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programmable restoration granularity constraint programming yong lin martin henz feb school computing national university singapore singapore linyong henz abstract constraint programming systems limited number search engines offered programming search algorithms requires efforts complicates deployment algorithms alleviate limitation concepts computation spaces developed computation spaces provide restoration mechanism store information contained search tree node granularities possible paper make case dynamically adapting restoration granularity search order elucidate programmable restoration granularity present restoration aspect constraint programming system using model programming implementation using gecode shows promising results introduction constraint programming solves combinatorial problems constraintbased search explores search tree certain order exploring search tree may encounter failure signals false search direction recover search failure restoration service called provide ever visited state switch search direction systems restoration implemented several manners system copies ever accessed state direct retrieval format computation space space short constraint logic programming clp systems chip clp ecli pse record state change information trail data strucutre roll back gecode system maintains generated constraints recompute newly proposed recollection stores constraint propagation affected variable domains recollect implemented techniques differ granularity stored information copying since computation space encapsulates speculative computation structures hand recomputation store specific data structures instructions committed guiding search trailing recollection space copying since store part information state actually granularity restored technique determines advantages limitations one program synthetically use techniques system efficient yong lin martin henz restoration expected early effort conducted proposal computation space enables user place state copies search tree intend however claimed space recomputation aware committed constraints therefore advocate study include techniques program powerful restoration scheme paper propose program restoration granularities aiming improving constraint programming system performance initial effort conducted integrating copying recollection recomputation especially propose program restoration granularity programming modular flexible restoration implementation rest paper organized follows section briefly goes notions systen implementation section present prototype programming restoration granularity subsequently section employ concept aspect illustrate landscape modular flexible restoration implementation taking gecode example last conclude section basics systems constraint propagation demonstrated insufficient solve problem reaches fix point search comes play branches fix point generate constraints commits one generated constraints proceed interleaving constraint propagation branching creates search tree node computation state search tree branches constraints internal nodes fix points leaf nodes either solutions search failures search failure indicates false search direction requires restoration previously visited internal state switch search directions constraint programming systems state restoration implemented several ways method stores one states modified reach conjunct state trail structure roll back previous performed operations method efficient problems imposing weak propagation however trailing couples tightly propagation engine search facilities potentially limits parallel exploration copying approach duplicates ever accessed fix point state restoration simple direct state retrieval technique causes neither runtime memory issues small medium size problem large problems copying may introduce memory management penalty account substantially occupied memory recomputation barely maintains previously committed constraints recompute state approach consumes rather limited memory runtime may dragged significantly problem computation expensive proposed recollection memorizes variables changed fix point reasoning steps moderate extra memory investment conducts restoration updating higher level state using stored variables restoration programmable restoration granularity constraint programming niques observe ways conduct restoration determined granularity information stored search specifically copying since stores information states contrast trailing recollection recomputation merely stores constraints used instruct search directions prototype section would present programs restoration granularity organization section follows section highlights traits problem spurs investigation programming restoration granularity describe prototype section evaluation conducted section motivation deep search tree may expanded solving hard problem table illustrates search tree statistics four problems exploring first solution queens problem modeled either set disequality constraints three global constraints constraints family denoted table column failures counts total number failures search first signals tree level first search failure emerges peak value deepest tree level first accumulates number failures occurs root first level first peak records number failures first peak problem failures first peak first first peak queens knights table search tree statistics problem search trees although restoration techniques capable accomplishing duty correctly respectively exhibit advantage towards certain case claimed search tree statistics perceive emergence first failure important signal intensive search failures therefore copying employed underlying restoration support solving problems space copies maintained root irst level contribute indicates ideal restoration customize problem search traits yong lin martin henz programming granularity deal skewed failure distribution displayed table straightforward method storing restoration information difference granularity search proceeds restoration adapt stored information conduct relevant process initially program prototype integrating copying recomputation recollection prototype take search tree level first failure emerges border level horizontally divides search tree upper bottom two parts search upper part search tree generated constraints kept neither state copies propagation modified variables stored search explores beyond border level copying recollection activated collaborate placing state copy alternates recollection explorations configurable set eithgt change information granularity restoration need switch different code segment specifically state upper part search tree expected restoration routine switches recomputation state bottom part required restoration routine first attempts recollect nearest state copy effort fails recomputes restore border state first update border state requested one recollection take optimization measure maintaining border state copy avoid possible recomputation restoring bottom state similarly one also employ measures optimize recomputation upper part evaluation evaluate programmed prototype four problems sought motivation program restoration granularity recomputation recollection enable adaptive function set copying distance eight evaluations conducted intel core quad processor system running ubuntu operating system four gigabyte main memory built sample top gecode version also served reference instance comparisons collected value arithmetic mean runs variation coefficient less memory numbers peak memory consumption compare prototype adaptive recomputation adaptive recollection table depicts evaluations results numbers reveal prototype significantly save memory two restoration alternatives problem memory saving significant account first failure comes earlier level tree peak depth meanwhile queens problems expose better runtime performances adaptive recomputation nevertheless deserves highlight take wall clock time programmable restoration granularity constraint programming problems queens knights recomputation recollection prototype time mem time mem time mem table comparisons prototype recomputation recollection knights problem almost halves memory consumption two techniques marginally improves runtime recomputation promising result confirms opportunities programming restoration granularity seek better performance programmability mentioned previous section way implement restoration actually determined granularity storing information restoration information implemented store search engine exploration steps hand restoration service provided search engine observation claims maintained restoration information cuts across two abstractions search engine restoration developing applications occurrence crosscutting abstraction rare transactions operation logging etc exemplify crosscutting abstractions facilitate programming crosscutting abstraction programming aop proposes encapsulate crosscutting abstraction aspect implementation aspect mainly consists two tasks advice pointcut advice means specifying code run joint point additional behaviors attempt add program pointcut determines matches executing specified advice joint point program restoration granularity one correctly implement switching restoration technique code segments prototype achieve communication code segments introducing signal however signal couples tightly specific program suppose one redefines criteria switch restoration techniques code quite likely change probably drastically therefore great significance implement flexible modular design program restoration granularity claim programming restoration aspect achieve design goal section propose adopt programming program restoration granularity gecode system modular flexible implementation specifically section briefly explains design search facilities gecode system section discuss issues related deploying aspect gecode search facilities yong lin martin henz search facilities gecode system defines class edge keep restoration information visited fixpoint edge objects pushed stack structure path search engine program outlines class declaration edge edge choice object encapsulates fix point generated two respectively represented currently committed constraint alternative recorded integer variable alternative choice object compulsory information component created edge object since instructs search direction committing constraint whereas space copy space optional specifically search engine wraps space edge restoration none edge stores space copy restoration hybrid variants obtained placing space copies edges occasionally adaptive recomputation gecode restoration related services defined structure path collects edge objects generated root current node program class edge class edge space space space copy choice choice fixpoint generated choice committed choice alternative unsigned int altternative doms variable domains public edge space choice doms alternative commit first alternative initialization statements claimed space data constraints metainformation instructing search introduce data structure dom collects variable domains affected fix point vector contains sufficient information conduct recollection one also introduce data structures explore restoration possibilities gecode search engine constructed programming computation spaces space provides interface status trigger internal constraint propagation returns status value search engine switches relevant code segment according space status depicted space return constraint propagation results failure search engine restrict discussion binary search tree insist finite domain integer domain problems discussion programmable restoration granularity constraint programming first requests adjust search direction calling adjust enter code segment restoration program search engine true switch status space query space status case fixpoint choice space return solution space push edge commit space commit constraint space case solution return space return solution space case failure adjust break problem solvable end restore recomputation code originally programmed recomputation end switch end restoration aspect program programming paradigm requires support underlying programming language currently quite programming languages implemented aop within language external library provided set extensions facilitate use aop gecode open source constraint programming library constructed thus following exploration provisioning restoration service gecode programming paradigm deploy restoration aspect requires address definition advice pointcut specifically define advice exploit information specific granularity individually collaboratively recomputation code restore ever visited fix point collaboration requires pointcuts advice matched applied execution either orignal recomputation code segment illustrated program prototype place code copying recollection place marked stressed currently use specific signal coordinate code switching thus conjecture modular flexible restoration built using aop return boolean false another search direction impossible restore method actually defined search engine expose body search engine facilitating explanation yong lin martin henz conclusion restoration key component search facilities constraint programming systems implementation significantly impact performance construction search engine although proposal computation space facilitates programming restoration since space encapsulates data structures state search tree paper proposed program restoration granularity implemented prototype integrates copying recollection recomputation evaluation gives promising results prototype employed specific signal coordinate execution restoration code segments couples tightly program thus propose provision restoration aspectoriented programming significance prospective effort one hand modularizing restoration code segments implement different techniques flexing assemble hand also potentially provides provisions construct search engines run wide spectrum search algorithms references abderrahamane aggoun david chan pierre dufresne eamon falvey hugh grant warwick harvey alexander herold geoffrey macartney micha meier david miller eclipse user manual philippe codognet daniel diaz compiling constraints clp journal logic programming mehmet dincbas pascal van hentenryck helmut simonis abderrahmane aggoun thomas graf berthier constraint logic programming language chip proceeding international conference fifth generation computer science pages joxan jaffar michael maher constraint logic programming survey journal logic programming pages special issue ten years logic programming gregor kiczales john lamping anurag mendhekar chris maeda cristina lopes loingtier john irwin programming springer yong lin prototypical implementation recollection based gecode may christian schulte comparing trailing copying constraint programming proceedings international conference logic programming pages cambridge usa massachusetts institute technology christian schulte programming constraint services volume lecture notes artificial intelligence project team homepage gecode project team generic constraint development environment june
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exploring spatial context semantic segmentation point clouds francis theodora alexander hermans bastian leibe computer vision group visual computing institute rwth aachen university feb engelmann kontogianni hermans leibe abstract deep learning approaches made tremendous progress field semantic segmentation past years however current approaches operate image space direct semantic segmentation unstructured point clouds still open research problem recently proposed pointnet architecture presents interesting step ahead operate unstructured point clouds achieving encouraging segmentation results however subdivides input points grid blocks processes block individually paper investigate question architecture extended incorporate spatial context build upon pointnet propose two extensions enlarge receptive field scene evaluate proposed strategies challenging indoor outdoor datasets show improved results scenarios context figure explore mechanisms extend spatial context semantic segmentation point clouds obtained visual slam approaches operating vehicle cameras finding approaches directly operate point cloud data highly desirable since avoids costly preprocessing format conversion steps however question best network architecture process unstructured point clouds still largely open paper take inspiration recent pointnet work currently defines state art semantic segmentation pointnet learns higher dimensional spatial feature representation point aggregates points within small volume typically occupancy grid cell order bring form neighborhood context however neighborhood context restricted grid cell processed independently paper investigate possible mechanisms incorporate context point cloud processing architecture focus spatial context identified important semantic segmentation introduce two mechanisms add spatial context existing pointnet first mechanism incorporates neighborhood information processing input data multiple scales multiple adjacent regions together inputlevel context second mechanism operates estimated point descriptors aims consolidating exchanging information larger spatial neighborhood introduction semantic segmentation important capability intelligent vehicles autonomous cars mobile robots identifying semantic meaning observed structure around vehicle prerequisite solving subsequent tasks navigation reconstruction consequently problem attracted lot attention notable successes achieved help deep learning techniques however semantic segmentation approaches operate images naturally lend processing convolutional neural networks cnns processing unstructured point clouds obtained lidar stereo sensors much harder problem recently first successful deep learning approaches proposed task point clouds obtained lidar sensors mounted top recording vehicle context authors contributed equally related work unstructured point clouds varied number sensors setups exist help obtain unstructured point clouds areal data airborne laser scanners laser scanners mounted dynamic setups configuration rotating lasers velodyne static lasers additionally indoor spaces scanned using devices microsoft kinect matterport cameras devices produce point clouds different quality density apply method indoor data synthetic urban outdoor data traditional methods hackel use traditional random forest classifiers features without color method based eigenvalues eigenvectors covariance tensors created nearest neighbors points main contribution efficient approximate nearest neighbors computation different scales munoz follow similar approach replace random forest classifier associative markov network random forest classifiers also used classify data images point clouds later fuse similarly fuse camera lidar sensor data xiong propose sequential parsing procedure learns spatial relationships objects lai introduce hierarchical sparse coding technique learning features synthetic data vosselman combine multiple segmentation methods achieve useful point cloud segmentations methods deep learning context point clouds represented regular volumetric grid order apply convolutions alternatively points mapped representation followed convolutions authors performing convolutions snapshots point cloud project labels back space deep learning framework learns semantic segmentation tracking point clouds use spectral cnns models represented shape graphs shape part segmentation recent methods operate directly raw point clouds fully convolutional layers mlp point features global feature mlp nxm nxd context mechanisms explore several possible realizations compare experimentally results show mechanisms improve semantic segmentation quality contributions key contributions work summarized follows present two mechanisms used incorporate spatial context semantic point cloud segmentation show mechanisms incorporated pointnet pipeline verify experimentally proposed extensions achieve improved results challenging indoor outdoor datasets figure simplified pointnet architecture work build upon pointnet architecture semantic segmentation short computes global feature summarizes set input points specifically network takes points input applies series transformations aggregates point features max pooling global feature global local features concatenated per point class scores returned mlp vertical stack concatenate see text details method section start reviewing pointnet model introduce mechanisms extending context finish describing two exemplary architectures pointnet pointnet deep neural network used semantic segmentation takes input point cloud outputs per point semantic class labels first splits point cloud blocks takes points inside block series mlp per point points mapped higher dimensional space called local applied aggregate information points resulting common invariant input permutations concatenated another series mlps combined features used predict output class scores figure shows simplified model caveats pointnet summarize context single block result aggregated information passed among points inside block context outside block equally important could help make informed class label predictions therefore introduce two mechanisms add context context operates directly input point clouds context consolidates output context context straightforward addition increase context network considering group blocks simultaneously instead one individual block time done pointnet context shared among blocks group groups blocks selected either nxd context mlp output score block feature nxd mlp nxd mlp mlp block feature consolidation unit nxo blocks position different scales nxm mlp max pool stack concatenate figure architecture input blocks consolidation units network takes input three blocks multiple scales one containing points separately scale learns blockfeature similarly pointnet mechanism concatenated appended transformed sequence consolidation units see section network outputs per point scores shaded fields represent position multiple different scales multiscale blocks see figure left neighboring cells regular grid grid blocks see figure left input block compute using mechanism pointnet version train scale individually obtain scaledependent case grid blocks block features computed shared blockdescriptor end approaches output set corresponding input blocks context stage consolidate obtained previous stage differ two consolidation approaches consolidation units consume set point features transform higher dimensional space using mlps apply generate common concatenated high dimensional input features see figure blue box procedure similar mechanism pointnet key point cus chained together sequence cus forming deeper network intuition behind setup follows beginning point sees features appending point additionally informed features neighboring points applying cus multiple times shared knowledge reinforced recurrent consolidation units rcu second type context consolidation employ rcus take input sequence originating spatially nearby blocks return sequence corresponding updated core idea create take consideration neighboring blocks well detail rcus implemented rnns specifically grus simpler variation standard lstms grus capability learn long range dependencies range either time speech recognition space case cells unrolled gru connected unsynchronized fashion see figure blue box means updated returned gru seen whole input sequence intuitively gru retain relevant information scene internal memory update according new observations use memory mechanism consolidate share information across input blocks example decision whether point belongs chair changed network remembers seen table room following describe two exemplary architectures combine previously introduced components provide detailed evaluation report improved results section architecture full architecture displayed figure learned blocks see section concatenated one blockfeature concatenated transformed input passed series cus see section applying final mlp results output scores input point specific architecture sampling procedure select positions blocks randomly pick point input point cloud center blocks group together randomly selected points fall within specified radius procedure repeated point multiple radii output score context nxm shared mlp shared mlp shared mlp nxd nxd mlp nxd nxd grid blocks different positions scale mlp mlp mlp mlp shared shared shared recurrent consolidation unit rcu block feature gru rnn unrolled max pool stack concatenate figure architecture grid input blocks recurrent consolidation unit network takes input four blocks grid structure one containing points learns using mlp weights block passed recurrent consolidation unit see section shares spatial context among blocks returns updated updated blockfeatures appended together original used compute output per point scores shaded fields represent omitted clarity grid architecture figure shows pipeline architecture grid input blocks consists following components input level context group four blocks see section fed series mlps transform point features weights shared among blocks passed rcu updates individual common context neighboring blocks updated concatenated original used along local features class predictions series fully connected layers output class scores computed point experiments experimental evaluation compare two architectures pointnet current semantic segmentation method directly operating point clouds produce quantitative results models baseline two challenging datasets stanford largescale indoor spaces virtual kitti vkitti additionally provide qualitative results point clouds obtained velodyne lidar scanner kitti dataset describe datasets detail stanford large scale indoor scenes dataset composed different large scale indoor areas mainly conference rooms personal offices open spaces contains dense point clouds scanned using matterport camera point labeled one semantic classes listed table using reference implementation