes inductive assertions update function unknown expressions etc associated applying update schemas valuations program location function finally assumeconstraints ensures modifications guards assume statements corresponding conditional statement consistent thus every pair updated assume assume statements labeling edges starting node transition graph uninterpreted functions constrained satisfy formula true extract models unknowns witness satisfiability formula crc assumeconstraints particular extract corresponding modified statements yield following theorem states correctness correct boolean program completeness algorithm repairing boolean programs partial correctness theorem given set specified given incorrect boolean program annotated assertions cost function repair budget exists method finds repair method finds proof note formula formula boolean variables boolean program variables copies unknown boolean expressions boolean variables inductive assertions expressions modified program statements sequences update schemas update functions corresponding sequences integer costs valuations number boolean variables finite hence number unknown boolean expressions finite finite number update functions drawn finite sequences update schemas finite set corresponding finite number functions set besides includes boolean operators operator finite number integer constants corresponding cost function clearly truth formula decidable particular formula finite number models given set specified completeness method follows completeness floyd inductive assertions method decidability formula soundness method follows soundness floyd inductive assertions method example boolean program fig tool modifies two statements guard stmt changed guard stmt changed concretization present second step framework computing conb follows assume alcrete repaired program ready extracted models recall denotes mapping boolean variables respective predicates mapping extended standard way map expressions boolean variables expressions concrete program variables goal concretization repaired boolean concretization program compute corresponding repaired concrete program involves computing mapping denoted modified statement corresponding modified statement concrete program follows define type modified statement let fix attention statement location denoting set concrete abstract program variables respectively whose scope includes let skip skip assume assume call call definition assignment statement fact case may empty set may contain multiple concrete assignment statements say assignment statement concretizable one compute expressions type concrete program variables respectively certain set constraints valid precise concretizable following formula true constraint essentially expresses concretization abstract assignment substitutions reflect new values concrete program variables concrete assignment formula true extract models exprq respectively witness satisfiability inner say exprq note practice expri may equivalent thereby generating redundant assignment parallel assignment compressed eliminating redundant assignment fact may possible infer without using analyzing dependencies concrete program variables predicates actually affected boolean assignment question exercise beyond current scope work recall associated concretization location denotes template specifying desired syntax expressions concrete modified statement henceforth use shorthand find helpful illustrate concretization using example template let assume concrete program variable fix linear arithmetic expressions program variables form assume call statements integer linear arithmetic terms assignment program variables form statements let assume parameters given let denote mapping abstract statements concrete statements compatible define type modified statement shown basic idea compute suitable values template parameters satisfy certain constraints note general may empty set may contain multiple concrete statements skip skip statement assume concretizable following formula true formula true extract values witness satisfiability inner say assume similarly statement call concretizable following formula true formula true extract values witness satisfiability inner generate concrete call statement call statement concretizable formula true convenience let formula true extract values witness satisfiability inner generate concrete assignment statement example example fig modified guards respectively concretized stmt stmt true respectively using concretization inductive assertions concretization inductive assertion simply experiments prototype tool built prototype tool repairing boolean programs tool accepts boolean programs generated predicate abstraction tool satabs version sequential programs experience found programs multiple procedures satabs generates single procedure boolean programs procedure calls inlined within calling procedure hence perform intraprocedural analysis version tool set update schemas handled currently assign assign assume assume int main int assert boolean program boolean program repair change guard stmt concrete program repair change guard stmt fig repairing program permit statement deletions set costs assign assign assume assume large number every location wish disallow statement modifications locations initialize tool repair budget also provide tool locations boolean program input given tool automatically generates smt query corresponding inner generating repairability query update schemas involving expression modifications stipulate every deterministic boolean expression modified unknown deterministic boolean expression described fig every nondeterministic boolean expression modified unknown nondeterministic expression form choose smt query fed version solver either declares formula satisfiable provides models unknowns declares formula unsatisfiable latter case choose increase repair budget repeat process solver provides models unknowns extract repaired boolean program currently next step concretization partly automated assignment statements manually formulate smt queries corresponding inner int main int skip else assert boolean program boolean program repair change guard stmt change guard stmt concrete program repair change guard stmt true change guard stmt fig repairing program feed queries relevant queries found satisfiable obtain repaired program queries unsatisfiable attempt concretization using templates manually formulate smt queries corresponding inner call experiments allowed degree flexibility guiding solver choose right template parameters fig fig fig fig present details repairing four programs first two programs handmade second one one shown fig next two programs mutations two programs drawn nec laboratories static analysis benchmarks int int foo int ptr ptr return int main foo foo assert boolean program ptr ptr boolean program repair change stmt concrete program repair change stmt fig repairing program emphasize repairs respective boolean programs shown due lack space obtained automatically concretization repaired boolean program fig trivial involved concretizing guard corresponding statement location concretization repaired boolean program fig involved concretizing two different guards corresponding statements locations respectively manually simplified concretized guards obtain concrete guards true respectively concretization repaired boolean program fig involved concretizing assignment statement location manually formulated smt query corresponding int main int int assert boolean program boolean program repair change stmt concrete program repair change stmt fig repairing program formula simplifying restricting lhs stmt concrete program remain unchanged query found satisfiable yielded rhs assignment statement concrete program repeated exercise concretize assignment statement location fig obtained repair concrete program unsatisfied repair formulated another smt query corresponding formula restricting rhs stmt template unknown query found satisfiable yielded table present results repairing four programs benchmark programs competition software verification complexity programs stems nondeterministic assignments function invocations within loops experiments run machine intel dual core unix desktop ram enumerate time taken individual step involved generating repaired boolean program columns labeled loc loc enumerate number lines code original program boolean program generated satabs respectively column labeled enumerates number variables boolean program column enumerates time taken satabs generate boolean program column enumerates time taken tool generate repairability query column enumerates time taken solve query columns asg asm count number assign assign assume assume update schemas applied respectively obtain final correct program notice implementation either produces repaired program quickly fails reasonable time whenever significant increase number boolean variables case example loop true smt solver might need search simultaneous assignments boolean variables every assignment statement order solve repairability query last two programs satabs main bottleneck satabs failing generate boolean program counterexample minutes particular experienced issues using satabs programs relied heavily character manipulation emphasize successful tool repair diverse set errors programs containing loops multiple procedures pointer array variables benchmarks able repair operators incorrect conditional statement repaired array indices incorrect assignment repaired modify constants program variables incorrect assignment repaired program variable also note many benchmarks repaired programs required multiple statement modifications discussion algorithm presented paper separates computation repaired boolean program concretization obtain separation necessary fact separation may table experimental results name loc loc asg asm infinite loop true array true true terminator true true true netbsd libc loop true vogal true count true may possible concretize modified statements computed may indeed exist concretizable solution directly search modified statements concretizable done combining constraints presented sec one particular set nknown modified include unknown parameters needed formulas sec crc modified include inner constraints formulas sec noted sec target total correctness repaired programs associating ranking functions along inductive assertions including termination conditions part constraints finally wish explore ways ensure repaired program unnecessarily restrict correct behaviors original program conjecture done computing weakest possible set inductive assertions least restrictive references alur benedikt etessami godefroid reps yannakakis analysis recursive state machines acm trans program lang syst arcuri automation fixing software bugs international conference software engineering icse acm ball bounimova kumar levin static driver verification false alarms formal methods computer aided design fmcad ball naik rajamani symptom cause localizing errors counterexample traces principles programming languages popl acm ball rajamani boolean programs model process software analysis tech msr ball rajamani automatically validating temporal safety properties interfaces international workshop model checking software spin bloem chatterjee henzinger jobstmann better quality synthesis quantitative objectives computer aided verification cav springer bouajjani esparza maler reachability analysis pushdown automata application international conference concurrency theory concur chandra torlak barman bodik angelic debugging international conference software engineering icse acm clarke kroening sharygina yorav satabs predicate abstraction tools algorithms construction analysis systems tacas springer verlag competition software verification loops benchmarks http debroy wong using mutation automatically suggest fixes faulty programs software testing verification validation icst dijkstra discipline programming prentice hall floyd assigning meanings programs mathematical aspects computer science american mathematical society goues forrest weimer systematic study automated program repair fixing bugs international conference software engineering icse ieee press graf construction abstract state graphs pvs computer aided verification cav springer verlag griesmayer bloem cook repair boolean programs application computer aided verification cav jobstmann griesmayer bloem program repair game computer aided verification cav jose majumdar cause clue clauses error localization using maximum satisfiability programming language design implementation pldi acm bloem automated error localization correction imperative programs formal methods computer aided design fmcad logozzo ball modular verified automatic program repair object oriented programming systems languages applications oopsla acm manna introduction mathematical theory computation moura efficient smt solver proceedings tools algorithms construction analysis systems tacas nec necla static analysis benchmarks http samanta deshmukh emerson automatic generation local repairs boolean programs formal methods computer aided design fmcad singh gulwani automatic feedback generation introductory programming assignments programming language design implementation pldi singh synthesizing manipulations storyboards foundations software engineering fse rabbah bodik ebcioglu programming sketching programs programming language design implementation pldi acm tancau bodik seshia saraswat combinatorial sketching finite programs architectural support programming languages operating systems asplos acm srivastava gulwani foster program verification program synthesis principles programming languages popl acm wei pei furia silva buchholz meyer zeller automated fixing programs contracts international symposium software testing analysis issta acm zaeem gopinath khurshid mckinley data structure repair using sat tools algorithms construction analysis systems tacas zeller hilebrandt simplifying isolating input ieee trans softw eng
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learning complex swarm behaviors exploiting local communication protocols deep reinforcement learning sep maximilian adrian gerhard swarm systems constitute challenging problem reinforcement learning algorithm needs learn decentralized control policies cope limited local sensing communication abilities agents although recent advances deep algorithms applied systems learning communication protocols simultaneously learning behavior agents still beyond reach deep algorithms however often difficult directly define behavior agents simple communication protocols defined easily using prior knowledge given task paper propose number simple communication protocols exploited deep reinforcement learning find decentralized control policies swarm environment protocols based histograms encode local neighborhood relations agents also transmit information shortest distance direction desired target framework use adaptation trust region policy optimization learn complex collaborative tasks formation building building communication link pushing intruder evaluate findings simulated environment compare implications different communication protocols introduction nature provides many examples performance collective limited beings exceeds capabilities one individual ants transport prey size single ant could carry termites build nests nine meters height bees able regulate temperature hive common phenomena fact individual basic local sensing environment limited communication capabilities neighbors inspired biological processes swarm robotics tries emulate complex behavior collective rather simple entities typically robots limited movement communication capabilities sense local neighborhood environment distances bearings neighbored agents moreover agents limited memory systems agents access short horizon perception consequence design control policies capable solving complex cooperative tasks becomes problem common approach program systems extracting rules observed behavior calculations research conducted lichtenberg high performance computer darmstadt maximilian gerhard neumann school computer science university lincoln lincoln mhuettenrauch gneumann adrian neering technische department electrical engidarmstadt darmstadt germany natural counterparts kube example investigate cooperative prey retrieval ants infer rules swarm robots fulfill task cooperative similar work found however extracting rules tedious complexity tasks solve via explicit programming limited paper want learn complex swarm behavior using deep reinforcement learning based locally sensed information agents desired behavior defined reward function instead simple controllers agents swarm systems constitute challenging problem reinforcement learning algorithm needs learn decentralized control policies cope limited local sensing communication abilities agents collective tasks require form active cooperation agents efficient cooperation agents need implement basic communication protocols transmit local sensory information neighbored agents learning communication protocols simultaneously learning behavior agents seems reach current reinforcement learning algorithms explains limited success deep reinforcement learning swarm systems however using prior knowledge given task simple communication protocols defined much easily directly defining behavior paper propose evaluate several simple communication protocols exploited deep reinforcement learning find decentralized control policies multi robot swarm environment communication protocols based local histograms encode neighborhood relation agent agents also transmit information shortest distance direction desired target histograms deal varying number neighbors sensed single agent depending current neighborhood configuration protocols used generate high dimensional observations individual agents turn exploited deep reinforcement learning efficiently learn complex swarm behavior base algorithm trust region policy optimization trpo deep reinforcement learning algorithm resulting algorithm provides integrated learning framework specifically tailored swarm setting approach able learn decentralized control policies fashion mappings local sensory input actions without need complex feature engineering summary method addresses emerging challenges decentralized swarm control following way homogeneity explicit sharing policy parameters agents partial observability efficient processing actionobservation histories windowing parameter sharing communication usage communication protocols simple features demonstrate framework formulate three cooperative learning tasks simulated swarm environment environment inspired colias robot see figure modular platform two wheel movement various sensing systems fig colias robot platform provides basis simulation experiments background fitted returns objective maximized subject fixed constraint divergence policy parameter update ensures updates new policy parameters bounded order avoid divergence learning process overall optimization problem summarized maximize subject dkl section provide short summary trust region policy optimization formalize learning problem domain problem approximately solved using conjugate gradient optimizer linearizing objective quadratizing constraint trust region policy optimization problem domain trust region policy optimization trpo algorithm optimize control policies reinforcement learning problems problems formulated markov decision processes mdp compactly written tuple mdp agent chooses action via policy based current state progresses state according transition function step agent assigned reward provided reward function judges quality decision goal agent find policy maximizes cumulative reward achieved certain period time trpo policy parametrized parameter vector containing weights biases neural network following denote parameterized policy reinforcement learning objective expressed finding new policy maximizes expected advantage function current policy trpo building upon theory reinforcement learning formulate problem domain swarm environments instead considering one single agent consider multiple agents type interact environment limited sensory input agent obtain local observation vicinity environment hence partially informed global system state global system state case comprised local states agents additional attributes environment order cope limitation agents need make use observation histories successfully solve global collaborative task considering homogeneous system assume agents execute distributed policy defined mapping histories past actions observations within finite horizon global task agents encoded reward function write denote joint action vector whole swarm estimate advantage function current policy defined vold herein value function typically estimated single trajectory rollout value function rather simple baselines used currently majority deep reinforcement learning literature focuses scenario approaches tackling problem one approaches found authors use variation deep deterministic policy gradient paper outline section review concepts trpo describe problem domain section iii show detail tackle challenges modeling observations policy partially observable swarm context adapt trpo setup section present model parameters agents introduce three tasks evaluate proposed observation models policies related work range histogram bearing histogram joint histogram local agent configuration fig schematic illustration observation model algorithm learn centralized policy accounts reasoning agents behavior linear increase dimensionality joint observation action spaces makes scaling algorithm many agents hard another algorithm tackling credit assignment problem found baseline agents behavior subtracted centralized critic reason quality single agent behavior however approach possible scenarios discrete action spaces since requires marginalization agents action space finally different line work concerning learning communication models agents found good example rule based behavior found group swarming robots transports object goal comparable work found aggregation flocking foraging iii ulti earning ocal ommunication rotocols section briefly discuss used model state representation single agent subsequently introduce different communication protocols based neighborhood histograms used combination solve complex swarm behaviors algorithm relies deep neural network policies special architecture exploit structure observation histories present network model subsequently discuss small adaptations make trpo algorithm order apply cooperative setting single agent model local state single agent modeled position orientation xmax ymax robot control speed wheels therefore apply force left right side agent similarly wheels real robot model single agent inspired colias robot underlying principles straightforwardly applied swarm settings limited observations communication protocols communication protocols based histograms either encode neighborhood relations distance relations different points interest neighborhood histograms individual agents observe distance bearing neighbored agents communicate agent assume agents constantly sending signal neighbored agents localize sources arising neighborhood configuration important source information used observations individual agents one arising difficulties case handle changing number neighbors would result variable length observation vector policy representations neural networks expect fixed input dimension one possible solution problem allocate fixed number neighbor relations agent agent experiences less neighborhood relations standard values could used high distance bearing however approach comes several drawbacks first size resulting representation scales linearly number agents system number parameters learned second execution learned policy limited scenarios exact number agents present training third fixed allocation neighbor relation inevitably destroys homogeneity swarm since agents longer treated interchangeably particular using fixed allocation rule requires agents must able discriminate neighbors might even possible first place solve problems propose use histograms observed neighborhood relations distances bearing angles representation inherently respects agent homogeneity naturally comes fixed dimensionality hence canonical choice swarm setting experiments consider two different types representations concatenated histograms distance bearing multidimensional histograms types illustrated figure representation advantage scalability grows linearly number features downside potential dependencies features completely ignored shortest path partitions many applications important transmit location point interest neighbored agents currently observe point due limited sensing ability assume agent observe bearing distance point interest within communication radius agent transmits observed distance agents agents see point interest might case observe message another agent containing distance point interest distance sending agent added received distance obtain distance point interest would use sending agent via point agent might compute several distances transmits minimum distance computed indicate length shortest path seen location neighbored agents including distance shortest path information important knowledge policy navigating point interest hence adapt histogram representation partition contains minimum received shortest path distance agent located position weight sharing policy networks policy maps sequences past actions observations new action use histories fixed length input policy deep neural network architecture architecture chosen initial experiments recurrent neural networks led poor results small history lengths often sufficient swarm behaviors communication protocols provide enough useful information single observation constituted multiple histograms partitions observation space already quite high dimensional using history observations adds dimensionality input network cope high input dimensionality propose weight sharing approach pair agent history first processed independently network using weights initial reduction dimensionality hidden states concatenated subsequent layer finally mapped output independent layers formally described weight matrices bias vectors activation function number independent hidden layers input first layer defined denotes concatenation vectors note shared whole length history reducing number parameters necessary process history let concatenation independent hidden states combining layers action fig diagram shows model proposed policy numbers inside boxes denote dimensionalities hidden layers plus sign denotes concatenation vectors policy model given number combined hidden layers output last layer used action vector agent homogeneity agents achieved using set parameters policies diagram architecture shown figure adaptations trpo order apply trpo setup small changes original algorithm made similar formulation first since assume homogeneous agents one set parameters policy shared agents since agents rely global state advantage function redefined order estimate function agent assigned global reward time step transitions treated executed single agent experimental setup section describe three experimental setups underlying robot model used simulation colias colias robot consists base module housing motion controller two movement three shortrange bump sensors several extension modules module six sensors camera module bluetooth communication furthermore ambient light sensor work rely solely data short long range sensors ambient light sensor agents able move maximum speed approximately case active object detection short range sensors range long edge task link task push task fig illustration three cooperative tasks used paper green dots represent agents green ring segments located next agents indicate short range front sensors outer green circles illustrate maximum range distances bearings agents observed depending used observation model edge task red rings show penalty zones agents punished outer green rings indicate zones legal edges formed link task red dots correspond two points need connected agents push task red dot represents intruder tries reach point marked black dot simulated light source agents try push far away possible range sensors range signals transmitted agents received reliably processed terms distance bearing estimation distance platform still development thus able deploy learned policies yet generally observation model comprised raw sensor readings short long range sensors later denoted sensor evaluations furthermore augment observation representation communication protocols developed section model also accounts noise estimation since precise positions provided agents consider following dimensionality observations short range long range ambient light sensor histogram distances histogram bearing distances bearing discretizing way leads resolution distance angle histogram representation note either histograms used time additional features added depending task explained experiments section simulation running realistic physics engine allowing correct physical interaction bodies agents policy architecture decided policy model three hidden layers first two layers process pairs timestep history individually map hidden layers size output second layer concatenated form input third hidden layer eventually maps two actions left right motor example case observation representation using histogram dimensionality previous timesteps instead process input vector size map dimensions single timestep dimensional space concatenate representation dimensional representation tasks focus experiments tasks agents need collaborate achieve common goal purpose designed following three scenarios task building graph first task goal agents find maintain certain distance kind behavior required example surveillance tasks group autonomous agents needs maximize coverage target area maintaining communication links formulate task graph problem agents nodes try maximize number active edges graph herein edge considered active whenever distance corresponding agent lies certain range setting visualized figure experiment provide positive reward edge range give negative feedback distances smaller accordingly reward function dim dim average returns average return trpo iterations sensor comparison different history lengths histogram trpo iterations comparison different observation models fig learning curves edge task curves show mean values plus minus one standard deviation computed eight learning trials legend histogram distances bearings two independent histograms distances bearing distance histogram bearing histogram sensor histogram denotes euwhere dim clidean distance centres agent agent else indicator function note omit dependence dim system state keep notion simple task establishing communication link second task adds another layer difficulty maintaining network agents locate connect two randomly placed points state space link established successfully communicating agents connecting two points figure shows example active link spanned three agents two points task resembles problem establishing connection two nodes wireless hoc network experiments distance two points chosen larger requiring least three agents bridge gap reward determined length shortest distance two points dopt straight line length shortest active link dsp spanned agents opt dsp link established otherwise task use shortest path partitions communication protocol agent communicates shortest path knows points interests resulting two partitions used observation input single time step task pushing intruder third task group colias robots defenders shall prevent another colias robot intruder reaching specified target position task similar problem challenging intruder adds additional dynamics problem target position indicated light source allows intruder determine euclidean distance target via ambient light sensor defenders however informed distance intruder light position within communication range light position part agents observation policy intruder modeled phototaxis behavior part learning process reward computed based current distance intruder target position distance given dlp position light given point intruder position reward function dlp addition standard sensor readings give agents ability distinguish intruder agents additional information encoded two extra dimensions range bearing observation representation task tested different observation protocols neighborhood histograms well shortest path partitions intruder position used point interest results evaluate task standardized environment size initialize ten agents randomly scene special interest amount information provided agents affects overall system sensor average returns average returns sensor trpo iterations link task trpo iterations push task fig learning curves link task push task obtained different observation models fixed history length curves show mean values plus minus one standard deviation based eight learning trials legend two dimensional histogram shortest paths histogram distances bearings two independent histograms distances bearing sensor histogram performance herein keep mind general local observations generally provide information global system state time result complex learning task recall information content mostly influenced two factors length history composition observation edge task first evaluate history length affects system performance figure shows evaluation weight sharing policy using histogram distances bearings interestingly observe longer observation histories show increase performance either increase information could counter effect increased learning complexity history length already sufficient solve task use findings set history length remainder experiments next analyze impact observation model figure shows results learning process different observation modalities first observation irrespective used mode agents able establish certain number edges naturally complete information distances bearing yields best performance however independent histogram representation yields comparable results two two dimensional histogram due aforementioned complexity higher amount information makes learning process difficult link task evaluate link task raw sensor measurements count based histograms distance bearing advanced shortest path histograms distance bearing based findings edge task keep history length figure shows results learning process observation model tested averaged trials since least three agents necessary establish link two points models without shortest path information struggle reliably establish connection chance spread wide possible thus cover area points interesting see independent histograms counts seem favorable histogram however versions surpassed histogram shortest paths yields information current state whole network agents currently connected points push task push task turned hardest challenge given information current distance bearing neighboring agents defenders unable find strategy prevent intruder reaching light position agents access shortest path intruder able push away goal however would like investigate improve agents behavior successfully execute task conclusions future work paper demonstrated histograms simple local features effective way processing information robot swarms central aspect new model ability handle arbitrary system sizes without discriminating agents makes perfectly suitable swarm setting agents identical number agents neighborhood varies time use protocols adaptation trpo swarm setup learn cooperative decentralized control policies number challenging cooperative task evaluation approach showed model leads agents reliably fulfill tasks interesting future directions include example learning explicit communication protocol furthermore expect assigning credit agents taking useful actions speedup learning algorithm eferences review swarm robotics tasks neurocomputing montijano schwager rus distributed formation control among obstacles geometric optimization approach consensus proceedings ieee international conference robotics automation pages arvin murray zhang yue colias autonomous micro robot swarm robotic applications international journal advanced robotic systems basu redi movement control algorithms realization hoc robot networks ieee network chen gauci strategy transporting tall objects swarm miniature mobile robots proceedings ieee international conference robotics automation pages correll martinoli modeling designing aggregation swarm miniature robots international journal robotics research foerster assael freitas whiteson learning communicate deep reinforcement learning advances neural information processing systems pages foerster farquhar afouras nardelli whiteson counterfactual policy gradients goldberg mataric robust control distributed collection tasks technical report university southern california los angeles united states lillicrap ghahramani turner levine policy gradient critic proceedings international conference learning representations gupta egorov kochenderfer cooperative multiagent control using deep reinforcement learning proceedings adaptive learning agents workshop hoff sagoff wood nagpal two foraging algorithms robot swarms using local communication proceedings ieee international conference robotics biomimetics pages kube bonabeau cooperative transport ants robots robotics autonomous systems lillicrap hunt pritzel heess erez tassa silver wierstra continuous control deep reinforcement learning lowe tamar harb abbeel mordatch multiagent mixed environments martinoli easton agassounon modeling swarm robotic systems case study collaborative distributed manipulation international journal robotics research mnih kavukcuoglu silver rusu veness bellemare graves riedmiller fidjeland ostrovski control deep reinforcement learning nature moeslinger schmickl crailsheim emergent flocking swarm robots pages nouyan gross bonani mondada dorigo teamwork robot colonies ieee transactions evolutionary computation oliehoek decentralized pomdps pages springer berlin heidelberg schulman levine moritz jordan abbeel trust region policy optimization proceedings international conference machine learning pages schulman wolski dhariwal radford klimov proximal policy optimization algorithms teh bapst czarnecki quan kirkpatrick hadsell heess pascanu distral robust multitask reinforcement learning witkowski habbal herbrechtsmeier tanoto penders alboul gazi network communication infrastructure systems disaster scenarios proceedings workshop robotics risky interventions environmental surveillance
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truth validation evidence papis wongchaisuwat diego klabjan department industrial engineering management sciences northwestern university evanston abstract modern era abundant information easily accessible various sources however sources reliable mostly contain unverified contents develop system validate truthfulness given statement together underlying evidence proposed system provides supporting evidence statement tagged false work relies inference method knowledge graph identify truthfulness statements order extract evidence falseness proposed algorithm takes account combined knowledge ontologies system shows good results provides valid concise evidence quality plays role performance inference method explicitly affects performance algorithm accessing information online expanding tremendously various domains reported study pew internet project research anderson perrin according study offline population declined significantly since extent adults use internet internet usage gives people opportunity extensively seek information online however posted contents available web pages necessarily reliable online information spreads rapidly quality considerably crucial misinformation potentially leads serious consequences significantly affecting internet users main motivation study validate truthfulness textual information obtained various sources well provide supporting evidence humans identify truthfulness statement particularly common fact cases nevertheless manually inspecting statements process becomes impossible data determining truthfulness statement automated fashion promising alternative solution problem highly challenging due lack encompassing comprehensive corpora true statements despite challenges draws lot attention prior studies develop systems previously proposed systems mainly rely web search engines verify whether statements true false additional information regarding sources statements extracted also taken account algorithms comparison systems work relies knowledge reliable sources rather web search engines statements gathered reliable sources various length may verbose thus represent statements triplets consisting subject entity object entity relation triplets capture main contents embedded within statements knowledge graph constructed triplets nodes entities arcs represent relationships nodes algorithm relation extraction method used extract triplets statement aim verify truthfulness call statement corresponding triplets lay statement lay truthfulness lay triplet verified based inference method corresponding constructed reliable sources identifying truthfulness lay triplets algorithm additionally provides supporting evidence determining evidence true triplets relatively straightforward compared identifying evidence falseness considering true triplet supporting evidence set paths subject object entities inferred associated reliable sources hand unclear obtain evidence false triplets reasonable evidence false triplet collection relevant triplets extracted specific condition explain key idea example consider false triplet property space rocket find triplets property case set possible candidates denoted bedroom kitchen bathroom roof garden shed swimming pool long proof evidence candidate evidence set fact space rocket drawback size large summarizing candidate collection concise meaningful evidence set challenging especially size collection large order overcome difficulty develop novel algorithm extract supporting evidence concepts ontologies false triplet rely idea representing candidate broader concepts ontologies given false triplet concept part broader concepts considering example ontology could provide fact first four terms related house remaining terms correspond finally evidence provide property house property backyard facts space rocket house space rocket given false triplet candidates matching concepts ontologies considered gather set potential evidence includes candidate concepts broader concepts satisfying conditions evidence various levels granularity constructed graph based algorithm subsumption tree ontology select optimal collection evidence potential evidence set optimized set evidence smallest subcollection potential evidence set assumption candidates covered broader concepts leads set covering problem rest paper consider following running example given false triplet google officelocationinus minneapolis generate evidence falseness based relevant triplets relevant triplets retrieved officelocationinus relation associated google minneapolis entity particular first find locations google offices atlanta chicago los angeles miami mountain view etc also companies whose office located minneapolis target corporation bancorp xcel energy considered retrieved entities used falseness evidence claim minneapolis part retrieved locations similarly google part set retrieved companies rely knowledge ontologies generate concise set evidence example ontology geography used state google offices many states across minneapolis located minnesota one states main contribution provide supporting evidence given lay triplet truthfulness identified triplet true paths provide evidence however false much challenging come concept evidence best knowledge prior work provided supporting evidence given false lay statement taking account ontologies proposed system combines knowledge ontologies predicate triplets contributes space specifically focus selecting complete set falseness evidence concise possible also system relies mainly ontologies constructed reliable sources instead knowledge unverified web pages algorithm provide supporting evidence along truthfulness lay triplet applicable various domains politics sciences news health care work focuses health care domain case study mainly abundant information available online importance information quality specifically large number medically related web sites easily accessible online half sites content reviewed professionals gottleb addition distorted information related health conditions potentially causes devastating effects summarize literature section section describe relevant background information problem definitions thoroughly discuss main algorithm data preparation results algorithm based case study reported section discussions provided section conclusion future work stated section work algorithm verifies truthfulness lay statement based thus survey prior work truth fields substantial work exists truth discovery determining veracity data particular truth discovery problem aims identify whether assertions claimed multiple sources true false reliability sources also determined waguih berti waguih provide extensive review evaluation truth discovery algorithms additional truth discovery methods proposed varying many aspects jointly estimate source reliability truth statements meng mukherjee xiao zhao zhi methods rely common assumption information provided reliable source tends trustworthy source providing trustworthy information likely reliable truthorrumor truth judgment system determines truth based results search engine liu considers reliability data sources based historical records copying relationship also implements currency determination techniques take account statements wang propose algorithm determine truthfulness given statement based combination support score credibility ranking value support score measures web search result supports statement credibility ranking computes reliability web pages system requires users specific parts statements verified systems take account additional information data set source determining truthfulness statement yin tan aim distinguish true false statements given small set ground truth facts yin tan graph optimization method used yin tan node graph represents statement edge connects pair relevant statements statements set ground truth facts labeled algorithm assigns truthfulness score ranging unlabeled statement scores unlabeled statements directly related labeled statements possibly close implies truthfulness statements remains undefined yamamoto tanaka propose system determine credibility lay statement extract aspects necessary verify factual validity web pages yamamoto tanaka whenever statement true order estimate validity lay statement system collects comparative fact candidates using web search engine fact candidates sentences retrieved search engine match pattern specified lay statement validity candidate computed based relation pattern entity contained candidate comparison previous work focus algorithm provide concise reliable supporting evidence addition identifying truthfulness lay statement algorithm proposed yamamoto tanaka similar system statement true particular yamamoto tanaka work use comparative candidate facts order assess credibility lay statement instead using web search engines rely inference method respect collect candidate triplets truthfulness score lay triplet computed compared scores candidate triplets order determine whether lay triplet true false none works provide evidence false statements main contribution work content commonly found textual documents especially online texts unreliable study aim identify whether given lay statement true false provide supporting evidence collect lay statements many web pages publicly available online use relation extraction algorithm rindflesch fiszman extract triplets consisting subject entity object entity relation embedded within lay statements problem scoped identifying truthfulness triplets representing original lay statements knowledge obtained reliable sources important factor determining trustworthiness lay triplets assure reliable resources structured form triplets used construct knowledge graph nodes edges represent entities relations respectively write mean nodes corresponds edge evidence falseness obtained based knowledge various ontologies related domains order properly discuss falseness evidence main algorithm first provide brief overview knowledge base ontology relevant background information according terminological knowledge elementary descriptions concept names atomic concepts role names atomic roles concept descriptions built concept role names concept role constructors concept names concept descriptions generally considered concepts deeper knowledge ontologies obtained franz let knowledge base tbox abox defined franz interpretation model corresponding ontology assume consistent assume concepts denote concept given entity corresponding concept define special concepts top universal bottom empty concepts concept constructors intersection union negation combined concept names used construct concepts let set concept names also define concepts concept classification algorithm computes partial order set concept names respect subsumption relationship concept names classification algorithm incrementally constructs graph representation form direct acyclic graph called subsumption tree partial order induced baader note paper use term tree use term consistent past literature underlying structure actually acyclic graph given set concepts computing representation order equivalent identifying precedence relation exists given precedence relation incremental method defined baader computes element method consists two main parts top bottom search top bottom search identify sets iteration arcs corresponding element well element added also existing arcs elements eliminated end arc constructed subsumption tree proposed system built pipeline involving two main steps denote lay statement triplet requires evidence determining truthfulness triplets rely mainly inference method called path ranking algorithm pra introduced lao verify whether triplet true false pra produces every pair nodes pra model trained relation level particularly pra model relation type trained retrieve nodes potentially relation given node retrieve related object candidates denoted subject candidate set extracted similar way triplet labeled true false otherwise addition paths corresponding high pra scores provided supporting evidences truthfulness true extracting evidence falseness assume labeled false step either extracting subject evidence falseness extracting object evidence falseness set set objects verify false case otherwise would true holds discuss detail object evidence subject evidence defined similarly validation false statements rely pra inference algorithm used found pra work best data next formally define evidence false triplets recall set objects important essence proof falseness provide together fact however many cases would provide long evidence since typically large instead want aggregate smaller set still claim wrapping intuitions ontology formalism yields following definition definition object evidence falseness collection concept names exists exists concept second condition rewritten turn equivalent words second condition equivalent requirement part element evidence set collection considered aggregated set define potential evidence exist object evidence falseness definition set object candidates relation given triplet first requirement definition assures candidate subsumed least one potential evidence aggregation elements example letting satisfies first requirement always subsumed according second requirement concept subsumed potential evidence ensures concept obtained false triplet belong evidence collection mimics among collections want find smallest one formalized later referring false triplet example google officelocationinus minneapolis let object candidates retrieved relation officelocationinus associated subject given ann arbor atlanta austin birmingham boulder cambridge chapel hill chicago irvine kirkland los angeles miami mountain view new york pittsburgh playa vista reston san bruno san francisco seattle sunnyvale washington selecting satisfies requirements definition second requirement satisfied associated minneapolis subsumed element evidence collection moreover collection west region northeast region south region michigan illinois example smaller evidence set satisfies requirements propose algorithm extract evidence falseness defined definition based subsumption tree originally defined concept names expanded negation concepts specific concepts set formally defined recall define concepts concept involves mandates even though infinitely many concepts constructed concept names concept constructors focus specific concepts ensure second requirement definition concept corresponds proper concept name concept name negation algorithm extracts potential evidence considering nodes along paths tree satisfy requirements definition order algorithm check satisfiability requirements also included subsumption tree hence standard tree consisting concept names expanded algorithm add existing standard tree provided appendix subsumption tree used algorithm form directed acyclic graph root top concept also define paths nodes associated subsumption tree used algorithm follows definition set possible paths node node subsumption tree denote node starting let set nodes along paths algorithm set set add else break remove duplicate nodes return setcover assume assumption implies part element statement true assures incremented step empty algorithm repeats steps compute potential evidence stores combined according step set equivalent set potential evidences note every contains due steps algorithm specifically constructed ensure every element one nodes along least one path root implies elements considered broader concepts candidate evidence first requirement definition requires least one corresponding subsumes hence algorithm considers nodes along possible paths root top concept every order extract potential evidence second requirement definition directly associated computed verified steps subsumption tree used algorithm necessary compute guarantee particularly implies therefore case considered potential evidence algorithm computes node path corresponding empty considered potential evidence added step elements correspond nodes thus concept names also correspond potential evidence computed algorithm considers concept names negation concept names definition considers concept algorithm consequently provides approximate evidence set exact algorithm discussed later figure illustrates algorithm path root highlighted blue nodes along red paths collected potential evidence subsumed nodes corresponding nodes empty figure illustration algorithm regarding run time analysis proposed algorithm consists nested loops steps let number nodes maximum number paths node root outer loop step considers element middle loop step processes path corresponding step also node along paths root considered inner loop step computing node requires hence computational complexity algorithm algorithm sped using bisection efficient version provided appendix referring running example let minneapolis consider mountain consider paths mountain view root well nodes along paths node mountain view added minneapolis mountain view empty particularly exists node belongs sets minneapolis minneapolis minneapolis mountain view according natural geography ontology associated example mountain view santa clara california west region retrieved note usa since minneapolis usa empty intuitively minneapolis location usa minneapolis part usa therefore usa counted evidence falseness obtaining set potential across possible evidences aim compute optimal set evidence smallest cardinality formally define object evidence falseness problem min object evidence falseness set covering problem proposed find optimal set evidence later give condition solves optimally set covering problem formulated follows setcover universe node define set covering problem reads min subject node know necessary feasibility set covering problem aims find minimum number set selected sets contain elements universe cover feasible solution set covering problem satisfies first requirement definition set specifically constructed based note set fact every contains construction elements added algorithm guaranteed satisfy second requirement definition according false triplet google officelocationinus minneapolis example consider minneapolis set given previously set generated yields feasible solution set covering problem given table left column lists elements table example feasible sets set covering problem west region boulder irvine kirkland los angeles mountain view playa vista san bruno san francisco seattle sunnyvale northeast region cambridge new york pittsburgh south region atlanta austin chapel hill miami reston washington michigan ann arbor birmingham illinois chicago propositions stated next establish relationship proofs propositions provided appendix proposition feasible feasible solution yields feasible solution smaller cardinality implies due proposition solution always feasible solution thus object evidence falseness obtained used representative evidence set consider either concept names negation concept names computed algorithm exact algorithm replaces defined step algorithm concept concepts constructed concept names concept constructors considered also observe checking equivalent checking satisfiability concept satisfiable stated baader proposition step algorithm substituted proof proposition show feasible solution also feasible solution replacing implies combined proposition yields define subject evidence way order identify subject evidence falseness algorithm applied definitions propositions defined similarly respect domain consideration multiple ontologies case implement proposed algorithms identify evidence falseness ontology minimum cardinality evidence selected across ontologies finally problem identify evidence falseness triplet formulated min min considering min ontologies object evidence subject min object min evidence ontologies subject evidence sets case study apply proposed algorithm health care domain case study reliable source case obtained biomedical publications stored medline database order construct semrep rindflesch fiszman used extract semantic predicate triplets biomedical texts semrep matches subject object entities triplets concepts umls metathesaurus matches relationship respect umls semantic network also takes account syntactic analysis structured domain knowledge hypernymic propositions extensively data contains extracted triplet corresponding sentence medline first train pra model based constructed semrep compare performance evaluation metrics reported lao pra trained nell data set average mean reciprocal rank mrr across different relation types reported lao average mrr pra semrep mrr computed based rank first correctly retrieved triplet however aim correctly retrieve triplets result additionally compute mean average precision map considers rank position triplet map based trained pra model implies average every retrieved result correct manually inspect original statements corresponding predicate triplets extracted semrep even though preliminary evaluation semrep reported rindflesch fiszman states precision extracted predicate triplets contain many errors based manual observation examples predicate triplets incorrectly extracted original statements provided appendix issue sentences clearly correct extracted triplets often wrong hence verifying extracted predicate triplet pra model additional relation extraction systems detailed explanations provided following data preparation section data preparation aim containing triplets high precision manually observing results trained pra model triplets high pra scores tend accurate low pra scores hence pra one models used verify triplets employ relation extraction systems filter incorrect triplets original ollie mausam open information extraction software aims extract binary relationships sentences according open information extraction schema relations need addition train recurrent neural network model called lstmer proposed miwa bansal miwa bansal publicly available training data set gold standard labels instance training data consists statement predicate triplet training data set used train model includes ade corpus gurulingappa hendrickx bionlp kim semrep gold standard annotation kilicoglu order original propose strategy combine triplets high pra scores triplets matching ollie models flow diagram proposed strategy order construct adjusted depicted figure figure flow diagram strategy according proposed strategy infer trained pra model rank results high low pra scores triplets positioned top percent ranked pra scores retrieved additionally collect possible matches triplets results ollie also use trained model infer possible relations statements associated triplets original triplet collected relation matches relation inferred model conduct preliminary experiment extracting matched triplets using ollie model based randomly selected triplets based experiment observe small proportion matching triplets among different relation extraction models detailed discussion experiment provided appendix additionally observe many statements original involve studies nonhuman subjects effects acetylcholine histamine serotonin infusion venous return order filter statements consider umls semantic type categorization concepts represented umls metathesaurus tagged statements particular eliminate statements contain amphibian animal bird fish mammal reptile vertebrate semantic types provide number nodes edges original adjusted table average mrr average map based pra model adjusted respectively table number nodes edges original adjusted original adjusted number nodes number edges results run whole pipeline algorithm validate truthfulness provide supporting evidence lay triplets adjusted based lay triplets consisting relation types collected web pages identify truthfulness triplet extract evidence candidates summarized appendix across relation types false triplets account percent lay triplets instead directly specifying thresholds step process identify rank threshold based ordered pra scores identify corresponding subject entity relation type retrieve set object entities relation identified pra model set denoted set also retrieved parameter defined used specify rank threshold follows parameter captures well pra model gives high ranks triplets higher value better pra model performs implies vary proportionally middle term hyper parameter calibrated experiment order obtain best performance last term formula takes account affects high indicates many object entities predicted respect subject therefore challenging pra model correctly rank retrieved object entities implies high leads low value threshold expressed formula extract entity whose rank based pra score higher candidates moreover specify max identify truthfulness rank higher specify true among false lay triplets first eliminate triplets whose object match concepts ontologies candidates used compute evidence set based remaining triplets using algorithm perform evaluations object candidates subject candidates done similarly average cardinality object candidates average cardinality corresponding evidence sets across relation types respectively histogram produced object evidence relation provided figure types based candidates distribution frequency figure histogram object evidence order evaluate performance proposed algorithm choose relation types representing high medium low map computed pra model treats diagnoses causes respectively relation type select cases compare evidence sets resulting algorithm denoted evidence sets constructed manually denoted illustrated table complete comparison elements evidence sets provided appendix table evidence sets obtained algorithm evidence sets constructed manually heparin treats fever amiodarone treats hepatitis stress management treats mitral capoten treats coughing losartan treats varicose ulcer platelet size diagnoses anemia esophageal monitoring caffeine causes gout hypercholesterolemia causes leukemia causes gout harpin causes cardiomegaly false triplets relation type treats valve prolapse average treats relation type diagnoses echocardiography diagnoses hyperlipidemia diagnoses malignant breast neoplasm cholesterol measurement test dianoses malignant breast neoplasm electrocardiogram diagnoses muscle strain average diagnoses relation type causes neuropathy ascorbic acid causes senile average causes average across relation types plaques future work work develop system validate truthfulness lay triplets provide supporting evidence system employs pra algorithm inferred reliable sources identify whether lay triplet true false experiment train pra model based constructed biomedical literature original contains incorrect triplets due relation extraction process attempt consisting accurate triplets verifying triplet original additional relation extraction algorithms trained pra model adjusted yields mrr map averaged across relation types performance pra model based adjusted improved however adjusted still contains errors due challenge complicated biomedical text limited resources training additional relation extraction algorithms use combination knowledge ontologies triplets adjusted extract concise supporting evidence set specifically algorithm aims find supporting evidence set overlap entity lay triplet evidence set aggregated candidates obtained triplets adjust using knowledge ontologies apply algorithm extract evidence sets based ontology repeatedly consider possible ontologies reasonable select ontology yields minimum cardinality evidence sets according algorithm first match object subject entity lay triplet concepts within ontologies ontologies taken consideration object subject candidates paired concepts ontologies assume candidates matched concepts ontology lay triplet disregarded extracted pra model compare consider number candidates cardinality evidence set resulted algorithm average larger average factor across relation types implies proposed algorithm provides valid concise evidence sets evaluate performance algorithm compare evidence set extracted proposed algorithm manuallyconstructed evidence set average number overlap evidence set algorithm manually constructed set across relation types proposed algorithm performs well especially specific relation types treats overlap problem challenging due limited resources construct complete accurate imperfect plays significant role inferior performance pra model directly impacts performance algorithm extract evidence sets better quality would lead higher performance proposed system hence future work focus improving relation extraction algorithms construct believe utmost importance work systems rely semrep references anderson perrin americans use internet retrieved http website baader description logic handbook theory implementation applications cambridge university press cambridge baader hollunder nebel profitlich franconi empirical analysis optimization techniques terminological representation systems applied intelligence https franz diego deborah daniele peter description logic handbook theory implementation applications cambridge university press cambridge gottleb health information internet often unreliable british medical journal gurulingappa rajput roberts fluck toldo development benchmark corpus support automatic extraction adverse effects medical case reports journal biomedical informatics https hendrickx kim kozareva nakov seaghdha pado pennacchiotti romano szpakowicz task classification semantic relations pairs nominals proceedings international workshop semantic evaluation association computational linguistics pages kilicoglu rosemblat fiszman rindflesch constructing semantic predication gold standard biomedical literature bmc bioinformatics https kim pyysalo ohta bossy nguyen tsujii overview bionlp shared task proceedings bionlp shared task workshop association computational linguistics pages lao mitchell cohen random walk inference learning large scale knowledge base proceedings conference empirical methods natural language processing association computational linguistics pages gao zhao demirbas fan han approach truth discovery data proceedings large data base endowment https meng verifying truthfulness fact statements proceedings ieee international conference data engineering institute electrical electronics engineers pages liu wang chen gao truthorrumor truth judgment web proceedings web technologies applications web conference apweb springer international publishing pages qiu gao zhi zhao han faitcrowd fine grained truth discovery crowdsourced data aggregation proceedings acm sigkdd international conference knowledge discovery data mining association commputing machinery pages mausam schmitz bart soderland etzioni open language learning information extraction proceedings joint conference empirical methods natural language processing computational natural language learning association computational linguistics pages meng jiang gao ding cheng truth discovery crowd sensing correlated entities proceedings acm conference embedded networked sensor systems association commputing machinery pages miwa bansal relation extraction using lstms sequences tree structures arxiv preprint mukherjee weikum people drugs credibility user statements health communities proceedings acm sigkdd international conference knowledge discovery data mining association commputing machinery pages rindflesch fiszman interaction domain knowledge linguistic structure natural language processing interpreting hypernymic propositions biomedical text journal biomedical informatics https waguih truth discovery algorithms experimental evaluation arxiv preprint wang zhu wang novel method fact statement verification proceedings web technologies applications web conference apweb springer berlin heidelberg pages xiao gao feng zhang towards confidence truth bootstrapping based truth discovery approach proceedings acm sigkdd international conference knowledge discovery data mining association commputing machinery pages yamamoto tanaka finding comparative facts aspects judging credibility uncertain facts proceedings web information systems engineering wise international conference springer berlin heidelberg pages yin tan truth discovery proceedings international conference world wide web association commputing machinery pages zhao cheng truth discovery data streams probabilistic approach proceedings acm international conference conference information knowledge management association commputing machinery pages zhi zhao tong gao han modeling truth existence truth discovery proceedings acm sigkdd international conference knowledge discovery data mining association commputing machinery pages appendix algorithm presented next adds negation nodes existing subsumption tree implementing incremental methods defined baader augmented tree form nodes concept names nodes negation concepts concept names nodes specific concepts ensure second requirement definition arc node following properties exists concept names concept constructors combined order construct concepts consequently yields significant number concepts nodes may relation many concepts added subsumption tree note add necessary concepts required algorithm therefore first property takes consideration able assume relation nodes nodes algorithm generate initially set concept name create node compute set compute set add arcs element element remove arcs elements compute set element create artificial node add arc connecting add arc connecting connecting extract evidence set algorithm computes order ensure second requirement definition set depends overlap two set nodes corresponding verify overlap two sets necessary algorithm adds standard subsumption tree includes concept names relies mainly incremental method involving top bottom search computed steps arcs tree represent subsumption relationship steps take account overlap case involves nodes appendix efficient version algorithm propose algorithm using bisection method based follows algorithm set set true add break else else remove duplicate nodes return setcover note algorithm contains due steps algorithm identifies position path bisection search method know concepts hence implies step adds nodes subsumed conditions steps satisfied run time analysis algorithm let number nodes maximum number paths node root similarly algorithm outer loop step middle loop step considered algorithm relies bisection search accounted log computing node requires therefore computational complexity algorithm log compared corresponding algorithm appendix proof proposition first argue feasible path according assumption exists node yields paths implies corresponds nonempty path hence set every implies feasible let feasible solution know exists let next argue object evidence falseness definition implies implies first requirement definition therefore satisfied next show second property definition added step satisfy conditions steps according algorithm path implies path path hence equivalent assured second requirement definition satisfied clear construction proof proposition let feasible requirement every exists tbox classification used construct subsumption tree know path hence path one paths step requirement corresponds computed substituted step algorithm verified checking satisfiability concept step considered evidence added step step algorithm thus hence exists since implies feasible since feasible solution yields feasible solution implies proposition directly followed appendix manually observe extracted triplets compare original statements based manual observation incorrectly extracted triplets provided table table examples incorrectly extracted triplets based semrep original statements extracted triplets six additional imino derivatives pyridoxal pyridoxal prolactin studied none new inhibiting hormone compounds effective pih significant different dehydrogenase levels activity subjects gda measurement quantitative uses gda gdb two methods removal erythrocytes erythrocytes produces human buffy coats production human leukocyte interferon leukocyte interferon appendix according experiment observe matches among ollie models table illustrates number matches among different models table number matches among ollie models based triplets number triplets ollie appendix first identify truthfulness lay triplet based pra model relation type proportion false triplets equivalent number false triplets divided total number triplets computed object candidates computed false triplets total number triplets false triplet proportion average number object candidates relation types summarized table table statistics truthfulness object candidates obtained pra model number triplets false triplets proportion isa predisposes treats causes affects prevents inhibits augments uses produces diagnoses disrupts precedes relation type appendix compare cardinality candidates cardinality evidence elements set corresponding table table complete comparison elements evidence sets false triplets heparin treats fever amiodarone treats hepatitis stress management treats mitral valve prolapse capoten treats coughing losartan treats varicose ulcer echocardiography diagnoses hyperlipidemia platelet size diagnoses anemia esophageal monitoring diagnoses malignant breast neoplasm system symptom anatomical entity system disease disorder finding resistance hypertrophy disease anatomical entity diseases process finding disorder neoplasm morphology digestive system disorder system symptom anatomical entity system disease disorder finding disease anatomical entity platelet disorders cardiovascular diseases finding nonneoplastic disorder neoplasm morphology cholesterol measurement test dianoses malignant breast neoplasm electrocardiogram diagnoses muscle strain caffeine causes gout leukemia causes gout disorder site finding physiological phenomena hypercholesterolemia causes neuropathy process mouse disorder site finding nonneoplastic disorder dependence disorder harpin causes cardiomegaly ascorbic acid causes senile plaques nonneoplastic disorder disease neoplasm disease processes phenomena processes category cerebral ventricles disorder processes category behavior behavior mechanisms signs symptoms digestive nonneoplastic disorder disease neoplasm disease death digestive system physiology nervous system physiology
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oct regular poset resolutions monomial ideals timothy clark alexandre tchernev abstract use natural homeomorphism regular face poset establish canonical isomorphism cellular chain complex result applying poset construction monomial ideal whose free resolution supported regular isomorphism allows free resolution ideal realized resolution conversely resolution monomial ideal gives rise resolution supported regular introduction plethora commutative algebra research centered search combinatorial topological objects whose structure exploited give explicit constructions free resolutions many approaches take advantage combinatorial data inherent grading monomial ideal produce resolution module particular search regular support resolutions quite active due nature variety construction methods topological spaces example diana taylor resolution may viewed resolution supported full simplex whose vertices labeled minimal generators bayer peeva sturmfels take approach resolves specific class ideals using canonical subcomplex full simplex generally criterion using regular support resolutions developed recently techniques connected discrete morse theory topological approach reducing length regular resolutions discussed furthermore individual techniques visscher sinefakopolous mermin use framework regular describe resolutions individual classes ideals despite richness results velasco showed exist minimal free date april timothy clark alexandre tchernev resolutions monomial ideals supported cwcomplex velasco result makes clear structural framework provided restrictive encompass entire spectrum structures needed supporting minimal free resolutions monomial ideals thus flexible structural framework needed main goal paper provide additional argument framework based posets introduced refered poset construction provides necessary flexibility main result theorem poset construction applied face poset regular recovers cellular chain complex demonstrated provides useful criterion checking given monomial ideal supported regular final evidence poset construction right tool study structure minimal free resolutions monomial ideals given shown every monomial ideal poset poset construction applied poset supports minimal free resoluton paper organized follows section extend connection combinatorics topology regular cwcomplexes category complexes vector spaces underlying combinatorial structure regular established way classify spaces whose poset cellular incidences reflects topology space first evidence extending correspondence useful described first author mermin result recovered using poset combinatorics correspondence theorem constructions mentioned describe topological structure resolution without explicitly using combinatorial interpretation described paper section use theorem reinterpret variety constructions free resolutions monomial ideals realize cellular resolution resolution lastly show general framework poset resolutions serves common construction method many classes monomial ideals whose minimal resolutions constructed using topological means value using combinatorial side correspondence lies ability use classify ideals admit cellular minimal resolutions indeed determining whether resolution supported regular amounts determining whether resolution supported generally regular poset resolutions result velasco makes clear limitations purely topological perspective constructing minimal resolutions propose shift focus toward general notions poset combinatorics constructing resolutions since many benefits topological approach appear naturally poset construction throughout paper notions poset order complex algebraic chain complex assumed familiar reader describing topological property poset implicitly describing property order complex poset poset write order complex regular write face poset complexes recall said regular attaching maps define incidence structure homeomorphisms class regular studied introduces investigates following related class posets definition poset called least element nontrivial one element open interval homeomorphic sphere gives following characterization providing connection poset combinatorics class regular cwcomplexes proposition proposition poset isomorphic face poset regular remark recall ranked poset property every maximal chains greatest element finite length ranked posets admit rank function case rank function takes form dim note cells regular cwcomplex correspond bijectively poset elements rank face poset example correspondence given figure remark result arises natural see lundell weingram refer timothy clark alexandre tchernev figure regular hasse diagram face poset sequel correspondence regular collection closed cells regular whose set closed cells alternately space may viewed simplicial complex whose vertices indexed cells fact abstract barycentric subdivision furthermore simplicial cellular homology groups subcomplexes corresponding open intervals form isomorphic applying acyclic carrier theorem theorem subdivision map describe algebraic objects provide setting results paper throughout denote regular standard write collection cells dimension cellular chain complex coefficients field precisely relative homology group dimension since nonempty consists empty cell convention consider onedimensional vector space generally basis elements vector spaces appearing chain complex correspondence cells furthermore cell regular poset resolutions differential takes form dim coefficients incidence numbers determined chosen orientations cells see chapter details let poset minimum element say covers write briefly recall construction sequence vector spaces vector space maps owes structure partial ordering homology open intervals see full description poset construction general necessarily complex vector spaces case indeed complex need exact remainder paper restrict discussion details construction applied sequence constructed using homology order complexes spherical open intervals form let element write order complex open interval vector spaces take form since sphere dimension space order describe maps write order complex interval take advantage decomposition next appeal sequence reduced homology triple timothy clark alexandre tchernev notational simplicity let map homology induced write inclusion connecting homomorphism homology sequence map defined componentwise composition border case map defined componentwise state main result paper theorem correspondence induces canonical isomorphism cellular chain complex sequence proof since homology taken coefficients field omit notation definition cellular chain complex furthermore realization relative homology homology quotient cellular chain complex relative cellular chain complex isomorphism induces isomorphism equality regular poset resolutions realizes relative homology direct sum quotients chain complexes appropriate direct sum relative homology groups lastly isomorphism given connecting map long exact sequence relative homology referred reindexing appendix thus every following sequence canonical isomorphisms vector spaces canonically isomorphic therefore isomorphism vector spaces appearing two chain complexes established remains show composition isomorphisms commutes differentials sequences consideration precisely must show following commutative diagram timothy clark alexandre tchernev reindexy yreindex commutativity bottom square immediate lemma general fact poset homology commutativity remaining squares straightforward verification turning machinery necessary lemma let write simplify notation write note next consider long exact sequence relative homology regular poset resolutions induced inclusions lemma let ranked poset diagram reindex reindex commutative proof orient face using ordering chain next suppose representative homology class generated image relative cycle since relative cycle must simplicial bound ary map therefore relative cycle reindexing class yields cycle next choose partition timothy clark alexandre tchernev property choice allows write therefore component given hand since cycle relative cycle connecting map long exact sequence homology upon reindexing becomes therefore image equal desired conclusion free resolutions monomial ideals let considered usual multigrading suppose monomial ideal vertices inherits minimal generators following correspondence let cell identify set vertices set lcm regular poset resolutions multidegree defined multidegree monomial clearly multigrading induces multigrading face poset cellular chain complex homogenized usual way produce chain complex rmodules precisely cell let rank one free module mhas multidgree complex differential dim write collection cells whose multidegrees comparable bayer sturmfels give following characterization support free resolutions proposition proposition complex free resolution acyclic multidegrees give details homogenization sequence first consider map map induced multigrading next define sequence free multigraded multigraded homomorphisms multigrading defined element differential sequence multigraded modules defined component takes form set define multigrading differential defined componentwise timothy clark alexandre tchernev established commutative algebra interpretation objects section state motivating result paper theorem let monomial ideal suppose resolution resolution supported regular poset resolution supported proof theorem homogenizations produce chain complex remark using language peeva velasco chain complexes isomorphic frames resolution apply theorem several resolutions monomial ideals corollary taylor scarf lyubeznik resolutions cwposet resolutions proof resolutions supported simplicial complex described mermin hence applying theorem face poset associated simplicial complex case recovers desired resolution recall monomial ideal said stable every monomial monomial max divides eliahou kervaire first gave construction minimal free resolution stable ideal mermin reinterpreted resolution supported regular using correspondence along theorem recover mermin result corollary resolution supported regular proof main result establishes resolution supported admissible symbols theorem therefore applies minimal resolution supported regular whose face poset isomorphic poset admissible symbols another combinatorial object associated monomial ideal also serves source support minimal free resolutions recall monomial ideal set least common multiples minimal generators along considered least common multiple regular poset resolutions empty set ordering given divides homological importance established gasharov peeva welker motivated work class ideals studied first author definition let minimal multigraded free resolution monomial ideal multigraded bases free modules fixed differential property coefficient extension correspondence complete notion may viewed topologically exhibits precise enough combinatorial structure corollary monomial ideal ideal furthermore ideal minimal cellular resolution proof write monomial ideal since open interval homeomorphic sphere every theorem therefore guarantees monomial corresponds exactly one free module minimal free resolution aiming contradiction suppose hence exists basis element basis element coefficient expansion differential covered hence must exist however interval homeomorphic interval homeomorphic since open interval also homeomorphic sphere however homeomorphic sphere dimension homeomorphic sphere dimension indeed order complex proper subcomplex sphere moreover contains complex proper subcomplex timothy clark alexandre tchernev hence must sphere contradiction therefore monomial exist means indeed write regular whose face poset isomorphic cells inherit multidegree directly corresponding monomial note since lattice regular unique cell top dimension property pair cells intersect cell said intersection property applying theorem minimal resolution supported may reinterpreted minimal cellular resolution supported remark fact said monomial ideals whose indeed ideals rigid sense general rigid ideal minimal free resolution unique multigraded basis case cwposet unique regular supports minimal multigraded resolution edge ideals complete bipartite graphs studied visscher class ideals whose closing note property cwposet necessary condition rigidity ideal given velasco whose resolution supported cwcomplex rigid monomial ideal hence admits minimal poset resolution many ideals whose failure topological methods completely encode structure free resolutions allows consider following natural questions subject ongoing research question resolutions supported cwcomplex realized poset resolutions every resolution realized poset resolution positive answer two questions given respectively wood woo clark tchernev references anders posets regular bruhat order european combin dave bayer irena peeva bernd sturmfels monomial resolutions math res lett regular poset resolutions dave bayer bernd sturmfels cellular resolutions reine agnew math ekkehard batzies volkmar welker discrete morse theory cellular resolutions reine angew math timothy clark poset resolutions monomial ideals algebra minimal poset resolution stable ideals progress commu tative algebra combinatorics homology christopher francisco lee klingler sean janet vassilev eds gruyter timothy clark sonja mapes rigid monomial ideals appear commut algebra timothy clark alexandre tchernev monomial ideals posets minimal support progress shalom eliahou michel kervaire minimal resolutions monomial ideals algebra vesselin gasharov irena peeva volkmar welker monomial resolutions math res lett albert lundell stephen weingram topology complexes van nostrand william massey singular homology theory graduate texts mathematics vol new york jeffrey mermin resolution cellular commut algebra jeff mermin three simplicial resolutions progress commutative algebra combinatorics homology christopher francisco lee klingler sean janet vassilev eds gruyter james munkres elements algebraic topology perseus books ryota okazaki kohji yanagawa complexes supporting type resolutions borel fixed ideals irena peeva mauricio velasco frames degenerations monomial ideals trans amer math soc achilleas sinefakopoulos borel fixed ideals generated one degree algebra diana taylor ideals generated monomials thesis university chicago mauricio velasco minimal free resolutions supported algebra daniel visscher minimal free resolutions complete bipartite graph ideals comm algebra woo daniel wood thesis university albany