pointnet able reproduce results reported see table throughout paper follow evaluation protocol used cross validation areas virtual kitti due lack semantically annotated outdoor datasets rely photorealistic synthetic vkitti dataset closely mimics kitti dataset consists different monocular video sequences urban settings fully annotated depth semantic labels total semantic class listed table purposes project given depth space obtain semantically annotated point clouds conveniently procedure results point clouds resemble varying density real world point clouds obtained velodyne lidar scanners see figure test training split original sequences subsequences final sets created choosing point clouds subsequence regular evaluation also follow cross validation protocol evaluation measures evaluate intersection union iou average per class accuracy overall accuracy intersection union computed iou number true positives number false positives number false negatives quantitative results section analyze effectiveness inputblock schemes consolidation units exemplary two previously introduced models input features differentiate geometry xyz geometry color geometry color first compare gridblocks combination recurrent consolidation block original pointnet using evaluation setup described able show improved results pointnet see table table proves hypothesis rcu able convey context among blocks thus improving results training room split blocks ground plane block extends whole room height neighboring blocks overlapping meters directions select four blocks simultaneously see figure left block contains points unrolled gru cells long input output memory size testing room split blocks evaluated groups blocks block evaluated next take look input block consolidation units model build multiscale blocks follow process described section radii choose distance metric choose generates axisaligned rectangular blocks middle scale block equal pointnet block regarding shape size using sampling necessary block construction diverge previous training procedure experiments new conditions validate influence architecture components adding pipeline evaluating step see table table first consider context block feature input pipeline skipping consolidation units shows performance benefits pointnet much one would expect figure qualitative results laser point clouds dataset velodyne laser scans kitti raw trained model vkitti point clouds without color applied laser point clouds far classes like road building car give decent results considering enlarged input context next take input blocks add one consolidation unit results show outperforms input blocks also shows cus provide simple technique boost network performance finally combine blocks appending another network full model depicted figure geometry input point described feature vector spatial coordinates point color normalized coordinated based size environment see details without doubt color strong input feature context semantic segmentation section pose question happen color information available like case point clouds obtained laser scanners simulate missing colors simply discard color information input feature experiments table show obtained results see caption discussion results qualitative results present qualitative results models applied indoor scenarios figure outdoor results figure along short discussion additionally applied model vkitti laser data results shown figure figure conclusion figure train network synthetic point clouds generated vkitti left apply onto realworld velodyne lidar point clouds right structure varying density comparable work investigated question incorporate spatial context neural network architecture semantic segmentation building upon pointnet proposed two extension context outputlevel context successfully applied onto indoor outdoor datasets still numerous combinations remain possible full exploration design space left future work floo wal bea colu win dow doo tab cha sofa boo boa clut ter kcas ceil pointnet rcu pointnet rcu mean iou dataset boa clut car van kcas boo truck sofa traffi misc traffi csign ail guar cha tab pole doo light win dow colu road bea vkitti dataset pointnet ing build wal ion vege tat mean iou floo tree pointnet mean iou ceil dataset terra ing table iou per semantic class dataset input features compare models different components original pointnet baseline adding different components see improvement mean iou obtain results mean iou individual class iou entries marked use random sampling input block selection instead discrete positions regular grid table iou per semantic class vkitti datasets xyz input features color methods outperform pointnet consistently two datasets improvements mean iou also considerable color available suggests network architectures able learn improved geometric features robust varying point densities occur outdoor vkitti dataset mean iou overall accuracy avg class accuracy dataset rgb pointnet vkitti dataset rgb pointnet table vkitti datasets xyz input features without rgb show improved results indoor outdoor vkitti datasets presented mechanisms even important color available dataset mean iou overall accuracy avg class accuracy pointnet rcu pointnet rcu table dataset input features comparison different context expansion techniques see sections singlescale grid consolidation unit rcu recurrent consolidation unit entries marked use random sampling input block selection instead discrete positions regular grid ceiling floor input wall beam column pointnet window door table chair sofa bookcase board clutter ground truth figure indoor qualitative results dataset input features left right input point cloud baseline method pointnet results using model see figure results using model see figure ground truth semantic labels models produce consistent less noisy labels terrain tree vegetation building road car truck van guardrail trafficsign trafficlight pole misc pointnet ground truth figure outdoor qualitative results dataset virtual kitti results obtained using xyz coordinates input color information used left baseline method pointnet center results using model illustrated figure right ground truth semantic labels outputs method less fragmented cars houses finer structures like street lights poles recognized better tree grass topiary ground obstacle unknown acknowledgment grateful colleagues providing valuable feedback paper fruitful discussions especially umer rafi paul voigtlaender work supported erc starting grant project cvsuper ground truth labels top prediction labels terrain tree vegetation guardrail trafficsign trafficlight figure qualitative results challenge trained model vkitti point clouds without color applied laser data training test datasets semantic labels despite common classes like trees successfully segmented plausible ones given otherwise terrain instead grass guardrail instead obstacle references armeni sener zamir jiang brilakis fischer savarese semantic parsing largescale indoor spaces cvpr boulch saux audebert unstructured point cloud semantic labeling using deep segmentation networks eurographics workshop object retrieval chen papandreou kokkinos murphy yuille deeplab semantic image segmentation deep convolutional nets atrous convolution fully connected crfs arxiv preprint cho van merrienboer bougares schwenk bengio learning phrase representations using rnn statistical machine translation emnlp engelmann leibe joint object pose estimation shape reconstruction urban street scenes using shape priors proc german conference pattern recognition gcpr engelmann leibe samp shape motion priors vehicle reconstruction wacv gaidon wang cabon vig virtual worlds proxy tracking analysis cvpr geiger lenz stiller urtasun vision meets robotics kitti dataset ijrr hackel savinov ladicky wegner schindler pollefeys new point cloud classification benchmark arxiv preprint hackel wegner schindler fast semantic segmentation points clouds strongly varying density isprs hochreiter schmidhuber long memory neural computation huang point cloud labeling using convolutional neural network icpr kasyanov engelmann leibe online slam relocalization iros klokov lempitsky escape cells deep recognition point cloud models arxiv preprint lai fox unsupervised feature learning scene labeling icra long shelhamer darrell fully convolutional networks semantic segmentation cvpr maddern pascoe linegar newman year oxford robotcar dataset ijrr maturana scherer voxnet convolutional neural network object recognition iros mottaghi chen liu cho lee fidler urtasun yuille role context object detection semantic segmentation wild cvpr munoz vandapel hebert directional associative markov network point cloud classification nathan silberman derek hoiem fergus indoor segmentation support inference rgbd images eccv noh hong han learning deconvolution network semantic segmentation iccv ondruska dequaire zeng wang posner tracking semantic segmentation using recurrent neural networks rss workshop limits potentials deep learning robotics pohlen hermans mathias leibe fullresolution residual networks semantic segmentation street scenes cvpr guibas pointnet deep learning point sets classification segmentation cvpr niener dai yan guibas volumetric cnns object classification data cvpr vosselman point cloud segmentation urban scene classification isprs shen van den hengel semantic segmentation using deep fully convolutional networks arxiv preprint xiong munoz bagnell hebert scene analysis via sequenced predictions points regions icra davoine bordes zhao denoeux information fusion oversegmented images application urban scene understanding mva guo guibas syncspeccnn synchronized spectral cnn shape segmentation arxiv preprint zhang candra vetter zakhor sensor fusion semantic segmentation urban scenes icra
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apr probabilistic interpretation filtering application polynomial systems polytopic bounding alessio benavoli dario piga idsia dalle molle institute artificial intelligence manno switzerland imt institute advanced studies lucca piazza san francesco lucca italy abstract estimation usually formulated context calculus probabilistic calculations necessary paper show estimation equivalently formulated probabilistic setting employing sets probability measures inference estimation thus carried computing expectations respect updated set probability measures probabilistic case particular shown inference performed solving particular linear programming problem special case truncated moment problem order moment known support writing dual linear programming problem shown nonlinearities measurement process equations polynomial bounding sets initial state process measurement noises described polynomial inequalities approximation linear programming problem efficiently obtained using theory polynomial optimization derive smart greedy procedure compute polytopic true computing polytope set includes means computed respect key words state estimation filtering estimation set probability measures polynomials introduction inferring value state dynamical system various time instants classical problem control estimation theory state estimated based noisy signal observations state transition model turn affected two sources uncertainty namely process disturbance uncertainty initial state conditions literature two main approaches dealing uncertainties noises acting system stochastic probabilistic approach assumes noises uncertainties unknown described known probability distributions approach assumes noises uncertainties unknown bounded compact sets probabilistic approach grounded bayesian filtering whose aim update measurements paper presented ifac meeting corresponding author alessio benavoli tel email addresses alessio alessio benavoli dario piga preprint submitted automatica propagate time probability density function pdf state inferences carried computing expectations respect pdf mean variance credible regions well known linear discretetime dynamical systems corrupted gaussian noises bayesian filter reduces kalman filter approach instead based construction compact set guaranteed include state values system consistent measured output assumed bounds compact set propagated time updated recursively output observations estimation computing inferences thus means determine compact set setmembership estimation first proposed ellipsoidal bounding state linear dynamical systems computed application ellipsoidal sets state estimation problem also studied authors example independently communications signal processing community starting works order improve estimation accuracy use convex polytope instead ellipsoid proposed unfortunately polytope may extremely complex november bounding region well probabilistic moments mean variance noises able deal gaussian measurement noise bounded known moments process noise etc moreover allow compute credible regions bayesian confidence intervals takes account deterministic probabilistic uncertainty well allows make decisions choosing action minimizes expectation loss function important instance control design context paper first attempt combining deterministic probabilistic uncertainty proposed proposed joint zonotopic gaussian kalman filter ltv systems simultaneously subject bounded disturbances gaussian noises work instead proposes bayesian approach estimation imposing uniform distribution similar idea proposed show approach different estimation since estimation interpreted bayesian framework framework set probability measures second probabilistic interpretation inferences estimation carried computing expectations respect set probabilistic case particular show membership set set includes state guarantee obtained computing union supports probability measures moreover prove minimum volume convex simply obtained computing set includes means computed respect probabilities proof constructive hence convenient description however show determine least conservative includes solving linear programming problem problem special case truncated moment problem order moment known support third writing dual linear programming problem show nonlinearities measurement process equations polynomial bounding sets initial state process measurement noises described polynomial inequalities feasible solution dual obtained simply checking polynomial compact set described polynomial inequalities approximation linear programming problem obtained reformulating semidefinite programming using theory sos polynomial optimization prove approximate solution robust sense computed guaranteed include membership set fourth provide procedure determine minimumvolume polytope bounding procedure based refinement algorithm originally proposed compute approximation polytope containing given semialgebraic set particular use monte carlo integration approach compute approximation volume polytope greedy ing polytopic updating algorithms may require excessive amount calculations storage without approximations number vertices polytope increases exponentially time reason suggested outer approximate true polytope simpler polytope possessing limited number vertices equivalently faces respect parallelotopic approximation set presented parallelotope generalisation parallelogram bounding parallelotopes used estimate state dynamical system polynomial complexity zonotopes proposed reduce conservativeness parallelotopes intuitively zonotopes polytopes parallel faces precise definition see parallelotope thus special zonotope zonotopes used build state bounding observer context linear discrete systems zonotopes also employed address problem estimation systems bounded description noise uncertainties sample time guaranteed bound uncertain state trajectory system calculated using interval arithmetic applied nonlinear functions mean interval extension theorem outer bound represented zonotope similar approaches estimation nonlinear systems presented ellipsoids used instead zonotopes recently randomized methods used approximate probabilistic guarantees uncertain state trajectory polynomial sublevel sets aim paper address problem estimation state dynamical system characterized polynomial initial state noises unknown bounded compact sets defined polynomial inequalities therefore context estimation address problem different way approaches presented reformulate setmembership probabilistic setting solve using theory moments positive polynomials precisely contributions following first exploiting recent results filtering sets probability measures show estimation equivalently formulated probabilistic setting employing sets probability measures particular show prediction updating steps estimation obtained applying equation bayes rule elements set probability measures unifies probabilistic approach bayes filter setmembership approach state estimation result enormous impact finally allow combine classical probabilistic uncertainty order obtain hybrid filters stochastic probabilistic filters instance able use information therefore cedure determine polytope intersection number added description minimize volume polytope including allows solve estimation problem polynomial systems efficiently convex optimization finally means numerical example involving lotka volterra model show effectiveness approach polynomial functions variable sets described similar manner paper addresses filtering problem aims recursively estimating time sample outer approximation state uncertainty set defined set values compatible available information namely system equations bounds initial state noises output observations formally filtering problem defined follows state system time measured output vector process noise measurement noise paper consider polynomial compact basic sets compact sets described polynomial inequalities consider uncertain dynamical system described difference equations assume available information initial state noises problem description problem filtering given system equations observations bounding sets noises initial state uncertainty set compute recursively state uncertainty set defined xdn vector monomials degrees less equal dimension known coefficient matrices resulting system referred paper uncertain polynomial system degree note general sets might nonconvex representation become complicated time index increases assumption bounded algorithms computing simple sets boxes parallelotopes zonotops ellipsoidal regions outerbounding state uncertainty sets proposed reduce complexity formulating setmembership filtering problem probabilistic setting paper presents algorithm computing approximation polytope sets example let consider polynomial system output equation given rewrite system probabilistic framework estimation estimation usually formulated context calculus show following upper bound expectation given solution optimization problem paragraph estimation equivalently formulated probabilistic setting employing sets probability measures consider constraint time index dropped brevity notation constraint translated probabilistic setting saying probabilistic information value variable belongs set equivalently sup problem determining upper bound expectation respect probability measure given knowledge support special case truncated moment problem order moment known support hence following result lemma support uniquely define probability measure indefinite number probability measures support hence equivalent constraint probability measure belongs set set probability measures variable support let define cumulative distribution function cdf probability measure instance definition easily extended easily characterize set probability measures follows proposition optimum obtained atomic measure arg note fact associated cdf linear program since finite number constraints infinite dimensional variable probability measure note use sup instead max indicate optimal solution might attained lower bound expectation obtained replacing sup inf probability measure precisely nonnegative borel measure words means know support probability measure variable definition supremum first integral must understood lebesguestieltjes integral respect cumulative distribution atomic measure means denotes distributional derivative cumulative distribution atomic measure case dirac measures hence second integral result follows probability measures gives lower upper bounds expectation atomic discrete measures order formulate filtering problem probabilistic framework useful exploit result derived karr proven set probability measures feasible semiinfinite linear program problem convex compact respect topology result expressed convex hull extreme points according proposition extreme points atomic measures integral integral respect hence equivalence borel probability measures cumulative distributions hereafter use interchangeably inference state state estimation interested making inferences equivalently computing expectations realvalued functions since indefinite number probability measures support compute single expectation however compute upper lower bounds expectation respect probability measures support instance clarify aspect consider experiment rolling dice assume probability outcomes dice completely unknown knowledge experiment equivalently therefore statement model epistemic uncertainty probabilities dice outcomes know sample space considering borel assumed element uniform distribution one one considering uniform distribution loose full equivalence means equivalent terms inferences expectations summing obtained far atomic measure measure accepts argument subset returns zero otherwise constraint equivalent inferences equivalent convex hull atomic measures applying equation probability measures obtain hence derive prediction updating step estimation applying chapmankolmogorov equation bayes rule set probability measures means reformulating constraints probabilistic way reformulate estimation realm stochastic probabilistic filtering applied set probability measures denotes indicator function exploited fact definition theorem follows propagating time updating set distributions start deriving filtering prediction step applying equation theorem prediction consider system equation assume probabilistic knowledge support follows probability measure value state time belongs set theorem shows applying chapmankolmogorov equation probability measures obtain set probability measures completely defined support whose support coincides one obtained estimation prediction step derive similar result updating step theorem updating consider measurement equation assume probabilistic knowledge described follows updated probability measures value state time belongs set equivalently pxk proof let consider time instant system equation follows proof observe time conditional set probability measures variable given value variable hence since set probability measures variable updating step consists applying bayes rule probability measures zero otherwise rrm initialize defined pxk defined steps prediction updating steps respectively note set probability measures pxk computed taking account observations respectively hence correctly denoted pxk respectively omitted notation brevity hence inequality holds chosen time bayes rule defined probability measures denominator strictly positive implies equality must satisfied equality satisfied together constraint implies remark assumptions set semialgebraic set described intersections semialgebraic sets formally projection space set constraint follows thus denominator equal one hence algorithm prediction updating note probability point nonzero since atomic measure order apply bayes rule need ensure denominator strictly greater zero given equivalently words support probability measure value state given output observation system equations nothing accordance formulation claims belongs state uncertainty set defined solve filtering applying recursively theorems described algorithm exploited fact augmented state vector polynomial functions equivalently defining rest paper use following notation describe set remark reformulation probabilistic framework important two main reasons first allows reinterpret operations performed estimation justifies terms probabilistic framework seen reinterpretation prediction updating terms equation bayes rule investigate interpretation next sections particular section show convex membership set computed estimation also interpreted set posterior means calculated hence updated probability measure values state time proves theorem theorem support updated probability measure value state time given way updating set probability measures proposed walley appendix name regular extension theorem assume compact convex set defined follows respect posterior set probability measures pxk bayesian setting know posterior mean optimal estimate respect quadratic loss function similar result holds set posterior means result also applied estimation probabilistic interpretation able compute expectations moreover section also highlight connection estimation theory moments duality second potentially able combine setmembership classical probabilistic uncertainty order obtain hybrid filters stochastic probabilistic filters instance able use information bounding region well probabilistic moments mean variance noises able deal gaussian measurement noise bounded known moments process noise etc first attempt direction described scalar systems plan investigate direction future work using theory sos polynomial optimization also use next sections arg results pxk proof follows minimum volume convex set includes thus convex hence equality immediately follows conversely assume convex since minimum volume convex set includes must equal means exist definition convex hull consider probability measure holds pxk conv probabilistic formulation filtering available information time encoded posterior probability distribution state given observations setting information encoded updated set probability measures pxk inferences expressed terms expectations computed respect set estimation problem instance reformulated follows pxk computing support inference set probability measures arg minn inf pxk solution set probability measures pxk support thus coincides since may convex problem general