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finding connected path decompositions polynomial time dariusz dorota feb abstract connected path decomposition simple graph path decomposition subgraph induced connected connected pathwidth minimum width connected path decompositions prove fixed connected pathwidth input graph computed answers open question raised fedor fomin grasta workshop since connected pathwidth equivalent connected monotone node search game keywords connected graph searching connected pathwidth graph searching pathwidth introduction since famous graph minor project robertson seymour started notions treewidth pathwidth received growing interest vast amount results obtained pathwidth informally speaking allows say closely arbitrary graph resembles path concept proved useful designing algorithms various graph problems especially case pathwidth input graph small fixed case quite often variant dynamic programming enumeration progresses along path decomposition input graph turns successful several modifications pathwidth proposed work interested connected variant one requires path decomposition graph satisfies vertices induce connected subgraph version classical pathwidth problem motivated several games including limited edge search node search mixed search precisely computing minimum number searchers needed clean given graph node search game computing node search number equivalent computing connected pathwidth moreover connected path decomposition easily translated corresponding node search strategy cleans vice versa references provide details correspondences connected path decomposition different variants search games see related work lot research done direction obtaining fpt algorithms pathwidth parametrized pathwidth one early algorithm running time first fpt algorithm running time due number constructive fpt algorithms developed algorithms order produce optimal solution approximate thus interest good approximations problems numerous works published direction leading currently fastest algorithm approximation treewidth working time single exponential treewidth exist exact algorithms computing pathwidth whose running times exponential order input graph pathwidth computed space use polynomial space using simple algorithm also faculty faculty electronics telecommunications informatics university technology poland mathematics information science warsaw university technology warsaw poland faster algorithm running time improved recently see experimental approaches pathwidth computation pathwidth known due set minimal forbidden minors obstruction set finite fixed pathwidth however significant difference pathwidth connected pathwidth latter one closed taking minors hence known set minimal forbidden minors connected pathwidth finite also point number results obtained connected pathwidth closely related connected graph edge search including algorithmic computational ones monotonicity structural properties distributed algorithms motivation connectivity constraint pathwidth natural useful graph searching games connectivity cases implied potential applications security constraints may enforce clean safe area connected necessity like distributed online versions problem second motivation comes connections pathwidth connected pathwidth specifically implies graph parameters differ multiplicatively small constant implies approximation algorithm connected pathwidth immediately provides approximation algorithm pathwidth asymptotically approximation ratio may potentially lead obtaining better approximations pathwidth since informally speaking algorithmic search space connected pathwidth graphs much smaller pathwidth hand know algorithm computing connected pathwidth time thus despite smaller algorithmic search space clear two problems algorithmically differ context designing exact algorithms grasta workshop fedor fomin raised open question whether verify polynomial time connected pathwidth given graph fixed constant paper answer question affirmative outline next section recall definition connected pathwidth related terms used work section provides algorithm determining whether connected pathwidth arbitrary input graph fixed integer section contains analysis algorithm correctness running time finish open problems section definitions simple graph set subgraph vertex set edge set denoted called subgraph induced write denote neighborhood defined definition path decomposition simple graph sequence exists iii holds width path decomposition width pathwidth denoted minimum width path decompositions say path decomposition connected subgraph connected connected pathwidth graph denoted cpw minimum width taken connected path decompositions finally connected partial path decomposition graph connected path decomposition subgraph words latter condition require vertex neighbor outside belongs last bag intuitively connected partial path decomposition potentially prefix connected path decomposition also note connected path decomposition connected partial path decomposition algorithm say path decomposition starts first bag single vertex belongs present algorithm deciding whether input graph exists path decomposition width starts generalization consider starting bag decomposition dictated fact algorithm recursive subsequent levels recursion content first bag arbitrary connected path decompositions opposed path decompositions observe interested finding connected path decomposition enough take follows simple observation connected path decomposition sequence also connected path decomposition starts following denote input graph vertices also fixed rest work refers maximum bag size connected path decomposition computed thus algorithm checks whether cpw input graph say subgraph connected component define set set called bottleneck number least observe connected component exactly one subset states potential state mean triple fsb consisting set subset cardinality chosen every function fsb chosen every additionally require fsb fsb observe set may chosen ways number choices every number chosen ways function fsb chosen ways therefore number potential states polynomial potential state fsb associate following notions bag denote set cover denote set vertices fsb denote subgraph induced cover say two states indistinguishable cover cover bag bag otherwise states distinguishable note two distinguishable states may hold cover cover bag bag let vertex cover neighbor cover argue bag otherwise bag belong bag thus either cover outside follows every vertex cover neighbor cover must belong bag let denote set vertices border proved observation potential state holds border bag say potential state state connected introduce boolean table indexed states state value set true algorithm exists connected path decomposition width starts bag use dynamic programming fill table conclude cpw additional constraint concerning true state cover observe final state exists since fsb cover fsb every however astute reader may notice representation might included connected partial path decompositions could possible find solution even though exists show cpw exists special type connected path decomposition width called structured path decomposition defined later found using algorithm argue table omit structured path decompositions initialize setting true every state cover bag remaining states initialize alse extension rules let introduce total ordering set states say resolve tie arbitrarily dynamic programming process states according ordering fill table using two extension rules step extension jump extension step extension distinguishable states true connected component bag either contains vertex bag neighbor vertex bag border bag cover cover bag bag cover bag set true jump extension distinguishable states fsb gsb true exists bottleneck set every every fsb gsb every exists connected path decomposition width starts set true let present intuitions behind extension rules step extension corresponds connected partial path decomposition bag corresponds connected partial path decomposition bag extend adding single bag namely bag jump extension state corresponds connected partial path decomposition state corresponds connected partial path decomposition path decomposition graph induced obtained concatenating path decompositions analysis let start introducing definitions additional notation let path decomposition say connected subgraph contained interval note definition valid since follows connected subset indices indeed interval contained denote endpoints interval clear context often write shortly set connected path decomposition say lemma gives lower bound number lemma every set path decomposition proof consider means first consider let minimum index let vertex exists definition recall neighbor vertex since path decomposition must bag containing since first bag appears observe thus suppose note since recall neighbor vertex however since vertex appear bag containing vertex path decomposition thus therefore bag contains least one vertex since observe consider let maximum index analogously previous case observe maximality thus bag contains least one vertex shows number therefore total number completes proof recall set bottleneck thus lemma implies following corollary path decomposition bottleneck following properties hold least one properties justify following definition definition bottleneck let respectively minimum respectively maximum index respectively sbranch note definition implies interval called interval often write shortly clear context bottleneck refine classification say denote set pre post clear context write instead denote set connected components example graph figure illustrates concepts sequence forms connected path decomposition bottleneck set consists two vertices denoted circles example take one eleven one interval equal component vertices neighbor component figure illustration bottleneck set corresponding subgraph say waits interval say path decomposition contained waits component waits main technical tool approach following result concerning structure connected path decompositions lemma exists connected path decomposition starts also connected path decomposition width width starts every bottleneck define cmin minimum index size set minimum also set set let set connected path decompositions set bottlenecks define function given transforms connected path decomposition width width simplicity notation cmin cmin whenever clear context let hlin ordered according first occurrences hlin let words sum lengths intervals contained path decomposition lin define following sequence since connected path decomposition straightforward observe connected path decomposition denote elements let define sequence follows lin denotes concatenation sequences observe length exactly denote elements define set vertices define path decomposition follows cmin cmin xcmin min min min min min cmin observe obtained modification interval prefix suffix copied without changes see conditions components apart covered bags positions cmin see position cmin see wait steps interval cmin cmin see interval cmin cmin used cover one one order appearance additional two bags without vertices added positions cmin cmin ensure bottleneck cmin see see figure illustrates conversion bottleneck notice interval given cmin cmin possibly different next lemma show necessary properties lemma connected path decomposition starts bottleneck connected path decomposition width width starts proof let recall notation moreover define also recall first want show satisfies conditions definition let edge since path decomposition suppose note means cmin cmin path decomposition transformation cmin cmin new bags inserted components apart wait interval wait figure illustration conversion bottleneck simplification assumed components denoted letters respectively appropriate indices otherwise finally consider case say previous case means vertex appears bag thus implies cmin cmin implies satisfies conditions definition let verify condition iii also satisfied every vertex indices clearly condition satisfied every since parts copied modifications situation similar included bags cmin cmin condition iii follows correctness finally vertex condition iii follows correctness observe show path decomposition connected analogous way next thing show width width let max let width max observe exact copy moreover cmin cmin obtained removal vertices finally cmin cmin min however definition observe therefore width width finally recall starts subset observe contains whole least one vertex specific since conclude therefore starts observe every connected component either contained means waits observation bottleneck path decomposition want define series transformations start arbitrary connected path decomposition transform connected path decomposition larger width every apply bottlenecks order need technical lemmas structure bottlenecks branches lemma bottlenecks exists let every different connected component proof let note vertex adjacent thus vertices belong connected component since true every may contained one connected component conclude see consider vertex assumption hence also vertex thus neighbor every thus consider every vertex adjacent vertex contradiction thus connected component see figure illustration figure illustration lemma bottlenecks subgraphs every different connected component lemma bottlenecks exists exactly one connected component proof let clearly intersects connected component suppose intersection two connected components thus since every connected observe contains vertex since implies number turn corollary however contradicts assumption bottleneck next remark straightforward consequence lemma lemma remark let bottlenecks let connected component ssuch exists exactly one connected component moreover possibly subgraphs figure illustration two cases lemma remark bottlenecks apart connected component may may subgraphs see figure illustration lemma remark say two bottlenecks iii observe ordering definition matters lemma connected path decomposition bottlenecks moreover either proof let connected path decomposition let suppose let assume show either case lemma exists every holds thus recall consider two subcases subcase since observe implies subcase first observe thus contradicts assumption analogously obtain therefore assume observe implies since waits particular since necessary otherwise thus recall waits summing either completes proof case case lemma exists exactly one connected component since observe observe equivalent thus assume connected component may still consider two subcases subcase lemma possibly except components subgraphs every possibly waits particular every wait thus every holds note second condition implies contradicts assumption therefore obtain note shows second claim lemma subcase let connected component holds whose existence guaranteed remark observe hand wait subgraphs wait every every since contained thus since conclude completes proof next lemma show apply series one bottleneck structure obtained previous destroyed subsequent lemma let bottlenecks let connected path decomposition let proof let connected path decomposition let moreover assume first let prove case assume observe lemma obtain lemma exists exactly one connected component observe otherwise would wait contradicts assumption going show every subgraph obtain immediately lemma let let connected component whose existence guaranteed remark recall might might sure contradiction thus remark every subgraph every subgraph conclude changes made transformation concern vertices every connected component apart waits see cases hold well notice prefix suffix copied without changes following lemma crucial path decomposition obtained applying transformation described particular bags added play crucial role ensuring path decomposition returned remains lemma let bottlenecks let connected path decomposition let proof let connected path decomposition let moreover assume connected path decomposition particular implies finally assume case lemma exists however recall applied change structure bags restricted therefore conclude assume construction cmin every either waits obtain case lemma exists exactly one connected component observe otherwise contradicts assumption thus facts observe waits remark exists connected component subgraphs change bags inside one hand assume construction cmin every either waits obtain case lemma exists observe otherwise contradicts assumption let means waits notice impossible construction cmin every either waits obtain ready prove lemma proof lemma let connected path decomposition starts let set bottlenecks define path decomposition following recursive way going prove every induction obviously thesis holds every lemma assume claim holds let every induction assumption lemma lemma lemma every bottleneck note lemma ensures connected path decomposition width width starts finishes proof ready show correctness algorithm prove two steps lemma connected path decomposition width starts true state cover proof suppose connected path decomposition width starts lemma exists connected path decomposition width starts bottleneck lemma set bottlenecks relation forms partial order assuming ties resolved arbitrarily let maximal elements respect partial order note bottleneck assume without loss generality maximal bottlenecks ordered according left endpoints intervals show arrive desired state end argue induction exists state cover bag true since starts clearly holds take assume claim true consider two cases first case suppose hence implies according lemma bottleneck either thus exists state cover bag step extension rule true holds induction hypothesis imply true required second case bottleneck definition set hence preceding index set definition lemma maximality respect partial order bottleneck set either satisfy least contained satisfy depending whether thus exists state cover bag consider jump extension rule constructed set take covered note bottleneck waits interval ensures condition condition holds decomposition exist certified decomposition namely thus true holds induction hypothesis ensures true finally observe cover true thus required state lemma state true connected path decomposition width starts proof proof induction position ordering first let cover notice states smallest according set true initialization step justified considering connected path decomposition consisting single bag proper connected path decomposition graph hand alse extension rules apply states also correct decomposition may start vertex suppose lemma holds states since true value must set one extension rules consider two cases case set step extension consider state bag bag since bag cover bag bag cover therefore cover cover bag implies means hand bag bag cover cover due however since distinguishable bag bag thus inductive assumption set properly exists connected path decomposition starts width bag let bag let claim connected path decomposition indeed cover bag cover consider edge belong cover appear inductive assumption belong bag done finally cover bag cover know border bag previous case suppose contradiction means bag contains vertex cover bag contradiction moreover since width width finally since connected according every connected component either contains vertex adjacent one observe also connected finally immediately obtain induction hypothesis starts justifies setting true case set jump extension let fsb gsb let defined definition jump extension simplify notation set observe since bottleneck thus least one since cover cover thus inductive assumption set properly true exists connected path decomposition width bag every connected path decomposition width contains neighbor claim xjhi denote concatenation appropriate sequences connected path decomposition width first observe cover cover due definition decompositions observe covers exactly cover consider edge vertices belong cover definition appear bag decomposition cover know border therefore vertices appear every bag containing finally observe edges joining vertices different third condition definition path decomposition follows directly definition fact subgraphs observe definition since every xjh width since starts according induction hypothesis finally connected since connected connected set contains neighbor completes proof combining lemmas obtain following corollary corollary algorithm correct value true state cover cpw let prove main result paper theorem every fixed algorithm determining cpw time functions depending time polynomial proof induction first observe connected graph cpw graph cpw caterpillar checked polynomial time assume claim holds run dynamic programming algorithm described section correctness algorithm follows corollary let estimate computational complexity recall total number states total number pairs states pair states check one extension rules applied observe state compute cover bag border polynomial time thus checking step extension applied also done polynomial time consider possible jump extension state state verifying first three conditions clearly done polynomial time check appropriate path decomposition exists calling algorithm recursively initial set inductive assumption done total time bounded functions gives total time complexity functions observe function theorem large particular exponential try optimize interested obtaining algorithm every fixed open problems pointed pathwidth connected pathwidth asymptotically arbitrary graph namely cpw however several open questions regarding complexity exact algorithms connected pathwidth one immediate question natural next step context work whether connected pathwidth fpt respect parameter also known connected pathwidth computed faster time arbitrary graph recall possible pathwidth notion connected pathwidth appeared context games called node search edge search mixed search challenging open question related games whether connected variants belong see details regarding question acknowledgements research partially supported national science centre poland grant number references eyal amir approximation algorithms treewidth algorithmica flocchini fomin fraigniaud nisse santoro thilikos connected graph searching inf fraigniaud santoro thilikos connected internal graph searching technical report technical report upc barcelona micah best arvind gupta dimitrios thilikos dimitris zoros contraction obstructions connected graph searching discrete applied mathematics therese biedl thomas benjamin niedermann martin roman prutkin ignaz rutter using determine pathwidth visibility representations graph drawings graph drawing international symposium bordeaux france september revised selected papers pages blin fraigniaud nisse vial distributed chasing network intruders theor comput hans bodlaender algorithm finding small treewidth siam hans bodlaender drange markus dregi fedor fomin daniel lokshtanov michal pilipczuk algorithm treewidth siam hans bodlaender fedor fomin arie koster dieter kratsch dimitrios thilikos note exact algorithms vertex ordering problems graphs theory comput hans bodlaender ton kloks efficient constructive algorithms pathwidth treewidth graphs algorithms borowiecki dereniowski kuszner distributed graph searching sense direction distributed computing david coudert note integer linear programming formulations linear ordering problems graphs technical report inria universite nice sophia antipolis david coudert dorian mazauric nicolas nisse experimental evaluation algorithm computing pathwidth directed pathwidth acm journal experimental algorithmics dereniowski connected searching weighted trees theor comp dereniowski pathwidth connected pathwidth siam discrete jonathan ellis ivan hal sudborough jonathan turner graph separation search number proc allerton conference communication control computing uriel feige mohammadtaghi hajiaghayi james lee improved approximation algorithms minimum weight vertex separators siam flocchini huang luccio contiguous search hypercube capturing intruder ipdps proc ieee inter parallel distributed processing symposium page washington usa ieee computer society fedor fomin complexity connected search number searchers small open problems grasta workshop graph searching theory applications fedor fomin dimitrios thilikos annotated bibliography guaranteed graph searching theor comput fomin thilikos todinca connected graph searching outerplanar graphs electronic notes disc fraigniaud nisse connected treewidth connected graph searching latin proc latin american symposium theoretical informatics pages valdivia chile fraigniaud nisse monotony properties connected visible graph searching inf martin faster computation combinatorial algorithms international workshop iwoca helsinki finland august proceedings pages ilcinkas nisse soguet cost monotonicity distributed graph searching distributed computing kenta kitsunai yasuaki kobayashi keita komuro hisao tamaki toshihiro tano computing directed pathwidth time parameterized exact computation international symposium ipec ljubljana slovenia september proceedings pages jens lagergren efficient parallel algorithms graphs bounded algorithms nisse connected graph searching chordal graphs discrete applied nisse soguet graph searching advice theor comput bruce reed finding approximate separators computing tree width quickly proceedings annual acm symposium theory computing may victoria british columbia canada pages neil robertson paul seymour graph minors excluding forest comb theory ser neil robertson paul seymour graph minors xiii disjoint paths problem comb theory ser neil robertson paul seymour graph minors wagner conjecture comb theory ser karol suchan yngve villanger computing pathwidth faster parameterized exact computation international workshop iwpec copenhagen denmark september revised selected papers pages yang dyer alspach sweeping graphs large clique number discrete mathematics
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jun interactive algorithms pool stream sivan sabato tom hess department computer science university negev abstract consider interactive algorithms setting streambased setting interactive algorithms observe suggested elements representing actions queries interactively select receive responses algorithms select elements order algorithms observe elements sequence select elements immediately observing assume suggested elements generated independently source distribution ask stream size required emulating pool algorithm given pool size provide algorithms matching lower bounds general pool algorithms pool algorithms show maximal gap two settings exists also special case active learning binary classification introduction interactive algorithms algorithms presented input form suggested elements representing actions queries iteratively select elements getting response selected element reward algorithm applicationspecific function final set selected elements along responses interactive algorithms used many application domains including instance active learning mccallum nigam interactive sensor placement golovin krause summarization singla promotion social networks guillory bilmes specific motivating example consider application elements represent web users algorithm select users present free promotional item selected user response observed behavior user received promotion next link user clicked final reward algorithm depends total amount promotional impact obtained measured function set selected users observed responses note algorithm use responses previous selected users deciding next user select consider two interaction settings interactive algorithms setting setting setting entire set suggested elements provided advance algorithm select elements order instance web promotion example might set users use website extended period time approached promotion setting elements presented algorithm sequence algorithm must decide immediately observing element whether select web promotion example consistent setting users access website sessions promotion must decided immediately user observed setting general weaker setting nonetheless important useful many scenarios possible postpone selection elements instance due storage retrieval constraints timing constraints especially pertinent data stream nature streaming document classification bouguelia spam filtering chu web streams twitter video surveillance loy active sensors krishnamurthy work goal study relationship two important settings settings widely studied many contexts active learning settings studied classic works cohn lewis gale works address mainly setting include instance balcan hanneke dasgupta balcan long sabato munos theoretical results hold equally settings balcan long hanneke yang several algorithms developed setting dasgupta golovin krause golovin hanneke sabato gonen cuong setting also heavily studied various active learning applications tong koller tong chang mitra gosselin cord cebron berthold guo general interactive algorithms also studied setting golovin krause guillory bilmes deshpande streambased settings demaine arlotto streeter golovin golovin note unlike works interactive algorithms streambased setting direct restriction timing selecting elements place restrictions storage space resources study relationship setting setting assume settings suggested elements along hidden responses drawn unknown source distribution ask conditions cost algorithm obtain output distribution given pool algorithm exact emulation advantageous allows direct application methods results developed setting setting especially algorithm succeeds practice analysis unknown limited exact emulation guarantees success transferred stream setting well discrete source distributions algorithm emulated streambased setting simply waiting long enough desired element shows challenge interactive algorithms thus achieve output distribution algorithm observing suggested elements possible clearly many cases desired require less suggested elements could result saving resources time money communication active learning well examples usually assumed cheap usually completely free respects study emulation algorithm two settings first consider fully general case provide stream algorithm emulate given pool algorithm uses uniformly bounded expected number observed elements bound expected number observed elements exponential number selected elements prove lower bound indicates exponential dependence necessary second consider interactive algorithm pool setting provide stream algorithm emulates pool algorithms using repeated careful solutions well known secretary problem dynkin gilbert mosteller ferguson expected number observed elements algorithm linear number selected elements case prove matching lower bound finally show lower bound applies active learning binary classification conclude even setting cases exists significant gap best algorithm best algorithm result generalizes previous observation gonen cal cohn classical active learning algorithm compared pool algorithms paper structured follows section formal definitions notations provided section discusses natural suboptimal solutions section provides algorithm lower bound general case section addresses case pool algorithms section provide lower bound holds active learning binary classification conclude section proofs provided appendix definitions predicate denote indicator function holds zero otherwise integer denote sequence member sequence denote concatenation sequences sequences one set one sequence use denote equality inclusion unordered sets elements let measurable domain elements let measurable domain responses pool interactive algorithm receives input integer pool elements assume response initially hidden denote given denotes pool round selects one elements selected yet receives response yit rounds terminates output set xiq yiq pool algorithm denote selp element selects round pool interacts selp random depend yik denote selp sequence elements selected first rounds pairsp pairsp similarly denote selected elements along responses final output set pairs sequence pairsp assume pairsp measurable assume pool algorithm permutation invariant permutation selp selp randomized output distributions pool drawn lose generality stream interactive algorithm receives input integer assume infinite stream iteration observes may select one following actions nothing select observe terminate termination algorithm outputs subset size set pairs observed denote sels element selects also output set denote sels sequence first elements selects also output set use pairss denote elements along responses output interacting set pairs sequence pairss assume pairss measurable total number elements selected interacting including discarded elements denoted nsel number iterations observed elements terminates denoted niter look stream algorithms emulate pool algorithms define equivalence stream algorithm pool algorithm follows definition let distribution let integer let pool algorithm stream algorithm total variation distance distributions pairsp pairss zero denote marginal unless specified otherwise assume probability observing single zero lose generality since case replaced distribution unif interactive algorithms ignoring second element pair simple equivalent stream algorithms let pool algorithm discrete distribution easy define stream algorithm let value define await alg algorithm algorithm await first iterations observe nothing repeat iteration observe element selp select observe end return set pairs stream algorithm equivalent discrete distribution nsel await however niter await bounded class discrete distributions hand stream algorithm anowait defined alg also equivalent niter await pool algorithm however also nsel anowait two simple approaches demonstrate possible tradeoff number selected elements number iterations emulating pool algorithm algorithm algorithm anowait input pool size pool algorithm iteration select observe return pairs pairsp equivalent algorithm uniform bound expected iterations present stream algorithm agen see alg emulate pool based algorithm using access algorithm emulates general pool algorithm making sure iteration probability selecting element identical conditional probability pool algorithm selecting element conditioned history elements responses selected observed far achieved repeatedly drawing remaining part pool keeping consistent elements already selected use partial pool draw element selected happens observed last algorithm algorithm agen input original pool size label budget pool algorithm repeat draw elements denote pairsp selp select get response end output show agen improves two stream algorithms presented selects exactly elements uniform upper bound expected number iterations source distribution first prove agen indeed emulates poolbased algorithm proof provided appendix theorem pool algorithm distribution integer agen next theorem provides upper bound expected number elements observed agen unlike await upper bound holds uniformly source distributions theorem pool algorithm distribution integer agen nsel niter proof first clearly nsel prove upper bound expected number iterations let denote let selp suppose expected number times steps repeated index inverse probability condition holds condition notation selp selp permutation invariance selp addition every draw selp selp since conditional one elements must selected round therefore probability condition step holds expected number times steps repeated index inverse round elements observed therefore expected number elements observed selection made conditioned unconditional expected number elements observed selection sels set indices denote sels selp selp hence sels follows expected number elements observed selection selection conclude niter completes proof existence agen conclude setting essentially equivalent number observed elements however expected number observed elements exponential next section show exponential dependence avoided general pool algorithms lower bound expected number iterations provide lower bound shows pool algorithm equivalent stream algorithm expected number observed elements least exponential indicates much improvement achieved agen class poolbased algorithms proof involves constructing algorithm last selected element determines identity previously selected elements easy pool setting since algorithm advance knowledge available elements stream setting however requires possibly long wait obtain matching last element stream algorithm allowed select elements different order pool algorithm additional care taken make sure case possible circumvent problem way proof theorem provided appendix theorem integer constant log exist pool algorithm marginal stream algorithm equivalent selects elements niter log pool algorithms agen gives uniform guarantee expected number iterations however guarantee exponential consider restricted class pool algorithms show allows emulation expected number iterations linear common approach designing interactive algorithms employed seung lewis gale tong koller guo greiner golovin guillory bilmes golovin krause gonen cuong define utility function scores element depending history selected elements responses far round algorithm selects element maximizes current utility function consider emulation class algorithms formally interactive pool algorithm defined utility function form score element given history pool algorithm selects round element assigned maximal score utility function given history assume simplicity ties interactive pool algorithm denoted aup defined alg stream algorithm pool algorithms propose stream algorithm aus emulates pool algorithms aup stress attempt maximize value selected elements emulate behavior pool algorithm uses assume specific relationship value utility function reward algorithm instance pool algorithm might empirically successful although analysis fully understood tong koller definition aus uses solution secretary problem dynkin gilbert mosteller ferguson classical formulation algorithm aup input elements budget select xit get yit xit yit end output set pairs lem algorithm sequentially observes stream real numbers selects single number goal algorithm select maximal number select number immediately observed observing numbers assumed numbers stream unknown selected adversary order appearance uniformly random goal select maximal number maximal probability known algorithm task optimally solved simple deterministic algorithm achieving success probability psp satisfies psp optimal algorithm observes first numbers selects next observed number least large first limit given stream size real values say secpr holds optimal solution secretary problem size selects observing stream prefix aus given alg uses repeated applications solution secretary problem retrieve selected elements solution succeeds probability less application might fail identified retrospect case new solution selected approach means aus usually selects elements however expected number selected elements constant factor make sure equivalence holds asu never selects element could pool previous elements selected achieved discarding elements round upper bound expected number observed elements bounds expected number elements discarded way first show aus indeed equivalent aup proof provided appendix theorem utility function distribution integer aus aup following theorem give upper bound expected number selected elements expected number observed elements used aus algorithm asu repeat repeatedly draw elements drawing element denote let secpr select get response end end max end output set pairs theorem utility function distribution integer nsel aus niter aus exp follows theorem expected number selected elements expected number observed elements theorem call full run loop starting step attempt element attempt element elements observed expected number attempts element since attempt run secretary problem success probability psp therefore expected number elements observed selected denote utility function let element added consider probability space defined input stream algorithm let random variables independent denote random variable since depends let since assume ties single positive probability conditioned distributed uniformly hence statistically independent define random variable expected number need drawn get single element elements therefore niter aus element maximizes function independent draws elements conditioned hence also maximizes therefore maximum independent copies hence hence dpq niter aus niter aus dml dml psp dml therefore niter aus exp concludes proof lower bound expected number iterations following lower bound shows expected number observed elements required alg significantly improved emulation general pool algorithms theorem holds stream algorithms select exactly elements alg selects approximately elements conjecture even allowing constant factor element selections one achieve constant factor improvement expected number observed elements proof lower bound follows constructing utility function effect allows one set selected elements interaction pattern forces stream algorithm select order pool algorithm given distribution let set distributions marginal equal proof theorem provided appendix theorem exists pool algorithm marginal stream algorithm equivalent pool algorithm selects elements niter log active learning binary classification active learning binary classification recent works provide relatively tight label complexity bounds hold settings balcan long tight upper lower bounds active learning homogeneous linear separators distributions provided bounds hold setting bound number unlabeled examples hanneke yang tight minimax label complexity bounds active learning provided several classes distributions bounds also hold streambased setting work restriction placed number unlabeled examples results leave open possibility distributions algorithm label complexity algorithm might require significantly fewer unlabeled examples example theorem show indeed case example given integers define includes arbitrary elements define following hypothesis class mod mod essentially least significant bits binary expansion equal binary expansion bits consecutive bits starting bit equal binary expansion theorem let log integers consider example setting log consider defined exist active learning algorithm uses pool unlabeled examples labels distribution consistent uniform marginal probability least hand active learning algorithm guarantee requires least log unlabeled examples expectation proof provided appendix result shows gap streambased settings exists general interactive algorithms also specifically active learning binary classification gap significant large unlabeled examples stream versus required pool previously observed gonen cases specific active learning algorithm halfspaces superior classical algorithm cal cohn theorem shows limitation specifically cal active learning algorithm upper bound theorem pool algorithms applied several deterministic algorithms use utility function golovin krause gonen cuong upper bound shows label budget relatively small gap stream pool settings significant instance consider active learning problem utilitybased pool active learner achieves label complexity close lower bound realizable setting kulkarni log passive learning sample complexity therefore active learner properties needs log unlabeled examples therefore case difference setting setting seen negligible conclusions work studied relationship interactive settings designing algorithms emulate behavior setting proving upper lower bounds stream sizes required emulation results concern mostly case label budget stream algorithm similar identical pool algorithm expect label budget grows smooth improvement expected stream length approach label budget approaches many open problems left work among whether possible emulate utility based pool algorithms linear stream size exactly labels relaxation requirement exact equivalence would perhaps allow using smaller streams acknowledgements work supported part israel science foundation grant references arlotto mossel steele quickest online selection increasing subsequence specified size arxiv preprint balcan long active passive learning linear separators distributions proceedings annual conference computational learning theory colt pages balcan beygelzimer langford agnostic active learning journal computer system sciences bouguelia belaid active learning approach document classification document analysis recognition icdar international conference pages ieee cebron berthold active learning object classification exploration exploitation data mining knowledge discovery chu zinkevich thomas tseng unbiased online active learning data streams proceedings acm sigkdd international conference knowledge discovery data mining pages acm cohn atlas ladner improving generalization active learning machine learning cuong lee adaptive active learning general loss conference uncertainty artificial intelligence dasgupta analysis greedy active learning strategy advances neural information processing systems nips dasgupta consistency nearest neighbor classification selective sampling colt pages demaine indyk mahabadi vakilian streaming communication complexity set cover problem distributed computing pages springer deshpande hellerstein kletenik approximation algorithms stochastic boolean function evaluation stochastic submodular set cover proceedings annual symposium discrete algorithms pages siam dynkin optimum choice instant stopping markov process soviet math dokl volume pages ferguson solved secretary problem statistical science gilbert mosteller recognizing maximum sequence journal american statistical association golovin krause adaptive submodularity theory applications active learning stochastic optimization journal artificial intelligence research golovin faulkner krause online distributed sensor selection proceedings international conference information processing sensor networks pages acm golovin krause ray bayesian active learning noisy observations advances neural information processing systems nips pages gonen sabato efficient active learning halfspaces aggressive approach journal machine learning research gosselin cord active learning methods interactive image retrieval image processing ieee transactions guillory bilmes interactive submodular set cover proceedings international conference machine learning icml pages guo greiner optimistic using mutual information ijcai volume pages guo silins stenius korhonen active information structure analysis full scientific articles two applications biomedical literature review bioinformatics hanneke teaching dimension complexity active learning proceedings twentieth annual conference computational learning theory colt hanneke rates convergence active learning annals statistics hanneke yang minimax analysis active learning journal machine learning research krishnamurthy algorithms optimal scheduling management hidden markov model sensors signal processing ieee transactions kulkarni mitter tsitsiklis active learning using arbitrary binary valued queries machine learning lewis gale sequential algorithm training text classifiers proceedings annual international acm sigir conference research development information retrieval pages new york loy hospedales xiang gong joint active learning computer vision pattern recognition cvpr ieee conference pages june mccallum nigam employing active learning text classification proceedings fifteenth international conference machine learning icml mitra murthy pal probabilistic active support vector learning algorithm pattern analysis machine intelligence ieee transactions sabato munos active regression stratification ghahramani welling cortes lawrence weinberger editors advances neural information processing systems pages sabato sarwate srebro auditing active learning query costs advances neural information processing systems nips seung opper sompolinsky query committee proceedings fifth annual workshop computational learning theory pages acm singla tschiatschek krause noisy submodular maximization via adaptive sampling applications crowdsourced image collection summarization conference artificial intelligence aaai active learning sentiment analysis financial domain information sciences streeter golovin online algorithm maximizing submodular functions advances neural information processing systems pages tong chang support vector machine active learning image retrieval proceedings ninth acm international conference multimedia multimedia pages acm tong koller support vector machine active learning applications text classification journal machine learning research jmlr additional proofs several proofs use following lemma lemma let let independent bernoulli random variables let random integer dependent entire sequence suppose proof minimized constraint therefore assume equality holds let random variable whose value smallest integer let largest integer expectation lower bounded subject cases therefore therefore definition largest integer hence log therefore log log log hence log log elementary calculus shows theorem consider probability space defined infinite sequence generates input stream algorithm independent sequence input pool algorithm denote every pairsp pairsp pairsp pairsp holds pairss show equivalence thus suffices show pairss pairss pairsp pairsp definition pairss pairss pairsp pairsp pairsp pairsp last equality follows since permutation invariant never selects index twice proves equivalence theorem denote set permutations let domain elements assume responses define pool algorithm follows call pool exactly one element pool rest good pool bad pools always selects elements elements good pool denote simplicity single element elements define mapping uniform permutations range equally likely behaves follows let first elements selects last element selects response previous elements otherwise define marginal range uniform probability good pool show lower bound expected number iterations stream algorithm let distribution probability let input proof follow series claims probability good pool given set permutation set first selected elements least emulates good pool selects element selecting elements therefore emulates good pool expected number observed elements selecting last element lower bounded overall expected number lower bounded start claim given set define set permutations follows expectedp number elements smaller let log define permutations first elements according permutation mapped elements ranks denote rank elements ordered value since selp determines choice selp selp good selp good good last inequality follows since selp uniform permutations hoeffding inequality good exp therefore using definition applying union bound get selp selp good completes proof claim turn claim consider stream algorithm consider runs input denote event output equal possible output good pool claim sels words simulating good pool elements selected element show claim note definition source distribution outputs set elements exactly one element output responses output elements probability suppose sels sels since one element output good pool consider running source distribution positive probability first selected elements responses therefore also sels positive probability response last element contradicting pool proves claim show claim completes proof claim conclude sels therefore claim sels sels therefore sels sels let element observed selecting first elements let set selected elements independent bernoulli random variables probability success definition log let number elements observes selecting selecting element lemma assumption theorem statement log hence large enough hence constant log since niter completes claim finalizes proof theorem consider probability space defined independent prove equivalence showing could selected pool algorithm pairsp pairsp pairss pairss given denote distribution generated drawing conditioned depends denote finite sequences pairs optimal secretary problem solution applied sequence succeeds optimal value score indeed selected definition asu pairss pairss argmax given sequence let permutation success optimal secretary problem algorithm depends ordering ranks input sequence hence set permutations argmax depends identity pairs depends order since elements two properties independent therefore argmax argmax therefore pairss pairss argmax argmax argmax argmax pairsp argmax pairsp pairsp pairsp prefix length since equality holds pairss theorem let log let uniform distribution assume pool size includes elements probability least exp consider utility function given history form assigns maximal score given history form assigns maximal score pool includes elements pool algorithm based behaves follows every round selected elements far received response selects round element otherwise selects element let distribution response deterministically zero distribution selects probability least denote response deterministically zero distribution algorithm must select elements probability least show lower bound probability selects order input sequence denote probability event occurs consider random process defined input sequence randomness let random variable smallest round algorithm selects round exists since exists consider distribution define sequence pairs elements order responses determined instead clearly distributed according consider run algorithm parallel run random bits algorithm selects elements sequences selection inclusive selection element include element selected round since selects exactly set probability least therefore hence let number elements observes selecting element observing next element let element observed selecting first elements let independent bernoulli random variables bwi lemma follows expected number iterations selections least log theorem let uniform let event define define let first algorithm achieve required accuracy follows let mod mod holds element selected pool algorithm obtained follows round selected element inductively strategy algorithm finds least significant bit binary expansion round thus use set round labels identified exactly happens probability uniform marginal let distribution uniform marginal labels consistent consider algorithm denote output input let random variable drawn uniformly random let hypothesis chosen uniformly random consider probability space defined run let examples receives labels gets order let let let entropy binary entropy taylor expansion binary entropy around therefore hence psx fano inequality last inequality follows definition exp setting noting log therefore follows psx argument holds round conditioned mod since case labels algorithm queries left needs select equivalent instead moreover mod well since holds every individually conclude every probability least follows probability least hence suppose let number elements observes selecting element observing next element let element observed selecting first elements let independent bernoulli random variables bwt lemma itk follows expected number iterations selections least log
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prime graph question integral group rings groups nov andreas leo margolis abstract study prime graph question integral group rings question reduced almost simple groups result kimmerle konovalov prove prime graph question affirmative answer almost simple groups socle isomorphic psl establishing prime graph question groups composition factors aforementioned form using determine exactly far help method take almost simple groups order divisible different primes introduction let finite group integral group ring comes natural augmentation map map ring homomorphism hence every unit mapped sign every unit lies units augmentation also called normalized units connections properties inspired lot interesting research culminating weiss results group rings nilpotent groups hertweck counterexample isomorphism problem jespers del comprehensive books among others concerning finite subgroups major open question today zassenhaus conjecture also known first zassenhaus conjecture zassenhaus conjecture finite order exist element unit rational group algebra case exist torsion unit elements said rationally conjugate conjecture proven nilpotent groups groups normal sylow subgroup abelian complement groups conjecture known simple groups psl primes see references general conjecture remains widely open first step towards zassenhaus conjecture kimmerle proposed following question prime graph question contains unit order element order different primes ask whether prime graph prime graph group graph whose vertices primes appearing orders elements two different vertices joined edge element order kimmerle probably motivated raise question proved true solvable groups class solution zassenhaus conjecture seems reach prime graph question became even approachable reduction result theorem mathematics subject classification key words phrases integral group ring torsion units projective special linear group prime graph question almost simple groups first author postdoctoral researcher fwo research foundation flanders second author supported marie curie grant project andreas leo margolis theorem kimmerle konovalov prime graph question affirmative answer almost simple images affirmative answer group called almost simple sandwiched simple group automorphism group inn aut case socle prime graph question confirmed frobenius groups psl prime furthermore known hold almost simple groups socle isomorphic one smaller sporadic simple groups work bovdi konovalov collaborators references therein alternating group degree theorem gives together theorem first time positive answer prime graph question finite groups composition factors infinite series simple groups theorem prime graph question affirmative answer almost simple groups socle psl theorem follows proposition theorem previously mentioned results achieved extensively using help method discussed section finite group denote set prime divisors order group called kimmerle konovalov obtained reduction result investigated article almost simple groups seemed promising eight simple groups kind classification finite simple groups help method turned entirely sufficient result completed authors using new method lattice method prime graph question groups settled natural consider question groups much bigger class groups lately prime graphs almost simple groups investigated simple groups first classified independently using classification finite simple groups arithmetic restrictions obtained theorem three potentially infinite series specific simple groups simple groups give rise specific series almost simple groups sum results relevant following proposition proposition shi huppert lempken bugeaud cao mignotte almost simple groups ones listed table page psl either mersenne prime case also prime psl pgl either prime open number theoretical problem whether actually infinitely many groups noted kourovka notebook problem study prime graph question groups divided two parts first part apply help method almost simple groups determine exactly help method suffices answer prime graph question bunch groups straight forward done gap package developed authors purpose instances data collected compact form section however cases left requiring detailed study different reasons arguments given detail analysis starting point part two study lattice method applied many remaining cases results groups part read follows theorem let almost simple group following table shows whether help method sufficient prove prime graph question simple group appearing bold indicates almost simple groups socle help method suffices answer prime graph question isomorphism type appears left column following table help method suffice prove units order prime graph question integral group rings groups group right column given italic parentheses help method sufficient prove psl prime help method sufficient prove psl psl psl psl psl psu psu psu psu psu psu psu psp psp psp psp psl psl psl psl psl psl psl psl psl psl psl psl psl psl psl psl psl psl psl psu psu psl psl psu psp psl psl psl psl psl psl psl psl psl psl pgl psl psl psl psl psu proof given section together reduction theorem obtain corollary let finite group assume almost simple images assume moreover almost simple image appears right column table theorem contains elements orders given parentheses prime graph question affirmative answer class groups turns good benchmark power help method encounter several difficulties limitations applying firstly possibly infinite series handled using generic ordinary brauer characters many cases needed manual computations character tables available gap atlas group representations gap character table library cases use induced characters special tables literature specific arguments one situation encounter limit representation theoretical knowledge namely projective symplectic group psp table known parameter show however value parameter influence ability help method answer prime graph question group article prime graph question proved almost simple groups question among others series psl aut psl psl psl psu using lattice method help method main notion study zassenhaus conjecture related questions partial augmenzg tations let conjugacy class andreas leo margolis called partial augmentation respect sometimes wepdenote also element note normalized unit summing conjugacy classes connection rational conjugacy partial augmentations provided theorem unit order rationally conjugate element conjugacy classes divisors known general torsion unit unless theorem proposition moreover order elements divide order theorem method study zassenhaus conjecture using ordinary characters introduced luthar passi later extended brauer characters hertweck subsequently became known help hertwecklutharpassi name coined konovalov let character algebraically closed field characteristic linearly extended restricted afterwards providing also called character defined elements torsion units whose order divisible see torsion units considered let torsion unit order divisible summing conjugacy classes obtain holds ordinary characters also brauer characters theorem next two paragraphs let always unit order denote fact diagonalizable matrix eigenvalues multiplicities assume one knows induction eigenvalues divisors apart one obtain restrictions possible eigenvalues following way power prime exist case prime power different prime divisors integers assume uaq ubr since uaq ubr simultaneously diagonalizable permutation uaq ubr usually character ring smallest ring containing values elements allows obtain information eigenvalues since lie comparing eigenvalues obtained way one obtains restrictions possible partial augmentations vice versa via discrete fourier inversion may formalized following way denote fixed primitive complex root unity complex root unity case still understood possible eigenvalue via fixed brauer correspondence roots unity denote number theoretical trace field extension multiplicity eigenvalue integer trq trq second sum runs conjugacy classes containing elements order coprime thus know partial augmentations divisors obtain system integral inequalities partial augmentations mostly convenient calculate actual eigenvalues representations formal formulation method given provides algorithm directly implementable computer program done help package written authors programmed computer algebra system gap uses prime graph question integral group rings groups program solve integral inequalities package character table library gap gap atlas group representations provide basis second part current paper give results way easily reproduced using package hand several results especially series groups given section complete proofs notation use following notation indicate certain eigenvalues occur greater multiplicity diagonalizable matrix write indicate eigenvalues multiplicity multiplicity multiplicity conjugacy classes elements fixed order write conjugacy class denote conjugacy class containing powers elements following lemma found remark lemma let finite group torsion unit prime every conjugacy class mod assume order different primes contains elements order using theorem obtain mod mod remark provide explanations understanding help method view criterion rational conjugacy torsion units elements important know partial augmentations torsion unit also powers divisor order reason consider possible partial augmentations elements order partial augmentations divisors since always partial augmentations included say group possesses two conjugacy classes involutions one elements order say none elements order typical tuple possible partial augmentations unit order looks like always list partial augmentations units might equal call tuple possible partial augmentations trivial coincides tuple partial augmentations element theorem case tuple consists numbers speak maximal possible results using help method include results listed far paragraph means torsion unit order conjugacy class consists elements order dividing satisfies help constraints given ordinary characters characters primes dividing satisfies lemma let torsion unit order different prime divisors assume representation character rational valued conjugacy classes elements order dividing method explained uaq ubr integers every fixed multiplicity andreas leo margolis primitive roots unity eigenvalues uaq trace rational hence uaq give situations diagonal form instead diagonal form uaq general much harder compute brauer tables group ordinary character tables reason often importance able exclude prime divisors order candidates whose tables provide new information help method character coincides conjugacy classes ordinary character called liftable surely provide new information theorem theorem character group liftable one need include tables computations case general results section prove general results independent special class groups results turn however useful class groups lemma let finite group normal subgroup prime assume contains exactly one conjugacy class elements order possesses representation field characteristic coprime center kernel units order rationally conjugate elements proof let unit order let conjugacy class elements order contained class elements order done assume classes elements order possible assumptions denote natural projection set representation degree say let character afforded let primitive root unity hence hand sum roots unity hence thus considering basis using obtain exactly one partial augmentations must others apply lemma frequently representations degree refer table along character inflated establish results almost simple groups containing psl need properties groups representation theory collect convenience reader following remark remark let pgl psl set gcd orders elements exactly divisors orders elements exactly divisors group possesses partition cyclic subgroups different cyclic subgroups order trivial intersection conjugacy classes elements order classes fuse order coprime conjugate hgi conjugacy class chapter xii see also prime graph question integral group rings groups outer automorphism group isomorphic outer automorphism group isomorphic group induced projection action frobenius automorphism let generator group group called full automorphism group group semilinearities projective line natural permutation representation projective line elements example thus modulo trivial representation obtain irreducible ordinary representation degree satz character corresponding representation known steinberg character facts given example character values may easily calculated ordinary character tables first computed particular every prime dividing fixed conjugacy class elements order exist characters degree fixed primitive root unity possible moreover order order divides general linear group acts via conjugation lie algebra trace kernel action exactly center giving say character let element order coprime computing eigenvalues find primitive root unity let odd prime divisor different character liftable character liftable mod odd exactly two irreducible ordinary characters whose inductions irreducible induced character since take different values conjugacy classes elements order reduction modulo irreducible implies brauer character also liftable proof following results found propositions proposition hertweck let psl torsion unit following statements hold prime order different rationally conjugate element order divisible order coincides order element affirmative answer generalization hertweck proofs propositions obtain following result pgl proposition let pgl torsion unit order coprime order element prime graph question affirmative answer thus almost simple group socle psl proof zassenhaus conjecture small values proved pgl pgl pgl pgl groups may also easily verified using help let first let order prime different elements order let twist steinberg character andreas leo margolis character degree let representation affording integral takes value classes involutions first assume sum eigenvalues integer since may assume next assume sum eigenvalues integer since let pgl let order coprime let representation described remark character remark may assume odd order primes let elements order respectively since sum exactly three roots unity obtain exactly one others rationally conjugate element applies thus exists primitive root unity since real eigenvalue two primitive roots unity eigenvalues inverses thus exists coprime hence consider gcd basis note primitive root unity coefficient sum respect basis primitive root unity coefficient sum recall denotes sum partial augmentations classes elements order comparing coefficient sums obtain using implies contradiction since odd theorem let almost simple group socle psl prime prime graph question affirmative answer proof alternating symmetric group degree case even zassenhaus conjecture holds alternating group degree prime graph question answered possibilities assume prime graph question integral group rings groups use facts given remark let involution pgl lying psl let automorphism induced frobenius automorphism outer automorphism group generated modulo inner automorphisms isomorphic klein four group see example use names groups given fig pgl hpsl hpsl figure almost simple groups containing psl indices successive groups pgl rrr rrr psl let different primes dividing order contains elements order divide divide moreover pgl contains element order remark contains elements order whenever since commutes exactly elements lying natural subgroup psl note different conjugacy classes psl elements prime order still different classes pgl remark fuse one class powers elements lie particular seen thinking elements diagonalized matrices necessary bigger field conjugating element power note elements prime order already lie clear elements odd order follows involutions end proof lemma let primes discussion suffices prove following units exist order order pgl order order consider cases simultaneously first let odd primes let brauer character degree induced brauer character pgl degree described remark let representation affording show first units order rationally conjugate elements order element order lies pgl twice value thus argue way proof proposition prove rationally conjugate element order let element order exists primitive root unity let conjugacy classes elements order viewing basis using fact sum exactly roots unity obtain rationally conjugate element andreas leo margolis assume unit order let primitive root unity order coprime order prj coprime consider using coefficient sum expressed respect basis exactly hand fact rationally conjugate elements conclude exists coprime coprime prj either sum four primitive roots unity sum two primitive roots unity two primitive roots unity two primitive roots unity comparing corresponding coefficient sum expressed respect sum computed find first case contradiction second case obtain left hand side equation odd right side even exist assume order prime denote conjugacy class elements order let primitive root unity use ordinary character psl degree described remark takes value conjugacy class elements order sum two primitive roots unity recall elements order let character induced computing several multiplicities eigenvalues representation realizing using obtain trq trq trq trq since mod lemma multiplicities imply since otherwise one multiplicities negative since also mod lemma get contradicting assumption assume unit order odd prime denote character obtained inducing sum galois conjugates thus degree order every order using prime graph question integral group rings groups obtain trq trq trq trq trq trq trq trq since mod mod multiplicities simultaneously integers completing proof prime graph question assume next pgl order use character induced character character degree described remark denote conjugacy class involutions lie already class involutions pgl unique class involutions contained class ignored following computations see discussion following fig denote class elements order character containing kernel mapping elements outside lemma rationally conjugate element assume first rationally conjugate element get since normalized implies denoting primitive root unity computing multiplicities certain roots unity eigenvalues representation realizing obtain trq trq trq trq trq trq since contradicts mod mod fact expressions andreas leo margolis assume rationally conjugate elements form get computing multiplicities obtain trq trq trq trq trq trq contradicts mod mod multiplicities finally let let order odd prime denote conjugacy class involutions lie conjugacy classes involutions group unique class involutions classes ignored following computations see discussion following fig let conjugacy classes elements order fixed elements order denote character induced character psl described remark character containing kernel sending elements outside since normalized get fixed root unity consider basis let representation realizing separating cases respectively case rationally conjugate element prove case implies existence coprime since get thus coefficient sum respect moreover eigenvalues deduce yields since normalized obtain thus hand using value expressed basis coefficient sum prime graph question integral group rings groups implying comparing coefficient sum computed means however divisor assumption divisor mod hence also impossible satisfies remark psl known whether integral group ring contains normalized unit order particular groups almost simple groups possible obtain maximal information prime graph question help method give using package reason mostly character brauer tables known yet available character table library gap gap atlas group representations package solve underlying inequalities series groups handled following lemmas lemma let group isomorphic psl psl pgl let different primes element order element order except possibly following cases pgl pgl psl psl pgl proof psl pgl pgl result follows table assume psl pgl let representation affording steinberg character remark positive integer let denote primitive root unity first assume psl proposition prime divisors proposition suffices consider elements order let unit order respectively case trace integral contradicting fact steinberg character takes integral values assume pgl proposition either prime divisors note case number also prime need consider elements order proposition suffices case consider elements order suppose order respectively sum eigenvalues integral andreas leo margolis remark possible partial augmentations normalized torsion units order psl possible partial augmentations normalized torsion units order pgl assume psl order note odd proposition maximal information obtain partial augmentations using help method provided characters given table since character classes written linear combination integers trivial character given ones character table moreover irreducible brauer character characteristic liftable remark table part character table psl let representation corresponding character denote primitive root unity since number smaller number attains maximal value thus using obtain hence minimize value thus hence since mod lemma may changed exactly integers congruent modulo legitimate possibilities calculations character give stronger bounds let pgl odd let order denote conjugacy class elements order conjugacy classes involutions class psl character table given iii maximal information help method extracted ordinary characters given table since irreducible brauer character characteristic liftable remark table part character table pgl prime graph question integral group rings groups using lemma involution rationally conjugate element assume first using fact normalized get arguing case psl get thus hand hence using lemma possibilities fixes also bounds obtained using give stronger restrictions partial augmentations assume rationally conjugate elements using hence via get gives another possible implying using lemma partial augmentations let psl odd number admissible possibilities partial augmentations normalized units order depends number possibilities units order show help method suffice show unit order verifying unit partial augmentations ruled unit would admissible partial augmentations pgl new obstructions arise irreducible characters stay irreducible induced pgl characters listed table table part character table psl primitive root unity let representation affording let fixed primitive root unity assume following eigenvalues normalized unit fulfills obstructions derived help method character fulfilled take images element gal eigenvalues image representation affording character also eliminate possibility andreas leo margolis lemma let almost simple group socle psl units order rationally conjugate elements order restrictions partial augmentations using help method brauer tables using ordinary character table proof let outer automorphism group generated automorphism induced frobenius automorphism involution pgl outside see remark two conjugacy classes elements order fuse one class contains elements order two classes elements order using brauer character called remark defined character induced performing computations proof proposition obtain units order rationally conjugate elements brauer tables available show new information deduced partial augmentations units using brauer characters compared ordinary character table elementary abelian brauer tables characteristic available result follows let contain elements order let field characteristic large enough admit representation let simple also looking brauer table comparing ordinary table obtain character associated could give information would generated simple module since even ordinary representation case ordinary characters suffice show elements order table however pgl possess simple module thus generated certain groups characters need directly accesible gap give expanded arguments lemma let psl units order proof character table contained contains exactly five conjugacy classes elements order two conjugacy classes elements order possesses character given table table part character table psl character may used show units order via package following code gap psl charactertable gap size conjugacyclasses gap positions ordersclassrepresentatives gap chi listwithidenticalentries gap chi gap chi sqrt chi sqrt gap chi classfunction chi gap chi number solutions elements order stored prime graph question integral group rings groups remark let pgl show help method sufficient prove units order five conjugacy classes elements order one class elements order looking character table given see maximal possible information using ordinary character table help method obtained characters given table table part character table pgl denotes fixed root unity assuming rationally conjugate elements partial augmentations class elements order satisfy help constraints imposed characters table seen help package via following gap code gap pgl charactertable gap size conjugacyclasses gap ordersclassrepresentatives positions positions gap chi listwithidenticalentries gap chi gap chi gap chi classfunction chi gap field gap galoisgroup generatorsofgroup gap psi list characters gap psi psi psi size psi psi classfunction psi gap gap concatenation chi psi since every character character liftable remark ordinary table provides maximal information finishes proof number possible partial augmentations elements order given table computed using program normaliz lemma let aut psl psl let different primes element order element order except possibly proof using calculate characters aut psl given table applying lemma linear character psl corresponding obtain involutions rationally conjugate elements let assuming order analogues calculations proof lemma prove andreas leo margolis table parts character table aut psl class stands class elements order dots indicate zeros blank spots entries needed assume recall denotes sum partial augmentations classes elements order using normalized thus denote representation character primitive root unity note hence implying contradicting lemma assume character values get denotes primitive root unity primitive third root unity yields contradiction assume considering character values obtain primitive root unity primitive root unity eigenvalues either multiplied whole block roots unity multiplicity eigenvalue impossible remark show psl aut psl constraints imposed help method allow exactly possible partial augmentations elements order let assume using help package way proof lemma characters table one obtains exactly tuples partial augmentations admissible according help constraints three characters need show maximal possible information available using also modular characters note consider characters decomposition matrices psl principle given get explicitly requires computational effort give arguments let aut psl psl sylow sylow cyclic thus investigate blocks using theory brauer tree algebras ordinary character table prime graph question integral group rings groups determined ordinary characters lie block given prime number conjugacy classes degrees ordinary characters lying block conclude contains two irreducible modular characters corresponding brauer tree edges thus character liftable apart main block contain exactly one irreducible ordinary character number conjugacy classes conclude main block contains six main block consist three irreducible modular characters irreducible ordinary representation deleted permutation representation coming natural permutation action projective plane thus beispiel reduction corresponding lattice trivial constituent implies vertices corresponding trivial character character degree connected brauer tree main block since trivial character always sits end tree tree form restricting characters characters obtain brauer tree main block looks follows conclude possesses one irreducible brauer character lifted possesses four four however fall pairs coinciding conjugacy classes elements order characters however exclude partial augmentation computed characters liftable arguments elaborated main block block possessing one irreducible brauer character namely three comparing character values get following decomposition matrix repeating lines omitted second line corresponds canceled natural permutation module points projective plane action short computations implies simple invariant space spanned vectors containing odd number thus fifth line corresponds steinberg character theorem get since decomposition matrix brought unitriangular form theorem obtain dimensions degrees obtain since determinant cartan matrix power theorem implies done prime also main block contains one irreducible brauer character dimensions representations main block follows apart irreducible brauer character corresponding character main block irreducible brauer character liftable character provide new information possible partial augmentations units order lemma let psp aut psp let different primes normalized units order exist elements order except possibly andreas leo margolis proof groups ordinary character tables one character characteristic available exclude existence normalized units order psp aut psp first let psp assume order note contains two conjugacy classes elements order distinguished sizes centralizers one conjugacy class elements order disprove existence use ordinary character degree uniquely determined values involutions together character coming isomorphism psp values characters given table gap charactertable gap ordersclassrepresentatives gap atlasgroup characteristic dimension gap conjugacyclasses gap list order representative gap phi listwithidenticalentries size irr gap phi brauercharactervalue representative position gap phi position brauercharactervalue representative position gap positions positions sizescentralizers order centralizer representative phi brauercharactervalue representative gap phi classfunction phi gap phi number solutions elements order stored gap chi first irr positions gap chi phi number solutions elements order stored table parts characters psp aut psp psp aut psp assume aut psp order character table available gap via command charactertable similar code one exclude existence units order using character elements order characters elements order relevant values given table note conjugacy classes elements order distinguished structures centralizers particular number involutions contained therein power maps class contains powers elements order remark let psp aut psp help method sufficient decide exist units order first let psp since one conjugacy class elements order need consider brauer characters irreducible brauer characters known every irreducible brauer character liftable theorem excatly two irreducible brauer characters theorem obtained gap via code given called irreducible brauer characters completly known knowledge determined theorem parameter take values constraints prime graph question integral group rings groups weaker thus suffice consider case since show existence solutions five irreducible brauer characters called theorem note liftable characters corresponding symplectic group characters projective symplectic group help restrictions provided characters leave exactly possible partial augmentations elements order thus following code demonstrates possible partial augmentations satisfying help constraints elements order gap charactertable gap irr irr gap irr irr brauer characters gap gap gap gap gap gap gap irr irr irr irr irr irr irr irr irr brauer characters gap gap gap concatenation irr number solutions elements order stored gap concatenation irr number solutions elements order stored aut psp analogues computations demonstrate possible partial augmentations satisfying help constraints irreducible brauer characters derived ordinary table arguments particular irreducible brauer character psp induced irreducible conjugacy classes elements order example give explicit example help package used check information given table gap loadpackage help gap charactertable gap number solutions elements order stored gap irr number solutions elements order stored gap irr number solutions elements order stored gap irr number solutions elements order stored example give example induced characters used package necessary groups specifically prove aut psu elements order gap charactertable gap psu gap automorphismgroup gap allcharactertablenames size size means character table automorphism group psu available gap gap filtered normalsubgroups order order group size generators gap subgroup isomorphic gap charactertablewithstoredgroup gap charactertable gap chi inducedclassfunction irr andreas leo margolis gap chi number solutions elements order stored proof theorem prove theorem give table complete information almost simple groups compact form results easy every simple group contains every subgroup aut isomorphism almost simple groups socle grouped together separated single line groups different socles separated two lines read following way first column contains one names group ordinary character table group available gap character table library name library given quotes instance character table aut psl obtained gap charactertable second column contains prime graph group third column records orders torsion units need checked obtain positive answer prime graph question given group critical order given biggest group checked case existence units order excluded group subgroup contain elements order order entry appear contain involutions elements order order appear appears since contains elements order column sometimes also contains primes prime appears partial augmentations units order needed obtain maximal restrictions units order obtain units order psl one needs know partial augmentations units order two cases prime appear column although appears factor happen either one conjugacy class elements order group thus partial augmentations clear characters used obtain information units order constant conjugacy classes elements order second case partial augmentations units order influence calculations elements order also marked fourth column possibly infinite series appearing table groups series contain elements order others reason orders appear parentheses edge dotted prime graph fourth column contains one obtain information units order given third column order composite information sufficient obtain maximal available information using help method information following types group handled literature reference given specific order handled reference given order several cases covered general results article case internal reference given explicit computations performed list characters given composite orders characters allow obtain maximal possible information available via help restrictions explained remark mostly characters taken gap character table library denotes ordinary character table character degree table table contains one character degree index omitted restriction obtained units order psl use two ordinary characters charactertable units order file check claimed data http research available website first author prime graph question integral group rings groups character used second character degree table see example use information implementation help method characters form refer brauer characters modulo available gap character table library lower indices read way ordinary characters character used psl refers third character charactertable mod situations characters available gap character table library sufficient obtain maximal possible information induced characters used happens two groups containing psu see example fifth column contains number possible partial augmentations obtained order third column composite orders minimal possible number obtainable using help method case set bold exactly critical cases remaining answer prime graph question units prime order information obtainable characters gap character table library may sometimes sufficient obtain maximal possible restrictions since brauer tables available library cases given number marked star situations however able prove prime graph question reason aim obtaining best possible information table results applied almost simple groups group psl psl psl psl lemma proposition theorem theorem theorem theorem continued lemma psl psl pgl solutions theorem psl psl psl psl characters psl andreas leo margolis group psl psl pgl psl psl psl psl pgl psl psl psl psl psl psl subgroup psl solutions lemma theorem theorem theorem theorem lemma lemma lemma lemma lemma lemma lemma lemma lemma psl psl psl pgl characters psl psl continued prime graph question integral group rings groups group psl pgl characters remark remark lemma proposition remark proposition lemma proposition remark lemma lemma proposition proposition remark lemma lemma proposition lemma proposition lemma lemma lemma psl psl pgl prime psl prime psl psl pgl psl psl psl psl psl psl subgroup psl subgroup psl proposition psl pgl psl subgroup solutions continued andreas leo margolis group psl psl psl pgl psl psl pgl psl psl psl psl solutions lemma lemma lemma lemma lemma remark remark psl psl psl characters psl psl psl psl pgl psl psl continued prime graph question integral group rings groups group psl characters lemma lemma lemma lemma indu indu indu indu psl pgl psl psl psl psu psu psu psu psu psu psu psu psu subgroup psu psu subgroup psu psu psu psu psu psu psu psu solutions psu continued andreas leo margolis group psu psu psu psu psu psu psu psu psu psu psu psu psu lemma lemma psu psu psu psu psu psp psp psu psp solutions psu characters psu psu psu psu psp psp continued prime graph question integral group rings groups group psp characters lemma remark continued psp psp psp psp psp psp psp lemma remark psp psp psp psp solutions andreas leo margolis group characters lemma solutions lemma remark note almost simple groups socle thus listed table namely automorphism group additional prime divisor automorphism groups psl psl degree psl group either appears series psl described proposition note case psl prime follows easily proposition acknowledgements grateful christof sebastian gutsche helping solve linear inequalities computers thank thomas breuer help gap character table library also want thank referee improving presentation article prime graph question integral group rings groups references allen hobby characterization units comm algebra caicedo prime graph question almost simple groups alternating socle internat algebra comput bugeaud cao mignotte simple algebra bosma cannon playoust magma algebra system user language symbolic comput computational algebra number theory london bovdi hertweck zassenhaus conjecture central extensions group theory bruns ichim power pyramid decomposition normaliz symbolic comput bovdi jespers konovalov torsion units integral group rings janko simple groups math comp bovdi konovalov integral group ring first mathieu simple group groups andrews vol london math soc lecture note vol cambridge univ press cambridge bovdi konovalov siciliano integral group ring mathieu simple group rend circ mat palermo margolis help gap package torsion units integral group rings submitted pages help method gap package version http rational conjugacy torsion units integral group rings groups proc edinb math soc press breuer gap character table library version http may gap package burkhardt die zerlegungsmatrizen der gruppen psl algebra caicedo margolis del zassenhaus conjecture groups lond math soc curtis reiner methods representation theory vol pure applied mathematics new york john wiley sons new york applications finite groups orders publication methods representation theory vol wiley classics library john wiley sons new york applications finite groups orders reprint original wileyinterscience publication dickson linear groups exposition galois field theory introduction magnus dover publications new york dipper decomposition numbers finite general linear groups trans amer math soc feit current situation theory finite simple groups actes international des nice tome paris gap group gap groups algorithms programming version http gorenstein finite groups centralizers whose involutions normal canad math huppert blackburn finite groups iii grundlehren der mathematischen wissenschaften fundamental principles mathematical sciences vol york hertweck counterexample isomorphism problem integral group rings ann math torsion units integral group rings algebra colloq partial augmentations brauer character values torsion units group rings pages arxiv zassenhaus conjecture proc indian acad sci math sci hiss number trivial composition factors steinberg module arch math basel huppert lempken simple groups order divisible four primes proc scorina gomel state university hughes pearson group units integral group ring canad math bull huppert endliche gruppen die grundlehren der mathematischen wissenschaften band york andreas leo margolis james decomposition matrices gln proc london math soc jespers del group ring groups volume orders generic constructions units berlin gruyter group ring groups volume structure theorems unit groups berlin gruyter jordan various types linear groups amer math kimmerle prime graph unit group integral group rings finite groups groups rings algebras contemp vol amer math providence kondrat khramtsov finite tetraprimary groups imm uro ran russisch englische version proc steklov inst math suppl kimmerle konovalov recent advances torsion subgroups integral group rings proc groups andrews kimmerle konovalov graph integral group rings finite groups internat algebra comput luthar passi zassenhaus conjecture proc indian acad sci math sci luthar trama zassenhaus conjecture comm algebra mazurov khukhro unsolved problems group theory kourovka notebook english version marciniak ritter sehgal weiss torsion units integral group rings metabelian groups number theory salim kimmerle conjecture integral group rings alternating groups acta math acad paedagog prime graph conjecture integral group rings alternating groups int group theory schur untersuchungen die darstellungen der endlichen gruppen durch gebrochene lineare substitutionen reine angew math simpson frame character tables psl psu canad math shi simple chinese science bull chinese team software package algebraic geometric combinatorial problems linear spaces available version weiss rigidity ann math torsion units integral group rings reine angew math white numbers odd algebra decomposition numbers primes dividing algebra brauer trees comm algebra wilson finite simple groups graduate texts mathematics vol london london wilson parker nickerson bray breuer atlasrep gap interface atlas group representations version http july refereed gap package zassenhaus torsion units finite group rings studies mathematics honor almeida costa instituto alta cultura lisbon vakgroep wiskunde vrije universiteit brussel pleinlaan brussels belgium address abachle departamento facultad universidad murcia murcia spain address