difficult solve however problem simplified restricting convex thus computing convex thus belongs vice versa theorem following fundamental implications convex set possible means computed respect probability measures pxk set also convex outerbounding support set probability measures pxk tightest convex support set probability distributions pxk set means computed respect probability measure pxk following theorem shows computing minimumvolume convex set equivalent obtain set includes possible means computed respect probability measure pxk thus union supports probability measures pxk equivalently including obtained max xdp thus use algorithm therefore modified include following additional steps refinement algorithm step defined redefine pxk proof let xdp point belonging let first prove first note sup xdp unfortunately theorem provide constructive way find set however restricting outerapproximation support simple form polytope theorem still exploited determine set following theorem provides results compute box therefore means xdp also belongs thus contains choosing obtain tightest defined normal vector includes theorem box approximation minimum volume box includes found solving following family optimization problems opt observed reduces element natural basis note problem optimization variables amount mass assigned point measure objective function constraint linear optimization variables therefore linear program infinite number decision variables finite number constraints exploiting duality linear program see instance write dual defined byrselecting opt min max obtain ther respectively define box proof theorem provided together proof theorem based theorem computing lower upper means components vector tightest box obtained following discuss efficiently solve optimization problems similar find less conservative box simplicity notation rest paper dependence state set time index dropped used necessary inf also linear program finite number decision variables infinite number constraints solution feasible problem provided hence checking feasibility equivalent check polynomial set exploiting duality remark probabilistic formulation setmembership estimation described far general enough valid also dynamical system polynomial system uncertainty sets semialgebraic compact sets assumptions polynomiality used following efficiently solve linear programming problem convex optimization section discuss efficiently solve optimization problems similar particular slightly modify order able determine general xdp polynomials theorem let fix normal vector defining halfspace tightest including sufficient condition polynomial semialgebraic set written terms sos polynomials see lies space means definition polynomial degree polynomial denoted written qqd linear equalities matrix coefficients besides enforcing sum square polynomials leads linear matrix inequality lmi constraints coefficients robust constraint appearing problem guaranteed satisfied matter fact definition furthermore sos polynomials always nonnegative thus left right side equation problem nonnegative since equality constraint gives sufficient condition follows therefore conservativeness introduced solving instead highlighted corollary however according putinar positivstellensatz see polynomial nonnegative compact semialgebraic set exactly always written combination sos polynomials provided degree sos polynomials large enough words make close increasing degree sos however practice often happens relaxed solution optimal one coincide small values sos degree real symmetric positive semidefinite matrix dimension vector monomials defined set sos polynomials degree less equal denoted given integer sufficient condition see instance polynomial nonpositive inequality constraints defining semialgebraic set order avoid confusion would like stress also polynomial variable fact remind augmented state defined following conservative optimization problem solved instead inf corollary set guaranteed belong halfspace proof proof straightforwardly follows theorem note rewriting sos polynomials problem also rewritten example let consider polynomial system described difference equations inf output equation given following conditions assumed initial state belongs process noise bounded measurement noise observed output time interested computing containing state uncertainty set equivalently time normal vector characterizing fixed equal order compute constant parameter defining sdp problem remarks problem semidefinite programming sdp problem thus convex fact checking polynomial equal leads consider following family goal choose normal vectors along constant parameters defining polytope minimum volume words also normal vectors optimized formulate problem aim solve fig true state uncertainty set dark grey region light gray region inf constrained polytope two main aspects making challenging problem polytope generic compact set might exist instance ellipsoid convex hull described infinite number namely supporting hyperplanes every boundary point problem computing exact volume polytope see interested reader also referred details problems although several algorithms proposed literature compute volume polytope triangulation gram relation laplace transform randomized methods approaches mentioned require exact description polytope terms vertex representation however case parameters defining unknown determining part problem solved sos degree sostools used easily handle sos polynomials appearing cpu time taken solver sedumi compute solution sdp problem intel pentium ram seconds computed plotted fig along true state uncertainty set according theorem corollary included note also although original robust optimization problem replaced sdp problem computed parameter defining hyperplane almost tangent set thus small level conservativeness introduced using sos following paragraph present greedy algorithm evaluate approximation polytope set approximation objective function already pointed previous paragraph one main problems solving analytical expression computation volume polytope available polytope unknown computing part problem order overcome problem monte carlo integration approach used approximate volume specifically given box set computed discussed theorem sequence random points independent uniformly distributed computation polytope containing previous section given normal vector defining shown compute convex optimization constant parameter integral algorithm polytopic outer approximation input list random points uniformly distributed box set compute defined contains minimum number points list included approximated volume box indicator function unknown polytope defined otherwise minn corresponding objective function problem new minimizes approximation volume polytope generated order approximate volume points list belong polytope discarded points belonging collected new list volume step approximated procedure repeated step means number samples belonging polytope equal number samples belonging polytope note constraint appearing optimization problem guaranteed contain set thus outer approximation finally would like remark case interested also bounding maximum number defining polytopic outer approximation algorithm stopped specified number iterations probability finite samples level accuracy approximation depends shape set well volume outer box reader referred details monte carlo integration methods basis volume minimization problem approximated min collect points belonging halfspace computed list let number elements step otherwise set step define polytope output polytope expectation taken respect random variable furthermore strong law large numbers remark worth remarking lim following subsection describe greedy procedure aiming computing approximation minimization problem greedy approach solving key steps approach proposed section compute polytopic set summarized algorithm example let consider example first steps algorithm visualized fig box true state uncertainty set dark gray region first computed fig set random points black dots uniformly distributed generated fig containing true state uncertainty set minimum number points computed points belong discarded gray dots fig new containing true state uncertainty set algorithm generates sequence follows first minimizes approximation volume polytope computed approximation due rthe fact volume given integral appn proximated constant minimum number black dots computed fig points belong discarded gray dots fig procedure terminates black points discarded rhj ihj fig indicator function ihj black solid line approximate function gray thin line overlapped equal tions convex functions minn theorem exists least one point list optimal solution problem hyperplane supporting hyperplane set fig first steps algorithm proof theorem proved contradiction let technical details step core algorithm provided following sections defined let suppose feasible solution problem supporting hyperplane approximation indicator functions let define note still feasible solution problem let value cost function note objective function problem noncontinuous nonconvex since sum indicator functions defined problem obtained given transform convex objective function indicator function approximated convex function defined similarly let value cost function problem obtained term given plot functions given fig since also problem thus relaxed replacing indicator equal zero hand following sdp problems equal either zero arg minn basis considerations follows since hypothesis exists least one point list follows therefore optimal solution problem contradicts hypothesis denoting first component vector optimizer problem given pair provides minimum value objective function remark fixed degree sos polynomials number optimization variables problems increases polynomially state dimension linearly number constraints defining set specifically number optimization variables problem fact number free decision variables matrices hand fixed size matrices increases exponentially degree sos polynomials order obtain conservative results practical experience authors suggests take denotes ceiling operator remind degree considered polynomial system roughly speaking memory requirement issues relaxed sdp problems solved commercial workstations general purpose sdp solvers like sedumi case polynomial systems state variables degree greater systems state variables considered case smaller values similarly systems higher degree considered case smaller number state variables constraints handled sosbased approach already discussed section specifically introducing sos relaxation problem replaced arg minn handling constraint theorem following interpretation among defined normal vector containing set optimization problem provides minimizes volume rof polytope even integral approximated constant indicator functions replaced convex functions arg minn note already discussed section constraint satisfied therefore guaranteed contain thus also set included finally note order deal nonconvex constraint problem splitted two remark already discussed algorithm computes iteration containing set thus also parameters computed solving problem replacing robust constraint sos constraint see problem note principles algorithm relaxation discussed section used compute instead complex semialgebraic set ellipsoid described polynomial inequality fig final polytope running algorithm approximation robust constraint pmconvex conservative constraint vector monomials variable parameters computed properly modifying problem instance case interested computing ellipsoidal outer approximation function quadratic form hessian enforced positive definite latter source approximation reduced increasing degree sos polynomials fact already discussed section according putinar positivstellensatz function written provided degree sos polynomials large enough hand theoretical result concerning accuracy approximation indicator functions problem convex functions appearing problem polytope obtained solving convex problems guaranteed minimize original nonconvex optimization problem algorithm used refine polytopic outer approximation provided algorithm example let continue example fig shows polytope obtained applying algorithm solving problems instead nonconvex optimization sdp problems solved degree sos polynomials equal solution polytope observed approximations introduced sos approximation indicator functions necessary efficiently solve optimizations bounding tangent computed region still include two black points therefore computed polytope polytope however already good next section describe refinement algorithm aiming computing tighter polytope according steps algorithm next time step setmembership filter repeat procedure compute new polytope difference instead linear inequalities define polytope fig procedure repeated recursively time main principle algorithm process one one points belonging polytopic initially given algorithm points including set containing point equivalently seeked way points belong minimum volume polytopic outer approximation discarded thus tighter complex polytopic outer approximation obtained refinement polytope important feature enjoyed refined polytope given following theorem summarizing approximate solution robust optimization problem computed solving convex sdp problems basis algorithm approximation set defined note solving instead two different sources approximation introduced theorem polytope computed algorithm global minimizer problem proof let polytope belonging set feasibility problem minimize means exists polytope point given input algorithm approximation indicator functions convex functions see fig algorithm refinement polytope input sequence random points provided input algorithm let number points belonging compute solution following optimization problem min fig exampe hyperplanes defining polytope black lines true state uncertainty set gray region problems format required used sdp solver numerical examples output polytope let consider lotka volterra preypredator model described difference equations thus optimal solution problem let defined obviously besides output algorithm contained hyperspace therefore since follows point polytope minimize optimization problem output algorithm denote prey predator population size respectively example following values parameters considered observed output sum population prey predator densities theorem mainly says exists polytope including containing less randomly generated points however worth remarking approximated solution problem computed robust constraint appearing handled techniques described previous section thus conservativeness could added step therefore main interpretation given theorem algorithm cancels effect approximating indicator function convex function measurement noise bounded initial prey predator sizes known belong box noise process bounded data obtained simulating model initial conditions corrupting output observations random noise uniformly distributed within interval example let continue example fig shows computed polytope along true state uncertainty set cpu taken proposed algorithm compute define polytope seconds however seconds spent solver sedumi solve sdp problems type seconds required sostools interface formulate times sdp problems format used sedumi therefore computational time required compute polytope drastically reduced using efficient sdp solvers also directly formulating sdp polytopic outer approximations state uncertainty sets computed algorithm random points used approximate volume polytope described section order limit complexity description polytopes maximum number halfspaces describing set means algorithm stopped iterations remind initial box already described fig example polytopes gray true state trajectory black dots fig example boxes gray true state trajectory black dots output algorithm polytope described less algorithms used refine polytopic outer approximation fig shows computed polytopes outer approximating state uncertainty sets along true state trajectory hybrid toolbox used plot polytopes fig average cpu time required compute polytope seconds including time required sostools interface formulate sdp problems format used solver sedumi sake comparison fig shows approximations state uncertainty sets boxes instead polytopes propagated time better comparison fig bounds state variable obtained propagating boxes polytopes plotted obtained results show expected propagating polytopic uncertainty sets instead boxes provides accurate state estimation finally would like remark small uncertainty noise process assumed since larger bounds would possible clearly visualize uncertainty boxes fig time fig example bounds state trajectories obtained propagating boxes black line bounds state trajectories obtained propagating polytopes gray line true state trajectory black dots lem obtained using theory polynomial optimization finally derived procedure compute polytopic true computing polytope set includes means computed respect worth remarking filtering approach discussed paper extended handle input signal observations uncertainty model parameters provided corresponding state uncertainty set remains set future works aim first speed proposed state estimation algorithm order able use applications systems fast dynamics aim dedicated numerical algorithms written fortran solving formulated sdp conclusions paper shown estimation equivalently formulated probabilistic setting employing sets probability measures inferences estimation thus carried computing expectations respect updated set probability measures probabilistic case formulated linear programming problem shown nonlinearities measurement process equations polynomial bounding sets initial state process measurement noises described polynomial inequalities approximation linear programming mization problems developed furthermore sdp problems directly formulated format required sdp solver thus avoiding use interfaces like sostools open source toolbox released second exploiting probabilistic interpretation setmembership estimation plan reformulate using theory moments developed lasserre allow ground totally estimation realm probabilistic setting give possibility combining two approaches order obtain hybrid filters filters include classical probabilistic uncertainties uncertainties walter results recursive polyhedral description parameter 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evolving deep convolutional neural networks image classification yanan sun bing xue mengjie zhang oct school engineering computer science victoria university wellington box wellington new zealand input data corresponding label respectively denotes architecture choosing function given data refers initialization method connection weights weight based chosen architecture measures differences true label label predicted cnn using weight typically gradient descend approaches stochastic sgd utilized minimize weight within given number epochs connection weight values iteratively updated although differentiable occasions methods preferred due effectiveness good scalability number connection weights increases cnn commonly huge number connection weights however countable functions discrete neither convex concave well addressed practice furthermore optimizers heavily dependent initial values weights including biases essential choose suitable help consecutive approaches escape local minima furthermore performance assigned architectures evaluated minimization finished minimization progress multiple iterations turn increases difficulty choosing ntroduction potential optimal therefore architecture design convolutional neural networks cnns demonstrated connection weight initialization strategy carefully exceptional superiority visual recognition tasks treated cnns traffic sign recognition biological image segmentatypically architecture design approaches tion image classification cnns originally initially developed deep learning algorithms early motivated computational model cat visual cortex date stacked saes specializing processing vision signal related tasks deep belief networks dbns grid search since proposed random search gaussion implementation cnns whose connection weights process bgp parzen estimators optimized algorithm various tpe sequential global optimizavariants cnns developed vggnet tion smbo theoretically exhaustively tests resnet variants significantly improve clascombinations parameters expectedly seize best sification accuracies best rivals image classification one however evaluate combinations within tasks diverse variants cnns differ architectures acceptable time reality moreover difficult optimize weight connections parameters continuous values could reduce mathematically cnn formulated exhaustive adverse absolute random severely context image classification task input challenges sampling behavior search space addition bgp incurs extra parameters kernels architecture weight architecture arduous tune tpe treats parameter independently key parameters cnns dependent minimize weight computation methods successfully applied neural networks since two decades ago methods scale well modern deep neural networks due complicated architectures large quantities connection weights paper propose new method using genetic algorithms evolving architectures connection weight initialization values deep convolutional neural network address image classification problems proposed algorithm efficient gene encoding strategy designed represent different building blocks unpredictable optimal depth convolutional neural networks addition new representation scheme developed effectively initializing connection weights deep convolutional neural networks expected avoid networks getting stuck local minima typically major issue backward optimization furthermore novel fitness evaluation method proposed speed heuristic search substantially less computational resource proposed algorithm examined compared existing algorithms nine widely used image classification tasks including methods experimental results demonstrate remarkable superiority proposed algorithm algorithms terms classification error rate number parameters weights index algorithms convolutional neural network image classification deep learning convolutional layer size stride details seen subsection methods mentioned shown good performance saes dbns suitable cnns success saes dbns due largely architecture construction approaches greedy stacking group building blocks structures neural networks building block methods utilized optimizing parameters number neurons corresponding layer however cnns method applied due architecture characteristics routine must confirm entire architectures time furthermore multiple different building blocks exist cnns different orders would result significantly different performance therefore architecture design cnns carefully treated recently baker proposed architecture design approach cnns based reinforcement learning named metaqnn employed graphic processing unit gpu cards days experiments dataset due drawbacks existing methods limited computational resources available interested researchers works cnns typically performed experts rich domain knowledge genetic algorithms gas paradigm evolutionary algorithms require rich domain knowledge adapt pattern motivated process natural selection optimization problems gas preferred various fields due characteristics insensitivity local minima promising properties collectively achieved repeated series selection mutation crossover operators therefore naturally utilized optimization architecture design connection weight initialization cnns indeed gas evolving neural networks traced back yao presented survey different approaches largely optimization connection weights fixed architecture topologies stanley miikkulainen proposed neuronevolution augmenting topology neat algorithm evolve architecture connection weights small scale neural network afterwards hyperneat neat combined compositional pattern producing networks proposed evolve larger scale neural network indirect encoding strategy motivated hyperneat multiple variants proposed evolve even larger scale neural networks however major deficiencies approaches suitable evolving deep neural networks global connection single building blocks saes dbns cnns local connection exists multiple different building blocks need evolved simultaneously hybrid weigh connections connections two layers may evolved contrast architectures cnns indeed views many years evolutionary algorithms incapable evolving architecture connection weights cnns due tremendous number related parameters recently google showed large evolution image classification leic method specializing architecture optimization cnns letc materialized gas without crossover operator implemented computers archives competitive performance dataset training days actually directly using gas architecture design cnns several issues would raise nature best architecture unknown performance received based however evaluating performance one individual takes long time appears severely entire population would require much computational resources speeding evolution optimal depth cnns one particular problem unknown therefore hard constrain search space architecture optimization regard variablelength gene encoding strategy may best choice assign crossover operation different building blocks newly resulted problem weight initialization values heavily affect performance confirmed architecture addressing problem involves good gene encoding strategy optimization hundreds thousands decision variables goal aim paper design develop effective efficient method automatically discover good architectures corresponding connection weight initialization values cnns first two formulae without manual intervention achieve goal objectives specified design flexible gene encoding scheme architecture constrain maximal length building blocks cnns gene encoding scheme evolved architecture expected benefit cnns achieve good performance solving different tasks hand investigate connection weight encoding strategy capable representing tremendous numbers connection weights economy way encoding approach weight connection initialization problem cnns expected effectively