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projected power method efficient algorithm joint alignment pairwise differences yuxin chen emmanuel dec september revised november abstract various applications involve assigning discrete label values collection objects based pairwise noisy data due hence problem computing optimal assignment maximum likelihood assignment becomes intractable first sight paper makes progress towards efficient computation focusing concrete joint alignment problem recovering discrete variables given noisy observations modulo differences mod propose procedure operates lifted space representing distinct label values orthogonal directions attempts optimize quadratic functions hypercubes starting first guess computed via spectral method algorithm successively refines iterates via projected power iterations prove broad class statistical models proposed projected power method makes hence converges maximum likelihood suitable regime numerical experiments carried synthetic real data demonstrate practicality algorithm expect algorithmic framework effective broad range discrete assignment problems introduction nonconvex optimization nonconvex optimization permeates almost fields science engineering applications instance consider structured recovery problem one wishes recover structured inputs noisy samples recovery procedure often involves solving optimization problem maximum likelihood estimation subject objective function measures well candidate fits samples unfortunately program may highly nonconvex depending choices measure well feasible set contrast convex optimization become cornerstone modern algorithm design nonconvex problems general daunting solve part challenges arises existence possibly exponentially many local stationary points fact oftentimes even checking local optimality feasible point proves despite general intractability recent years seen progress nonconvex procedures several classes problems including matrix recovery phase retrieval dictionary learning blind deconvolution empirical risk minimization name example learned several problems kind provably enjoy benign geometric structure sample complexity sufficiently large sense local stationary points except global optimum become saddle department statistics stanford university stanford electrical engineering princeton university princeton department mathematics stanford university stanford department points difficult escape problem solving certain random systems quadratic equations phenomenon arises long number equations sample size exceeds order denoting number unknowns also learned possible minimize certain random associated famous phase retrieval may multiple local minima problems one find reasonably large basin attraction around global solution method converges geometrically fast importantly existence basin often guaranteed even challenging regime minimal sample complexity take phase retrieval problem example basin exists soon sample size order motivates development efficient paradigm consists initialization scheme enter basin followed iterative refinement procedure expected converge within logarithmic number iterations see also related ideas matrix completion present work extend knowledge nonconvex optimization studying class assignment problems represented finite alphabet detailed next subsection unlike aforementioned problems like phase retrieval inherently continuous nature work preoccupied input space discrete already nonconvex start would like contribute understanding possible solve setting joint alignment problem paper primarily focuses following joint discrete alignment problem consider collection variables variable take different possible values namely imagine obtain set pairwise difference samples index set noisy measurement modulo difference incident variables mod example one might obtain set data consistent truth mod goal simultaneously recover based measurements unrecoverable global tackle problem one often led following program maximize subject function evaluates consistent observed sample corresponds candidate solution instance one possibility may mod else program seeks solution maximizes agreement paiwise observations recovery throughout rest paper set whenever joint alignment problem finds applications multiple domains begin binary case deserves special attention reduces graph partitioning problem instance community detection scenario one wishes partition users two clusters variables recover indicate cluster assignments user represents friendship two users allows model example haplotype phasing problem arising computational geometric property alone sufficient ensure rapid convergence algorithm say symmetric implies specifically impossible distinguish sets inputs even obtain perfect measurements pairwise differences mod genomics another example problem separation magnetic resonance imaging precisely dixon imaging crucial step determine image pixel phasor associated field inhomogeneity two possible candidates represented respectively task takes input pairwise cost functions provides information whether pixels see details moving beyond binary case problem motivated need jointly aligning multiple arises various fields imagine sequence images physical instance building molecule represents orientation camera taking ith image variety computer vision tasks reconstruction multiple scenes structural biology applications microscopy rely upon joint alignment images equivalently joint recovery camera orientations associated image practically often easier estimate relative camera orientation pair images using raw features problem boils jointly aggregate pairwise information order improve collection camera pose estimates contributions work propose solve problem via novel nonconvex procedure informally procedure starts lifting variable higher dimensions distinct values represented orthogonal directions encodes measure matrix way representation allows recast constrained quadratic program equivalently constrained principal component analysis pca problem attempt optimization means projected power iterations following initial guess obtained via suitable lowrank factorization procedure proves effective broad family statistical models might interesting many boolean assignment problems beyond joint alignment algorithm projected power method section present nonconvex procedure solve nonconvex problem entails series projected power iterations space follows algorithm termed projected power method ppm matrix representation formulation admits alternative matrix representation often amenable computation begin state represented vector canonical basis vectors addition pair one introduce input matrix encode given possible input combinations take choice example mod else words cyclic permutation matrix obtained circularly shifting identity matrix positions convention take preceding notation enables quadratic form representation notational simplicity stack concatenated vector matrix rnm respectively representing states altogether consequence problem succinctly recast constrained quadratic program maximizez subject representation appealing due simplicity objective function regardless landscape allows one focus quadratic optimization rather optimizing possibly complicated function directly families also lead problem single simple yet important family obtained enforcing global scaling offset specifically solution remains unchanged replaced ali numerical values another important instance family debiased version ldebias follows ldebias essentially removes empirical average block algorithm one interpret quadratic program finding principal component subject certain structural constraints motivates tackle constrained pca problem means power method assistance appropriate regularization enforce structural constraints precisely consider following procedure starts suitable initialization follows update rule scaling parameter represents projection onto standard simplex namely vector rnm projection onto standard simplex particular reduces rounding procedure specifically largest entry strictly larger second largest entry one lim ali azi given denoting index largest entry see fact justification key advantage ppm computational efficiency expensive step iteration lies matrix multiplication completed nearly linear time time log arises fact block circulant compute product using two ffts projection step performed log flops via sortingbased algorithm figure hence much cheaper multiplication given occurs applications one important step towards guaranteeing rapid convergence identify decent initial guess accomplished factorization follows compute best approximation input matrix namely arg min rank lkf represents frobenius norm pick random column set initial guess remark alternatively one take best approximation debiased input matrix ldebias defined computed slightly faster manner remark natural question arises whether algorithm works arbitrary initial point question studied special stochastic block models shows suboptimal conditions critical points correspond truth hence arbitrary initialization works however condition presented therein much stringent optimal threshold moreover unclear whether local algorithm like ppm achieve optimal computation time without proper initialization would interesting future investigation main motivation comes approximate structure input matrix shall shortly see many scenarios data matrix approximately rank samples therefore approximation serves denoised version data expected reveal much information truth factorization step performed efficiently via method orthogonal iteration also called block power method section power iteration consists matrix product form well decomposition matrix matrix product computed log flops assistance ffts whereas decomposition takes time summary power iteration runs time log consequently matrix product constitutes main computational cost log decomposition becomes bottleneck log noteworthy initialization refinement propose make assumptions data model whole algorithm summarized algorithm course question sequence use defer section proposed algorithm based proper initialization followed successive projection onto product simplices new paradigm solving class discrete optimization problems detail next section provably effective family statistical models hand remark exist many algorithms similar flavor tackle generalized eigenproblems including limited sparse pca separation hidden clique problem phase synchronization pca automatic network analysis algorithms variants projected power method combine proper power iterations additional procedures promote sparsity enforce feasibility constraints instance deshpande show simple models pca efficiently computed using generalized projected power method provided cone constraint convex current work adds new instance growing family nonconvex methods throughout standard notion mean exists constant means lim means exists constant means exist constants algorithm projected power method input input matrix scaling factors initialize defined random column best approximation loop defined output index largest entry block statistical models main results section explores performance guarantees projected power method assume obtained via random sampling observation rate pobs included independently probability pobs assumed independent measurement noise addition assume samples independently generated independence noise assumption may hold reality serves starting point develop quantitative theoretical understanding effectiveness projected power method also common assumption literature assumptions mind mle exactly given representing equivalent function candidate solution given outcome key finding ppm much practical computing mle also capable achieving nearly identical statistical accuracy mle variety scenarios proceeding results find convenient introduce block sparsity metric specifically block sparsity vector defined denoted indicator function since one hope recover global offset define misclassification rate normalized block sparsity estimation error modulo global shift mcr min shiftl shiftl shiftl rmn shiftl obtained circularly shifting entries positions additionally let log represent natural logarithm throughout paper random corruption model goal accommodate general class noise models helpful start concrete simple random corruption model mod probability unif else unif uniform distribution term parameter rate since probability observation behaves like random noise carrying information whatsoever model one write log mod log else finding mle hard problem general solved within polynomial time practically one might attempt compute via convex relaxation much expensive ppm apart mathematical simplicity random corruption model somehow corresponds worstcase situation since uniform noise enjoys highest entropy among distributions fixed range thus forming reasonable benchmark practitioners additionally algorithm certainly implemented using formulation mod log log else recommend taking input matrix case easy verify equivalent global scaling offset parameter free hence practically appealing show ppm guaranteed work even rate vanishingly small corresponds scenario almost acquired measurements behave like random noise formal statement theorem consider random corruption model input matrix given fix suppose pobs log sufficiently large constants exists absolute constant probability approaching one scales iterates algorithm obey mcr provided rate log mnpobs remark throughout ith largest singular value fact one often replace usually good choice unless employ debiased version instead typically corresponds direct current component could excessively large addition note computed spectral initialization result result extra computational cost remark seen section stronger version error contraction arises mcr mcr mcr uniform result sense occurs simultaneously obeying mcr regardless preceding iterates statistical dependency particular one hence forms sequence feasible iterates increasing accuracy case iterates become accurate whenever mcr remark contraction rate actually small npobs log fixed condition holds according theorem convergence ground truth expected log iterations together cost order since cyclic permutation matrix shows computational complexity iterative stage log nearly optimal since even reading data likelihood values take order happens time log soon corruption rate exceed mnpobs uncovering remarkable ability ppm tolerate correct dense input errors shall see later section theorem holds long algorithm starts initial guess obeying mcr irrespective whether independent data therefore often suffices run power method constant number iterations initialization stage completed theorem continues hold replace constant flops fixed broader implication algorithm remains successful one adopts initialization enter basin attraction finally result sharp sure capability projected power method statistically optimal revealed following converse result theorem consider random corruption model fixed suppose pobs log sufficiently large constant log mnpobs minimax probability error inf max mcr infimum taken estimators vector representation mentioned binary case bears similarity community detection problem presence two communities arguably popular model community detection stochastic block model sbm two vertices within cluster resp across different clusters connected edge probability resp asymptotic limits exact partial recovery extensively studied note however primary focus community detection lies sparse regime logarithmic sparse regime log contrast joint alignment problem measurements often considerably denser however theoretical results cover dense regime facilitate comparison consider case sbm reduces random corruption model one easily verify obs limit log derive matches recovery threshold given theorem proposition general noise models theoretical guarantees develop random corruption model special instances set general results subsection cover far general class noise models ind mod additive noise random variables supported follows define distribution instance random corruption model special case noise distribution simplify notation set mod throughout paper unless otherwise noted take restrict attention class symmetric noise distributions obeying largely simplifies exposition theorem continues hold replace constant model studied sbm vertices vertices belonging cluster therefore threshold chacterization proposition read exp applied setting note key metrics feasibility accurate recovery necessarily depends noise distribution precisely distinguishability output distributions given distinct inputs particular distributions like emphasize represents distribution conditional alternatively also distribution noise given write mod interchangeably whenever clear context adopt cyclic notation quantity taking form would like quantify distinguishability distributions via distance metric one candidate divergence defined log plays important role main theory performance guarantees proceed main findings simplify matters shall concern primarily kind noise distributions obeying following assumption assumption miny bounded away remark one replace miny miny assumption however allowed scale case section prefactor dropped words assumption ensures noise density exceedingly lower average density point reason introduce assumption begin enables preclude case entries equivalently wild instance log resulting computational instability reason simplify analysis exposition slightly making easier readers note however assumption crucial dropped means slight modification algorithm detailed later another assumption would like introduce subtle assumption klmax bounded klmin min klmax max roughly speaking assumption states mutual distances possible output distributions lie within reasonable dynamic range one find pair considerably separated pairs alternatively understood variation ratio show later often governed divergence two corresponding distributions point view assumption tells submatrix significantly volatile remaining parts often leads enhanced stability computing power iteration assumptions place positioned state main result hard see theorem immediate consequence following theorem theorem fix assume pobs log sufficiently large constants assumptions exist absolute constants probability tending one scales iterates algorithm input matrix obey mcr provided klmin theorem log npobs remains valid replace constant remark alternatively theorem stated terms divergence metrics like squared hellinger distance specifically theorem holds minimum squared hellinger distance obeys min log npobs see later lemma klmin justifies equivalence recovery condition takes form minimum divergence criterion consistent understanding hardness exact recovery often arises differentiating minimally separated output distributions within log projected power iterations ppm returns estimate absolutely error soon minimum divergence exceeds threshold threshold remarkably small pobs large equivalently many pairwise measurements available theorem accommodates broad class noise models highlight examples illustrate generality begin random corruption model belongs class klmax beyond simple model list two important families satisfy assumption receive broad practical interest list however means exhaustive class distributions obey klmin klmin says output distributions closest two corresponding inputs minimally separated class unimodal distributions satisfy says likelihood decays distance truth increases lemma fix suppose assumption holds noise distribution satisfying either obeys assumption proof see appendix algorithm works pause gain insights algorithm particular minimum divergence emerged key metric without loss generality assume simplify presentation recall algorithm attempts find constrained principal component enable successful recovery one would naturally hope structure data matrix reveal much information truth limit large samples helpful start looking mean given pobs log pobs log pobs log pobs throughout log entropy functional kll thus write pobs denoting circulant matrix easy see largest entries lie main diagonal due fact kll consequently column knowledge largest entries block reveals relative positions across take column example first blocks column attain maximum values entries telling given noisy nature acquired data one would need ensure true structure stands noise hinges upon understanding serve reasonably good proxy projected power iterations since interested identifying largest entries signal contained essentially mean separation largest second largest size pobs min log log pobs min kll pobs klmin total signal strength thus given npobs klmin addition variance measurement bounded max log log max kll klmax inequality demonstrated later lemma semicircle law perturbation controlled npobs klmax spectral norm exceed size signal namely npobs klmin npobs klmax condition reduces klmin npobs assumption consistent theorem logarithmic factor optimality preceding performance guarantee turns information theoretically optimal asymptotic regime fact divergence threshold given theorem arbitrarily close information limit level every procedure bound fail minimax sense formalize finding converse result theorem fix let sequence probability measures supported finite set inf bounded away suppose exists min klmin sufficiently large pobs log sufficiently large constant klmin log npobs minimax probability error inf max mcr infimum possible estimators vector representation usual extension removing assumption return assumption mentioned exceedingly small might result unstable loglikelihoods suggests regularize data running algorithm end one alternative introduce little entropy samples regularize noise density namely add small level random noise yield probability ind unif else appropriate small constant distribution new data given thus given effectively bumps min min propose run algorithm using new data leading following performance guarantee theorem take sufficiently small constant suppose algorithm operates upon theorem holds without assumption proof see appendix extension case far study focused case alphabet size scale however shortage situations large treated fixed constant encouraging news algorithm appears surprisingly competitive case well begin random corruption model analysis developed fixed immediately applies theorem suppose log poly pobs sufficiently large constant theorem continues hold probability least long replaced npobs universal constant arbitrary replaced constant theorem continues hold replace constant main message theorem error correction capability proposed method improves number unknowns grows quantitative bound implies successful recovery even overwhelming fraction measurements corrupted notably exceedingly large theorem might shed light continuous joint alignment problem particular two cases worth emphasizing random corruption model converges following continuous spike model scales mod prob unif else coincides setting studied orthogonal group name synchronization shown leading eigenvector certain data matrix becomes positively correlated truth long pobs addition generalized power equivalent projected gradient provably converges solution nonconvex estimation long size noise threshold comes exact recovery wang prove semidefinite relaxation succeeds long theorem constant threshold irrespective contrast exact recovery performance operates lifted discrete space rather allowing arbitrarily small sufficiently large hand model reminiscent general robust pca problem consists recovering matrix fraction observed entries corrupted learned literature perfect reconstruction feasible tractable even though dominant portion observed entries may suffer random corruption consistent finding theorem preceding spike model probability density measurement experiences impulse around truth variety realistic scenarios however noise density might smooth rather spiky smoothness conditions modeled enforcing rule sharp jump satisfy condition take npobs view theorem sense uncovers resolution estimator smooth noise model constrain input domain unit interval letting represent grid points respectively ppm recover variable resolution npobs notably discrete random corruption model investigated prior literature best theoretical support derived convex programming specifically shown convex relaxation guaranteed work soon npobs comparison stringent recovery condition develop ppm logarithmic factor furthermore theorem immediate consequence general result theorem assume log poly pobs sufficiently large constant replaced ldebias computing initial guess ldebias sufficiently large constant theorem continues hold probability exceeding provided replaced log universal constant constant log npobs log replaced brief interpretations order discussed quantity klmin represents strength signal contrast term maxl log matrix precise controls variability block data kli max log demonstrate proof thus side regarded snr experienced block recovery criterion thus terms lower threshold snr vanishingly small regime considered theorem note general alignment problem studied well although focus therein show stability semidefinite relaxation presence random vertex noise caution however performance guarantees presented subsection general optimal instance shown mle succeeds long log mnpobs regime npobs log performance mle improves increases noteworthy none polynomial algorithms proposed prior works achieves optimal scaling remains seen whether arises due drawback algorithms due existence inherent gap numerical experiments section examines empirical performance projected power method synthetic instances real image data statistical assumptions noise model underlying theory typically hold practical applications shape alignment graph matching numerical experiments show ppm developed based statistical models enjoy favorable performances applied real datasets synthetic experiments begin conduct series monte carlo trials various problem sizes random corruption model specifically vary number unknowns input corruption rate alphabet size observation rate set pobs throughout tuple monte carlo trials conducted trial draw uniformly random generate set measurements according record misclassification rate mcr algorithm mean empirical misclassification rate calculated averaging monte carlo trials fig depicts mean empirical misclassification rate accounts two choices scaling factors particular solid lines locate asymptotic phase transitions exact recovery predicted theory cases empirical phase transition curves come closer analytical prediction problem size increases another noise model studied numerically modified gaussian model specifically set random noise generated way exp controls flatness noise density vary parameters take pobs experiment two choices scaling factors mean misclassification rate ppm reported fig empirical phase transition matches theory well theory theory number variables theory input corruption rate theory theory theory number variables number variables number variables number variables input corruption rate input corruption rate input corruption rate input corruption rate input corruption rate number variables theory theory number variables standard dev gaussian density standard dev gaussian density theory theory number variables theory number variables figure empirical mean misclassification rate algorithm modified gaussian model number variables number variables theory number variables standard dev gaussian density standard dev gaussian density standard dev gaussian density standard dev gaussian density figure empirical mean misclassification rate algorithm random corruption model cumulative distribution cumulative distribution figure performance ppm chair dataset shapes left first input shapes right first shapes alignment ppm sdp absolute angular estimation error ppm sdp absolute angular estimation error figure cumulative distributions absolute angular estimation errors left plane dataset right chair dataset joint shape alignment next return motivating joint work validate applicability ppm two datasets drawn shapenet repository chair dataset plane dataset specifically shapes taken dataset randomly sample points shape input features shape rotated plane random continuous angle since shapes datasets high quality low noise perturb shape data adding independent gaussian noise coordinate point use perturbed data inputs makes task challenging instance resulting snr chair dataset around since mean square values coordinate samples apply projected power method discretize angular domain points represents angle following procedure adopted compute pairwise cost using distance metric precise set average squared distance samples ith jth shapes rotated respectively pairwise cost functions widely used computer graphics vision one regard assuming average distance follows gaussian distribution careful readers might remark specified experiment practically oftentimes access pairwise functions rather fortunately need run algorithm proxy fig shows first representative shapes joint alignment chair dataset one see shapes aligned reasonably good manner quantitatively fig displays cumulative distributions absolute angular estimation errors datasets also reported fig performance semidefinite programming sdp matchlift algorithm presented note angular errors measured distance angles hence somewhat continuous see ppm resp estimates https plane resp chair dataset error lower proportion resp sdp formulation recall resolution discretization would mean estimates error less sense perfect recoveries computationally takes around seconds run ppm sdp implemented using alternating direction method multipliers admm runs seconds experiments carried macbook pro equipped ghz intel core memory joint graph matching ppm applicable combinatorial problems beyond joint alignment present example called joint graph matching consider collection images containing feature points suppose exists correspondence feature points pair images many algorithms able compute feature correspondence points two images joint matching problem concerns recovery collection globally consistent feature matches given noisy pairwise matches put mathematically one think ground truth permutation matrices representing feature mapping image reference true feature correspondence ith jth images represented provided pairwise matches features two images encoded noisy version goal recover global set pairwise observations see detailed problem formulations well theoretical guarantees convex relaxation problem differs joint alignment ground truth permutation matrix light make two modifications algorithm maintain iterates zit matrices replace projects zit set permutation matrices via algorithm corresponds hard rounding power iterations initial guess taken projection random column block approximation first apply ppm two benchmark image datasets cmu house consisting images house cmu hotel consisting images hotel image contains feature points labeled consistently across images initial pairwise matches obtained algorithm mismatching rates resp house resp hotel dataset algorithm allows lower mismatching rate resp house resp hotel representative results dataset depicted fig next turn three shape datasets hand dataset containing shapes fourleg dataset containing shapes human dataset containing shapes drawn collection set feature points hand fourleg human datasets respectively follow shape sampling pairwise matching procedures described evaluate matching performance report fraction output matches whose normalized geodesic errors see threshold ranging sake comparisons plot fig quality initial matches matches returned projected power method well matches returned semidefinite relaxation computation runtime reported table numerical results demonstrate projected power method significantly faster sdp achieving joint matching performance competitive sdp preliminaries notation starting section turn attention analyses main results proceeding gather preliminary facts notations useful throughout http http https initial pairwise matches cmu house optimized matches cmu house initial pairwise matches cmu hotel optimized matches cmu hotel figure comparisons input matches outputs ppm cmu house hotel datasets representative images shown dataset yellow dots refer manually labeled feature points green resp red lines represent set matches consistent resp inconsistent ground truth fraction correspondences fraction correspondences inputs output projected power method output sdp inputs output projected power method output sdp fraction correspondences geodesic error threshold inputs output projected power method output sdp geodesic error threshold geodesic error threshold hand fourleg human figure fraction correspondences whose normalized geodesic errors smaller threshold sdp ppm hand sec sec fourleg sec sec human sec sec table runtime sdp implemented using admm ppm benchmark datasets carried macbook pro equipped ghz memory draw graphs page figure illustration facts standard simplex vector obeying one sufficiently large projection onto standard simplex page firstly algorithm involves projection onto standard simplex light single several elementary facts concerning follows throughout kak norm vector fact suppose obeys kak proof feasibility condition requires therefore easy check hence fact vector value one proof hence arg arg fact vector let largest second largest entries respectively suppose proof convexity since see words fact claims global offset alter projection fact reveals large scaling factor results sufficient separation largest entry remaining ones see fig graphical illustration properties likelihood ratios next study ratio statistics first result makes connection divergence properties ratio two distributions supported total variation distance defined lemma consider two probability distributions finite set log min min addition log hold proof see appendix particular small one almost attains equality stated lemma consider two probability distributions finite set bounded away zero one log functions satisfying universal constant proof see appendix block random matrices additionally data matrix assumed independent blocks thus crucial control fluctuation random block matrices following lemma proves useful lemma let random symmetric block matrix independently generated suppose poly maxi kmi pobs pobs log probability exceeding npobs proof see appendix lemma immediately leads upper estimate fluctuations ldebias lemma suppose poly define log log pobs log debias matrices given respectively satisfy probability exceeding debias debias log npobs proof see appendix notation vector rmn denote jth component matrix matrix kronecker product defined iterative stage establish performance guarantees algorithm reverse order specifically demonstrate section iterative refinement stage achieves exact recovery provided initial guess reasonably close truth analysis initialization deferred section error contraction section mainly consists establishing following claim concerns error contraction iterative refinement presence appropriate initial guess theorem conditions theorem theorem exist absolute constants probability exceeding min holds simultaneously klmin min kxk klmax provided npobs klmin sufficiently large constant iteration ppm produces accurate estimate long iterates stay within reasonable neighborhood surrounding term neighborhood basin attraction fact initial guess successfully lands within basin subsequent iterates never jump see observe obeying klmin kxk klmax klmin klmax inequality implies error contraction moreover since kxk one min min kxk kxk klmin klmax numerical constant arbitrary replaced constant precluding possibility leaves basin result invoking preceding theorem iteratively arrive indicating estimation error reduces zero within logarithmic iterations remark fact contraction rate small scenario considered theorem maxl log npobs case studied theorem furthermore emphasize theorem uniform result namely holds simultaneously within basin regardless whether independent data consequently theory analyses remain valid initialization schemes produce suitable first guess rest section thus devoted establishing theorem proofs two fixed case large almost identical arguments hence shall merge analyses analysis outline key steps proof theorem continuing helpful introduce additional assumptions notation used throughout assume without loss generality shall denote rnm rnm set khk khk kxk min min one key metrics play important role proof following separation measure min defined vector metric important fact projection block onto standard simplex returns correct long sufficiently large aim show vector given obeys klmin index set size bounded away specified later taken collectively fact implies every result min kxk provided scaling factor obeys npobs klmin organize proof claim based size block sparsity leaving two separate regimes deal regime min klmin klmax regime min one take small positive constant independent follows input matrix takes either original form debiased form version tailored random corruption model discussed section regime suppose falls within regime order control decompose terms easier work specifically setting expand allows lower bound separation ith component min mind attention naturally turns controlling kri first quantity admits expression one sees pobs kxj pobs klmin giving formula pobs pobs klmin faced problem estimating kri end make following observation holds uniformly residing within regime lemma consider regime suppose poly pobs log log npobs sufficiently large constant probability exceeding index set kri npobs klmax min klmax cardinality exceeding bounded away min arbitrarily small constant particular fixed assumption holds replaced klmax sufficiently large constant npobs proof see appendix combining lemma preceding bounds obtain pobs klmin klmax min klmax klmin klmin given provided min klmax bounded iii max sufficiently small sufficiently large concludes treatment regime regime turn second regime obeying similarly find convenient decompose lower bound separation measure controlling separately kqi start obtaining uniform control separation components lemma suppose assumption holds pobs log sufficiently large constant fix let sufficiently small constant condition one klmin probability exceeding exp log absolute constants exist constants npobs klmin probability log provided log npobs maxl log klmin log npobs proof see appendix next step comes controlling kqi accomplished using similar argument lemma summarized lemma consider regime lemma continues hold replaced remark notably lemma rely definition regime putting inequality lemma together yields klmax min klmax npobs klmax klmax npobs klmin klmin high probability follows definition regime recall klmax bounded according assumption picking sufficiently small constants applying lemma arrive summarize established claim hence error long fixed condition satisfied conditions hold interestingly one simplify case pobs leading matching condition theorem lemma suppose log poly pobs sufficiently large constant inequalities hold condition addition assumptions proof see appendix choice scaling factor far proved result scaling factor condition given theorem conclude analysis theorem theorem remains convert conditions terms singular value begin follows pobs pobs leading upper estimate pobs kpobs npobs pobs kkk since circulant eigenvalues given exp fact except one simplify kli exp eigenvalues see well leads upper bounds mklmax mklmin mklmin follow assumption addition immediate see remain valid replace ldebias take pobs ldebias instead bound remaining terms side divide two separate cases fixed follows kkk mklmax log combined lemma yields pobs kkk pobs log pobs npobs klmax npobs last inequality follows putting together using assumption get npobs mklmin pobs npobs klmax npobs klmin thus one would satisfy taking sufficiently large poly consider ldebias set ldebias log thus indicating debias pobs according one pobs kkk ldebias log together lemma gives log pobs kkk kldebias ldebias log npobs npobs klmin npobs condition combine derive ldebias mnpobs klmin npobs klmin mnpobs klmin thus justifying long ldebias sufficiently large constant consequences random corruption models obtained qualitative behavior iterative stage general models specialize random corruption model continuing straightforward compute two metrics log log klmin klmax log log log log fixed small hard see klmin klmax taken collectively leads log poly condition reduces npobs coincides fact one also easily verify condition assuming pobs improves slightly upon condition pobs required general theorem next demonstrate algorithm input matrix undergoes trajectory version using avoid confusion shall let lrcm denote matrix set wrcm lrcm discussed constants lrcm ali indicating wircm awi numerical values view fact projection remains unchanged global shift justifies equivalence two input matrices running algorithm finally one would adjust scaling factor accordingly straightforward show scaling factor condition translated npobs input matrix employed observe continues hold long set also verify kkk npobs last inequality follows lemma taken collectively lead npobs pobs npobs npobs condition justifies choice advertised spectral initialization come back assess performance spectral initialization establishing theorem similar definition introduce counterpart distance modulo global offset dist min shiftl theorem fix suppose pobs log assumptions universal constants probability least initial estimate algorithms obeys dist mcr following scenarios random corruption model poly provided given pobs iii general model fixed provided given klmin pobs iii general model poly provided replaced ldebias given log pobs main reason success spectral initialization approximation resp ldebias produce decent estimate resp ldebias discussed resp ldebias reveals structure truth follows first prove result general specialize three choices considered theorem usual suppose without loss generality begin set pobs write pobs pobs first term side rank let best approximation matrix perturbation theory gives kpobs hence triangle inequality yields pobs pobs pobs kkk follows together facts rank rank gives pobs pobs let resp first column resp taken random column straightforward verify pobs pobs expectation randomness picking column see section apply markov inequality deduce probability least dist pobs simplicity presentation shall assume pobs dist pobs shall pay particular attention index set pobs pobs dist pobs consists blocks whose estimation errors much larger average estimation error easily seen set satisfies hence contains blocks comes fact pobs pobs pobs happen holds error block also bounded pobs pobs dist pobs error sufficiently small projection operation recovers truth blocks falling specifically adopting separation measure defined obtain pobs pobs pobs pobs pobs pobs constant would follow fact long pobs taken collectively fact reveals dist mcr claimed result everything boils proving view condition would hold sufficiently small constant finish establish three scenarios considered theorem random corruption model given simple calculation gives follows npobs thus condition would hold assumption general model given observe klmin recall pobs npobs klmax thus necessarily condition assumption iii general model poly replaced ldebias klmin shown log npobs consequence establish assumption finally repeating analyses scaling factor section justifies choice suggested main theorems finishes proof minimax lower bound section proves minimax lower bound claimed theorem done apply random corruption model immediately establishes theorem using exactly calculation section prove theorem suffices analyze maximum likelihood rule minimizes bayesian error probability impose uniform prior possible inputs continuing provide asymptotic estimate tail exponent likelihood ratio test proves crucial bounding probability error decoding lemma let two sequences probability measures fixed finite set minn minn bounded away let triangular array independent random variables define log given constant exp log exp hold long log log exp exp log proof lemma consequence moderate deviation theory see appendix remark asymptotic limits presented lemma correspond gaussian tail meaning sort central limit theorem holds regime shall freeze input consider conditional error probability rule without loss generality assume minimally separated namely klmin follows suppress dependence whenever clear context let xml represent estimate claim suffices prove theorem boundary regime klmin log npobs klmin log npobs fact suppose instead error probability mcr xml tends one regime bounded away one klmin log npobs indicates regime one always add extra noise decrease klmin significantly improving success probability results contradiction moreover bounded away follows lemma log log klmin log npobs obs log consider set small constant first single subset local likelihood ratio restricted samples subgraph induced sufficiently large precisely take log throughout set notational simplicity recall rule favors resp resp log would happen log log thus conditional lower bound probability error log log log log instance let unif probability controlling noise level else last identity comes definition pause remark facilitates analysis begin depends samples lying within subgraph induced thus independent log importantly statistically independent across rely distinct scores log samples allow derive conditional log klmin exp npobs klmin exp exp npobs last line results elementary inequality see holds note according chernoff bound number samples linking obs npobs high probability provided sufficiently large sufficiently log large taken collectively lemma yield log klmin exp npobs thus justifying establish theorem would need show hence lower bounded equivalently exp npobs klmin condition would hold npobs klmin log since two hypotheses one exp npobs klmin exp log exp log log first condition consequence long sufficiently small remains verify second condition npobs log sufficiently large constant connected least pobs vertices high probability meaning number random variables involved sum log concentrates around lemma thus implies log log exp klmin exp klmin constants provided sufficiently large gives rise upper bound log exp klmin arises condition result markov inequality implies probability approaching one equivalently finishes proof theorem discussion developed efficient nonconvex paradigm class discrete assignment problems numerous questions leave open might interesting future investigation instance seen fig fig algorithm returns reasonably good estimates even information limits natural question characterize accuracy algorithm one satisfied approximate solutions addition work assumes index set pairwise samples drawn uniformly random depending application scenarios might encounter measurement patterns modeled random manner example samples might come nearby objects hence sampling pattern might highly local see determine performance algorithm general sampling set moreover functions incorporate data matrix might imperfect study could help understand stability algorithm presence model mismatch returning assumption remark assumption imposed primarily computational concern fact klmax exceedingly large might actually favorable case information theoretic viewpoint indicates hypothesis corresponding klmax much easier preclude compared hypotheses would interesting establish rigorously performance ppm without assumption case becomes suboptimal shall modify algorithm adaptive general class noise models moving beyond joint alignment interested seeing potential benefits ppm discrete problems instance joint alignment problem falls category maximum posteriori map inference discrete markov random field spans numerous applications including segmentation object detection error correcting codes specifically consider discrete variables given set unitary potential functions prior distributions well collection pairwise potential functions likelihood functions graph goal compute map assignment xmap arg max arg max log log similar one introduce vector represent use matrix encode pairwise function log unitary potential function also encoded diagonal matrix log log enables quadratic form representation map estimation maximizez subject expect ppm effective solving many instances map inference problems one key questions amounts finding appropriate initialization allows efficient exploitation unitary prior belief leave future work acknowledgements partially supported nsf via grant math award simons foundation supported award thank qixing huang motivating discussions join image alignment problem chen grateful qixing huang leonidas guibas nan helpful discussions joint graph matching proof theorem concentrate proving case assumption violated set ymax arg maxy ymin arg miny since fixed seen ymax also ymin denoting max ymin dynamic range introducing metric log obtain max kpl max log ymax ymax log log ymin elementary inequality together second pinsker inequality lemma reveals klmax max log making use assumption get klmin klmax log next single element arg maxy log important fixed log element log result would happen ready prove theorem replaced defined one exceeds threshold log npobs long pobs log sufficiently large constant addition since bounded away easy see hence assumption remains valid invoking theorem concludes proof proof lemma suffices prove case klmin suppose maxj view pinsker inequality lemma maxj addition maxj comes eqn consequence assumption follows since fixed combining establishes klmin klmax suppose miny klmin applying pinsker inequality gives max max follows unimodality assumption last line results facts maxj minj taken collectively finish proof proof lemma since log get log else min min min last inequality comes pinsker inequality using inequality well definition obtain log log note min min exactly divergence satisfies proposition substitution concludes proof proof lemma notational simplicity let log log first recall calculation max min last identity follows since bounded away notation means universal constant fact tells thus indicating log log allows one write log arises due difference absorbed prefactor adjusting constant appropriately last line follows since proposition max max max furthermore follows fact log log max maxy maxy thus establishing view seen log proof lemma tempting invoke matrix bernstein inequality analyze random block matrices loses logarithmic factor comparison bound advertised lemma turns would better resort talagrand inequality starting point use standard moment method reduce case independent entries studied specifically standard symmetrization argument section gives kbk obtained inserting standard gaussian variables front order upper bound kbk recognize expanding sum cycles length conditioning gij cyclic notation summands distinct edge visited even number times summands obey gij result kmij gij make observation side equal kmi following argument section setting log one derives log obtain maxi kmi upper bound maxi kmi putting together kbk log last inequality follows since log long combining undoing conditional expectation yield kbk log furthermore markov inequality gives hence median log controlled expected spectral norm obtain concentration results means talagrand inequality see pages introduction proposition talagrand inequality let form product probability measure equipped norm supxl kxl holds define kxl let convex function respect exist absolute constants median exp let represent sample spaces respectively take spectral norm clearly one kmi consequently talagrand inequality together implies probability median log log finally pobs pobs log chernoff bound combined union bound indicates kmi npobs probability turn gives npobs npobs together well assumption pobs log concludes proof proof lemma first step see ldebias ldebias ldebias debiased version given shown recognizing log log fixed constant irrespective main advantage work ldebias entry ldebias written linear combination ratios specifically ldebias log log log log log circulant spectral norm bounded norm column since ldebias ldebias ldebias log log log finish apply lemma arrive debias log debias max kldebias npobs npobs proofs lemma lemma proof lemma view one kri kgi follows look term side separately term side feasibility constraint implies