optimized proposed develop associated selection including environmental selection crossover mutation operators cope designed gene encoding strategies architectures connection weights propose effective fitness measure individuals representing different cnns require intensive computational resources input feature maps feature maps feature maps feature maps full connection layer flatten convolution pooling convolution pooling fig general architecture convolutional neural network investigate whether new approach significantly outperform existing methods classification accuracy number weights organization feature map input data padding zeros convolutional operations categorized two types valid without padding padding specifically element feature map sum products element filter corresponding elements filter overlaps input data multiple channels say one feature map also require different filters filter convolves channel results summed reminder paper organized follows filter feature map background cnns related works architecture stride input input design weight initialization approaches cnns reviewed section framework details fig illustration convolutional operation step proposed algorithm elaborated section iii experiment design experimental results proan example valid convolutional operation illusposed algorithm shown sections respectively trated fig filter width height next discussions made section finally equal input data size conclusions future work detailed section vii stride height width equal shown fig generated feature map size background elated ork shadow areas input data different colors refer architecture convolutional neural network overlaps filter different positions input bone cnns fig exhibits extensive architecture data shadow areas feature map respective one cnn two convolutional operations resulted convolutional outcomes numbers filter two pooling operations resulted four groups feature connection weight values generally convolutional results maps full connection layer tail last layer regarding filter updated adding bias term full connection layer receives input data nonlinearity rectifier linear unit flattening elements fourth group feature maps relu stored feature map obvigenerally convolutional layers pooling layers ously involved parameters one convolutional operation mixed stack together head architecture filter width filter height number feature full connection layers constantly stacked maps stride width stride height convolutional tail architecture numbers fig type connection weight filter refer sizes corresponding layer particularly pooling intuitively pooling operation resembles input data output convolution operation except product numbers denote feature map configurations resulted values corresponding feature map example implies feature maps briefly pooling operation employs predefined window size kernel collect average value maximal following details convolutional layer value elements slides slide size pooling layer associated convolution also called stride convolutional operation pooling operations respectively documented better understanding example pooling operation detail full connection layer intended illustrated fig kernel size describe stride width height input data convolution given input image size size shadows different colors order receive feature map generated refer two slide positions resulted pooling values convolutional operations filter must defined advance example maximal pooling operation employed actually filter also simply seen matrix evidently pooling operation involved parameters randomly initialized predefined size filter width kernel width kernel height stride width filter height filter travels leftmost stride height pooling type rightmost input data step size equal stride stride width travels cnn architecture design algorithm subsection leic concerned due moving downward step size equal stride stride height reach right bottom input specific intention architecture design cnns image depending whether keep sizes end review algorithm point first initialization method second initialization approach shortage first one solved major culty exists choosing parameter distribution range uniform distribution mean value kernel stride feature map well standard derivation gaussian distribution input input solve problem xavier initializer presented range uniform sampling based neuron saturation prior using fig illustration pooling operation sigmoid activation function supposed number neurons two adjacent layers values limitations used highlight necessity weights connecting two layers initialized within corresponding work proposed algorithm range uniformly leic employed evolve architectures cnns sampling although xavier initializer works better individuals evolved scratch initialization methods two categories couple ual encoded chromosome major issues exist highly relies architectures phase evolution different kinds layers cnns particularity number neurons layer porated individuals mutation operation networks formulation optimal expectation individuals promising performance architectures networks found resulted generated note crossover operation initialized parameters perform badly well designed local search gas investigated way evaluate desired performance architectures main part leic without crossover operator may mislead adjustment architectures gas typically require large population size reducing xavier initializer presented usage sigmoid adverse impact therefore letc adopted population activation widely used activation function cnns size magnitude order relu general setting community addition best knowledge computers employed leic ing evolutionary algorithm searching connection caused fitness assignment approach utilized leic weight initialization deep learning algorithms including leic individual evaluated final image cnns main reason tremendous numbers weights classification accuracy typically took long time due difficult effectively encoded chromothe iterative nature optimizers however leic somes efficiently optimized due reported external experiment crossover operation global optimization nature used exchanging mutation probabilities iii proposed algorithm connection weights trained sgd although section proposed evolving deep convolutional difficult reach actual reason crossover used evolving architectures letc obvious neural networks evocnn image classification docuat least crossover operations easy achieve mented detail chromosomes different lengths would algorithm overview complicated cnns multiple different building blocks summary main deficiency leic high algorithm framework evocnn putational complexity mainly caused fitness evaluation population proposed gene encoding strategy strategy adopts well without crossover operations huge computational resource leic requires makes termination criterion satisfied intractable academic environment evaluate fitness individuals connection weight initialization select parent solutions developed slack binary tournament selection typically initialization methods classified three generate offsprings designed genetic different categorises first employs constant numbers operators initialize connection weights zero initializer selection one initializer fixed value initializer second distribution initializer using gaussion distribution uniform distribution initialize weights third end covers initialization approaches prior select best individual decode corresponding convolutional neural network edge famous xavier initializer belongs category numerous connection weights existalgorithm outlines framework proposed ing cnns necessary connection weights start values major deficiency evocnn method firstly population initialized based max max convolutional pooling full connection fig example three chromosomes different lengths evocnn table encoded information cnn unit type convolutional layer pooling layer full connection layer encoded information filter width filter height number feature maps stride width stride height convolutional type standard deviation mean value filter elements kernel width kernel height stride width stride height pooling type average maximal number neurons standard deviation connection weights mean value connection weights proposed flexible gene encoding strategy line evolution begins take effect predefined termination criterion maximum number generations satisfied lines finally best individual selected decoded corresponding cnn line final training evolution individuals evaluated first based proposed efficient fitness measurement line parent solutions selected developed slack binary tournament selection line new offspring generated designed genetic operators line next representatives selected existing individuals generated offsprings form population next generation participate subsequent evolution line following subsections key steps algorithm narrated detail gene encoding strategy table commonly hundreds thousands connection weights may exist one convolutional full connection layer explicitly represented chromosome effectively optimized gas therefore evocnn use two statistical real numbers standard derivation mean value connection weights represent numerous weight parameters easily implemented gas optimal mean value standard derivation received connection weights sampled corresponding gaussian distribution details population initialization evocnn given next subsection based gene encoding strategy introduced population initialization algorithm population initialization input population size maximal number convolutional pooling layers ncp maximal number full connection layers output initialized population ncp uniformaly generate integer ncp ncp uniformly generated number initialize convolutional layer random settings else initialize pooling layer random settings end end uniformaly generate integer initialize full connection layer random settings end end return introduced subsection three different building blocks convolutional layer pooling layer full connection layer exist architectures cnns therefore encoded parallel one chromosome evolution optimal depth cnn solving one particular problem unknown prior confirming architecture gene encoding strategy suitable occasion employed proposed evocnn method furthermore performance cnns highly affected depths convenience elaboration chromosome gene encoding strategy makes proposed evocnn method chances reach best result due separated two parts first part includes convolutional layers pooling layers part constrains architecture search space particular example three chromosomes full connection layers based convention cnn ferent lengths evocnn illustrated fig architectures first part starts one convolutional layer represented information three layers detailed second part added tail first part addition length part set randomly evaluation evocnn evocnn concerns solving choosing number within predefined range image classification tasks classification error best algorithm shows major steps population strategy assign fitness number connection ization cardinality operator lines show weights also chosen additional indicator measure generation first part one chromosome lines individual quality based principle occam show second part initialization razor first part convolutional layer added first conventions represented cnn trained convolutional layer pooling layer determined training set dtrain fitness estimated coin tossing probability added end another dataset itness cnns frequently deep repeated predefined length part met architectures thus thoroughly training receiving second part full connection layers chosen final classification error would take considerable expenditure added note convolutional layers pooling layers computing resource long time due full connection layers initialized random large number training epochs required epochs settings information encoded randomly invariably treated fully training cnns make specified stored corresponding part much impracticable due two parts finished combined returned gas multiple generations individual take one chromosome approach population full training generation designed efficient individuals generated method address concern method individual trained small number epochs say fitness evaluation epochs based architectures connection weight initialization values mean value standard derivation classification error calculated batch algorithm fitness evaluation itness last epoch mean value input population training epoch number standard derivation classification errors employed measuring accuracy tendency training fitness one individual obviously smaller mean value set dtrain fitness evaluation dataset itness better individual compared individuals batch size num batch mean values less standard derivation indicates output population fitness better one individual summary three indicators used fitness ation mean value standard derivation eval steps itness batch number parameters several motivations behind fitness evaluation strategy sufficient train connection weights cnn ing tendency performance individuals represented individual better performance first several training epochs cnns probably still better performance accy list following training epochs greater confidence mean value standard derivation statistical eval steps cance indicators thus suitable investigating tendency accyj evaluate classification error final classification error received optimizing batch data itness best individual evolved proposed evocnn accy list accy list accyj method cnn models less number connection weights preferred smart devices details end discussed section calculate number parameters mean value standard derivation slack binary tournament selection accy list assign individual update develop one slack version standard binary end nament selection documented algorithm select parent solutions crossover operations end proposed evocnn method briefly two sets comparisons end employed comparisons mean values return individuals involves threshold comparisons parameter numbers involves another threshold fitness evaluation aims giving quantitative measure determining individuals qualify serving parent solutions algorithm manifests framework fitness original training set randomly split train itness itness unseen cnn training phase give good indication generalization accuracy test set units convolutional layer pooling layer full connection layer firstly collected three different lists based orders corresponding chromosome refers unit collection phase three lists aligned top units positions performed crossover phase named crossover phase finally unit restore phase employed crossover operation completed units lists restored original positions associated chromosomes three consequential phases crossover two chromosomes different lengths could easily exchange gene information crossover crossover operation performed unit lists units types loaded proposed crossover operation natural origins remaining units perform crossover operations due paired ones kept position mutation operations may perform position units one chromosome selected mutation point unit could added deleted modified determined probability case unit addition convolutional layer pooling layer full connection layer added taking probability mutation modify existing unit particular modification dependent unit type encoded information unit would changed encoded information unit type seen table note encoded formation denoted real parent solution selected comparisons numbers therefore simulated crossover sbx individual smaller standard derivation chosen polynomial mutation employed proposed practice tremendous number parameters exist deep evocnn method due notable show real number cnns would easily cause overfitting problem gene representations therefore difference mean values two individuals smaller threshold environmental selection sider number connection weights slight change parameter numbers highly affect performance algorithm environmental selection cnns consequently also introduced input elistsm fraction current population iteratively performing selection parent solutions selected stored mating pool proposed output selected population evocnn method size mating pool set calculate number elites based population size predefined population size algorithm offspring generation select individuals best mean values steps generating offspring given follows step randomly select two parent solutions mating pool randomly select two individuals step use crossover operator selected solutions generate offspring employe algorithm select one individual step use mutation operator generated offspring step store offspring remove parent solutions mating pool perform steps mating end pool empty return proposed crossover operation seen fig achieve crossover design method called unit alignment recombining two individuals different chromothe environmental selection shown algorithm dursome lengths crossover operation three different ing environmental selection elitism diversity algorithm slack binary tournament selection input two compared individuals mean value threshold paramemter number threshold output selected individual individual larger mean value individual mean values standard derivations parameter numbers return else return else return else return else return randon one end end end convolutional pooling full connection chromosome convolutional unit list chromosome pooling unit list full connection unit list convolutional unit list pooling unit list full connection unit list unit collection crossover crossover crossover crossover crossover crossover crossover unit aligh crossover offspring offspring unit restore fig example illustrate entire crossover process example first chromosome length including three convolutional layers two pooling layers three full connection layers one length including two convolutional layers three pooling layers four full connection layers first step crossover unit collection convolutiona layers pooling layers full connection layers collected chromosome stacked orders chromosome see fig second step unit lists unit types aligned top two convolutional layer lists picked aligned operations two lists unit lists finish alignment units positions two lists paired performed crossover operation shown fig last units unit lists restored based positions see fig note fig units experienced crossover operations highlighted italics underline fonts units perform crossover operations remain explicitly elaborately addressed specific fraction individuals promising mean values chosen first remaining individuals selected modified binary tournament selection demonstrated subsection two strategies elitism diversity considered simultaneously expected collectively improve performance proposed evocnn method note selected elites removed tournament selection gets start work line algorithm individuals selected purpose diversity kept current population next round tournament selection lines algorithm based convention community decoded cnn deeply trained larger number epochs sgd future usage xperiment esign order quantify performance proposed evocnn series experiments designed performed chosen image classification benchmark datasets compared peer competitors following benchmark datasets briefly introduced first peer competitors given finally parameter settings proposed evocnn method participating experiments documented benchmark datasets best individual selection decoding experiments nine widely used image classification end evolution multiple individuals benchmark datasets used examine performance promising mean values different architectures proposed evocnn method fashion tion weight initialization values regard rectangle rectangle images convex multiple choices select best individual example sets mnist basic mnist concerned best performance background images mbi random background could neglect architecture configurations consider mrb rotated digits mrd plus classification accuracy otherwise emphasis background images mrdbi benchmarks based types classification objects smaller number parameters corresponding decision could made best individual confirmed benchmarks classified three different categories corresponding cnn decoded based encoded first category includes fashion recognizing architecture connection weight initialization information fashion objects trousers coats training images test images second final classification accuracy invisible public one composed mnist variants including architecture design approaches introduced section mbi mrb mrd mrdbi benchmarks among others scalable cnns classifying digits suitable directly compared well mnist easily achieved mnist variants experiments algorithms arbitrarily added different barriers random reported promising classification errors chosen benchgrounds rotations mnist improve complexity marks collected peer competitors proof classification algorithms furthermore variants posed evocnn method specific peer training images test images petitors fashion benchmark collected challenges classification algorithms due mush less dataset training data test data third category googlenet recognizing shapes objects rectangle alexnet mlp rectangle benchmarks convex perform experiments raw convex benchmark obviously category covers dataset without preprocessing peer competitors rectangle benchmarks contain benchmarks tirbm training images respectively softthem include test images compared rectangle max benchmark generated adding randomly sampled nnet backgrounds mnist increasing difficulties literature recently published provider classification algorithms addition image benchmarks size examples benchmarks shown parameter settings fig reference furthermore another reason using parameter settings set based conventions benchmark datasets different algorithms communities evolutionary algorithms deep reported promising results convenient learning addition maximal length basic comparisons performance proposed evocnn layer specifically population size total generation method host algorithms details seen number set distribution index sbx tion polynomial mutation set associated probabilities specified respectively maximum lengths convolutional layers pooling layers full connection layers set examples first category left right trouser maximal depths cnns experiments pullover dress coat sandal shirt sneaker bag ankle boot moreover proportion elitism set based pareto principle note data randomly selected training images fitness evaluation dataset implementation proposed examples second category left right two images evocnn method constrain values width one group group mbi mrb mrd mrdbi respectively images refer digits height filter stride kernel width height respectively stride convolutional layer set pooling layer specified value kernel convolutional type fixed proposed evocnn method implemented tensor examples thrid category left right first three images flow copy code runs computer rectangle benchmark following four ones benchmark remaingings convex benchmark specifically equipped two gpu cards identical model number examples index positive samples final training phase individual fig examples benchmarks chosen imposed batchnorm speeding weight decay unified number preventing peer competitors overfitting due heuristic nature proposed ideally algorithms neural network architecture evocnn method independent runs performed sign discussed section collected peer benchmark dataset mean results estimated competitors however metaqnn leic comparisons unless otherwise specified furthermore highly rely computational resources impossible experiments take days fashion benchmark reproduce experimental results academic run days benchmarks ronment furthermore benchmark dataset investigated https two algorithms employed different data preprocessing http augmentation techniques highly enhance xperimental esults nalysis classification error rates difference proposed evocnn method decreases error rate section experimental results analysis furthermore mean performance proposed evocnn method peer competitors shown evocnn even better best performance eight subsection weight initialization method competitors little worse best googlenet proposed evocnn method specifically investigated however evocnn much smaller number subsection connection weights evocnn uses million weights overall results googlenet uses million vgg uses million experimental results fashion benchmark dataset weights evocnn also employs half numbers shown table mrd mrb mbi training epochs used results show mrdbi rectangle convex benchmark datasets proposed evocnn method obtains much better performance shown table iii tables iii last two rows architecture design connection weight initialization denote best mean classification errors received cnns fashion benchmark significantly reduces proposed evocnn method respectively rows classification error evolved cnn generated show best classification errors reported peer proposed evocnn method according table iii evocnn best one among order conveniently investigate comparisons terms provided denote whether best different methods specifically evocnn achieves result generated proposed evocnn method better best performance five eight datasets second worse best result obtained corresponding peer best three datasets mrd convex competitor term means available result datasets tirbm softmax reported provider counted achieve lowest classification error rate respectively furtion number parameters number training epoch thermore comparing mean performance evocnn peer competitors fashion benchmark dataset best methods evocnn best four available therefore information evocnn also datasets mrb mbi rectangle second best two shown table comparisons however mrd convex third best fourth best two benchmarks table iii information presented datasets mrdbi particularly mrb mbi available peer competitors lowest classification error rates others note results peer competitors mean error rates evocnn proposed evocnn method without data augmentation respectively summary best classification error proposed evocnn method wins preprocessing benchmarks comparisons best results peer competitors mean classification error evocnn table classification errors proposed cnn method