enables express ith block pobs khj defined letting denoting resp lth column resp row see recall feasibility constraint since matrix one sees first term side whereas second term result lth entry bounded magnitude max max klmax klmax klmax last identity arises since setting obtain kfi klmax nklmax nklmax prove min arrive upper bound kpobs npobs klmax npobs klmax min see holds observe constraint implies revealing max khk remains bound term side making use lemma gives kgk kzk log npobs probability let kgk kgk kgk denote order statistics kgn pobs log defined addition constant regime hence min substitution yields log kgk pobs log pobs consequently denote index set blocks satisfying pobs log kgi one ready upper bound kri defined npobs klmax min log npobs klmax min klmax arbitrarily small constant proviso log npobs klmax npobs pobs sufficiently large constant since assumed fixed positive constant condition satisfied pick log npobs sufficiently large constant furthermore order guarantee one would need log pobs sufficiently large constant noteworthy fixed minl bounded away log log log minl kpl klmax comes pinsker inequality thus case would follow klmax npobs proof lemma part shown using similar argument proof lemma specifically definition kqi npobs klmax min last inequality due similar get log npobs log npobs npobs log arises since khk results fact one thus find index set cardinality npobs log side identical putting bounds together repeating argument complete proof proof lemma definition one log sum independent ratio statistics main ingredient control establish following lemma lemma consider two sequences probability distributions finite set generate independent random variables log exp suppose log min varyi log log log sufficiently large constant probability least log min start case fixed pobs log sufficiently large follows chernoff bound npobs probability least small constant taken together lemma set union bound give equivalently probability exceeding exp npobs result would follow probability least exp log long log log npobs remains translate results version based divergence assumption comes fact orderwise equivalent allows rewrite klmin constant addition lemma fact reveal klmin klmin klmin constants result would hold klmin klmin log log npobs log log klmin npobs log sufficiently small hard show consequence klmin obs log npobs finally second part lemma straightforward combining lemma union bound proof lemma taking chernoff bound obtain exp log log exp exp last identity follows since exp log claim follows observing exp taking expectation gives eyi log assumption log bernstein inequality ensures existence constants log varyi log log max log log probability least taken collectively assumption min varyi log log log establishes proof lemma begin immediate consequence assumption next klmax maxl log klmax log log npobs npobs arises together assumption follows soon pobs log establishes second property next turn first condition maxy log holds derive log log log log log log together implies log log log log npobs log obeys maxy log suppose maxy maxj hold preceding inequality maxy assumption finally consider complement regime maxy long follows divergence lower bounded klmin kll log moreover assumption taken collectively assumption ensures max log log allowing one bound log max log combining inequalities obtain log long pobs log log npobs log claimed proof lemma sake conciseness prove shown using argument recognize established demonstrating moderate deviation principle respect log precise main step invoke theorem deduce log constant fact one connect event likelihood ratio test log log reveals equivalent log exp claimed first identity moreover lemma seen regime considered herein leading second identity hence suffices prove order apply theorem need check double indexed sequence satisfies conditions required therein first independence assumption gives sup var sup var second var third follows lemma sup supi log log var min assumption gives log moreover making use bound well assumptions nan log log log derive log conditions place invoke theorem establish references abbe bandeira hall exact recovery stochastic block model ieee transactions information theory abbe sandon community detection general stochastic block models fundamental limits efficient recovery algorithms arxiv preprint berglund johansson kullberg dixon method flexible echo times magnetic resonance medicine bandeira boumal voroninski approach semidefinite programs arising synchronization community detection arxiv preprint bandeira charikar singer zhu multireference alignment using semidefinite programming conference innovations theoretical computer science pages brito dumitriu ganguly hoffman tran recovery rigidity regular stochastic block model symposium discrete algorithms pages blake kohli rother markov random fields vision image processing mit press bhojanapalli neyshabur srebro global optimality local search low rank matrix recovery arxiv preprint boumal nonconvex phase synchronization arxiv preprint bandeira van handel sharp nonasymptotic bounds norm random matrices independent entries arxiv preprint chen candes solving random quadratic systems equations nearly easy solving linear systems communications pure applied mathematics may chang funkhouser guibas hanrahan huang savarese savva song xiao shapenet model repository arxiv preprint chen guibas huang joint object matching via convex relaxation international conference machine learning icml pages chaudhuri graham tsiatas spectral clustering graphs general degrees extended planted partition model journal machine learning research chen jalali sanghavi caramanis matrix recovery errors erasures ieee transactions information theory chen kamath suh tse community recovery graphs locality international conference machine learning june chi kaczmarz method solving quadratic equations ieee signal processing letters cai optimal rates convergence noisy sparse phase retrieval via thresholded wirtinger flow arxiv preprint wright robust principal component analysis journal acm jun candes soltanolkotabi phase retrieval via wirtinger flow theory algorithms information theory ieee transactions graph partitioning via adaptive spectral techniques combinatorics probability computing chin rao stochastic block model community detection sparse graphs spectral algorithm optimal rate recovery arxiv preprint chen suh goldsmith information recovery pairwise measurements ieee transactions information theory chandrasekaran sanghavi parrilo willsky incoherence matrix decomposition siam journal optimization chen sanghavi clustering sparse graphs nips december chen wainwright fast estimation projected gradient descent general statistical algorithmic guarantees arxiv preprint deshpande montanari finding hidden cliques size nearly linear time foundations computational mathematics deshpande montanari richard principal component analysis advances neural information processing systems pages dragomir upper lower bounds terms distance applications inequalities information theory duchi singer chandra efficient projections onto learning high dimensions international conference machine learning pages gao brodzki mukherjee geometry synchronization problems learning group actions arxiv preprint giorgi biasotti paraboschi shape retrieval contest watertight models track shrec competition lee matrix completion spurious local minimum arxiv preprint globerson roughgarden sontag yildirim hard inference structured prediction international conference machine learning pages vershynin community detection sparse networks via grothendieck inequality probability theory related fields pages golub van loan matrix computations volume jhu press ganesh wright candes dense error correction matrices via principal component pursuit international symposium information theory pages hein inverse power method nonlinear eigenproblems applications clustering sparse pca neural information processing systems pages huang chen guibas scalable semidefinite relaxation maximum posterior estimation international conference machine learning huang guibas consistent shape maps via semidefinite programming computer graphics forum hernando kellman haldar liang robust separation presence large field inhomogeneities using graph cut algorithm magnetic resonance medicine huang guibas labeling large shape collections acm transactions graphics hajek achieving exact cluster recovery threshold via semidefinite programming ieee transactions information theory jalali chen sanghavi clustering partially observed graphs via convex optimization international conf machine learning icml javanmard montanari phase transitions semidefinite relaxations proceedings national academy sciences nesterov sepulchre generalized power method sparse principal component analysis journal machine learning research feb jain netrapalli sanghavi matrix completion using alternating minimization acm symposium theory computing pages acm jonker volgenant shortest augmenting path algorithm dense sparse linear assignment problems computing kim mitra diverdi funkhouser exploring collections models using fuzzy correspondences acm transactions graphics tog keshavan montanari matrix completion entries ieee transactions information theory keshavan montanari matrix completion noisy entries journal machine learning research karrer newman stochastic blockmodels community structure networks physical review ling strohmer wei rapid robust reliable blind deconvolution via nonconvex optimization arxiv preprint lee simchowitz jordan recht gradient descent converges minimizers conference learning theory pages liu yue estimation performance convergence rate generalized power method phase synchronization arxiv preprint community detection thresholds weak ramanujan property symposium theory computing pages acm mei bai montanari landscape empirical risk losses arxiv preprint mossel neeman sly consistency thresholds binary symmetric block models arxiv preprint peligrad rio bernstein inequality moderate deviations strong mixing conditions high dimensional probability luminy volume pages institute mathematical statistics wang chi chen implicit regularization nonconvex statistical estimation gradient descent converges linearly phase retrieval matrix completion blind deconvolution arxiv preprint netrapalli jain sanghavi phase retrieval using alternating minimization advances neural information processing systems pages oymak hassibi finding dense clusters via low sparse decomposition arxiv preprint park kyrillidis bhojanapalli caramanis sanghavi provable projected gradient descent class constrained matrix optimization problems arxiv preprint pachauri kondor singh solving matching problem permutation synchronization advanced neural information processing systems nips ravikumar lafferty quadratic programming relaxations metric labeling markov random field map estimation international conference machine learning pages shechtman beck eldar gespar efficient phase retrieval sparse signals ieee transactions signal processing seginer expected norm random matrices combinatorics probability computing shen huang srebro sanghavi normalized spectral map synchronization neural information processing systems nips singer angular synchronization eigenvectors semidefinite programming applied computational harmonic analysis sun luo guaranteed matrix completion via nonconvex factorization symposium foundations computer science focs pages ieee sun wright complete dictionary recovery using nonconvex optimization proceedings international conference machine learning pages sun wright nonconvex problems scary arxiv preprint sun wright geometric analysis phase retrieval sason inequalities arxiv preprint vikalo vishwanath haplotype assembly information theoretic view information theory workshop itw ieee pages ieee talagrand concentration measure isoperimetric inequalities product spaces publications institut des hautes etudes scientifiques tao topics random matrix theory volume ams bookstore boczar soltanolkotabi recht solutions linear matrix equations via procrustes flow arxiv preprint tropp introduction matrix concentration inequalities appear foundations trends machine learning tsybakov introduction nonparametric estimation springer science business media wang giannakis eldar solving systems random quadratic equations via truncated amplitude flow advances neural information processing systems wang sun automated network analysis projected power method international conference information automation pages ieee wang singer exact stable recovery rotations robust synchronization information inference park chen caramanis fast algorithms robust pca via gradient descent advances neural information processing systems yuan zhang truncated power method sparse eigenvalue problems journal machine learning research apr zhang chen bao alley pauly hargreaves vasanawala resolving phase ambiguity dixon imaging using projected power method magnetic resonance medicine zhang chi liang provable phase retrieval outliers median truncated wirtinger flow international conference machine learning june zheng lafferty convergent gradient descent algorithm rank minimization semidefinite programming random linear measurements nips pages zhang liang reshaped wirtinger flow solving quadratic systems equations advances neural information processing systems zhao wang liu nonconvex low rank matrix factorization via inexact first order oracle advances neural information processing systems
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work submitted ieee possible publication copyright may transferred without notice version may longer accessible processing enables bioplausible stdp compound binary synapses xinyu student member ieee vishal saxena member ieee abstract learning mechanisms spike timing dependent plasticity stdp enable agile fast adaptation capability spiking neural network incorporating emerging nanoscale resistive memory nvm devices power consumption highdensity integration capability spiking neural network hardware would result several orders magnitude reduction energy consumption small form factor potentially herald autonomous learning machines however actual memory devices shown intrinsically binary stochastic switching thus impede realization ideal stdp continuous analog values work processing architecture proposed addition novel cmos neuron circuits utilization spike attenuations delays transforms traditionally undesired stochastic behavior binary nvms useful leverage enables biologicallyplausible stdp learning result work paves pathway adopt practical binary emerging nvm devices neuromorphic computing index computing crossbar neuromorphic computing machine learning memristor emerging memory rram silicon neuron spiketiming dependent plasticity stdp spiking neural network introduction neuromorphic computing attracting lot interest recently deep neural networks deep learning quickly shaping modern computing industry human society outstanding performance imaging pattern recognition speech recognition natural language processing autonomous driving flight however running modern cpu gpu fpga platforms enabled advanced complementary metaloxide semiconductor cmos technologies computing machines power hungry still require several orders magnitudes higher energy compared biological analogs well need specialized programming recently xinyu vishal saxena electrical computer engineering department university idaho moscow usa email vsaxena neuromorphic hardware demonstrated impressive power performance implementing convolutional neural networks leveraging massive parallelism spiking neural processing techniques however adjust synaptic weights operation moreover view foreseeable physical limitations cmos based computing integrated circuits ics amenable accommodate neural network comparable level human cortex terms synaptic density power consumption past decade discovery stdp mechanisms emergence nanoscale memory nvm devices opened new avenue towards realization computing prior research suggests stdp used train spiking neural networks snns resistive memory rram synapses without parallelism devices shown consumption change states compact layout footprint hybrid analog integrated vlsi circuits proposed achieve dense integration cmos neurons emerging devices neuromorphic neusoc illustrates neusoc architecture three layer spiking neural network envisioned input layer encodes inputs spatiotemporal spike patterns subsequent layers process inputs using unsupervised semisupervised learning neurons layers connected higher layers using synapses hold weights snn shown nvm crossbar array used form synaptic connections two layers neurons spiking neurons second higher layers implement competitive learning using shared bus mechanism neuron spikes first input pattern inhibits rest neurons layer using combination stdp learning rules synapses locally updated interaction neuron spikes results network learning form weight conductance adaptation synapses presented work submitted ieee possible publication copyright may transferred without notice version may longer accessible output output layer backward spike labels output neurons spikes hidden layer spikes input layer forward spike synaptic crossbar array input data patterns input neurons wta bus synapse neuron neuron input neurons data flow crossbar array hidden neurons output neurons envisioned neuromorphic soc architecture spiking neural network showing input hidden output layers comprised spiking neurons synaptic connections shown one neuron hidden output layers section neural network architecture implemented using rram crossbar memory array cmos neurons wta bus architecture competitive learning single synapse input output neurons adjusts weight using stdp architecture leverages arrays peripheral circuits used memory technology achieve spiking neural network hardware motifs architecture similar existing memory architectures devices arranged dense array input output neurons forming peripheral circuitry laid matching pitch memory array see layout forms repeatable motif scaled deeper snn architectures thus neusoc architecture achieve highest area density aided nanoscale nvm devices compact neurons extension larger architectures using tsvs provides natural pathway scaling high integration density network complexity without resorting overhead incurred asynchronous communication protocols aer ideally weights required effective stdp learning however majority practical smallsized rram devices exhibit abrupt switching behavior consequently limits stable synaptic resolution binary bistable furthermore switching probability switching times typically depend upon voltage applied across device well duration voltage pulse circumvent binary resolution devices compound memristive synapse multiple bistable devices parallel recently proposed emulate analog weights average however show compound synapse yields simple linear stdp learning function work proposes new concept processing added compound synapse along necessary modifications cmos neuron circuit architecture introduces two additional set parameters spike waveforms amplitude modulation additional temporal delays turn enable nonlinear stdp learning functions exponentially shaped window appears biological neural electrophysiology found critical guaranteeing computing stability efficiency theoretical analyses therefore important step towards integrating practical binary nvm devices employed stochastic operating regime realize synapses resolution breakthrough lead practical realization spiking neural networks chip leveraging bistable probabilistic switching specifically demonstrate work proposed compound synapse processing realize resolution plasticity synapses complex stdp learning functions including highly desired exponential one needed learning therefore inclusion even simple dendritic behavior unlocks computing effectiveness stdp without limitations realistic synaptic device behavior novelty work combines bistable rram devices inherently stochastic used parallel simple dendrites simple circuit structures implementing delay attenuation cmos neurons realize higher resolution weights simulation results shown stdp learning behavior similar biological plasticity recreated proposed concept rest article organized follows section reviews stdp learning realization emerging nvm devices section iii introduces proposed dendriticinspired processing architecture respective circuitry section presents experimental setup results finally section estimates architecture discusses current limitations presents future investigations stdp learning emerging nvm devices dependent plasticity stdp mechanism uses relative timing spikes postsynaptic neurons modulate synaptic strength first formulated spiking neural network simulations without considering biologically plausible mechanism observed vivo experiments cortical pyramidal cells many neuroscience experiments conducted later stdp states strength synapse connection modulated according relative timing neuron firing illustrated fig spike pair spike arrives spike results increasing synaptic strength potentiation spike postsynaptic spike results decreasing synaptic strength work submitted ieee possible publication copyright may transferred without notice version may longer accessible vpre vpre vpost synapse stochastic switch vpost abrupt switch vnet vpost vpre vnet vpost vpre veff veff spike timing dependent plasticity stdp stdp learning window shows change synaptic connections function relative timing spikes spike pairings redrawn curves show typical stdp learning function pair spikes applied across synapse create dependent net potential vnet portion veff may cause rram resistance switch abrupt switching stochastic switching typical rram device adapted depression changes synaptic strength plotted function relative arrival timing spike respect spike called stdp function learning window typical stdp function function parameters control shape curve experiments shown relative voltage spike pair fundamental spike timing theoretical studies revealed stdp learning rule enables unsupervised local learning spiking neural networks realizing bayesian algorithm hidden markov models moreover nonlinear stdp learning function exponentially shaped window appears biological neural systems critical guaranteeing computing stability efficiency mentioned earlier emerging nvm devices considered enabler realizing neuromorphic hardware emerging nvms including phase change memory pcm resistive memory rram memory demonstrated realize switching characteristics enabling highly desired advantage small silicon area feature size semiconductor fabrication process operation per switching event cmos compatibility dense crossbar crosspoint arrays integration specially rram devices behave like biological synapses several aspects besides conductance equivalent synaptic strength weight conductance modulated voltage pulses rrams realize stdp directly identical spikes illustrated fig net potential vnet created presynaptic spike late arriving spike produces portion veff causes increase conductance typical bipolar rram contrary spike earlier arriving spike produces veff crosses negative threshold thus causes decrease conductance consequently natural envision integrated vlsi neuromorphic soc built using cmos neurons rram synapses research groups including developed prototype chips demonstrated neural motifs spiking neural networks works conductance modulation capability supports continuous weight change required effective stdp learning however experimental studies suggest nanoscale rrams exhibit stochastic process filament formation well abrupt conductance change filament formed fig illustrates experimental data consecutive switching cycles typical bipolar rram device clearly shows abrupt switching high resistance state hrs low resistance state lrs variations switching threshold voltages variations tend stochastic nature intrinsic stochastic switching rram consequence limits stable synaptic resolution bistable behavior circumvent issues compound memristive synapses multiple bistable devices parallel recently proposed emulate analog weights average concept using several bistable synaptic devices represent synapse proposed concept reinvigorated neuromorphic computing research several earlier works demonstrated analog memristors extremely difficult realize regime contrary binary memristors easier fabricate robust offer larger dynamic range showed connected form array achieve resolution thus surprising researchers turn seek viable solution constructing memristive neural network using binary synapses compound memristive synapse used first time alternative solution analog synapse work submitted ieee possible publication copyright may transferred without notice version may longer accessible vleaky vrefr vrest vthr mleaky foreward spikes cmem neuron vpost vcpr vmem backward spike phase controller spike generator vpre neurons rrams compound synapse dendritic processing bus interface vtch vmode vrst vwtab proposed concept hybrid neural network dendritic processing single layer spiking neural network rram synapses organized crossbar architecture neuron connects neuron several rram synapses parallel spikes dendritic processing modulate rram stdp learning rule schematic diagram proposed cmos neuron dendritic processing architecture parallel attenuators reduce voltage amplitude forward spikes applied parallel rram devices cmos soma fires integration mode parallel rram devices connected current summing point neural network computing simulation demonstrated compound synapse memristors parallel able achieve comparable accuracy neural network mnist handwritten digits dataset time simulation work also suggested possible create biological plausible stdp learning window using compound binary synapse however circuit implementation practical method control shape learning window provided thus order unlock learning snn hardware solution enable especially exponential stdp learning function binary rrams highly desired investigated work iii dendritic processing architecture circuits role active dendrites neural cells computing ability studied shown nonlinear dendritic processing enhances ability learn patterns low resolution synapses recent work neuromorphic computing implemented reservoir readout layer used dendritic concept provided improve classification performance nonlinear dendritic processing circuits used achieve learning analog synapses realized using cmos transistors binary storage however best knowledge processing nanoscale nvms associated spike waveform engineering explored prior work leverage rram compound synapse nonlinear switching probability processing stage applied presynaptic neuron output shown fig neural network implemented using crossbar architecture rram devices electrical synapses synapses interconnect neurons crosspoint configuration use nonlinearity active dendrites instead introduce simple circuit modifications delay spike amplitude variations hence terming processing novelty work combines bistable binary rram devices inherently stochastic used parallel simple dendrites varying delay attenuation realize higher resolution nonlinear stdp learning also use dendrites nanoscale nvm devices natural fit incur area penalty several memory devices laid pitch cmos neurons also separating attenuating buffers dendrite helps reduce resistive loading thus current drive needed drive load neuron presented neuron fig adapted previous experimental demonstration spiking neuron integrated circuits contrast fig original spike waveform neuron runs multiple dendritic branches reaching binary synaptic devices spike amplitudes reduced branches depending attenuation since synaptic devices switching depends voltage pulse duration applied across two terminals spike waveforms produce several different voltage amplitudes cause respective synaptic devices switch different probabilities assuming pulse duration device greater spike amplitude higher probability switching lower spike amplitude lesser probability switching statistical standpoint average conductance compound synapse binary rram devices parallel given voltage written mathematical expression set switching probability ith device work submitted ieee possible publication copyright may transferred without notice version may longer accessible fig stdp learning compound binary resistive synapse simple spike waveform respective double linear stdp window simulated resistance variations spike waveforms dendritic processing voltage amplitude attenuated factors creates parallel spikes respective stdp window dot density presenting probability resistance variations exponential curve fit well panel ron roff rram resistances states respectively generally roff greater ron several orders magnitude thus neglected overall conductance furthermore assuming ron devices simplified expression noting devices operated probabilistic switching regime device switches differently respect time difference thus combined effect parallel rrams could approximate nonlinear function linear functions time therefore proposed dendritic processing provides additional degrees freedom manage amplitudes relative timing spikes appropriate design proposed concept able approach exponential functions see example appendix terms circuit realization dendritic branch implemented adding attenuator cmos soma output using compact circuitry shown fig several possible realizations resistor diodeconnected mosfet ladder followed source follower buffers parallel buffers varying attenuations detail noting spike waveform generated spike generator dendritic processing generates spikes vpd vpd positive negative amplitudes ith dendritic spike attenuation factor ith dendritic branch besides amplitude attenuation time delays could also introduced dentritic processing noting spike waveform generated spike generator dendritic processing generates spikes ron actual rram could stochastic well see example delay dendritic branch cmos soma implemented circuit wta mechanism architecture based single opamp previously presented configuration cmos soma designed provide constant voltage neuron current summing input allows reliable linear spike integration charging membrane capacitor inflowing currents flowing passive resistive synaptic devices configuration cmos soma generates compatible spike drives spikes propagating forward well backward directions high energyefficiency soma circuitry supports local learning several neurons organized group becomes selective input patterns competition lateral inhibition shared wta bus worthwhile note dendritic attenuation delay natural phenomenon intrinsic properties biological neurons biological dendritic tree much higher resistance metal interconnection semiconductor chips yield larger attenuation spike amplitude examples biophysical simulation synaptic potential close center soma whereas potential approximately end dendritic tree attenuation simulation work use attenuation factors less experiments setup stochastic switching rram synapse modeled cumulative probability normal distribution experimentally demonstrated set reset switching probability net potential applied across two terminals rram work submitted ieee possible publication copyright may transferred without notice version may longer accessible fig details compound binary resistive synapse dendritic processing effective potential veff parallel devices versus relative arrival timing spikes levels created compound synapse shows equivalent levels weight stdp leaning dot represents possible resistance value switching probability devices versus relative pre spikes timing asymmetric shape created dendritic processing neuron probability function relative timing compound synapse conductance state state probability curve panel spaced increasing exponential manner vth mean threshold voltage standard deviation distribution demonstration purpose choose positive negative thresholds respectively simulation timescale change conductance normalized convenience simulation step used delay circuits ignored except delay implemented dendritic processing unit described earlier total epochs performed generate stochastic data simulation whole neural circuit dendritic neurons synapses modeled simulated python linux mint environment intel cpu running simulation took approximately seconds stochastic stdp exponential learning function first simulation circuit spike waveform shown fig selected hrht spike waveform constant positive shape linearly rising negative tail demonstrated cmos neuron chip authors positive tail spike amplitude spans time unit negative tail peak amplitude spans time units without dendritic processing pair spike applied compound synapse rram devices parallel scenario attenuating factors simply terminals devices connected one node simulation results stochastic stdp learning shown fig dot represents state conductance compound synapse lrs variation rrams modelled normal distributed random variable standard deviation connecting binary devices parallel simple compound synapse achieves weight resolution stdp leaning however learning curves linear linear best fit due switching probability devices parallel dendritic processing applied spike attenuating factors dendritic attenuators set values linearly spanning produced positive negative levels shown fig postsynaptic spike remained single waveform one shown fig simulation result shown fig discerned stdp learning windows significantly different previous one fig interestingly panel conductance change shows relationship relative time exponential curve fits well another significant difference two schemes flat plateau fig range almost disappeared plateau region narrowed panel close since maximum net potential rram devices portion spike tall positive head overlaps spike negative tail switching probabilities parallel devices also equal design ideally plateau eliminated using narrow positive tail spike waveform corresponds faster switching characteristics rram device provides closer look impact dendritic processing overall stdp learning fig shows effective potential veff parallel devices versus relative arrival times spikes voltage levels created respectively result produce levels conductance change shown fig plot dot represents probability conductance change without taking values probability consideration ignoring lrs variations fig plots individual switching probability devices versus attenuated amplitudes dendrites switching probabilities devices decreased entered status switching probability successively panel shown appendix successive shifting switching probability zero introduces quadratic term compound synapse thus produces approximated exponential curve one also find rrams switching probability curves dense panel due smaller amplitude spike negative tail positive head dense probability curves yield narrow span combined work submitted ieee possible publication copyright may transferred without notice version may longer accessible double rectangurlar spike double linear spike double exponential spike spike stdp window dendritic processing stdp window dendritic processing stdp window dendritic processing stdp window dendritic processing stdp window dendritic processing stdp window dendritic processing stdp window dendritic processing stdp window dendritic processing fig spike waveform shapes respective stdp learning functions without dendritic processing top row spike waveforms various pulse shapes dendritic attenuation processing original spike corresponding waveform largest amplitudes turns group spikes progressively reduced amplitudes middle row shows respective stdp windows created single original spikes compound binary synapse observe visibly linear respect relative timing spike arrival stdp windows bottom row ones created spike group shown top row show clear exponential curves compared counterparts respectively moreover stdp learning function customized tuning attenuation delays dendritic pathways distribution easier saturate especially positive head partially overlaps creates smaller change net potential respect simulation almost half rram devices saturated range stdp another view shown fig depicts probability compound synapse occupy normalized conductance states shows probability curves spaced exponential increments time stochastic stdp learning functions several spike waveform shapes also simulated results illustrated fig figure spike waveforms dendritic processing shown first row waveforms without dendritic processing correspond ones largest swing respective stdp learning functions without dendritic processing shown second row corresponding stdp learning functions dendritic processing shown third row hrht spike waveform used simulation easy realize cmos circuits produces single exponential curve stdp learning function however positive half curve fit follows straight line linear decrease time small exponential decreasing tail far end consequently waveform shapes explored realize widely employed simple rectangular spike produces rectangular stdp learning widow dendritic processing shown fig easy observe switching probability rram devices remains unchanged learning window amplitude spike waveform independent relative timing hence quadratic term introduced design work submitted ieee possible publication copyright may transferred without notice version may longer accessible fig row spike waveform shapes dendritic attenuation delays row respective stdp learning functions incorporating dendritic delays dendritic processing yields stdp learning window plausible measured biological stdp shown fig double sawtooth double exponential spike waveforms simulated results shown fig similar waveform fig dendritic processing introduced nonlinearity stdp learning window due sharp curves spike waveforms produced curves stdp learning functions eliminated plateau approximated function shown fig formulated waveform amplitudes attenuation factors interestingly spike waveform created stdp learning function similar vivo measured biological stdp seen fig waveform dendritic delays used realize stdp response highly faithful biological measurements biologically plausible spike also simulated shown fig dendritic processing introduced nonlinearity stdp learning window waveform well whereas conductance changes also found around region due smooth change positive negative tails waveform energy efficiency primary motivation exploring memristive rrambased spiking neural network potential energy efficiency saving comparing conventional computing paradigms goal could achieved two aspects asynchronized architecture snn memory devices snn spike shape parameters state resistance ron memristive devices roff generally several orders magnitude greater ron thus neglected contribute energy computation one spike event total energy consumption also decided percentage rram devices state spike activity power consumption neurons taking hrht waveform tall thin positive rectangular head short fat negative triangular tail example pulse shape defined case stdp learning function doubleexponential curve desired vlsi implementation sawtooth waveform good choice easier realize circuits compared exponential biologicalplausible spike waveforms incorporating dendritic delay propagation delay another parameter implemented dendritic processing architecture find impact dendritic delay delay used simulation accounts total waveform span shown fig hrht double sawtooth double exponential biological plausible spike waveforms applied time delays produce conductance change probability around dots seen two sides wider distribution dots towards time delays also reduce probability large conductance change due shift pairwise spike peaks problem resolved choosing appropriate discussion amplitude waveform positive negative tail duration respectively energy consumption one spike memristor device resistance ron given work submitted ieee possible publication copyright may transferred without notice version may longer accessible total snn energy consumption one event formulated neuron activity ratio rram ratio snn number synaptic connections number neurons pneuron neuron power spike duration instance alexnet convolutional neural network deep learning built million synaptic connections thousand neurons conservative estimation based spiking neuron chip realization compound binary synapse energy consumption processing one image shown table comparing today advanced gpu nvidia deep learning could provide better noting great room improvement cmos design memristor devices fast stdp demonstrated penitential low neuron activity ratio energy efficiency improvement could reasonable target snn hybrid chip table energy efficiency estimation memristor based snn duration spike amplitude aon state resistance ron single spike energy espk neuron baseline energy eneuron neuron act ratio state rram ratio single event energy esnn images sec watt acceleration ratio gpu conservative medium aggressive aggressive estimation power consumption neuron ron kept unchanged considering steep slowdown cmos process evolution ratio snr fundamentally limited thermal noise however faster spike duration much sparse neuron activity could possible scale spike duration neuron activity could expect energyefficiency improvement limitations future work work proposes dendritic processing architecture provides potential solution implementing biologically plausible stdp realization using cmos neuron circuits stochastic binary rram devices although mathematical analysis hrht waveform proves dendritic attenuation introduces second order approximation exponential function performed terms average conductance instead maximum probability work simulations used demonstrate produced stdp learning functions closely mimic measured biological stdp order provide analytical guidance spike waveform design precise mathematical analysis terms probability conductance state required establish relationship maximum probability relative timing could future theoretical work besides variations lrs noise waveform amplitude also practical parameter considered amplitude noise impact switching rram devices shift device switching probabilities cause switching probability saturate higher end turn switching lower end result amplitude noise create undesired conductance change states skip conductance states simulations show normal distributed amplitude noise standard deviation significantly distorted stdp learning window noise standard deviation less created additional conductance states retaining shaped similar ideal stdp learning windows finally theoretical studies impact stdp learning function learning simple neural networks however detailed study performed neural network practical size multilayer perceptron would interesting understand impact stdp function overall network learning performance including stability learning speed convergence time resulting classification accuracy metrics sensitivity variations stdp function step stdp applied networks like deep convolutional neural networks recurrent neural networks impact stdp function remains completely unknown form bulk future study conclusion proposed compound synapse dendritic processing realizes biologically plausible exponential stdp learning using practically feasible bistable nonvolatile memory devices probabilistic switching potentially create breakthrough significantly compact machine learning hardware spiking neural networks require synaptic plasticity multibit resolution bottleneck taking next leap practical rram memristive devices immediate applications include practical realization deep neural networks compact form factor orders magnitude reduction energy consumption compared gpus digital asics architectural exploration using proposed compound probabilistic synapses help benchmark expected behavior emerging rram devices nanoscale rram devices large resistances help realize lower power consumption work submitted ieee possible publication copyright may transferred without notice version may longer accessible appendix approach exponential stdp curve approximation exponential function exponential function expressed using taylor expansion solved sufficiently small normalizing value one convenience rearranging obtain model stochastic switching rram device linear stochastic switching characteristics shown described equation model dendritic processed spike several spike shapes studied stdp learning possess different levels biological mimicry hardware realization complexity hrht waveform tall thin positive rectangular head short fat negative triangular tail selected formulated spike shape widely used widely used simulation cmos neuromorphic implementations time convenient mathematical analysis proposed dendritic processing applied prespike net potential created ith amplitude waveform step amplitude change average conductance value greatest interest analysis maximum likelihood value conductance versus relative time difference however difficult derive analytic results general case alternative average conductance assumed reasonable approximation without providing explicit proof work derive relationship average conductance relative timing substituting switching probability voltage across device vth minimum threshold voltage constant represents slope switching function last term expressed constant given factor term expressed substituting equation difficult find vth factor second order term positive numbers positive number well thus shown average conductance stdp learning window dendritic processed hrht waveforms approximation exponential learning function references waveforms switching probability less one index waveforms smallest switching probability linear function substituting allocating terms according order get esser convolutional networks fast neuromorphic computing proc natl acad masquelier memristive devices building visual cortex front vol march querlioz zhao dollfus bioinspired networks nanoscale memristive devices combine unsupervised supervised learning approaches international symposium nanoscale architectures nanoarch jeyasingh electronic synapse device based metal oxide resistive switching memory neuromorphic computation ieee transations electron devices vol kuzum jeyasingh lee wong nanoelectronic programmable synapses based phase change materials computing nano vol wang review memristor applications acta autom vol jun kang jun ryoo jeong sohn emulation work submitted ieee possible publication copyright may transferred without notice version may longer accessible dependent plasticity phase change memory neurocomputing vol sengupta azim fong roy torque induced dependent plasticity appl phys vol mandal alexander rajendran jha novel synaptic memory device neuromorphic computing sci vol saxena zhu member saxena zhu homogeneous spiking neuromorphic system pattern recognition ieee emerg sel top circuits vol saxena zhu balagopal cmos spiking neuron neural networks resistive synapses learning ieee trans circuits syst express briefs vol masquelier guyonneau thorpe spike timing dependent plasticity finds start repeating patterns continuous spike trains plos one vol diehl neil binas cook liu pfeiffer fastclassifying spiking deep networks weight threshold balancing int conf neural networks january diehl cook unsupervised learning digit recognition using plasticity front comput vol lee training deep spiking neural networks using backpropagation front vol november kheradpisheh ganjtabesh thorpe masquelier spiking deep neural networks object recognition tavanaei maida spiking convolutional neural network using sparse coding stdp learning neftci augustine paul detorakis random enabling neuromorphic deep learning machines arxiv vol kim programmable resistance switching nanoscale devices nano vol soni stochastic nature resistive switching doped based memory devices appl vol guan wong switching parameter variation metal oxide model corroboration device design strategy ieee trans electron devices vol suri stochastic computing using binary cbram synapses ieee trans electron devices vol gaba sheridan zhou choi stochastic memristive devices computing neuromorphic applications nanoscale vol khiat salaoru prodromakis stochastic switching memristive devices identical initial memory nanoscale res vol bill legenstein compound memristive synapse model statistical learning stdp spiking neural networks front vol december singha muralidharan rajendran analog memristive time dependent learning using discrete nanoscale rram devices international joint conference neural networks ijcnn sprekeler michaelis wiskott slowness objective plasticity plos comput vol jun toyoizumi pfister dependent plasticity mutual information maximization spiking neuron model advances neural information processing systems nips nessler pfeiffer buesing maass bayesian computation emerges generic cortical microcircuits plasticity plos comput vol apr gerstner ritz van hemmen spikes hebbian learning retrieval excitation patterns biol vol markram regulation synaptic efficacy coincidence postsynaptic aps epsps science vol poo synaptic modifications cultured hippocampal neurons dependence spike timing synaptic strength postsynaptic cell type vol abbott nelson abbott synaptic plasticity taming beast nat vol song miller abbott competitive hebbian learning synaptic plasticity nat vol turrigiano nelson rate timing cooperativity jointly determine cortical synaptic plasticity neuron vol poo synaptic modification correlated activity hebb postulate revisited annu rev vol markram gerstner plasticity comprehensive overview front synaptic vol july kappel nessler maass stdp installs circuits online approximation hidden markov model learning plos comput vol chang ebong bhadviya mazumder nanoscale memristor device synapse neuromorphic systems nano vol apr seo analog memory plasticity characteristics nanoscale titanium oxide bilayer resistive switching device nanotechnology vol jun krzysteczko memristive magnetic tunnel junction nanoscopic system adv vol ultrafast synaptic events chalcogenide sci vol synaptic plasticity chalcogenide electronic synapse neuromorphic systems sci vol chen review emerging memory nvm technologies applications solid state vol saxena zhu cmos spiking neuron dense connectivity computing international joint conference neural networks ijcnn prakash hwang multilevel cell storage resistance variability resistive random access memory phys sci vol likharev mayr muckra crossnets high performance neuromorphic architectures cmol circuits ann acad vol poirazi mel impact active dendrites structural plasticity memory capacity neural tissue neuron vol roy banerjee basu liquid state machine dendritically enhanced readout neuromorphic vlsi implementations biomed circuits syst ieee vol roy basu online structural plasticity rule generating better reservoirs neural vol vishal saxena mitkova addressing challenges neuromorphic computing memristive synapses neuromorphic computing workshop architectures models applications spruston pyramidal neurons dendritic structure synaptic nat rev vol gao fang kang wong stochastic learning oxide binary synaptic device neuromorphic computing front vol october nvidia new pascal gpus accelerate inference data center online available https energy scaling advantages resistive memory crossbar based computation application sparse coding front vol seung learning spiking neural networks reinforcement work submitted ieee possible publication copyright may transferred without notice version may longer accessible stochastic synaptic transmission neuron vol
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classifying space artin monoids feb giovanni paolini abstract theorem proved dobrinskaya shows strong connection conjecture artin groups classifying space artin monoids recently ozornova obtained different proof dobrinskaya theorem based application discrete morse theory standard model classifying space artin monoid ozornova work hints deeper connections model salvetti complex complex arises combinatorial study artin groups work show connections actually exist consequence derive yet another proof dobrinskaya theorem introduction beginning study artin groups dates back introduction braid groups artin groups defined general tits brieskorn relation theory coxeter groups singularity theory since general properties studied many questions remain open one questions called conjecture states certain space homotopy type finite complex classifying space corresponding artin group conjecture proved particular families artin groups see general case every artin group special submonoid called artin monoid whose groupification dobrinskaya proved conjecture holds artin group natural map classifying spaces homotopy equivalence revealed interesting connection conjecture geometry beneath artin monoids concrete proof theorem given recently ozornova using technique discrete morse theory new proof gave hints left open questions connection standard model salvetti complex finite model space introduced salvetti work explore connection showing standard model collapsed sense discrete morse theory obtain complex naturally homotopy equivalent salvetti complex results also answers open questions concerning computation homology artin groups gives another proof dobrinskaya theorem giovanni paolini preliminaries section going recall concepts known results concerning coxeter artin groups artin monoids salvetti complex conjecture discrete morse theory coxeter artin groups let finite set let square matrix indexed satisfying following properties symmetric matrix called coxeter matrix coxeter matrix define corresponding coxeter group follows given subset let subgroup generated also coxeter group natural structure deriving coxeter matrix also define set subsets finite coxeter groups naturally endowed length function namely function maps element minimal length expression product generators since generators order relations also written notation stands even odd instance relation written sts tst consider set natural bijection define artin group follows clearly coxeter matrix natural projection corresponding artin coxeter groups sending finite say artin group finite type artin monoids artin monoid corresponding coxeter matrix monoid presented reason take freedom use generating set given following theorem theorem natural monoid homomorphism injective classifying space artin monoids view theorem consider contained artin monoid also called positive monoid elements precisely written product positive exponents generators immediate consequence previous theorem artin monoid left right cancellative originally proved since relations involve number generators left hand side right hand side well defined length function sends element length representation definition given say exists similarly say exists also say left divisor left divides left divisible right divisibility partial order relations definition let subset left common divisor element left divides elements greatest left common divisor left common divisor left multiple left common divisors similarly left common multiple element left multiple elements left least common multiple left common multiple left divisible left common multiples define obvious way analogous concepts right divisibility proposition greatest common divisor least common multiple exists set unique proposition let subset admits left resp right common multiple also admits least left resp right common multiple proposition subset admits greatest left common divisor greatest right common divisor going introduce fundamental element artin monoid exists significantly important recall theorem artin monoid following conditions equivalent finite type admits least left common multiple admits least right common multiple moreover satisfied least left common multiple least right common multiple coincide definition coxeter matrix corresponding artin group finite type least left right common multiple called fundamental element usually denoted following theorem summarizes properties fundamental element two definitions required definition element squarefree written form giovanni paolini definition let rev bijection sends element reverse easy check well defined theorem let coxeter matrix corresponding artin group finite type admits fundamental element following properties hold rev element left divides right divides iii element squarefree left right divisor least left right common multiple squarefree elements squarefree uniquely determined squarefree element maximal length longest element vii element written form consider subset let let subgroup generated theorem natural homomorphism sends injective words artin group corresponding coxeter matrix theorem least left right common multiple exists coxeter matrix finite type coxeter matrix finite type makes sense consider fundamental element artin monoid corresponding element denoted lemma precisely least left right common multiple finally introduce normal form elements artin monoid define set words set elements right divides theorem exists unique tuple subsets salvetti complex conjecture salvetti complex first defined salvetti arrangements affine hyperplanes thus including coxeter graphs finite affine type later generalized arbitrary coxeter graphs see going define quote known results definition given poset derived complex simplicial complex set vertices simplex every chain classifying space artin monoids definition let element unique element smallest length coset wwt uniqueness element proved consider set following partial order omit proof indeed partial order relation lemma let set call geometric realizations derived complexes respectively pair homeomorphic pair definition salvetti complex coxeter matrix denoted sal geometric realization derived complex lemma structure one cell dimension cell coxeter group acts first coordinate thus also acts sal action free properly discontinuous cellular quotient map sal sal covering map moreover covering map induces structure quotient space sal sal complex sal one cell dimension let describe detail combinatorics cells complexes sal sal sal correspondence elements coxeter group reason also denote simply since joining vertices notice joins vertices different orient exists exists vertices wst see also figure representation cell case quotient complex sal one therefore fundamental group sal admits representation generator relation relation turns exactly form means fundamental group sal naturally isomorphic corresponding artin group giovanni paolini wts wsts wtst wst figure example complex sal case given coxeter matrix particular representation corresponding coxeter group gives rise natural way certain topological space conjecture due brieskorn groups finite type arnold pham thom full generality following conjecture conjecture space classifying space artin group corresponding discuss definition space simply need know homotopy equivalent salvetti complex sal therefore conjecture equivalent following conjecture complex sal classifying space corresponding artin group conjecture would important consequences theory artin groups instance existence finite model classifying space implies homology properties known general conjecture proved families artin groups important result regard probably following theorem conjecture holds artin groups finite type discrete morse theory discrete morse theory powerful tool simplifying complexes mantaining homotopy type first developed forman presented combinatorial analogue morse theory forman version discrete morse theory based concept discrete morse function later reformulated chari batzies terms acyclic matchings breafly present latter formulation use later let complex recall cell characteristic map attaching map definition face poset set open cells together partial order defined definition let dim dim say face say regular face addition two following conditions hold set dim let attaching map classifying space artin monoids homeomorphism homeomorphic definition regular complex attaching maps injective remark regular faces regular definition cell graph hasse diagram directed graph set vertices edge written face denote set edges definition matching subset regular face cell occurs one edge given matching define graph obtained inverting edges definition matching acyclic corresponding graph acyclic aim discrete morse theory construct complex acyclic matching simpler complex called morse complex homotopy equivalent fewer cells definition let acyclic matching cell occur edge definition let poset poset map given denote subcomplex consisting cells definition compact compact definition let acyclic matching say compatible words matching written union matchings matching fiber theorem let complex acyclic matching compact compatible exist complex correspondence ncells homotopy equivalence moreover construction natural respect inclusion let subcomplex diagram commutative restriction complex called morse complex respect acyclic matching finally going prove lemma useful later come apply discrete morse theory giovanni paolini lemma let matching let compatible let restriction fiber acyclic also acyclic proof suppose contradiction graph contains cycle since edges increase dimension whereas others lower dimension cycle must form edges labelled belonging since moreover since indices taken modulo therefore first last term chain inequalities equal terms equal element cycle contained graph therefore acyclic contradiction view lemma possible weaken hypothesis theorem removing requirement acyclic asking instead acyclic restriction fiber way used obtain compactness acyclicity classifying space monoids going introduce notion classifying space monoid particular case classifying space small category viewing monoid category one object definition classifying space monoid geometric realization following simplicial set given sequences elements denoted symbol face maps send simplices degeneracy maps send shown geometric realization simplicial set complex therefore classifying space monoid complex simplices notice also one denoted definition groupification monoid group together homomorphism satisfying following universal property group homomorphism exists unique homomorphism makes following diagram commutative remark presentation given groupification group presentation classifying space artin monoids remark fundamental group classifying space monoid given groupification easily seen using presentation fundamental group complex one generators given relations given attaching maps case generator set relation corresponding given indeed presentation groupification remark focusing case artin monoids going give explicit construction universal cover monoid injects groupification natural map injective construction generalizes one example let geometric realization simplicial set whose given face map sends degeneracy map sends notice vertices bijection group vertices form group acts freely simplicially left multiplication element sends simplex simplex thus quotient map covering map lemma naturally homeomorphic proof simplex identified equivalence class simplex identification bijective respects face maps degeneracy maps homeomorphism proposition space universal cover natural covering map obtained composing quotient map homeomorphism lemma proof already seen indeed covering map therefore enough show simply connected choose basepoints respectively becomes covering map element represented signed sequence sign indicates whether arc travelled positive negative direction lift path covering space obtain path passing vertices notice isomorphism remark path corresponds precisely xkk means lifts path since injective conclude trivial space particular subcomplex consisting cells analogy case group prove contractible giovanni paolini proposition space deformation retracts onto vertex particular contractible proof simplex face possibly degenerate simplex also deformation retraction onto vertex slides point along line segment line segment exists well defined attaching maps simplices linear salvetti complex artin monoids dobrinskaya proved conjecture reformulated follows theorem conjecture holds artin group natural map homotopy equivalence dobrinskaya theorem particularly interesting since relates conjecture problem determining natural map monoid groupification induces homotopy equivalence corresponding classifying spaces phenomenon known happen cases see general problem still open prove theorem dobrinskaya also proved following result theorem space homotopy equivalent classifying space artin monoid corresponding quite easy deduce theorem theorem indeed natural map homotopy equivalence conjecture holds artin group hand conjecture holds spaces classifying spaces natural map must homotopy equivalence since induces isomorphism level fundamental groups rest section present new proof theorem based discrete morse theory ideas taken recent work ozornova prove stronger statement space collapsed sense discrete morse theory obtain complex naturally homotopy equivalent salvetti complex sal particular answers questions left open section let coxeter matrix corresponding artin group finite type able show contractible recall following classical result deduced corollary lemma let complex let family contractible subcomplexes also contractible theorem artin group finite type space contractible classifying space artin monoids proof make notation readable denote space subcomplex subcomplex homeomorphic therefore contractible proposition notice also consists simplices theorem fundamental element union subcomplexes since apply lemma conclude contractible corollary artin group finite type classifying space classifying space proof result follows immediately remark proposition theorem going construct acyclic matching essentially combination two matchings used difference entirely topological level set fundamental element first going describe definitions results lead construction two matchings cell product lies cell given min convention exists notice cell define unique subset property cell lemma define acyclic matching essential cells given cells defined construct second matching assume set carries total order notice cell completely characterized sequence subsets defined cell max cell given min cell max max giovanni paolini lemma define acyclic matching essential cells given essential cells cells consider matching slight abuse notation define length cell cell define also function follows lemma function compact grading equipped lexicographic order proof first prove poset map enough prove cell cell face suppose contradiction cells since possibility means particular whereas since cell must form clearly also obtain contradiction remains prove compact immediate since contains cells length finite number proposition matching acyclic compatible compact grading proof first let prove compatible definition hand case means compatible consider fiber simultaneously contain edges edges value determines whether cells must since acyclic restriction also acyclic true fibers therefore lemma conclude also acyclic previous proposition allows apply theorem obtaining smaller complex call essential cells matching precisely cells notice cell uniquely identified set means cells correspondence classifying space artin monoids call cell corresponding set dim construction every oriented path graph starting cell ends cell also satisfying therefore attaching map image contained union cells thus subset closed inclusion implies corresponds subcomplex particular holds similar way subcomplexes salf sal subsets closed inclusion remark reduced complex natural respect inclusion set indeed consider coxeter matrices generating sets fixed total order induces total order obtain reduced complexes naturally identified subcomplex true simplex subcomplex bam faces well matched cell also belong subcomplex indicated artin groups corresponding respectively let recall results homotopy theory used later lemma proposition pair attaching maps homotopic rel corollary complex two attaching maps homotopic rel proof follows previous lemma finally ready prove complexes sal homotopy equivalent order prove main theorem following lemma required lemma orientation boundary curve given even odd proof remark sufficient treat case consists one two one suppose case result reversed orientation set even odd moreover set essentially product alternating elements ending ending example set also giovanni paolini cells correspond cells following cell call restriction matching cells respect partial order induced acyclic graph since also acyclic matching compatible compact grading consider complex obtained collapsing along matching simplicity call cell name corresponding cell want prove induction following two assertions boundary curve boundary curve base steps start case whose boundary given whereas matched therefore essential means boundary also given case similar boundary also boundary matched true case excluded condition want prove step case whose boundary given particular matched finally matched whose boundary induction hypothesis thus boundary given see left part figure picture case finally want prove step case whose boundary given matched notice true thus also left analyze matched whose boundary induction boundary given see right part figure picture case classifying space artin monoids figure left induction step case right induction step case boundary curve denoted clockwise starting black vertex light arrows indicate morse collapse induction argument complete end proof consider boundary matched whose boundary matched whose boundary therefore boundary orientation starting point boundary curve exactly stated lemma theorem coxeter matrix exists homotopy equivalence sal every subset closed inclusion restriction image contained salf salf also homotopy equivalence proof going construct simultaneously map statement homotopy inverse sal together homotopies idy sal sal idsal consider chain subsets closed inclusion define recursively subcomplexes yfi salfi starting subcomplexes consisting one extending one cell time construct maps way image contained salfi image contained yfi hold homotopies time simultaneously prove induction subset closed inclusion giovanni paolini constructed maps yfi salfi salfi yfi restricted salf image contained salf respectively constructed homotopies yfi yfi salfi salfi restricted salf image contained salf respectively particular means restrictions salf homotopy inverses one homotopies obtained restricting subspaces salf respectively define sending sal inverse moreover define homotopies constant maps assume induction already defined maps simplify notation set salfi yfi let element belongs moreover let corresponding respectively want extend simply send homeomorphically preserving orientation inverse homeomorphism homotopies extended constant new cells apply lemma observe boundary curve via boundary curve extend sending homeomorphically preserving boundary similarly extend homotopies extended new cells corollary going deal case consider following subsets closed inclusion salf salf notice set closed inclusion element following inclusions complexes classifying space artin monoids obtained let attaching map cell notice attaching cell image actually contained thus argument deduce setting attaching map image contained coxeter matrix corresponding finite coxeter group let artin group corresponding space type corollary similarly complex salf sal also space type theorem induction know homotopy inverses one homotopies restrictions homotopy inverses one restricted homotopies maps extended since maps extend map idx attaching extended map yields space construction map homotopy extend similarly homotopy identity map identity map finally extend gluing two maps coincide extend gluing two homotopies coincide call extended maps construction homotopy idx homotopy idx complete induction argument need prove subset closed inclusion restrictions image contained salf respectively analogous property hold similarly restrictions claim follows induction assume set claim holds induction cell corresponding construction restrictions image contained salf respectively since closed inclusion therefore restrictions image contained salf salf salf respectively paper ozornova found algebraic complex computes homology artin groups algebraic version discrete morse theory left open question whether complex related complex computes cellular homology salvetti complex homotopy equivalence theorem induces isomorphisms sal sal two cellular chain complexes dimension notation indicates complex naturality property proved giovanni paolini theorem sends sign standard generators namely corresponding sal seen generators sal sal moreover naturality cellular boundary maps following diagram commutative sal sal sal sal therefore isomorphism algebraic complexes words proved ozornova algebraic complex coincides algebraic complex computes cellular homology salvetti complex natural expect since two complexes rank dimensions acknowledgements note based material master thesis written supervision mario salvetti therefore would like thank introducing topic giving good advice throughout final year master degree university pisa references bourbaki fasc xxxiv groupes lie chap groupes coxeter tits chap chap racines hermann brieskorn sur les groupes tresses arnol bourbaki vol pages springer brieskorn saito und inventiones mathematicae batzies welker discrete morse theory cellular resolutions journal fur die reine und angewandte mathematik charney davis hyperplane complements associated infinite reflection groups journal american mathematical society chari discrete morse functions combinatorial decompositions discrete mathematics callegaro moroni salvetti problem affine artin group type cohomology eur math soc jems deligne les immeubles des groupes tresses inventiones mathematicae dobrinskaya configuration spaces labeled particles finite complexes proceedings steklov institute mathematics forman morse theory cell complexes advances mathematics hatcher algebraic topology cambridge university press hendriks hyperplane complements large type inventiones mathematicae milnor geometric realization complex annals mathematics mcduff segal homology fibrations theorem inventiones mathematicae okonek das die affinen wurzelsysteme vom typ mathematische zeitschrift classifying space artin monoids ozornova discrete morse theory reformulation arxiv preprint paris artin monoids inject groups commentarii mathematici helvetici paris conjecture artin groups arxiv preprint salvetti topology complement real hyperplanes inventiones mathematicae salvetti homotopy type artin groups math res lett segal iterated inventiones mathematicae van der lek homotopy type complex hyperplane complements phd thesis katholieke universiteit nijmegen
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journal latex class files vol august traffic sign timely visual recognizability evaluation based measurable point clouds oct shanxin zhang cheng wang senior member ieee zhuang yang member ieee chenglu wen member ieee jonathan senior member ieee chenhui yang timely provision traffic sign information drivers essential drivers respond ensure safe driving avoid traffic accidents timely manner proposed timely visual recognizability quantitative evaluation method traffic signs transportation environments achieve goal first address concept visibility field reflect visible distribution threedimensional space construct traffic sign visibility evaluation model vem measure traffic sign visibility given viewpoint based vem proposed concept visual recognizability field vrf reflect visual recognizability distribution space established visual recognizability evaluation model vrem measure traffic sign visual recognizability given viewpoint next proposed traffic sign timely visual recognizability evaluation model tstvrem combining vrem actual maximum continuous visual recognizable distance traffic big data measure traffic signs visual recognizability different lanes finally presented automatic algorithm implement tstvrem model traffic sign road marking detection classification traffic sign environment point cloud segmentation viewpoints calculation tstvrem model realization performance method traffic sign timely visual recognizability evaluation tested three road point clouds acquired mobile laser scanning system riegl according road traffic signs markings china showing method feasible efficient index sign visibility visibility field visual recognizability field recognizability mobile laser scanning point clouds ntroduction raffic signs include number important traffic information speed restrictions driving behavior restrictions changes ahead road conditions information timely provision information drivers zhang fujian key laboratory sensing computing smart city school information science engineering xiamen university xiamen china also xizang key laboratory optical information processing visualization technology information engineering college xizang minzu university xianyang china email wang yang wen yang fujian key laboratory sensing computing smart city school information science engineering xiamen university xiamen china cwang zhuangyng clwen chyang moe key laboratory underwater acoustic communication marine information technology school information science engineering xiamen university xiamen china also department geography environmental management faculty environment university waterloo waterloo canada email junli increases likelihood drivers respond timely manner ensure safe driving avoid traffic accidents detecting reading roadside sign driver involves complex series sequentially occurring events mental physical include message detection processing intervals eye head movement alternating sign roadway environment finally active maneuvering vehicle lane changes deceleration turning destination required response stimulus provided sign complex procedures paramount importance traffic signs clearly visible driver however signs damaged humans nature signs occluded objects traffic environment may lead sign low visibility invisible thereby decreasing visual recognizability increasing probability traffic accident efficient traffic sign timely visual recognizability evaluation method needed judge whether traffic signs recognized driving process many facts affect traffic sign visual recognizability given traffic sign given traffic environment summarize two aspects objective factors subjective factors objective factors traffic sign size placement mounting height aiming depression angle shape damage degree traffic sign occlusion degree road curve road surface visual continuity surrounding environment factors affect driver ability achieve visual recognition known reasonable legitimate traffic sign provide driver good retinal imaging help visual recognizability subjective factors driver vehicle speed sight direction viewer reaction time vrt factors affect visual recognizability drivers geometric field view gfov decreases progressively increasing vehicle speed direction line sight determines whether traffic sign falls within gfov obviously occlusion frequency visual continuity interrupted also one factors affects traffic signs recognized drivers sign recognized humans actual maximum recognizable distance less viewer reaction distance vrd course facts included objective factors subjective factors also affect traffic signs visual recognizability mentioned example weather condition light influence caused solar elevation angle age sight driver cognitive journal latex class files vol august object occlusion object occlusion plants occlusion tilted signs wrong angle wrong height fig traffic signs low visual recognizability burden traffic density among others unify factors interface left tstvrem model regarding factors influence research future although promising results achieved areas traffic sign detection classification far works focused computing visibility traffic signs given viewpoint smart driver assistance systems transportation facilities maintenance purposes works based mainly sign images videos using computer vision methods works research occlusion traffic signs based point clouds acquired mobile laser scanning mls limitation lighting conditions fixed viewpoint position view angle getting images cameras way get information compute traffic sign visual recognizability position road surface therefore feasible use image video calculate visual recognizability position traffic sign development mls technology makes possible evaluate traffic sign visual recognizability measurable point clouds unlike optical imaging addition taking pictures mls provide complete point cloud entire roadway scene without limitations lighting conditions one extract information point cloud needed compute traffic sign visual recognizability appearance provides new way research visibility recognizability measurable real traffic environment fig illustrates examples traffic signs low visual recognizability caused object occlusion plants occlusion tilt wrong depress angle height respectively fig place different viewpoints road course height influence traffic sign visibility obvious influences brings exist paper present automatic traffic sign timely visual recognizability evaluation model based human visual cognition theory using measurable point clouds acquired mls system summarize main contributions follows addressed conception visibility field reflect visible distribution space viewpoint space presented vem model based human visual cognition theory compute traffic sign visibility vem combines principle retinal imaging actual human driving situation measure clarity traffic signs given viewpoint comparing vem addressed conception visual recognizability field reflect visual recognizable distribution space viewpoint space established vrem model measure traffic sign visual recognizability given viewpoint evaluate traffic sign visual recognizability propose tstvrem model combining vrem actual maximum continuous visual recognizable distance traffic big data presented automatic algorithm realize tstvrem includes extracting traffic sign point clouds segmented surrounding point clouds front traffic signs right roadway computing viewpoints according different lanes based extracted road marking point clouds pipeline traffic sign timely visual recognizability evaluation illustrated fig firstly segment mls point clouds ground point clouds nonground point clouds extract traffic sign point clouds nonground point clouds secondly split surrounding point clouds front traffic sign nonground point clouds according designed visual cognition distance roadway thirdly extract road marking point clouds ground point clouds get viewpoints different lanes finally use algorithm realize tstvrem model based extracted point clouds viewpoints get visual usability traffic sign paper organized follows review previous work given section sections iii describe definition tstvrem model implementation respectively section shows experiments section concludes paper elated ork rapid development laser radar especially application mls systems able collect accurate reliable point clouds point clouds provide geometric radiometric information infrastructure facilities makes simple efficient people survey urban roadway environment wen state attributes mls system data acquisition process according main focus paper divide related work three categories traffic sign detection classification road marking detection classification visibility research journal latex class files vol august fig pipeline traffic sign timely visual recognizability evaluation traffic sign detection classification goal traffic sign detection classification find locations types traffic signs existing traffic sign detection classification methods based extracting color shape information images videos method mainly uses color space design segment sign candidate area uses shape feature edge feature extract traffic sign detection methods include shape matching hough transform hog feature svm classification hog feature convolutional neural networks cnns among others however detection performance methods heavily effected weather conditions illumination view distance occlusion recently researchers developed various methods detect traffic signs point clouds yang proposed method extract urban objects include traffic signs mls urban scenes based supervoxels semantic knowledge lehtom took spin images ldhs account recognize objects including traffic signs roadway environment using machine learning method wen presented traffic sign detection process inventory purpose researchers also proposed detected method combining point cloud images getting good detecting performance road marking detection classification road markings paved roadways critical features traffic management systems important functions providing guidance information drivers pedestrians guan proposed algorithm extract road markings using segmentation morphological closing operation mentioned method rapidly extracting road marking generating georeferenced images point clouds method extracting road marking converting point clouds georeferenced feature images lead incompleteness incorrectness feature extraction process extracted road marking directly point clouds relying reflective properties classified using deep learning visibility research doman proposed visibility estimation method traffic signs part driving safety support systems preventing provision much information driver improved method considering temporal environmental changes integrated local global features driving environment use different contrast ratio distance counted pixel numbers different area image compute visibility traffic sign way limited position viewpoint weather condition consider traffic sign placement occlusion road curve subjective factor mentioned section model katz proposed hidden point removal hpr operator get visible points given viewpoint applied improving visual comprehension point sets researched properties transformation function hpr operator satisfy based hpr operator traffic signs occlusion detection point cloud researched paper measured occlusion occluded distribution index occlusion gradient index however factors including occluded area proportion influence driver speed vision road curve number journal latex class files vol august actual traffic sign visibility field vem model traffic sign actual traffic environment visibility viewpoint fixed height road surface driving direction lanes front traffic sign constitutes actual traffic sign visibility field related orientation traffic sign panel traffic sign height observation distance road curve road surface viewpoint position degree occlusion sight line deviation among others call influence traffic sign panel orientation traffic sign height observation distance road curve road surface viewpoint position geometric factor vem model constructed geometric factor geo occlusion factor occ sight line deviation factor sight viewpoint visibility traffic sign visibility defined follows visibility geo occ sight fig framework tstvrem model lanes others considered factors important traffic sign influence visibility recognizability traffic sign beside hpr operator detect surface points occluded traffic sign detect occlusion point clouds occlusion point cloud composed many objects plans following part describe evaluate geometric factor occlusion factor sight line deviation factor respectively geometric factor evaluation order make visibility calculation consistent human visual recognition theory use principle retinal imaging consider impact geometric factor evaluation geometric factor geo given blow geo aview type iii efinition tstvrem odel framework tstvrem model shown fig paper phrase viewpoint visibility means visibility traffic sign given viewpoint phrase viewpoint recognizability means recognizable degree traffic sign given viewpoint framework divided three parts visibility field definition vem model section visual recognizability field definition vrem model section traffic sign timely visual recognizability evaluation section definition visibility field vem model visibility field definition given environment around target object visibility viewpoint space around object constitutes visibility field reflects visible distribution target object space visibility field divided actual visibility field ideal visibility field according actual traffic sign set real road environment corresponding ideal traffic sign ideal road environment respectively take traffic sign example hemispherical visibility field traffic sign shown environment occlusion actual visibility field ideal visibility field separately traffic sign yellow white indicates visibility black indicates visibility fig aview retinal imaging area traffic sign observed given viewpoint standard retinal imaging area astandard type corresponding ideal traffic sign given viewpoint normal ideal traffic sign panel passes center panel fixed standard distance dstandard panel order make geo take standard distance dstandard less traffic sign disappeared driver view field meaning compute visibility traffic sign observation distance less dstandard meter different types traffic signs different standard retinal imaging areas obviously geo inversely proportional angle line connecting viewpoint traffic sign panel center normal passing traffic sign panel center orientation factor observation distance height difference viewpoint traffic sign panel center occlusion factor evaluation also use principle retinal imaging consider impact occlusion area ratio apart introduce occlusion distribution factor evaluation occlusion degree given blow kcocc csign aocc aocc aview dmax aview journal latex class files vol august viewpoints actual visibility field ideal visibility field fig hemispherical visibility field traffic sign aocc aview retinal imaging occluded area ratio aocc retinal imaging area occluded traffic sign region given viewpoint kcocc csign occlusion distribution kcocc csign distance center point occluded traffic sign region cocc traffic sign panel center point csign dmax maximum length csign vertex boundary polygon traffic sign panel weights satisfy condition adding punishment item evaluation occlusion factor occ given occ order make sure occ nearly zero nearly one meet conditions obviously degree occlusion constant occ decreases penalty parameter increases penalty parameter constant occ decreases degree occlusion increases therefore formula meets expectations effect visibility decreases degree occlusion increases sight line deviation factor evaluation object see fixed viewpoint clear look front view look oblique view factor reflects different imaging positions retina may lead different visibilities furthermore driver road different drive speeds lead different gfovs combine sight line deviation factor evaluation sight established sight sight line deviation angle line sight line connecting viewpoint csign actual gfov depends actual percentile speed comes traffic big data denoted function parameter used punishment item sight line deviation angle bigger half actual gfov sum factors discussed given traffic sign viewpoint ith lane visibility visibility equals visibility geo sight occ aview astandard type aocc aview occ kcocc aview dmax sight traffic sign ideal visibility field vem model traffic sign ideal traffic environment visibility viewpoint fixed height surface driving direction lanes front traffic sign constitutes ideal traffic sign visibility field ideal traffic environment environment ideal traffic sign installed specified suitable placement beside straight horizontal roadway according traffic design installation rules meets condition object around roadway addition traffic sign therefore vem model ideal environment degenerates product geometric factors geoi sight line deviation factors sighti formula viewpoint visibility ideal traffic environment visibilityi shown visibilityi sighti geoi geoideal evaluation geometric factor ideal traffic environment sightideal evaluation sight line deviation factor ideal traffic environment geoi aviewi type journal latex class files vol august retinal imaging area traffic sign aviewi observed given viewpoint ideal traffic environment sighti sight line deviation angle line sight line connected viewpoint csign ideal traffic environment ideal gfov depend design speed design road design used punishment item sight line deviation angle big half ideal gfov definition visual recognizability field vrem model although vem model uses retinal imaging area estimate viewpoint visibility accordance natural recognition process still difficult problem determine viewpoint recognizability viewpoint visibility example two viewpoints visibility traffic sign near viewpoint occlusion far viewpoint without occlusion sign recognized near viewpoint much effective information lost occlusion sign may recognized far viewpoint blurred silhouette order conquer problem introduced corresponding viewpoint visibility ideal traffic environment standard evaluate viewpoint recognizability beside advantage introducing ideal viewpoint visibility used standard judge whether traffic signs meet design installation specifications evaluate whether traffic sign recognizable given viewpoint visual recognizability field definition given environment around target object visual recognizability viewpoint space around object constitutes visual recognizability field reflects visual recognizable distribution object space vrem model viewpoint recognizability related actual viewpoint visibility corresponding ideal viewpoint visibility factors mentioned section denote intersection point polyline made viewpoints intersects perpendicular line driving direction passing traffic sign center pintersect remember intersection point line perpendicular driving direction passing traffic sign center intersects right road marking outline proutline remember distance viewpoint pintersect along polyline made viewpoints distance pintersect proutline dlength dwidth respectively corresponding viewpoint ideal traffic environment dlength dwidth viewpoint actual traffic environment recognizability visual recognizability given follows recognizability visibility visibilityi weights meet condition paper leave research factors influence cognition future threshold used judge whether traffic sign viewpoint recognized traffic sign timely visual recognizability evaluation according manual uniform traffic control devices mutcd united states road traffic signs markings china sight distance length road surface point driver see traffic sign acceptable level clarity traffic sign dsightdistance given based driver ability visual recognition designed vehicle speed use visual recognizability field composed viewpoints within length forward direction area road surface evaluate visual recognizability traffic sign traffic sign timely visual recognizability evaluated according different lanes forward direction area road surface related actual maximum continuous visual recognizable distance dmaxcoglength maximum continuous length viewpoints recognized along lane also actual vehicle speed roadway vrt vrt simply time necessary driver detect read react message displayed approaching sign lies within cone vision vrt ascertained vrd dvrd given sign location distance vehicle travels vrt interval calculated therefore dvrd tvrt relationship dmaxcoglength dvrd determines whether traffic sign enough time recognize evaluation traffic sign visual recognizability eirecognizability given condition return condition ture else return eirecognizability dmaxcoglength dvrd dmaxcoglength tvrt tstvrem odel mplementation construct automatic algorithm implement tstvrem model point clouds acquired mls system input road point clouds trajectory output visibility field recognizability field traffic sign timely visual recognizability lane first detect classify traffic signs extracted road marking input point clouds step called preliminary work second abbreviate surrounding point clouds right side roadway front traffic sign traffic sign surrounding point clouds segment traffic sign surrounding point clouds road marking point clouds along roadway according traffic sign step journal latex class files vol august called segment process viewpoints position road surface forward direction region computed form segmented road marking point clouds step called viewpoints computing end use traffic sign panel point cloud traffic sign surrounding point clouds viewpoints together extract information compute traffic sign timely visual recognizability using tstvrem model step called traffic sign timely recognizability computing following part introduce processes separately symbols used describe tstvrem model implementation illustrated fig preliminary works input point clouds road adopt method proposed wen traffic sign detection classification