better best error methods peer competitors fashion benchmark comparisons dataset performance regarding weight initialization clearly shown table comparing best performance proposed evocnn method outperforms ten peer competitors two algorithms googlenet achieve respectively convention deep learning community best result reported weight initializaed evocnn weight initialized xavier fashion mrd mrb mbi mrdbi benchmark datasets rectangle googlenet alexnet mlp evocnn best evocnn mean epochs parameters error classification error classifier convex fig performance comparison proposed evocnn method cnn using evolved architecture xavier weight initializer experiments performed examine effectiveness connection weight initialization method proposed evocnn approach comparing different weight initializers would also investigate whether architecture connection weight initialization values affect classification performance achieve initialize another group cnns architectures table iii classification errors proposed cnn method peer competitors mrd mrb mbi mrdbi ectangle onvex benchmark datasets classifier mrd mrb mbi mrdbi rectangle convex tirbm softmax nnet evocnn best evocnn mean evolved evocnn weights initialized widely used xavier initializer details subsection comparisons illustrated fig xaxis denotes benchmark datasets denotes classification error rates fig clearly shows proposed connection weight initialization strategy evocnn improves classification performance benchmarks widely used xavier initializer specific proposed weight initialization method reaches classification accuracy improvement fashion mrd mbi convex benchmarks mrdbi benchmark comparing results tables iii also concluded architectures cnns contribute classification performance connection weight initialization proposed evocnn method gained promising performance automatically evolving initial connection weights architectures cnns operation context crossover operations easily performed genes different origins therefore new crossover operation introduced proposed evocnn method well designed complete crossover operation enhances communications encoded information expectedly leverages performance pursuing promising architectures cnns typically used approaches optimizing weights cnns based gradient information well known optimizers sensitive starting position parameters optimized without better starting position algorithms easy trapped local minima intuitively finding better starting position connection weights gas intractable due huge numbers parameters elaborated huge number parameters efficiently encoded chromosomes effectively optimized proposed urther iscussions evocnn method indirect encoding approach employed encodes means standard derivations section discuss encoding stratthe weights layer fact employing standard egy architectures weights related parameters derivation mean initialize starting position fitness evaluation proposed evocnn method connection weights compromised approach extra findings experimental results discussed ubiquitously seen deep learning libraries well could provide insights applications motivation design proposed evocnn proposed evocnn method method strategy two real numbers used gas crossover operators play role exploitation denote hundreds thousands parameters search local search mutation operators act could save much computational resource encoding exploration search global search optimization local search global search complement well designing could remarkably existing techniques searching architectures promote performance generally crossover operators cnns typically take final classification accuracy operate chromosomes lengths fitness individuals final classification accuracy typically proposed evocnn method employs requires many epochs training ones furthermore multiple different basic layers exist process order complete architecture cnns improve difficulties designing crossover design fitness evaluation approach natural employ lot computation resource perform task parallel speed design however computational resource necessarily available interested researchers furthermore also requires extra professional assistances task schedule synchronization context beyond expertise fact necessary check individual final classification accuracy tendency could predict future quality would sufficient therefore employ small number epochs train individuals kind fitness measurement proposed evocnn method highly reply computational resource summary yet simplicity encoding strategies fitness evaluation method researchers without rich domain knowledge could also design promising cnn models addressing specific tasks academic environment computational resources typically limited table classification accuracy parameter numbers individuals fashion benchmark dataset classification accuracy parameters moreover interesting findings also discovered proposed evocnn method terminates multiple individuals similar performance significant different numbers connection weights see table produced due nature proposed evocnn method recently various applications developed car interesting mobile applications due limited processing capacity battery devices trained cnns similar performance fewer parameters much preferred require less computational resource energy consumption addition similar performance individuals also found different lengths basic layers multiple pieces hardware especially designed speed primitive operations cnns specialized hardware convolutional operations regard proposed evocnn method could give manufacturers choices make decision based preferences traditional approaches employed difficult find models single run vii onclusions uture ork goal paper develop new evolutionary approach automatically evolving architectures weights cnns image classification problems goal successfully achieved proposing new representation weight initialization strategy new encoding scheme chromosomes new genetic operator slacked binary tournament selection efficient fitness evaluation method approach examined compared peer competitors including algorithms nine benchmark datasets commonly used deep learning experimental results show proposed evocnn method significantly outperforms existing algorithms almost datasets terms best classification performance furthermore mean classification error rate proposed algorithm even better best others many cases addition model optimized evocnn much smaller number parameters yet promising best classification performance specifically model optimized evocnn employs epochs reach lowest classification error rate fasion dataset employs epochs classification error rate findings experimental results also suggest proposed evocnn method could provide options manufacturers interested integrating cnns products limited computational battery resources paper investigate proposed evocnn method commonly used benchmarks however data also widely exist current era big data evaluating one epoch largescale data would require significant computational resource take long period proposed fitness evaluation method suitable unless huge amount computational resources available future put effort efficient fitness evaluation techniques addition also investigate evolutionary algorithms recurrent neural networks powerful tools addressing data voice video data eferences meier masci schmidhuber deep neural network traffic sign classification neural networks vol ning delhomme lecun piano bottou barbano toward automatic phenotyping developing embryos videos ieee transactions image processing vol krizhevsky sutskever hinton imagenet classification deep convolutional neural networks advances neural information processing systems hubel wiesel receptive fields binocular interaction functional architecture cat visual cortex journal physiology vol lecun boser denker henderson howard hubbard jackel backpropagation applied handwritten zip code recognition neural computation vol lecun bottou bengio haffner learning applied document recognition proceedings ieee vol rumelhart hinton williams learning representations errors cognitive modeling vol simonyan zisserman deep convolutional networks image recognition arxiv preprint zhang ren sun deep residual learning image recognition proceedings ieee conference computer vision pattern recognition bourlard kamp multilayer perceptrons singular value decomposition biological cybernetics vol hinton zemel autoencoders minimum description length helmholtz free energy advances neural information processing systems hinton salakhutdinov reducing dimensionality data neural networks science vol bergstra bengio random search optimization journal machine 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feb isomorphism classes four dimensional nilpotent associative algebras field marco antonio pellegrini abstract paper classify isomorphism classes four dimensional nilpotent associative algebras field studying regular subgroups affine group particular provide explicit representatives classes finite field real field algebraically closed field introduction aim paper classification isomorphism classes four dimensional nilpotent associative algebras field result achieved exploiting properties regular subgroups affine group worth recalling affine group agln identified subgroup consisting invertible matrices first column follows agln acts right set affine points subgroup agln called regular acts regularly namely every exists unique element first row writing elements regular subgroup agln give rise functions matn instance translation subgroup agln regular subgroup corresponding choice equal zero function focus attention set regular subgroups agln property linear function notice abelian see however set also contains nonabelian groups see also remark unipotent contains nontrivial translations see main motivation studying set exists bijection set isomorphism classes nilpotent associative algebras dimension field set conjugacy classes subgroups see proposition fact classification four dimensional nilpotent associative theorem follow classification conjugacy classes regular subgroups paper organized follows section recall useful results concerning following sections classify conjugation regular subgroups see theorems detail section deal abelian regular subgroups section consider nonabelian mathematics subject classification key words phrases regular subgroup congruent matrices nilpotent algebra finite field marco antonio pellegrini subgroups corollaries propositions give explicit classification algebraically closed fields finite fields classification real field given corollary proposition describe results obtained need fix notation sets complete sets representatives projective congruent classes respectively symmetric asymmetric matrices see definition given field denote transversal also denote fixed set representatives equivalence classes respect following relation defined given write either char fix two additional sets set transversal subgroup set set representatives equivalence classes respect following relation defined given write convenience assume state main result theorem distinct isomorphism classes four dimensional nilpotent associative algebras field represented algebras described second column table abelian case table nonabelian case particular exactly algebraically closed char algebraically closed char finite nonisomorphic abelian nilpotent associative dimension four point classification given theorem abelian case already obtained using techniques graaf poonen algebraically closed fields also interesting observe finite field exactly odd even nonisomorphic nonabelian nilpotent associative dimension four see proposition preliminary results classification regular subgroups basically relies key results recall section proposition theorems let unipotent subgroup conjugation agln center contains nontrivial element coincides jordan form diag jnk upper unitriangular jordan blocks jni respective sizes isomorphism classes four dimensional nilp assoc algebras conditions char char char char table four dimensional abelian nilpotent associative falgebras abelian regular subgroups linear function regular subset agln subgroup see also given also recall two subgroups conjugate agln exists matrix diag gln see proposition hand useful mainly prove two regular subgroups conjugate introduce following three parameters marco antonio pellegrini conditions char table four dimensional nonabelian nilpotent associative falgebras nonabelian regular subgroups let may define max deg minf max dim ker last parameter justified since set subspace see proposition proceed considering possible values working centralizer jordan form described proposition unfortunately case require different approach based isomorphism classes four dimensional nilp assoc algebras classification regular subgroups obtained particular make use following observation written regular subgroup square matrix dei denotes canonical basis see particular take case square matrix satisfies set study subgroups justified mainly connection projectively congruent matrices definition two matrices matn said projectively congruent element invertible matrix gln lemma lemma given two matrices subgroups conjugate agln projectively congruent furthermore easy see zero diagonal otherwise given nilpotent associative algebra dimension embedded via denote respectively coordinate row vector matrix right multiplication respect fixed basis identifying image split local subalgebra jacobson radical subset consists invertible matrices closed multiplication since hence regular subgroup lying lemma hand given regular subgroup split local dimension theorem notice set marco antonio pellegrini nilpotent associative dimension proposition let following conditions equivalent subgroups conjugate agln split local algebras isomorphic nilpotent associative algebras isomorphic sake brevity write indicate regular subgroup denote matrix obtained taking finally elementary matrix position elsewhere regular subgroups classifying regular subgroups field consider possible values proposition subgroup unipotent center contains element conjugate one following jordan forms diag diag diag diag diag cases already studied recall results obtained lemmas first abelian particular conjugate subgroup two conjugacy classes regular subgroups conjugate conjugate isomorphism classes four dimensional nilp assoc algebras abelian conjugating subgroups obtained lemma diag obtain conjugate may assume diag diag working using linearity obtain conjugate subgroup may assume center contains element diag diag working using linearity obtain conjugate subgroup finally abelian lemma abelian case section assume abelian view results recalled section left determine conjugacy classes subgroups type symmetric matrix start subgroups lemma let abelian regular subgroup conjugate exactly one following subgroups char char proof previous argument assume first suppose char set write unique define epimorphism algebras setting marco antonio pellegrini ker subgroup conjugate proposition quite easy see two subgroups conjugate consider epimorphism defined whose kernel ker proposition conjugate conjugate next suppose char take exists polynomial reducible case define epimorphism setting proving conjugate obtain ker conjugate follows conjugate exactly one subgroup finally assume observe write unique epimorphism defined taking conjugate ker furthermore given subgroup conjugate see notation described introduction conclude observe subgroups conjugate subgroups classify abelian regular subgroups theorem let field distinct conjugacy classes abelian regular subgroups represented subgroups described first column table proof let abelian regular subgroup clearly seen section conjugate conjugate either suppose apply lemma char conjugate unique char conjugate either unique unique conjugate subgroup suppose apply lemma obtaining char conjugate zero matrix notice three cases conjugate subgroup lemma statement theorem proved considering set representatives projective congruent classes symmetric matrices isomorphism classes four dimensional nilp assoc algebras provide explicit representatives algebraically closed fields finite fields corollary let algebraically closed field char exactly distinct conjugacy classes abelian regular subgroups represented char exactly distinct conjugacy classes abelian regular subgroups represented proof since algebraically closed char set obtained applying instance theorem hence statement proved noticing observed proof lemma char subgroups conjugate corollary exactly distinct conjugacy classes abelian regular subgroups represented proof first take set obtained applying theorem noticing two matrices matn projectively congruent congruent corollary let finite field exactly distinct conjugacy classes abelian regular subgroups represented odd fixed irreducible polynomial even fixed irreducible polynomial proof odd even set determined theorem marco antonio pellegrini nonabelian case section assume nonabelian reduced study cases recall conjugate subgroup shape conjugate clearly consider case notice since nonabelian must furthermore lemma exactly conjugacy classes nonabelian regular subgroups conjugate classes represented subgroups proof previous considerations may assume first suppose case consider algebra defined span function defined isomorphism proposition conjugate suppose case conjugate consider algebra span function defined isomorphism conjugate consider case instead working diag convenient write see regular subgroups classified theorem section let field distinct conjugacy classes regular subgroups represented char char isomorphism classes four dimensional nilp assoc algebras suppose follows abelian furthermore hence consider six subcases corresponding lemma exactly distinct conjugacy classes nonabelian subgroups classes represented subgroups corresponding subgroups conjugate proof let hypothesis nonabelian obtain moreover condition implies conjugate via suppose consider algebra span function defined isomorphism conjugate suppose case consider algebra span function defined isomorphism follows conjugate finally suppose consider algebra span marco antonio pellegrini function defined isomorphism hence conjugate notice otherwise furthermore conjugate direct computations holds subgroups since subgroups conjugate conclude observe conjugate consider next subcase recall following notation given introduction denote fixed set representatives equivalence classes respect following relation defined given write either lemma distinct conjugacy classes nonabelian subgroups represented furthermore proof let nonabelian subgroup hypothesis nonabelian taking diag diag obtain conjugate via subgroup let quite easy verify subgroups conjugate exists element diag holds isomorphism classes four dimensional nilp assoc algebras provided get case gives notice taking diag hence may suppose case gives provided taking follows conjugate either words holds remark let conjugate conjugate conjugate either conjugate remark useful give presentation following algebras span span lemma assume char exactly distinct conjugacy classes nonabelian subgroups classes represented subgroups corresponding furthermore proof let hypothesis nonabelian conjugate marco antonio pellegrini via consider algebra span clearly condition omitted define isomorphism split local algebras following way hence conjugate conjugate conclude exactly conjugacy classes lemma let nonabelian subgroup conjugate one following subgroups proof let conjugate via easy see conjugate detail diag function defined isomorphism proposition conjugate subgroup function defined isomorphism conjugate isomorphism classes four dimensional nilp assoc algebras values write unique function defined isomorphism hence conjugate left consider case suppose char write unique function defined isomorphism hence conjugate char function defined isomorphism hence conjugate set function defined isomorphism hence conjugate write unique function defined isomorphism hence conjugate lemma assume char let nonabelian subgroup conjugate one following subgroups proof let must conjugate via easy see conjugate case diag suppose let notice case assume consider function given marco antonio pellegrini since isomorphism obtain conjugate assume write unique function given isomorphism conjugate lemma let nonabelian subgroup conjugate one following subgroups let conjugate via easy see conjugate case suppose may also assume namely suffices conjugate diag conjugate matrix diag conjugate matrix diag set suppose case write unique diag char diag char diag isomorphism classes four dimensional nilp assoc algebras conditions char table representatives conjugacy classes nonabelian subgroups diag hence may suppose write unique diag need distinguish two cases diag diag corollary let nonabelian subgroup suppose conjugate conjugate exactly one subgroups listed table proof view lemmas conjugate one subgroups listed table since follows lemma subgroups conjugate direct computations show pair subgroups conjugate give classification nonabelian subgroups marco antonio pellegrini theorem let field distinct conjugacy classes nonabelian regular subgroups represented subgroups described first column table proof let nonabelian regular subgroup considerations given section following possibilities conjugate either lemma conjugate either one subgroups table subgroup corollary conjugate subgroup statement theorem proved considering set representatives projective congruent classes asymmetric matrices would like give explicit sets least algebraically closed real field finite field observe every field characteristic remark fixing define following polynomials lemma suppose either reducible conjugate proof clearly statement holds assume since reducible take obtain conjugate provided suppose observe implies since may suppose root also root clearly implies take case implies whence follows contradicting assumption following given subset denote sets respectively corollary let field quadratic extensions char fixed subset char proof follows lemma remark proposition let algebraically closed field distinct conjugacy classes nonabelian regular subgroups represented isomorphism classes four dimensional nilp assoc algebras char char char subsets char subset proof theorem corollary left determine set applying theorem elements asymmetric matrices obtained diagonal sum following matrices char also allow char must omit matrices diag lemma let char suppose exists element proof show every subgroup conjugate unique subgroup statement clearly holds suppose keeping notation let root follows conjugate provided notice suppose take case follows conjugate clearly holds even suppose let take follows conjugate remark finally easy computation verify conjugate clearly previous lemma gives alternative set algebraically closed field characteristic similar way prove following marco antonio pellegrini lemma let char suppose lemma suppose char irreducible reducible conjugate proof since reducible take obtain conjugate provided first suppose lemma implies particular get whence case obtain absurd since irreducible next suppose root root root let resultant respect get since irreducible implies condition implies reducible contradiction corollary let proof result follows lemmas remark observing reducible provided theorem theorem obtain following classification proposition distinct conjugacy classes nonabelian regular subgroups represented lemma suppose char let let irreducible conjugate respective proof since irreducible polynomials first consider statement obvious assume let since discriminant exists denote resultant isomorphism classes four dimensional nilp assoc algebras respect obtain get absurd obtain contradiction initial assumption hence conjugate next consider statement obvious assume let since discriminant exists clearly contradicting assumption irreducible suppose denote resultant respect since obtain conjugate lemma let finite field keep previous notation suppose even take every fixed element particular suppose mod take particular suppose mod take particular proof let suppose even either reducible conjugate unique lemma remark suppose irreducible also irreducible obtain conjugate conjugate chosen way marco antonio pellegrini suppose mod set given lemma consider case either reducible conjugate unique lemma remark assume irreducible reducible conjugate lemma recall remark conjugate either conjugate conjugate finally reducible conjugate lemma conjugate furthermore conjugate suppose mod set given lemma consider case proceeding way similar case mod conjugate subgroup one following shape unique conjugate either conjugate conjugate furthermore conjugate conjugate proposition let finite field odd exactly distinct conjugacy classes nonabelian regular subgroups represented fixed irreducible polynomial take mod even exactly distinct conjugacy classes nonabelian regular subgroups represented fixed irreducible polynomial isomorphism classes four dimensional nilp assoc algebras proof set determined theorem result follows theorem lemma conclude recalling theorem follows immediately theorems applying proposition references caranti dalla volta sala abelian regular subgroups affine group radical rings publ math debrecen graaf classification nilpotent associative algebras small dimension sep horn sergeichuk canonical matrices bilinear sesquilinear forms linear algebra appl pellegrini regular subgroups nilpotent algebras projectively congruent matrices appear int group theory pellegrini tamburini bellani regular subgroups affine group linear algebra appl pellegrini tamburini bellani regular subgroups affine group translations appear algebra http poonen isomorphism types commutative algebras finite rank algebraically closed field computational arithmetic geometry contemp amer math providence tamburini bellani remarks regular subgroups affine group int group theory williams projective congruence results math dipartimento matematica fisica cattolica del sacro cuore via musei brescia italy address