output traffic sign panel point clouds type using sign type speed limit roadway get traffic sign dsightdistance type according country traffic signs design specifications extracted traffic signs panel point clouds combine used segment traffic sign surrounding point clouds adopt algorithm presented road markings detection classification extracted road markings segmented extracted traffic signs position every traffic signs type road markings used distinguish road surface forward direction region different lanes segmented road marking point clouds segment process using traffic sign panel point clouds compute traffic sign center csign find nearest trajectory point ptraj csign vector csign ptraj denoted hsign road markings point clouds extracted use different length clusters along roadway direction distinguish solid dashed lines length threshold smallest meters solid line continuous partially missing low reflectivity use attribute approximately parallels trajectory line complete solid lines right ptraj sorted distances solid line ptraj perpendicular driving direction solid line maximum distance right roadside outline rroutline solid lines left ptraj sorted distance solid line ptraj perpendicular driving direction solid line nearest distance left outline rloutline road surface forward direction region vector hsign intersects rroutline point proutline vector hsign intersects rloutline point ploutline remember distance ploutline proutline ddrivingw idth road surface forward direction region detected road markings wear tear road marking get rloutline rroutline left right move trajectory minus height mls device every trajectory point method get ploutline proutline solid lines follows xoy plane select nearest point traffic sign trajectory two points near trajectory denoted segment solid lines slice along hsign compute centers sliced point clouds every cluster ploutline proutline selected distance side centers line using ploutline proutline start points constantly cutting pieces road marking clusters along trajectory interval remember intersection respectively compute center denoted distance accumulated bigger traffic sign remember point line last point second last point position last point minus extra length longer ptmp last point adjustment ratio ptmp line last point second last point construct two rectangles vertical plane segment traffic sign surrounding point clouds solid road marking point clouds rectangles horizontal edge always perpendicular horizontal length vertical height edges change according octree segment method used segment traffic sign environment point cloud solid road marking point cloud along vector denoted horizontal unit vector perpendicular denoted remember coordinate four coordinates rectangle used segment traffic sign environment point cloud constructed use method get rectangle segment solid road markings four coordinates rectangle used segment road marking point cloud constructed kploutline proutline kploutline proutline viewpoints computing order reduce effect reflectivity extraction traffic marks use solid road marking lines calculate number lanes lane standard width denoted dlane get number lanes ddrivingw idth actual lane width dlanew idth computed formula dlanew idth ddrivingw idth next step get lane dividing lines rroutline rloutline denote dividing lines rlane rlane rlane rlane right left routline lane loutline rlane rlane lane known depressed arrays respectively unit vector denoted hab firstly get split points lane dividing lines point rlane dlanew idth hab get every lane dividing line expressed arrays use interpolation sampling method get point lane method interpolation sampling journal latex class files vol august actual environment ideal environment fig illustration tstvrem model implementation tstvrem model compute traffic sign timely visual recognizability details include compute retinal imaging area point cloud given viewpoint get occlusion point cloud projected traffic sign panel get angle sight line deviation set ideal traffic environment top view side view fig viewpoints computation result determined got column points maybe column points along lane point pcol used denote lane lane point column get viewpoint pviewpointo plus observation height heye col coordinate observation height usually set meters road surface result calculated viewpoints shown fig traffic sign timely visual recognizability computing three steps traffic sign get traffic sign panel point cloud traffic sign environment point cloud viewpoints along lanes remaining work use extract information input retinal imaging area computing first aspect computing retinal imaging area rotate coordinates point clouds human view line traffic sign panel point cloud center csign viewpoint pviewpointo denoted lcp group traffic sign panel point clouds traffic sign surrounding point cloud viewpoint rotate coordinates line lcp using quaternions rotation method move origin coordinate system rotated csign remember coordinate transformed traffic sign panel center csignt csignt remember coordinate transformed pviewpointo pviewpoint traffic sign panel point cloud traffic sign environment point cloud denoted ssign senvironment respectively next project ssign xoy denote ssignp rojected edges ssignp rojected computed alpha shape algorithm parameter alpha algorithm remembered dalpha remember polygon composed edges epolygon last use polygon area formula compute edge area map area retinal imaging area using human retinal imaging principle distance center eye entrance pupil retina dretina set journal latex class files vol august millimeters occlusion point cloud obtaining compute distances csignt every vertex epolygon select vertex largest distance dmax remember line vector pviewpoint csignt line vector viewpoint gmax angle max equals arctan dmax csignt viewpoint environment point senvironment vector labeled symbol angle line every point senvironment compute intersection point xoy plane add intersection point point cloud sintersection add penvironment occlusion point cloud socclusion every point sintersection inside polygon epolygon add occluded point cloud soccluded else delete corresponding point socclusion retinal imaging area soccluded computed alpha shape algorithm human retinal imaging principle sight line deviation computing sight line driver point pviewpoint line esight defined viewpoint neighboring points pviewpoint sight line deviation angle angle angle esight sight sight angle ideal traffic environment setting kinds traffic signs included state traffic system need build traffic sign panel point cloud library class traffic signs contains one choose one traffic sign class actual traffic environment point clouds constitute library coordinates every traffic sign library transformed normal vector parallel yaxis center origin traffic sign design manual roadway height road surface depression angle angle direction road users pass road shoulder width wshoulder specified coordinate system ideal traffic environment set follows according whether traffic sign detected divided two parts road marking detected use origin proutline rrooutline normal vector traffic sign panel nsigni cos sin cos cos sin traffic sign right roadway traffic sign panel center csigni equals wshoulder traffic signs hanging roadway coordinates csign xsign sign sign proutline xroutline routline routline known distance dsign csign routline xoy plane computed formula dsign sign xroutline sign routline coordinate signi rotate corresponding traffic sign panel point cloud according nsigni csigni using quaternions rotation method coordinate translation transformation coordinate corresponding viewpoint pviewpointi pviewpoint dlength heye among width distance xoy plane pviewpoint rroutline dlength accumulated distance pviewpoint pviewpoint table descriptions two mobile lidar datasets dataset points length speed limit actual speed cabr mph mph lhsr mph mph wpr mph mph road marking detected see traffic sign ideal place change proutline coordinate proutline wshoulder hsign left work situation detected road marking xperiments iscussions mls system datasets riegl mls system used study acquire datasets area within xiamen island system integrated two laser scanners four digital cameras gnss imu dmi two laser scanners installed configuration pattern rotate emit laser beams maximum valid range measurement rate accuracy scanned point cloud data within four cameras installed four corners get pictures surroundings order prove practicality models algorithm urban roads mountain roads one survey including three roads performed mls obtain data required research zengcuoanbei road zcabr longhushan road lhsr wenping road wpr zcabr lhsr urban roads wpr mountain road three roads information presented table taxi travel records last year extracted traffic big data library xiamen china use driving speed taxi car estimate actual driving speed road parameter sensitivity analysis parameters geometric factor evaluation viewpoint traffic sign occ sight ascertained viewpoint visibility inversely proportional size parameter dstandard parameters occlusion factor evaluation occ circumstances occlusion occ nearly equals circumstances half occlusion relationship among parameters occlusion factor evaluation part function shown fig upper three lines generated occlusion ratio occlusion distribution equals figure see increment occlusion ratio weight occ decreases gradually occlusion distribution weight increasing occlusion ratio weight decreases occ decreases increase parameters ascertained nearly equal shown parameter sets meet demand model seemingly upper three lines lower three lines generated occlusion ratio equals occlusion distribution equals occ nearly equal journal latex class files vol august table descriptions parameters datasets fig occlusion value different weights occlusion ratio punishment item parameters zcabr lhsr wpr dstandard ield ieldi dalpha dretina heye dwidth dlength wshoulder tvrt mph mph mph fig occlusion value different occlusion ratio occlusion distribution meets model demand fig shows occ decreases occlusion ratio increases occlusion distribution near center traffic sign conditions parameters used test three datasets table dlength table viewpoint interval along road unit number trajectory points trajectory points number equals vehicle speed mph research shows traffic signs fall gfov equals recognized accurately mph velocity traffic signs fall gfov equals recognized accurately mph velocities linear interpolation used calculate gfov different velocities value sight changed value sight line deviation angle shown fig angle less gfov angle sight else diver needs turn head see traffic sign sight drastically reduced angle greater vehicle passed traffic sign equals middle values different design speeds listed used dsightdistance type experiments chinese traffic sign setting standard types change mounting height fig sight line deviation value different angle road side traffic signs height overhead signs set middle value state standard vrt vehicles traveling mph less environments estimated vehicles traveling mph complex environment considering driving maneuver made sign location tvrt set vehicles speed mph mph respectively experiment observing images actual viewpoint visibility ideal viewpoint visibility within vrt time students men women got threshold students recognize images journal latex class files vol august calculation result discuss example calculated results viewpoint visibility traffic sign illustrated fig yellow point cloud traffic sign panel green point cloud traffic sign surrounding objects road surface point cloud common vertex blue line cluster viewpoint fig closed yellow line traffic sign panel point cloud edges pink closed line traffic sign panel occlusion part occluded billboard visibility field results saved text fig shows viewpoint line composed column viewpoints along road visual recognizability field results saved text fig shows cognitivedouble figure ratio actual viewpoint visibility ideal viewpoint visibility normal situation cognitivedouble value smaller sometimes cognitivedouble value bigger road curve road surface may lead aiming distance sight line viewpoint traffic sign actual traffic environment better ideal traffic environment matter discriminating recognizability viewpoint cognitivedouble value bigger recognizability equals else equals traffic sign timely visual recognizability results shown fig maxcognitivedistance mincognitivedistance figure dmaxcoglength dvrd respectively figure see recognizability viewpoint line traffic sign viewpoint visibilities shown fig viewpoint recognizability shown fig fig shows graphical interface evaluation traffic sign timely visual recognizability generated algorithm section wenping road includes detected traffic sign yellow viewpoint visibility result mesh plane color red green viewpoint recognizability result mesh polygon color black occlusion point cloud red pictures position corresponding mesh actual environment color mesh planes change red green means values viewpoint visibility change big small symbol polygon means traffic sign recognizable polygon plane proposed traffic sign timely visual recognizability model implemented using running intel core computer computing times processing step every section recorded table iii seen table iii among traffic sign detection classification computing time stime road marking detection classification computing time mtime traffic sign timely visual recognizability evaluation computing time rtime traffic sign detection classification computing time least road marking detection classification takes computing time traffic sign detection classification computing time tstvrem model related number signs time complexity proposed method fast enough meet demand scale implementation benefited segmentation along right road outline step dramatically reduces quantity point cloud data processed taking wpr dataset approximately million points length road traffic signs example total computing time proposed method took evaluate traffic sign timely visual recognizability raw mls point clouds therefore method efficient capable rapid implementation transportation environment onclusion paper presented traffic sign timely visual recognizability evaluation model traffic sign inventory management purpose based human visual cognition theory traffic big data using measurable point clouds collected mls system process building model considered number factors traffic sign size position placement mounting height panel aiming depression angle shape damage occlusion actual vehicle speed sight line deviation gfov vrt road curve road surface gradient different lanes conception visibility field addressed reflect traffic sign visibility distribution space visibility evaluation model presented based human visual cognition theory compute traffic sign visibility given viewpoint comparing concept visibility field addressed conception visual recognizability field reflect visual recognizable distribution surface proposed visual recognition evaluation model compute traffic sign visual recognizability given viewpoint order evaluate traffic sign timely visual recognizability different lanes propose tstvrem model combining visual recognizability field actual maximum continuous cognitive distance traffic big data finally constructed automatic algorithm implement tstvrem model algorithm includes traffic sign road marking point clouds extraction classification traffic sign surrounding point clouds segmentation viewpoints computation different lanes tstvrem model realization addition traffic signs also traffic devices timely visual recognizability evaluated model based mls point cloud example traffic light different form traffic sign needed detect traffic light point cloud think plane model based three key ideas first evaluate viewpoint visibility angle human retinal imaging gfov changes different driving speeds roadway condition occlusion degree works second get actual driving speed traffic big data evaluate timely visual recognizability traffic sign third use latest equipment obtained measurable point clouds roadway environment use latest point cloud processing algorithm enable implementation model future facts may considered model example light influence caused solar elevation angle background influence cognitive burden traffic density among others journal latex class files vol august top view side view occlusion area fig occluded point cloud obtaining result fig visibility field results table iii computing time different datasets roads sections length pointsnumber stime mtime rtime totaltime zcabr lhsr wpr fig traffic sign timely visual recognizability result fig visual recognizability field results natural science foundation china natural science foundation xizang autonomous region china acknowledgment eferences would like thank anonymous reviewers valuable comments work supported grants liu wang song cognitive processing traffic signs immersive virtual reality environment erp study journal latex class files vol august fig graphical interface traffic sign timely visual recognizability results neuroscience letters vol kirmizioglu comprehensibility traffic signs among urban drivers turkey accident analysis prevention vol shinar effect context drivers age highway traffic signs comprehension transportation research part traffic psychology behaviour vol bertucci sign legibility rules thumb united states sign counc mourant ahmad jaeger lin optic flow geometric field view driving simulator display displays vol tieri tidoni pavone aglioti mere observation body discontinuity affects perceived ownership vicarious agency virtual hand experimental brain research vol belaroussi gruyer impact reduced visibility fog traffic sign detection intelligent vehicles symposium proceedings ieee ieee lambilliotte spitzenstetter giselbrecht muzet influence age speed duration monotonous driving task traffic drivers useful visual field vision research vol costa simone vignali lantieri bucchi dondi looking behavior vertical road signs transportation research part traffic psychology behaviour vol zhu liang zhang huang sign detection classification wild proceedings ieee conference computer vision pattern recognition riveiro arias traffic sign detection mls acquired point clouds geometric semantic inventory isprs journal photogrammetry remote sensing vol wen guan luo wang hierarchical deep models traffic sign detection recognition mobile laser scanning data isprs journal photogrammetry remote sensing vol garrido llorca alcantarilla parra herranz bergasa sotelo automatic traffic signs panels inspection system using computer vision ieee transactions intelligent transportation systems vol doman deguchi takahashi mekada ide murase sakai estimation traffic sign visibility considering local global features driving environment intelligent vehicles symposium proceedings ieee ieee khalilikhah heaslip analysis factors temporarily impacting traffic sign readability international journal transportation science technology vol wen luo cai wang wang spatialrelated traffic sign inspection inventory purposes using mobile laser scanning data ieee transactions intelligent transportation systems vol marinas salgado nieto detection tracking traffic signs using recursive bayesian decision framework intelligent transportation systems itsc international ieee conference ieee traffic sign segmentation classification using statistical learning methods neurocomputing vol sun liu wang novel traffic sign detection method via color segmentation robust shape matching neurocomputing vol qin wang zheng unified approach based hough transform quick detection circles rectangles journal image graphics vol greenhalgh mirmehdi detection recognition road traffic signs ieee transactions intelligent transportation systems vol yang luo towards traffic sign detection classification ieee transactions intelligent transportation systems vol yang dong zhao dai hierarchical extraction urban objects mobile laser scanning data isprs journal photogrammetry remote sensing vol jaakkola lampinen kaartinen kukko puttonen object classification recognition mobile laser scanning point clouds road environment ieee transactions geoscience remote sensing vol tan wang wang pan weakly supervised metric learning traffic sign recognition vehicle ieee transactions intelligent transportation systems vol tsai automated sign retroreflectivity condition evaluation methodology using mobile lidar computer vision transportation research part emerging technologies vol guan wang chapman yang using mobile laser scanning data automated extraction road markings isprs journal photogrammetry remote sensing vol guan wang using mobile lidar data rapidly updating road markings ieee transactions intelligent transportation systems vol riveiro arias segmentation classification road markings using mls data isprs journal photogrammetry remote sensing vol guan jia wang learning hierarchical features automated extraction road markings mobile lidar point clouds ieee journal selected topics applied earth observations remote sensing vol journal latex class files vol august doman deguchi takahashi mekada ide murase tamatsu estimation traffic sign visibility toward smart driver assistance intelligent vehicles symposium ieee ieee estimation traffic sign visibility considering temporal environmental changes smart driver assistance intelligent vehicles symposium ieee ieee katz tal basri direct visibility point sets acm transactions graphics tog vol acm katz tal improving visual comprehension point sets proceedings ieee conference computer vision pattern recognition visibility point clouds proceedings ieee international conference computer vision huang cheng chen luo wang traffic sign occlusion detection using mobile laser scanning point clouds ieee transactions intelligent transportation systems parkes geometric field view manipulations affect perceived speed driving simulators adminstration manual uniform traffic control devices yang liu road traffic signs markings banks introduction transportation engineering new york vol kuipers quaternions rotation sequences princeton university press princeton vol edelsbrunner kirkpatrick seidel shape set points plane ieee transactions information theory vol kaiser calculation visual angle joy visual perception web book york university available http yorku htm guan chapman wang automated road information extraction mobile laser scanning data ieee transactions intelligent transportation systems vol shanxin zhang received degree computer software theory shandong university science technology qingdao china assistant professor xizang minzu university currently student computer science technology fujian key laboratory sensing computing smart city school information science engineering xiamen university china current research interests include computer vision machine learning deep learning mobile lidar point clouds data processing cheng wang received degree information communication engineering national university defense technology changsha china currently professor associate dean school information science engineering executive director fujian key laboratory sensing computing smart city xiamen university china current research interests include remote sensing image processing mobile lidar data analysis fusion chair isprs integration fusion council member china society image graphics coauthored papers referred journals top conferences including aaai zhuang yang received degree computational mathematics guilin university electronic technology guilin china currently pursuing degree information communication engineering fujian key laboratory sensing computing smart cities department communication engineering school information science engineering xiamen university xiamen china current research interests include machine learning computer vision matrix analysis optimization algorithm chenglu wen received degree mechanical engineering china agricultural university beijing china currently assistant professor fujian key laboratory sensing computing smart city school formation science engineering xiamen university xiamen china coauthored research papers published refereed journals proceedings current research interests include machine vision machine learning point cloud data processing secretary isprs system calibration jonathan received degree geomatics engineering university cape town south africa currently professor head mobile sensing geodata science lab department geography environmental management university waterloo canada current research interests include information extraction lidar point clouds earth observation images publications published refereed journals including ieeejstars ijrs rse chair isprs working group lidar airborne spaceborne sensing chair ica commission mapping associate editor chenhui yang received degree mechanical engineering zhejiang university zhejiang china currently professor school information science engineering xiamen university china visiting scholar university national laboratory university southern california research interests focus computer vision computer graphics data mining applications sciences industries including transportation security medicine among others
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dropping activation outputs localized deep network enhancing user privacy data security hao dong chao zhen wei yike guo nov department computing imperial college london learning methods play crucial role anomaly detection prediction supporting decision making applications like personal pervasive body sensing etc however current architecture deep networks suffers privacy issue users need give data model typically hosted server cluster cloud training prediction problem getting severe sensitive medical data fmri body sensors measures like eeg signals addition also security risk leaking data data transmission user model especially internet targeting issues paper proposed new architecture deep network users reveal original data model method propagation data encryption combined one process migrate first layer deep network users local devices apply activation functions locally use dropping activation output method make output resulting approach able make model prediction without accessing users sensitive raw data experiment conducted paper showed approach achieves desirable privacy protection requirement demonstrated several advantages traditional approach encryption decryption ntroduction deep learning also known deep neural networks proved successful classification regression tasks well established deep learning models exhibited promising performance desired many prediction applications like automatic speech recognition image recognition natural language processing etc applications due hardware limitations computation capability due power consumption limitation expensive implement deep learning algorithms entirely users local devices sensors mobiles even laptops result typical approach collect data locally first transmit data remote server cluster apply deep networks training prediction however approach suffers important privacy issue especially sensitive data patients clinical data like fmri images sensitive environment data like video monitoring public space kind data becoming important big data analytics personalized brings privacy concern sending data directly remote model example would cause severe consequence company model provider lot users data like dna ehr fmri etc invaded malicious hackers also causes another security risk leaking data transmission user model especially internet problematic transfer users sensitive data directly deep learning model hosted remote server researches conducting neural network prediction encrypted data rather original data situation users sensors share remote servers however approach bring extra computation cost would safe key encryption hacked targeting issue proposed novel deep network architecture users need reveal original data model instead tradition approach shown fig split layers local device server migrate first layer local device shown fig apply activation functions first layer locally transfer outputs first layer server use dropping activation output method make output randomly drop outputs activation function method propagation data encryption combined one process investigated invertible activations performance privacy protection proved method make sure original data recovered transmitted data concretely main contributions paper listed follow introduced new deep network architecture combines neural network data encryption solve privacy data security problem privacy preserving feedforward propagation server model provider serve user without directly accessing user data proved evaluated dropping activation outputs encrypt data invertible activation function also evaluated ramp function better rectifier term encryption proposed privacy preserving propagation server train neural network user without accessing user data method data compression data transmission available number neurons less size input data paper organized follows section firstly introduces new architecture deep network discusses invertible activation functions mainly focusing invertible activation function encrypt data give details dropping activation output method figure works propagation without additional computation section iii provides mathematics proven explanation method section presents experiment result conclude work section ethod ropping activation utputs ocalized layer eep etwork firstly introduce new architecture localized firstlayer deep network introduce proposed encrypt methods invertible activation like sigmoid noninvertible activation like rectifier main content focus method encrypting data invertible activation function localized deep network fig illustrates traditional deep network architecture sends data local device server encryption decryption processes server uses original data compute result general equation single layer written row vector original input data weight matrix row vector biases row vector activation outputs reflects activation function sigmoid hyperbolic tangent softplus rectifier etc security aspect activation outputs captured network sniffer weight matrix bias acquired hacking software result given input data fully reconstructed equation inverse matrix inverse activation function therefore encrypting activation outputs desired data privacy security fig implement propagation server input data output value probability given new architecture decomposes layers neural network local device server shown fig local device transfers activation outputs first hidden number activation outputs equals greater number input data local server fig spliting neuron network propagation layer server server use activation outputs compute result discuss architecture enable privacy protection noninvertible invertible activation functions activation functions several activation functions example rectifier commonly used deep neural network able reach network best performance without unsupervised unlabeled data output linear function inputs vanishing gradient problem reduced rectifier shown equation sets negative activation outputs zero otherwise activation functions naturally encrypt activation outputs degree activation functions best way reconstruct input data use approximated inverse activation function transposed weight matrix instead equation describes mathematical explanation found section iii however due linearity rectifier activation outputs highly related scale feature patterns inputs general features original data decrypted equation alleviate problem introduce ramp activation function equation shows reflects small value smaller cut scale information activation outputs contain lesser scale information information features combination set small feature identifiers rectifier ramp functions evaluated section view information outputs result information loss original data fully reconstructed however neural network perspective function selects features therefore information lost accuracy even improved based principle rectifying layer even one activation output turned zero negative value original input data reconstructed completely practice due sparse property rectifying layer large portion rectifying outputs zero therefore reconstructed data distorted completely hence data encryption privacy realized activation functions input hidden layer reconstructed even parameters model known original inputs still reconstructed server hence privacy preserved users apart rectifier ramp functions activation function binary step noninvertible invertible activation functions different activation functions reconstruction solvable activation function invertible sigmoid hyperbolic tangent softplus etc sigmoid inverse sigmoid shown equation method would lead noticeable performance decreases small dropping probabilities enough encryption sparse behavior neural network achieved methods dropout dropconnect autoencoder paper proposed idea dropping activation outputs dropping connections definitions dropping activation outputs dropping connections different dropout dropconnect applied propagation whereas dropout dropconnect applied propagation figure dropping activation outputs dropping connections shown fig dropping activation outputs process encrypting data propagation process purposed method called dropping activation output mathematically modified activation output written therefore ways encrypt modifying modification bring uncertainty neural network therefore key find modification method affect final predicting result due sparse behavior neural network accuracy affected slightly changing activation outputs weights first training proposed method would compromise learning results dropout dropconnect way avoid overfitting testing inferencing noninvertible activation would affect result invertible activation dropping activation outputs also would harm final result theory point view dropout training considered training many job result averaged one sort ensemble learning recently proved gaussian process hand reasonable uncertainty neural network testing would affect testing result network uncertainty training experience point view dropping probability required method quite low compared dropping probability training usually would affect final averaged result subnetworks table show effect different dropping probabilities using sigmoid function expect multiplication hadamard product denoted binary vector randomly setting activation outputs zero setting activation output zero equivalent removing connections neuron input data describe detail section iii equation used reconstruct data found similar activation equation better reconstruct data dropping connections process another purposed method called dropping connections dropping connections elements weight matrix set zero modified weight matrix modified activation output written binary matrix elements zero however experiment shows method effective enough encrypt data iii athematical xplanation activation linear discuss dropping connections encrypt good dropping activation figure left figure right server local server local fig dropping activation output left dropping connection right propagation dropping connections assume different replace equation equation let represent new calculated changed new written note entries hence product small means neglected reason behind product identity matrix means sum product entries row corresponding entries column sum position diagonal sum elsewhere approximates identity matrix means input data nearly dropping connection result written therefore encrypt performance poor mathematically corresponds affine transformation change circumstances result applies matrix matrix entries change means terms affected summation terms therefore hence approximate identity matrix means give example matrix entries zero let changing row row changing entries different row corresponding entries rows inverse therefore xperimental tudies however represents change entries means terms affected summation five terms therefore hence approximate identity matrix means used mnist dataset evaluate methods training set validation set test set image digit classes total adopted accuracy classification task evaluate model standard way evaluating machine learning algorithms classification definition accuracy percentage correct prediction test set evaluated dropping activation outputs dataset challenging dataset compared mnist training images test images image rgb image classes total airplane automobile bird cat deer dog frog horse ship truck dropping activation outputs dropping activation outputs linear activation also used explanation let modified activation output corresponding modified input data written hence activation output vector size weight matrix size entries change modified vector written product original input data matrix diagonal entries necessary therefore necessary change circumstance affine transformation example let written written rectifier experimental network three rectifying hidden layers layer neurons model represented input layer output layer dropout applied training prevent overfitting dropout probability input layer first layer probabilities layers weights decay used neural networks trained using adam gradient descent size epochs accuracy found null hypothesis pairwise test evaluate rectifier equation used reconstruct approximate input data fig shows original input data reconstructed input data activation outputs first hidden layer using equation clear reconstruction failed figure activation functions means case unknown restriction means number hence identity matrix fig reconstructed input data rectifying output using right original input data left activation output max divergence digit input data better reconstructed equation middle column fig demonstrated even figure reconstructed data totally distorted outline original input data still recognized reconstructed data encryption applied transferring activation outputs second hidden layer encrypting data twice right hand side fig shows reconstructed input data equation outline digit recognized however applying local rectifying layers lead higher local computation fig reconstructed input data ramp activation outputs using middle right original input data left divergence digit smaller invertible activation functions dropping activation outputs fig reconstructed input data rectifying outputs original input data left reconstruct outputs first hidden layer middle divergence layer digit reconstruct outputs second hidden layer right divergence layer higher layer ramp let activation outputs first hidden layer contain lesser scale information ramp activation function applied two testing networks previous rectifier network except activation function first layer ramp respectively accuracies two networks approximately previous pure rectifier neural network according fig smaller better encrypt input data topology input data recognized reconstructed data experiments summarized purposed architecture matter kind activation function used without knowing model parameters local device original input data reconstructed activation output even knowing model parameters local device single rectifying layer able encrypt input data better encryption done applying rectifying layers transferring activation outputs encryption data done ramp activation function reduces scale feature pattern invertible activation function first experiment dropping activation outputs consider networks hidden layers units activation function sigmoid invertible training methods set previous rectifier neural network activation outputs set zero inverse sigmoid equation becomes insolvable since number divided zero approximate reconstruction achieved equation range approximate inverse number set zero otherwise fig shows reconstructed input data dropping probabilities clear reconstructed input data distorted distortion found dropping probability increase therefore dropping activation outputs enough encrypt input data lead noticeable performance decreases table shows addition adding noise activation outputs also lead result choose equation reconstruction reconstructed input data show fig worse fig used convolutional neural network cnn task network architecture follows expect zero following constants explored found giving similar results fig shows table andomly setting activation outputs zero first hidden layer feed forward propagation trails accuracy standard derivation accuracy table andomly setting activation outputs zero first convolutional layer feed forward propagation trails accuracy standard derivation accuracy sigmoid lrn relu lrn relu relu softmax represents cnn filters size stride represents maxpooling size stride represents layer unit lrn represents local response normalization model trained epochs learning rate adam gradient descent randomly dropped activation outputs convolutional layer result seen table experiments summarized follow invertible activation function dropping activation outputs propagation provide data encryption privacy make function insolvable activation value needs set reasonable range example setting value sigmoid hyperbolic tangent instead setting activation outputs reasonable range adding small noise values activation output similar impact invertible activation functions dropping connections randomly setting part weight values zero indirectly modify activation outputs however according experiment using mnist dataset combining sigmoid function dropping activation outputs process encryption appear reconstructed data almost original input data autoencoder small number activation output reduce computation communication cost local device case autoencoder applied even invertible activation function number activation outputs smaller number input data original input data reconstructed correctly activation outputs uses smaller dimension represent original data input data approximately reconstructed equation fig shows input data reconstructed autoencoder sigmoid activation function case number neuron half number input data words half information original input data reconstructed noting shape digit image still identified clearly therefore autoencoder noninvertible activation function dropping activation output recommended provide data encryption data compression privacy preserving analyze privacy preserving property method discussing works bruteforce attack model assume drop neurons discussed computation raised exponentially attackers precompute possible activation results take sigmoid function example outputs neurons continuous values assume attacker possible output values algorithm drop neurons outputs number possible combinations sigmoid experiment dropping probability define number possible combinations millions reconstructing millions images attacker also need algorithm recognize good image used rectifier attacker would know many neurons dropped method mean number possible combinations become significantly larger illustrate conducting bruteforce experiment mnist classifier sigmoid function outputs separated gaps therefore possible inputs takes hours compute possibilities titan pascal gpu note attacker needs extra algorithm select one data data attacker know content data difficult define criteria model invertible activation function intact activation outputs obtained local device inputs data network multiple times dropping positions overlap probability dropping position overlap second time first time relatively smaller original input data high probability decrypted data input network twice figure figure fig equation used reconstruct input data first sigmoid hidden layer dropping activation outputs first column original input data dropping probabilities increase second forth column note neurons layer neurons divergence similar fig reconstructed input data sigmoid outputs using equation right original input data left number neuron half number input data model work speed verify impact implementing sigmoid function user local devices system test measure used metal framework develop sigmoid function gpu swift language fig reconstructed input data first sigmoid hidden layer drop activation outputs using equation original input data first column second third columns dropping probabilities increase divergence higher result practice prevent situation drop position data used max value data random seed select dropping position need mention activation function activation outputs dropped inherently even without using dropping activation outputs kernel void sigmoid const device float invector buffer device float outvector buffer uint outvector exp device used iphone ios sdate used measured time lapse computing first layer activations input used vector pixels grayscale images https applied sigmoid size size neurons sample output time lapses measured calculated average latency induced onactivation begin onactivation onactivation end onclusion paper proposed new architecture deep network localized first layer network investigate architecture support better privacy protection model prediction invertible activation function dropping activation outputs propagation proved able encrypt original input data preserving privacy whole encryption process improved combining propagation data encryption one process means need specialized data encryption process local device data decryption process server encryption process invertible noninvertible activation functions discussed mathematically proved possible encryption propagation splitting neural network local device server provide data privacy training words server able provide model learning service using propagation without accessing original input data local device acknowledgment authors would like thank charles johnson william mary college xinman university cambridge helpful comments suggestions manuscript hao dong supported optimise portal eferences deng deep learning methods applications publishers jan redmon divvala girshick farhadi yolo look unified object detection cvpr yuan privacy preserving neural network learning made practical cloud computing ieee transactions parallel distributed systems bonde akib etc review techniques data privacy cloud using back propagation neural network international journal emerging technology advanced engineering bansal chen zhong privacy preserving neural network learning arbitrarily partitioned data neural computing applications bost popa goldwasser machine learning classification encrypted data crypto eprint archive graepel thore etc confidential machine learning encrypted data icisc springer khalid sayood introduction data compression edition elsevier glorot bordes bengio deep sparse rectifier neural networks aistats hinton etc improving neural networks preventing coadaptation feature detectors srivastava nitish hinton etc dropout simple way prevent neural networks overfitting jmlr wan zeiler etc regularization neural networks using dropconnect icml vincent etc stacked denoising autoencoders learning useful representations deep network local denoising criterion jmlr gal ghahramani dropout bayesian approximation representing model uncertainty deep learning icml gal yarin uncertainty deep learning university cambridge giryes sapiro bronstein deep neural networks random gaussian weights universal classification strategy ieee transactions signal processing blundell cornebise etc weight uncertainty neural networks icml goodfellow etc maxout networks jmlr hochreiter hochreiter etc long memory neural computation bengio learing dependencies gradient descant difficult ieee transaction neural networks gers schraudolph schmidhuber learning precise timing lstm recurrent networks jmlr greff srivastava steunebrink schmidhuber lstm search space odyssey nips chung gulcehre cho bengio empirical evaluation gated recurrent neural networks sequence modeling nips krizhevsky hinton imagenet classification deep convolutional neural networks nips kingma diederik jimmy adam method stochastic optimization iclr bastien lamblin etc theano new features speed improvements nips bergstra etc theano cpu gpu math expression compiler proceedings python scientific computing conference scipy hazewinkel michiel affine transformation encyclopedia mathematics springer isbn
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