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cient textualrepresentation structure brenton chapin jun abstract paper attempts formal approach legibility text based programming languages presenting proof minimum possible ways representing structure text interleaved information presumes minimalist approach best purposes human readability data storage transmission machine evaluation several proposals given improving expression interleaved hierarchical structure instance single colon replace pair brackets bracket types need repeated opening closing symbols words historic customary uses punctuation symbols guided chosen form nature improvements keywords programming language design structured programming human readability syntax notation history data compression minification acm reference format brenton chapin efficient textual representation structure proceedings acm conference washington usa july conference pages doi introduction information almost always useful organized structure key therefore efficient clear representation structure paramount importance structured programming languages one use structure organize one kind information source code make unprecedentedly elaborate use structure exposed deficiencies methods expression languages programming math developed evolutionary manner much borrowing earlier work reasons various decisions became buried custom history studies choices characters legibility patchy questions overlooked thought relatively unimportant pioneers programming languages hurriedly made expedient adaptations existing notations similar problems heavily borrowed mathematical notation many important questions needing settling creation first programming languages issues symbology mostly passed unimportant arbitrary bob bemer permission make digital hard copies part work personal classroom use granted without fee provided copies made distributed profit commercial advantage copies bear notice full citation first page copyrights components work owned others acm must honored abstracting credit permitted copy otherwise republish post servers redistribute lists requires prior specific permission fee request permissions permissions conference washington usa acm doi noted much documentation lost characteristic times nobody seemed think character sets important feature computers many studies readability properties various fonts color combinations discussed relation source code term readability refers comprehensibility punctuation class symbol closely associated showy interleaved structure positioning major method used indicate structure among intended purposes control characters attempted provide ways position text currently python popular programming language relies text positioning rather punctuation indicate structure visual programming goes yet replacing textual indicators structure flow graphical ones programming language wars still hot today new languages emerging gaining followers one cause passionate debates tendency language designers resort evangelical approach justify choices design elements little compelling technical reason sometimes designers make overlarge unsubstantiated claims many programming languages one defining features choice usage symbols choices modifiable programmers changes desired whole new programming language may need created another factor proliferation programming languages algol designers hoping avoid ideas paper aim foundation symbolic representation structure structure chosen crucial concept must addressed improve representation minimalism chosen guide much minimalism certainly possible instance expecting people work information minimized data compression techniques transform data compact human unreadable form minification techniques removing unnecessary whitespace shortening variable names another example target minimization efforts representation structure source code source code rewriting rearranging place information efficient structures making representation efficient regardless structure chosen punctuation always minimal using smaller less obtrusive symbols used represent letters language syntax structural elements follows pattern items note splits textual representations used source code versus used markup html data organization xml yaml within programming languages dot arrow notation object oriented programming completely different notations structured programming curly braces yet splits conference july washington usa seem artificial hierarchical structure used many programming languages needlessly poor expressing data several improvements touch issue allowing flexible constructions constants initializations arrays objects one intent json bridge divide also interleaved formats meaning symbols denote structure interleaved symbols data goals data storage minimal size fast access obvious common method achieve exclude complex structure data using external representation disadvantage require connection often expressed fixed sizes padding end using space interleaved method years attention paid brevity varied cost availability storage american standard code information interchange ascii set fixed size bits though least two variable codes morse code huffman coding existed time one goals xml human readability minimalism thought orthogonal possibly even antithetical goal human readability resulting language ironically suffers excessive verbosity obscures essentials rendering result less human readable xml cobol show negative attitude towards minimalism terseness xml markup minimal importance regarding minimalism unrelated even impediment comprehension correct minimalism also central information theory demonstrated crude redundancy repeating information wasteful poor preserving fidelity data errors transmission repetition poor means ensuring fidelity data perhaps also poor means representing structure ways easy humans read another demonstration limited usefulness repetition fat file system despite allocating room copy directories file names actually one fragile easily corrupted file systems currently use particular note programming language many programming languages adopted syntax tagged moniker languages perhaps one reasons curly brace syntax eclipsed pascal algol use single character rather words begin end delimit blocks designers restrict curly braces also used square brackets parentheses even angle brackets array indexing lists function parameters macros respectively choice symbol assignment use parentheses rely context means distinguish parameter list block code doubt possible use parentheses lisp programming language proof copy fortran use parentheses array indices one kind answer different kinds brackets serve sigils distinguish identifiers functions arrays variables begs question sigils still answer particular symbol chosen particular use brenton chapin history answers one must dig history computation mathematics case chain preceding languages roughly bcpl basic cpl cpl combined programming language finally algol algorithmic language paper algol says choice use square brackets delimit array indices subscripted variables term used algol today call array variable simply array designate quantities components multidimensional arrays complete list subscripts enclosed subscript brackets pick square brackets fortran oldest programming language achieve wide acceptance uses parentheses square brackets matter use bracket one says seems likely would rather used actual subscripted text like mathematical notation early computers could square brackets notational device indicate subscription without actually presenting text apart computer limitations big problem subscripting notation nest well levels becoming small human eye read one surmise use term subscript another borrowing linear algebra matrix denoted square brackets indeed original name algol international algebraic language deviation use square brackets array indexes algol bcpl among many simplifications cpl introduced attempted repurpose square brackets code blocks using pointer arithmetic access array elements ascii codified glyphs used nearly programming languages notable exception apl makes use mathematical symbols mainly set theory vector calculus put ascii unlike ebcdic ascii least organized alphabet contiguous block exact set punctuation symbols unclear ranging symbols letters numbers control characters used clarify structure meaning sentences text formal ordered centuries old lists punctuation symbols ascii ordering choice punctuation derived qwerty keyboard layout dates late century notion qwerty deliberately arranged slow typists popular wrong myth morse code many factors considered years small changes made accommodate new uses instance double quote mark many older keyboards today keyboards could back ask mathematical notation uses parentheses functions square brackets matrices customary canonical expression function particular use parentheses bracket independent variable history mathematical notations cajori credits euler first use parentheses bracket variable function paper paper numbering scheme created refer euler works however examining several others euler papers one finds use parentheses exact phrase formula efficient textual representation structure cajori quoted present euler uses parentheses group parts equations separate function names variables euler notation reader understand variables function note also choice letter first letter word function custom followed many places decision fortran use first letter variable name indicate integer name begins integer floating point name begins desire match functionality first letter appropriate term taken extremes one early game employed lucky placement keys qwerty keyboard plus refer west east south north respectively major work number theory regarded also important paper modern definition function dirichlet used parentheses around independent variable mathematicians pick symbols format engaged expedience adopting idea parentheses still earlier scholars mathematical notation long evolutionary history fascinating main point many choices symbols syntax made long possible use programming languages conceived also year babbage proposed analytical engine arguably first computer functioning computation machinery would built many years later therefore symbols syntax certainly could chosen based experiences computer programming far early pioneers went exploring questions best symbolize code data none terms areas study semiotics symbology linguistics grammar lexicology punctuation readability typography legibility notation expressiveness rubrication quite address questions studies notation syntax get closest even syntax confined issues context programming languages use hierarchical structure organize code possibly earliest simplest formally specified language expressing hierarchy dyck language object oriented programming functional programming abandon fundamental organization added declarative programming represented prolog sql first glance seems need much structure point confusion order structure hierarchy declarative programming needs structure order necessarily hierarchy hierarchic structure programs data efficiently represented several changes advent markup languages revived interest hierarchical data storage introduced relational database model longer interleaving structural symbols programs harnessed organize data traditionally data organized fixed size elements symbols need reserved explicit denotation structure even importantly random access quick taking time retrieve one element also true era used tables extensively carefully lining columns aid human eye inadequate expedient method used small fixed size field conference july washington usa hold value size variable length field packet networking example organization data roughly analogous method writing technique employing variety superscripted symbols asterisk dagger indicate footnote xml html well known markup languages like programming languages history also evolutionary trace back standard generalized markup language sgml standardized predating world wide web like many decisions languages creators web seized upon sgml expediency sgml turn descends gml ibm effort manage technical documentation data based upon ideas first articulated circa many complained years markup languages undesirable features among biggest extreme verbosity rules force upon users come closer goal human readability often opposite effect scales minimalism xml relatives extremely poor representations highly redundant must brackets balanced proper html xml matching tags must repeat tag name designers ironically rules done much add clutter thereby reduce human readability intended goal yaml yaml markup language motivated part recognition xml burdened design constraints little purpose data serialization lightweight markup languages markdown acknowledgment human readability html could better popular programming languages poor expressing data examples illustrate list first chemical elements encoded javascript array like simple trick yields much cleaner representation split sort trick makes programming needlessly difficult professional programmers unfamiliar arcana particular language one problem default unquoted meaning alphanumeric sequence treat name variable double quote mark changes mode mode support structural elements simple string encoded programmer forced change modes entering string mode give short string leaving string mode impart tiny amount structure entering string mode give next string programmer use clever trick split function create function parse string complicated object even employ library yaml another example family tree python means node def init name else self child conference july washington usa name familytree grandmother older uncle cousin fat older niece nephew son granddaughter daughter grandson younger terrible encoding littered alternating brackets kinds well double quote marks commas shows python even worse lisp thought python use indentation lead clean code cases lisp many parentheses get clean looking code expert programmer resorts using functions read simple string may data file complicated object employing data serialization library yaml common method handling issue preferred method programming languages able better native syntax native handling regular expressions made perl popular improvements representation structure applicable coding data representation eliminating runs brackets first change addresses problem languages perhaps obvious lisp criticized backronym lots idiotic spurious parentheses often brackets cluster several structures start end simultaneously add visual clutter without adding ease comprehension many solutions problem among operator precedence postfix notation also known reverse polish notation first conceived limitation polish notations make brackets unnecessary number operands must fixed inflexibility insufficiently general structures used programming popular short cut use context knowledge permitted sensible content subtrees instance html paragraph indicator nested often used omit matching closing bracket next structure another paragraph something else inside paragraph header omissions officially sanctioned html popular web browsers support anyway obvious problems approach knowledge every exception rules indicating nesting may large may change brenton chapin approach taken perl allow kinds shortcuts greatly complicate parser compared perl brackets longer required particular parentheses statements optional effectively change recognition enough indicate structure fact part set symbols used denote structure one could employ sets brackets perhaps scheme closing bracket closes matching opening bracket open brackets kind example becomes becomes idea work direction becomes even works directions becoming however best idea issue symbols needed employ one symbol eliminating one excess opening closing brackets still clean clutter call symbol system eliminates excess closing brackets closing symbol system eliminates excess opening brackets opening using colon symbol closing system approximately matches traditional use colon written natural languages changes becomes becomes becomes additionally brackets still balanced equal numbers opening closing brackets systems slightly larger example consider ackermann function classic textbook structure interpretation computer programs exercise cond employing closing system suggested gives cond else colons replaced opening brackets matching closing brackets removed indeed never need multiple adjacent closing brackets proven next theorem given sequence arbitrary symbols alphabet symbols opening closing symbol reserved denote hierarchy format interleaves data structure properly balanced hierarchy always represented system reserved symbols runs sequences length greater closing symbol proof wlog let parentheses represent opening closing symbols systems let colon efficient textual representation structure represent symbol symbol system allow elimination runs closing symbols assign meaning opening subtree except matching closing symbol already necessary matches existing precedes instances sequence arbitrary sequences may include balanced occurrences may replaced replacement symbols sufficient represent relationships symbols still indicate parent preserve relationships sequences none need change additional context needed preserve relationships contained within also none change add contextual dependencies replacement applied repeatedly reduce number adjacent closing brackets closing bracket replacement preserves property balance remaining parentheses exactly one pair matched parentheses replaced single colon corollary runs opening bracket needed opening system obvious pushdown automaton easily transform opening closing data processed reverse order course pushdown automaton easily reverse string practice style comment delimited slashes takes work detect back front start style comment simply determined working back front within comment natural question use symbol system originally outlined sets bracket symbols eliminate runs opening closing brackets simply put additional savings significant seen help examples possible employ finite set symbols represent infinitely many numbers symbol matter form representation takes representation takes form opening brackets followed closing brackets one kind bracket collapsed preserved collapsed must represented way idea using sets brackets work reduce runs opening closing brackets thus see idea replacing run closing brackets single closing bracket really removal redundancy redundancy specifying depth twice first opening brackets equal number closing brackets redundancy longer available remove one kind bracket reduced symbol system need exclusive mix symbol usage practice coding likely preferable use symbol subtrees known advance terminal child uses minification javascript json may want use symbol everywhere possible conference july washington usa removing redundancies symbol system value data compression since amount information encoded ideal data compression algorithm produce size compressed output whether symbol system used practice output sizes vary sometimes better symbol system sometimes worse better test whether efficient representation helps data compression try much larger example biologists organized millions species tree life using newick format interleaved hierarchical format tests upon grafted version file highest percentage interleaved structural symbols relative data containing parentheses characters total show opening system reduce size even compression compression none gzip system original symbol opening final note whether prefer opening closing system closing better fit customs instance curly brace languages name array given index desired element arr arr function names name comes first universal bracket sequence different sets brackets interwoven almost always error valid mainstream language analogous sequence html could valid even though meaning case make sense html specification calls misnesting invalidity used hoc fashion reduce html redundancy common case closing outer structure implies inner structure must also close example omissions require knowledge structure allowed instance valid element paragraph direct child another element therefore always implies closing tag preceding opening tag usage acknowledged still recommended closing tag considered optional never rely might produce unexpected results errors forget end combination opening closing tag one called selfclosing tag meant empty element xml start version html adopted variation idea xml tag requires slash character immediately closing angle bracket element types singled making sense empty void html use permit penultimate slash character tags another solution html verbosity omit name end tags using works fine since misnesting allowed often sensible anyway sgml conference july washington usa feature shorttag constructs calling empty end tag html allow idea universal closing bracket alternatively universal opening bracket employed language containing sets bracket symbols interweaving invalid eliminates misnesting interweaving longer possible reduces alphabet size choose square bracket universal closing symbol sequence could become closing parenthesis symbol would unused could repurposed note change reduce number closing brackets still example reduces required size alphabet still need closing characters unindent invisible control character universal closing bracket could still used would want invisible case converting back forth representation uses closing brackets match opening brackets representation uses universal closing bracket easily done pushdown automaton type node preserved choice opening bracket type repeated matching closing bracket merely redundant established universal bracket symbol workable several questions naturally arise make code easier understand human readable many expressed sentiment requiring closing tag repeat type given opening tag helps prevent human mistakes therefore good issue confused practice entirely omitting tags specific situation means representing structure tiresomely redundant ugly short cuts made unnecessary types nodes often representational capability hierarchical structure enhanced adding means representing different kinds children example tree tree nodes assigned additional property color often done independently structure means additional data item another popular method sigils form different kinds brackets ascii sets symbols meant solely brackets parenthesis square bracket curly braces one set angle brackets doubles mathematical symbols greater less reason used gingerly sets contrived instance matter course two arbitrary characters could chosen serve brackets obvious complaint sets far even dozen plus unicode added still enough case programming language designers used ascii symbols meant brackets early curly brace languages employ curly braces denote blocks code square brackets denote array indices parentheses parameter lists dual purpose symbols angle brackets unused brackets employed preprocessor later markup languages sgml xml html brenton chapin sgml markup languages expanded number bracket types infinitely allowing multiple character brackets although solves problems caused finite quantities different bracket symbols means requirements chosen add greatly verbosity common criticism often expressed abuse rules rather words possible desire visual match opening closing bracket led sgml requirement opening closing brackets contain copies string employed give type despite obvious redundancy efficient way designate one bracket sets typed start multicharacter bracket allows brackets remain bare used traditionally used programming languages still allows infinite bracket types symbol best employed positioned bracket name name combined symbol bracket indicates name follow since one kind bracket available possibly efficient use existing symbols keep parentheses bare designate square bracket curly brace indicator child structure type name follow similar html another method reserve symbol indicate type name structure often used similarly whichever method chosen indicate start type name name ended name could like variable names programming languages letters numbers underscore character allowed name symbol space character automatically ends name method using special symbol done closing angle bracket html also workable designers html let tag names names crammed additional structured information attributes example content information could system instance something like attr class content even attr class val val content purposes alternate subsystem really serves visual distinction less verbosity though claimed maintain distinction data metadata html evolved towards lighter use attributes moving much formatting information tags css also less redundant representing siblings cousins list well known long history item list considered sibling item traditionally item line separated comma lisp means list processor built around idea making lists fundamental building block organize data code comma separated values csv notation simple data format based one list items separated course commas one notorious departures use commas multidimensional arrays syntax access element index efficient textual representation structure idea separating items list single symbol word seems simple turns several surprisingly tricky issues consider represent list language symbol analogous comma dyck language interleaved data sibling relationship expressed first note convention place parent child although opposite expressive one way wrap brackets around individual data item number brackets needed represent relationship must increased depths means siblings means cousins array would although works far verbose additionally spoils abbreviation allowing siblings separated child must instead better way always separate siblings child using null child older sibling children array represented expanding cousins still problem addition comma sibling separator becomes sequence still comma help reduce obvious extension introduce another symbol say semicolon separate cousins sequence become cousins add brackets semicolon employ yet another symbol replace many symbols employed ascii committee settled ascii characters though proposed called information separators better several issues information separators merit careful consideration first consider sequence siblings descendants cousin several different precise meanings could semicolon given higher precedence brackets three children case particular sequence invalid children cousins children parent must siblings another interpretation allow single opening bracket separate variable number generations instead always one generation since grandchildren cousins one another must grandchildren idea big disadvantage adding context dependency grammar whether child grandchild known characters opening closing brackets scanned semicolon found level grandchild even deeper separators great grandchild even distant descendant none found child best consider semicolon combined open close bracket precedence bracket case sis descendant sis nephew meaning add context invalid strings instance simple sequence invalid brackets balanced conference july washington usa second consider combine separators colons colon bracket precedence sequence means parent parent sibling uncle cousin parent null perhaps easiest way see reverse colon transform get reverse separator transform get rare supposed siblings child correct way represent use semicolon colon transforms colon separator mostly complementary cases compete reduce redundancies following table shows results transforming dyck words length replacing comma rather semicolon greater visual clarity dyck word colon separator last column shows result applying semicolon transform followed colon transform applied second transform colon blind presence separators correct achieve maximum reduction separator acts bridge colon start list natural looking use rather opening last item list separator transform second achieve maximum reduction well correct transformation done awareness colons colon may opening last item list moved head become replacing open close pair brackets separator replacing opening bracket previous item list colon repeated colon migrated front list separator transform done blindly sequence colons incorrect replacing separator gives correct correct undoing shows sequence actually list items applying transforms ackermann function given earlier replaces total bracket pairs either single separator comma used example single colon define cond else conference july washington usa third types separated items types traditional meaning comma separator untyped data text adjacent comma interpreted type name way resolve question provide another means add type desired let separators remain separators example mentioned section types character could used indicate alphanumeric sequence type name deeper separators would need list types restrictions elements specify new type notations course invented little point opening brackets accomplish reasonable efficiency without additional rules fourth different potential meanings runs separator symbol adjacent semicolons could mean empty element middle like could mean separation data elements either side deeper cousins instead cousins semicolons mean former widely used meaning latter accomplished limited method information separator symbols neatly handle great depths seems useful clear concise ways express either meaning one way information separators one siblings one cousins wrinkle repetition symbols would different meanings sibling separator mean siblings omitted list cousin separator mean cousins nth cousins cousins approach combined efficient way express quantities discussed next express meanings however another way use system expressing quantities assign different meanings quantities use typing semicolon could followed integer indicate depth divide means adjacent elements cousins run semicolons mean cousins middle run commas means middle siblings makes slightly harder support typed separators course still done point moot sticking traditional meaning separators typeless minor matter separators inherent issue comma separated list usually contains one fewer commas data items often specifications allow meaningless trailing comma present convenience programmers big reason support efficient representation cousin relationship even reserve symbols especially rather rely brackets natural way map multidimensional arrays hierarchical structure another reason people familiar like separators efficient representation arbitrary quantities infinitely many numbers represented single symbols finite set symbols brenton chapin though collapse arbitrary quantities single symbols however better using symbols represent employing principle used arabic numbering system replaced unary numbering systems roman one hash marks well known binary numbering system minimum number symbols needed represent quantities log symbols even better log represent arbitrary quantity size even fewer symbols question pigeonhole principle applies data compression able represent quantities amount fewer log symbols finite set symbols quantities must represented log symbols amounts averaged quantities size log greater numbering systems employed represent structure rather come symbols represent greater greater quantities ascii committee separator symbols employ symbols binary code obviously one symbol may repeated made one member set symbols used binary encoding many symbols may repeated finding enough symbols becomes problem since quantities useful unused symbols precious better idea reserve symbols binary code quantities symbol may repeated example instead using kinds open bracket symbol binary code something like represent open brackets mean open brackets mean asterisks still better use escape character decimal representation backslash used backslash escape sequence uses number mean null character ascii open brackets represented one desirable additional symbol allow minus sign negative quantities integers allowed need overload meaning escaped period decimal point character sort representation well known idea encoding rle rle simple easy even relatively easy person understand without computer aid course minor problem numeric symbols object rle escape sequence several easy ways resolve issue easiest simply support repetition digit characters forcing use traditional method repetition digit wanted perhaps next easiest employ terminal symbol keep representation one character shorter terminal symbol optional used needed remove ambiguity rle limited express keeps dirt simple easy read perhaps expressiveness desirable uses repeating patterns single characters one simple example sequence pair trivial amount additional syntax possible efficiently encode repeating patterns use repetition escape suppose one string consisting many perhaps half characters must escaped one traditional method inflate double quantity characters preceding special character escape character efficient textual representation structure get difficult programmer read seen perl regular expressions quantity applied indicate many characters escaped supersede traditional escape character method notion fairly obvious proposed number occasions instance rivest draft called verbatim representation hollerith constants fortran perl regular expressions similar mechanism trivial extend run length encoding handle repeating patterns still many highly redundant strings extended rle encode yet simple describe question far much complexity really useful still easily encoded would still human readable perhaps efficient way represent larger combinations also desirable encode things required simple idea support encoding lists quantities vector rather single quantity escape character employed separator needed agreement meanings assign multiple quantities example encode repetitions string length abcd something else vector quantities good idea tree quantities takes symbols represent structure tree however additional complexity almost certainly much remain human readable question uses could make tree quantities one use vector quantities sizes dimensions multidimensional array usage creeps domain external representation structure interleaving reduced single character used separator removed entirely instance array variable sized elements could notated using separator symbol every time division known deeper rest info given vector quantities array fixed sized elements could notated means represent something analogous hollerith constant provided probably use complicated objects serialize data use total length resulting string size supporting runs symbol blocks data analogous hollerith constants provides enough elaboration desired keeping notation simpler get away unary representations stick small simple structures seldom count higher almost never higher tally marks work fine check firefox source code reveals programs reached nesting depth never exceeding rle opening brackets colons quantities indicate depths separators going remove much clutter perhaps flatter structuring chosen avoid clutter would result deeper nestings block escapes still viable use quantities conference july washington usa representing structure positioning ascii control characters still really useful indicating new line text next used tab ambiguous meaning easily often replaced spaces except special cases makefiles rest ascii control characters seldom seen present modern applications may simply ignore meanings text files firefox source code ascii control characters common tab mere files use ansi escape sequences start escape character ascii set text colors ascii minimal means positioning text sufficient efficient neat one worst inefficiencies repetitive use spaces indent lines text ansi escape sequences address issue well series text terminals became popular large part adopted extended ansi escape sequences yet much used outside terminal control grow beyond niche become common within text files colored text ansi escape sequence used yet rare one common uses colored text highlighting source code use ansi even though editing may still done terminal supports ansi rather text editors parse source code edited compute colors assign updating real time user makes changes html css specify colors directly limited tiny color palette word processor options become way set text colors documents ascii ansi must use fixed width font position text accurately consequently source code almost always viewed fonts sort positioning information would useful means clear easy minimal description position best supports useful structures leading contenders indentation popular way express hierarchy position alone common even though curly brace languages use indentation coders exhorted use proper indentation anyway source code readable perhaps prominent distinctive feature python programming language use pure positioning indicate code structure another major use alignment columns usually tables ascii tab character either well superscripting subscripting considered kind positioning major limitation scale successively deeper nesting requires progressively smaller text soon becomes small read proposal reassign ascii control characters indentation increase indent push revert indent pop boost indent analogous brackets colon closing system characters invisible width zero directly affecting position text change level indentation character mean forward next indentation replacing leading spaces indented lines text could also mean advance next line conference july washington usa would less flexible support situations text line number wanted indentation characters specify size indentation number levels would allow individual users set indentation size without affecting others could also make variable width fonts usable problem unit use specify indentation sizes entirely avoided add one item state text editor viewer must maintain stack indentation levels indentation characters could ease editing source code would need shift blocks code several spaces right left changing number leading spaces line whether manually adding deleting space using smart editor function assigned tab key emacs columns tables need better tab functionality would desirable rely use monospace font recreate intended horizontal alignments limitation dump sort tiny maximum distance spaces independent indentation setting set control characters contains several intended tabular positioning enough one problem state set global another still implicitly depend upon fixed font using cursor position fixing location tab stops basically copy ideas means used advanced typewriters limitations means html provides laying tables fairly comprehensive flexible proven many years use better handling tables desired plain text copying html handling subset capabilities concerning tables control character functionality seems good approach lightweight markup languages markdown bulletin board code bbcode arose satisfy desire able create lists tables text based forums easily html shows many users like text editing capabilities conclusion paper proposed several changes standard textual notations eliminate redundancy may hampering human readability structured documents source code proving human readability improved attempted instead paper surmises kinds redundancy merely add clutter showed redundancy lurks proposed ways eliminate good answers lots idiotic spurious parentheses desired long time perhaps satisfactory notation scales adds expressiveness allows much brevity without sacrificing clarity especially preferred punctuation long histories natural languages especially english tapped guide part uses familiar people literate languages first proposed change add kind bracket symbol roughly equivalent meaning colon english relationship represented instead represented proof given symbol system brenton chapin collapse runs closing brackets single closing bracket idea universal closing bracket presented represented reducing number different symbols required longer needed use separators proposed replace sequences closing brackets followed opening brackets represented ways adding types structure discussed positioning recognized important way denoting structure observed means expressing position neglected markup languages limit data much used writing structured information source code moreover using visible text express position requiring translation special tools web browser fail goal using position alone express structure means provided ascii work monospace fonts require much wasteful redundancy repurposing unused control characters better support indentation tabular structure proposed together changes reduced number quantity symbols needed improved amount data compression obtained general purpose data compression programs reduced size source code whether goal greater human readability also achieved studied surmised removing redundancies notation help readability references leonhard euler infinitis curvis eiusdem generis seu methodus inveniendi aequationes pro infinitis curvis eiusdem generis commentarii academiae scientiarum petropolitanae original available http english translation http dirichlet beweis des satzes dass jede unbegrenzte arithmetische progression deren erstes glied und differenz ganze zahlen ohne gemeinschaftlichen factor sind unendlich viele primzahlen enthlt abhand wiss berlin english translation available http florian cajori history mathematical paragraph vol jan lukasiewicz uwagi aksjomacie nicoda dedukcji uoglniajfbcej ksiga pamifbtkowa polskiego towarzystwa filozoficznego lww oliver efficient bell system technical journal perlis samelson preliminary report international algebraic language communications acm july kenneth iverson programming language john wiley sons new york usa richards martin bcpl tool compiler writing system programming winett ebcdic codes mapping ascii rfc doi july http abelson harold sussman gerald jay sussman julie structure interpretation computer felsenstein joe newick tree format http bemer robert great curly brace trace chase http fortran reference manual sunsoft part revision november https tim bray extensible markup language xml working draft https rivest network working group internet draft may section http efficient textual representation structure goldfarb charles sgml reason first published journal american society information science volume number july annotated reprint goldfarb charles mosher edward peterson theodore online system integrated text processing journal american society information science volume oct http oren clark evans brian ingerson yaml markup language yaml final draft http wall larry synopsis blocks statements http markstrum shane staking claims history programming language design claims shafranovich common format mime type values csv files rfc doi october http buse raymond weimer westley metric software proceedings international symposium software testing analysis anonymous authors bulletin board code http yasuoka kiochi yasuoka motoko prehistory zinbun https ecma json data interchange format https html elements http macfarlane john commonmark spec http leonard guidance markdown design philosophies stability strategies select registrations rfc doi march http open tree life http conference july washington usa
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theoretical results tensor elliptical distribution sep arashi department statistics school mathematical sciences shahrood university technology shahrood iran abstract multilinear normal distribution widely used tool tensor analysis magnetic resonance imaging mri diffusion tensor mri provides statistical estimate symmetric diffusion tensor voxel within imaging volume article tensor elliptical distribution introduced extension multilinear normal mln distribution properties including characteristic function distribution affine transformations given integral representation connecting densities mln distributions exhibited used deriving expectation measurable function variate key words phrases characteristic generator inverse laplace transform stochastic representation tensor vectorial operator ams classification primary secondary introduction nowadays analysis data sets become quite common medical sciences since collected data arrays example medical imaging become possible collect magnetic resonance imaging mri data used infer apparent diffusivity water tissue vivo regard need consider parallel extensions bilinear namely tensor matrices tensor matrices commonly used approximate diffusivity profile images approximation yields diffusion tensor magnetic resonance imaging data set processing data sets scientific significance clinical sciences figure shows tensor filed diffusion mri image image analysis characteristic precision matrix underlying model tensor observations distribution eigenvalues play deterministic roles hence underlying tensor distribution influences respective inference use tensor associated distributional structure statistics dates back mccullagh mccullagh already introduced tensor notation statistics particular reference computation polynomial cumulants selective papers tensors applications statistics refer sakata pronounced studies statistical tensor analysis tensor normal multilinear normal distribution employed underlying distribution observations however slight change specification distribution pointed basser pajevic may play havoc bilinear form array component represents vector observations figure visualization tensor filed brain figure visualization data set tensor resulting inferences broaden scope distributions achieve reasonable inferential conclusions order accommodate heavier tailed distributions reasonable way produce robust inference procedures applications tensor employed related analysis broader view point one may define class tensor elliptical distributions includes latter distribution special one article define new class tensor elliptical distributions study statistical properties preliminaries section introduce related notation study give definitions adhere notation ohlson let tensor order kth tensor tensor parlance dimension direction figure shows special case indeed tensor matrix tensor vector tensor scalar connection figure figure shows collected data interpreted tensor assessment cardiac ventricular helical structure done figure helical structure cardiac ventricular anatomy vectorial representation tensor make related inference much simpler let vec denote vectorization tensor according definition kolda bader given vec ekik unit basis vectors size respectively ekik ekik denotes kronecker product index set defined ohlson authors concentrated estimation kronecker structured covariance matrix order three called double separable covariance matrix generalizing work srivastava multilinear normal mln distributions let denote space vectors vec tensor order note tensor space described using vectors however define tensor spaces using matrices given following definition definition let iii theorem ohlson tensor mln order denoted elements independent standard normally distributed note written kronecker product indeed theorem configures mln distribution using stochastic representation vector methodology mimicked extend result elliptical models revealing main result paper need consider definition matrix elliptical distributions tensor elliptical distributions let denote random vector distributed uniformly unit sphere surface characteristic function hereafter using theorem fang propose definition tensor elliptical distribution methodology behind definition distribution comes two facts random matrix matrix elliptical distribution vec elliptical distribution used tensor see gupta difference elliptical lies structure parameter space generated definition random tensor order denoted vec square root independent cumulative distribution function cdf related following relation question arises whether parameters definition uniquely defined theqanswer see assume positive constants define tensor elliptical distribution using vector representation conveniently write probability distribution function pdf extending pdf mln distribution following result gives pdf random tensor elliptical possesses density extension ohlson theorem assumptions definition pdf distribution given function density generator say satisfying designate similar fashion following result theorem let characteristic function form eis remark since taking exp definition gives pdf mln distribution given theorem ohlson exp positive definite following result gives distribution affine transformations variates theorem let vec nonsingular proof let vec stochastic representation definition proof directly follows following result direct consequent theorem gupta tensor elliptical distributions theorem assumptions definition pdf following theorem reveals distribution quadratic form special case theorem let pdf given forthcoming section provide weighting representation pdf variate using laplace operator weighting representation although proposed theorems previous section obtained conventionally easy achieve statistical properties distributions definition straightforwardly however mild conditions one make connection densities mln pdfs derive properties distributions using mln distributions section propose weighting representation connects densities mln distributions result given following theorem theorem let also assume differentiable sufficiently large vanishes faster pdf represented integral series mln pdfs given fnp fnp pdf weighting function proof let laplace transform operator noted regularity condition inverse laplace transform exists fnp proof complete thus variable integral mln variables covariance subject different scales since pdf using fubini theorem obtain fnp dtdx fnp dxdt sample space hence positive weighting functions weighting representation distributions interpreted scale mixture mln distributions however sometimes negative note distribution completely defined matrix scalar weighting function theorem enables describe properties distributions via mln distributions done using following important result theorem let weighting function borel measurable function exists enp examples section provide examples distributions based definition respective weighting function defined theorem firstly consider examples weighting function always positive resulting scale mixture multilinear normal distributions multilinear normal distribution ohlson weighting function form dirac delta impulse function property every function multilinear normal distribution say random tensor multilinear normal distribution following density exp exp concluded weighting function given iii tensor say random matrix tensor following density corresponding weighting function form tensor cauchy distribution obtained setting much interest consider cases weighting function always positive kind distributions scale mixture multilinear normal distributions item tensor distribution however tensor distribution distribution following density weighting function given sin inference theorem suppose tensor variables jointly distributed following pdf suppose pdf finite positive maximum see anderson existence suppose estimator obtains solving following equations see ohlson eir mle given proof let epikk also write since likelihood written maximum attained mle given hand get substituting using theorem ohlson gives result conclusion article purpose robust inferring diffusion tensor magnetic resonance imaging observations proposed class tensor elliptical distributions class includes many heavier tail distributions tensor normal multilinear normal mln distribution important statistical properties including characteristic function along distribution affine transformations derived weighting representations also exhibited connects densities mln distributions references basser pajevic normal distribution random variables applications diffusion tensor mri ieee transactions medical imaging fang kotz symmetric multivariate related distributions chapman hall london gupta varga bondar elliptically contoured models statistics portfolio theory springer new york kolda bader tensor decompositions applications siam review mccullagh tensor methods statistics chapman hall london mccullagh tensor notation cumulants polynomials biometrika ohlson rauf ahmad von rosen kronecker structured covariance matrix communications statistics theory methods ohlson rauf ahmad von rosen multilinear normal distribution introduction basic properties journal multivariate analysis sakata applied matrix tensor variate data analysis springer japan srivastava von rosen von rosen models kronecker product covariance structure estimation testing mathematical methods statistics
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nov evaluation trackers scenarios yucao tang bilodeau university electronic science technology china chengdu china tang yucao litiv lab polytechnique canada gabilodeau ptz camera research topic computer vision many years compared tracking still camera images captured ptz camera highly dynamic nature camera perform large motion resulting quickly changing capture conditions furthermore tracking ptz camera involves camera control position camera target successful tracking camera control tracker must fast enough able predict accurately next position target therefore standard benchmarks allow assess properly quality tracker ptz scenario work use virtual ptz framework evaluate different tracking algorithms compare performances also extend framework add target position prediction next frame accounting camera motion processing delays assess predicting make tracking robust may help slower algorithms keeping target field view camera results confirm speed robustness required tracking ptz scenario index tracking performance evaluation tracking algorithms ntroduction tracking ptz camera research topic computer vision many years compared tracking still camera images captured ptz camera highly changing nature camera perform large motion resulting quickly changing capture conditions furthermore tracking ptz camera involves controlling camera online process therefore standard benchmarks allow evaluate performance tracker ptz scenario account camera control neither drop frames caused spending much time processing frame track target specific benchmark required evaluation scenario work used virtual ptz framework developed evaluate recent trackers ptz tracking scenario goal evaluation assess performance trackers online tracking scenario combination speed robustness required ptz scenarios include many changes scale illumination large motion evaluated tracking algorithms compared analyzed performances many trackers tested vot benchmark also extended framework add target work conducted yucao tang mitacs globalink internship polytechnique position prediction next frame accounting camera motion processing delays assess prediction help make tracking robust help slower algorithm keeping target field view camera valuation framework metrics chen proposed framework evaluate reproducible way trackers ptz scenarios online nature scenario authors proposed use ptz camera simulator pans tilts zooms based spherical video captured offline simulator also includes relevant delays result drop frames tracking takes long target large motion image plane delays categorized execution delay motion delay communication delay simulator configuration use evaluation framework indicated chen however since tracker codes others matlab adjusted delays calculated ensure fairness execution time evaluation used chronometer functions based execution time calculate time elapsed processing frames trackers instead clock program running time spent read image disk drive included execution time motion delay calculated time takes simulated camera tilt pan decided would communication delays supposing camera networked codes written originally matlab use matlab engine call matlab function word tracker interfaces minor changes made matlab source codes eliminating display drawing functions since affect speed processing totally unnecessary thus taken irrelevant delays besides practical experiments turns calling matlab engine actually time scale milliseconds overhead neglected since time process frame trackers much longer delay performance evaluation chen defined four performance metrics evaluate trackers metrics calculated image plane current camera viewpoint viewed subregion image sphere projected camera image plane let cgt target center predicted target center agt target bounding box predicted bounding box respectively center camera image plane words field view fov target point error bort box overlapping ratio evaluate quality target localization defined move experimented three motion models obtain target motion two instants model object moving constant speed uses velocity last instant bort agt agt target point offset track fragmentation evaluate quality camera control defined cgt invalid otherwise indicates whether target inside camera fov invalid assigned target outside fov overall metrics bor average metrics valid tracked frames sum divided number processed frames experiments report bor significant metrics besides bor similar purpose iii target osition rediction authors made remark tracker successful either fast use kind target position prediction keep target close fov camera assess practicality predicting target position based previous track information implemented three target motion models since delay frames necessary predict object position image sphere move camera accordingly target appears fov therefore prediction must account motion target motion camera calculation let target first frame appear point spherical image next frame target moves spherical image calculate speed third frame target moved speed would thus basic classical mechanics model used estimate next position fourth frame position prediction locate target near image center much possible knowing motion target possible predict later delay account processing time current frame addition time takes camera model object moving constant speed uses mean velocity last two instant model object accelerate acceleration ested rackers among trackers tested six trackers variations correlation filters kcf srdcf swcf dsst dfst skcf two trackers combine correlation filter outputs color staple one based structured svm struck two trackers based purely color dat asms one tracker based normalized crosscorrelation ncc two trackers based boosting mil boosting one tracker based optical flow medianflow one tracker includes detector tld two trackers categorized dpcf ctse another one combines many trackers ensemble kfebt briefly describe trackers details found original papers describing kernelized correlation filter tracker kcf kcf operating hog features localizes target equivalent kernel ridge regression trained sample patches around object different translations version kcf tracker includes support peak estimation model update linear interpolation spatially regularized discriminative correlation filter tracker srdcf tracker derived kcf introduces spatial regularization function penalizes filter coefficients residing outside target area thus solving problems arising assumptions periodicity learning correlation filters size training detection samples increased without affecting effective filter size selecting spatial regularization function sparse discrete fourier spectrum filter optimized directly fourier domain srdcf employs also color names greyscale features spatial windowing correlation visual tracking swcf tracker derived kcf predicts spatial window observation object correlation output correlation filter well windowed observation improved moreover estimated spatial window object patch indicates object regions useful correlation discriminative scale space tracker dsst dsst extends minimum output sum squared errors mosse tracker robust scale estimation dsst also learns discriminative scale filter used predict target size intensity features used mosse tracker combined representation hog features dynamic feature selection tracker dfst dfst visual tracking algorithm based selection locally temporally discriminative features dfst provides significant gain accuracy precision respect kcf use dynamic set features improvement given making predicted position according best template matching scalable kernel correlation filter sparse feature integration skcf tracker derived kcf introduces adjustable gaussian window function model scale estimation deals fixed window size limitation kcf sum template learners staple staple combines two image patch representations sensitive complementary factors learn model robust color changes deformations combines scores two models dense window translation search scores two models indicative reliability improved staple tracker multiple feature integration based staple tracker improves integrating multiple features extracts hog features color probability map exploit color information better final response map thus fusion scores obtained different features structured output tracking kernels struck framework adaptive visual object tracking applies support vector machine learned online introduces budgeting mechanism prevents unbounded growth number support vectors would otherwise occur tracking distractor aware tracker dat approach based appearance distinguish object surrounding areas discriminative model using color histograms applied adapts object representation beforehand distractors suppressed risk drifting reduced scale adaptive mean shift asms tracker optimizing hellinger distance template histogram target image optimization done gradient descent asms addresses problem scale adaptation scale estimation also introduces two improvements original make scale estimation robust presence background clutter histogram color weighting consistency check normalized ncc ncc follows basic idea tracking searching best match static grayscale template image using normalized multiple instance learning tracker mil mil uses approach multiple instance learning instead traditional supervised learning methods shows improved robustness inaccuracies tracker incorrectly labeled training samples boosting based mil realtime object tracker based novel version adaboost algorithm classifier uses surrounding background negative examples update step avoid drifting problem medianflow tracker uses optical flow match points frames tracking performed forward backward time discrepancies two trajectories measured proposed error enables reliable detection tracking failures selection reliable trajectories video sequences tld combines tracker detector tracker follows object frame frame using medianflow detector localizes target using appearances observed far corrects tracker necessary learning estimates detector errors updates avoid errors future deformable tracking coupled global local correlation filters dpcf tracker derived kcf relies joint interactions global filter local part filters local filters provide initial estimate used global filter reference determine final result global filter provides feedback part filters way handles partial occlusion part filters also scale changes global filter contextual object tracker structural encoding ctse tracker uses contextual structural information specific target object appearance model first achieved including features complementary region correlated motion target object second local structure represents spatial constraints features within target object included sift keypoints used features encode information kalman filter tracker tracker combines result two trackers asms using color histogram kcf using kalman filter tracker works cycles prediction correction firstly motion model predicts target next position secondly trackers results fused predicted position model updated esults analysis section report results video sequences video sequences consist tracking persons faces objects include difficulties motion blur scale change rotation fast motion cluttered background illumination variation low resolution occlusion presence distractors articulated objects results reported whole dataset ranking method testing discovered since different trackers different tracking speed using four metrics section enough example trackers bor metrics good track slowly simulation means track frames correctly frames invalid ignored calculation metrics circumstance tracker succeed tracking every processed frames first frames capture extent lack performance verifies target fov thus consider another essential metric processed frame ratio number processed frames total number frames contains part information since stands whether object inside fov processed frames low high since tracker able track object correctly processed frames long interval however high caused also poor robustness considering ptz camera nature tracker test results formulated ranking formula formula stands euclidean distance point defined pair bor obtained tracker ideal tracker point figure score thus score bor selected bor consider conveys similar information bor conveys similar information results without processing delays first set execution ratio zero order compare different trackers performances rotations rotations drastic scene changes caused camera motion looking robust trackers neglecting processing times set perform speed note experiment camera motion delays considered reflects robustness trackers tracker performs poorly cause unnecessary camera motion result drop frames table racker performances execution ratio set video sequences rackers ranked based score equation tracker names asms dpcf matlab tld dat matlab matlab staple matlab dsst struck skcf kcf swcf matlab mil dfst matlab boosting medianflow srdcf matlab ncc ctse score bor table figure give results trackers based ranking ptz camera scenario difficulties rotations amplified application dynamic nature surprisingly trackers adopted scale adaptation function srdcf necessarily perform better trackers due great part slow execution speed asms dpcf best performers experiment vot benchmark drop frames caused delays less viewpoint changes caused camera motion thus reasonable difference ranking framework vot ranking however ranking still quite similar vot benchmark trackers like staple dat good vot benchmark benchmark however benchmark performance asms surprisingly best ranked middle vot trackers like srdcf behave well ptz framework probably output bounding boxes tracking fails result ptz camera controlled correctly vot system check every five frames verify whether tracker failed failed reinitialized framework intervene tracking process early failures thus penalized results processing delays set execution ratio track objects real world objects keep moving tracker processing frame result task tracking harder case since intervals processed frames caused time processing frame camera motion delay decline trackers table racker performances execution ratio set video sequences rackers ranked based score equation tracker names asms staple matlab dat matlab tld dsst kcf struck skcf boosting matlab srdcf matlab dpcf matlab dfst matlab medianflow swcf matlab ncc mil ctse score bor lose targets easily table figure give results trackers compared previous case ranking trackers changes asms still ranks first dpcf degrades slow value declines previously trackers relative rankings change much average score trackers higher means performances worse execution delay tracking process still compared vot benchmark performance asms still surprisingly best means tracker particularly good handling viewpoint changes finally performance vot better task testing online tracking camera control ptz scenario requires tracking people different illumination variation different scale rapidly changing viewpoints reasons make results unique compared benchmarks table iii racker performances execution ratio set various prediction methods video sequences tracker names kcf kcf kcf kcf boosting boosting boosting boosting prediction none model model model none model model model score bor tracker lose target target tracked speed updated leading even worse situation camera rotate less randomly highspeed trackers affected wrong prediction process ratio decline therefore conclude experiment although appealing theory compensating slow tracking position prediction easy apply practice may work objects far away mostly move plane work target closer moving toward away camera cases motion target predicted best results thus preferable design fast tracker onclusion paper presented benchmark recent trackers ptz tracking scenario surprisingly trackers medianflow ncc necessarily behave better others however since predicting target position shown difficult slow trackers avoided ptz tracking task results test indicate top performing tracker ptz scenario asms tracker tracker performed well accuracy well robustness tests impossible conclusively determine whether performance asms trackers come image features approach nevertheless results top trackers show features play significant role final performance target position prediction finally tested target position prediction investigate help trackers perform better table iii gives results two trackers results similar trackers proposed models see section iii predicting next position target improving results due fact speed target difficult estimate moves estimate motion therefore predicted speed accurate calculating speed framework use speed predict target position next frame may cause unnecessary large motion camera example predicted target motion may large camera rotating wrongly predicted position add delays tracking process adds possibility eferences kristan matas leonardis felsberg cehovin vojir hager nebehay pflugfelder visual object tracking challenge results proceedings ieee international conference computer vision workshops chen bouachir bilodeau bergevin reproducible evaluation tracking image processing icip ieee international conference ieee henriques caseiro martins batista tracking kernelized correlation filters process ratio bor kcf boosting medianflow mil dfst srdcf stapleplus colorkcf dpcf swcf dat asms tld staple struck dsst skcf ncc ctse without processing delay process ratio bor processing delay fig bor scatter plot without left processing delays right video sequences legend indicates symbols representing trackers plots ideal results bor ieee transactions pattern analysis machine intelligence vol danelljan hager shahbaz khan felsberg learning spatially regularized correlation filters visual tracking proceedings ieee international conference computer vision gundogdu alatan spatial windowing correlation filter based visual tracking image processing icip ieee international conference ieee danelljan khan felsberg accurate scale estimation robust visual tracking british machine vision conference nottingham september bmva press bolme beveridge draper lui visual object tracking using adaptive correlation filters computer vision pattern recognition cvpr ieee conference ieee roffo melzi online feature selection visual bmvc montero lang laganiere scalable kernel correlation filter sparse feature integration computer vision workshop iccvw ieee international conference ieee bertinetto valmadre golodetz miksik torr staple complementary learners realtime tracking proceedings ieee conference computer vision pattern recognition hare golodetz saffari vineet cheng hicks torr struck structured output tracking kernels ieee transactions pattern analysis machine intelligence vol possegger mauthner bischof defense tracking proceedings ieee conference computer vision pattern recognition vojir noskova matas robust tracking pattern recognition letters vol comaniciu ramesh meer tracking objects using mean shift computer vision pattern recognition proceedings ieee conference vol ieee lewis fast normalized vision interface vol babenko yang belongie robust object tracking online multiple instance learning ieee transactions pattern analysis machine intelligence vol grabner grabner bischof tracking via kalal mikolajczyk matas forwardbackward error automatic detection tracking failures pattern recognition icpr international conference ieee ieee transactions pattern analysis machine intelligence vol akin erdem erdem mikolajczyk deformable tracking coupled global local correlation filters journal visual communication image representation vol chakravorty bilodeau granger contextual object tracker structure encoding image processing icip ieee international conference ieee
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image editing recurrent attentive models nov jianbo yelong jianfeng jingjing xiaodong university california microsoft jianbochen yeshen jfgao jingjl xiaodl abstract investigate problem image editing lbie work given source image natural language description want generate target image editing source image based description propose generic modeling framework two lbie image segmentation image colorization framework uses recurrent attentive models fuse image language features instead using fixed step size introduce region image termination gate dynamically determine inference step whether continue extrapolating additional information textual description effectiveness framework validated three datasets first introduce synthetic dataset called cosal evaluate performance lbie system second show framework leads performance image segmentation referit dataset third present first colorization result flowers dataset laying foundation future research introduction work aim develop automatic image editing lbie system given source image sketch grayscale image natural image system automatically generate target image editing source image following natural language instructions provided users system wide range applications design cad virtual reality illustrated figure envision fashion designer presents sketch pair new shoes source image customer provide modifications style color verbal description taken lbie system change original design final output target image revised enriched design would meet customers requirement figure showcases use lbie systems still using touchscreen interface lbie provides natural user interface future systems users easily modify base environment via natural language instructions lbie cover broad range tasks image generation shape color size texture position etc paper focuses two basic lbie segmentation colorization shapes colors example figure given grayscale image expression flower red petals yellow stigmas middle segmentation model identify region image petals stigmas colorization model paint pixel suggested color intertwined task segmentation colorization distribution target images sense pixel definitive ground truth segmentation necessarily color example pixels petals figure red based textual description specific numeric values red color rgb space uniquely specified system required colorize petals based knowledge another uncertainty lies fact input description figure interactive design interface sketch shoes presented customer gives verbal instruction modify design insole shoes brown vamp heel purple shining system colorize sketch following customer instruction images figure image left initial virtual environment user provides textual description afternoon light flooded little room window shining ground front brown bookshelf made wood besides bookshelf lies sofa cushions blue carpet front sofa clock dark contours system modifies virtual environment right image might cover every detail image undescribed regions leaves given example rendered based past memory summary ultimate goal generate images guided constraint natural language expressions also align common sense real world image segmentation studied previously however task far challenging textual description task often contains multiple sentences figure expressions simple phrases best knowledge colorization studied systematically previous work images generated either solely natural language expressions based another image instead want generate target image based natural language expressions source image related tasks discussed detail section unique challenge image editing complexity natural language expressions correlation source images like example shown figure description usually consists multiple sentences sentence might refer multiple objects source image many details also need inferred multiple sentences human edits source image based textual description often keep mind sentences related image back sentences multiple times editing region behavior going back often varies region region depending complexity description region investigation problem carried cosal dataset introduced purpose detailed section figure left sketch image middle grayscale image right color image image editing system take either first two images input generate third color image following natural language expression flower red petals yellow stigmas middle goal design generic framework two image editing diagram model shown figure inspired observation aforementioned introduce recurrent attentive fusion module framework fusion module takes image features encoding source image via convolutional neural network textual features encoding natural language expression via lstm outputs fused features upsampled deconvolutional network target images fusion module recurrent attentive models employed extract distinct textual features based spatial features different parts image termination gate introduced region control number steps interacts textual features reparametrization trick used training entire network details models training process described section contributions summarized follows define new task image editing lbie present generic modeling framework based recurrent attentive models two lbie image segmentation colorization introduce synthetic dataset cosal evaluate performance lbie system achieve new performance image segmentation referit dataset present first colorization result flowers dataset laying foundation future research related work task image editing studied community taken significant steps several related areas including language based object detection segmentation lbs translation iit generating images text git image captioning visual question answering vqa machine reading comprehension mrc etc summarize types inputs outputs related tasks table recurrent module termination gate termination gate spatial feature map decoder spatial feature map fusion spatial feature map output fusion image encoder attention attention attentive feature map attentive feature map language feature vectors flower red petals yellow stigmas middle language encoder figure diagram model composed convolutional image encoder lstm text encoder fusion module deconvolutional upsampling layer optional convolutional discriminator mrc vqa iit git lbs lbie inputs text image yes yes yes yes yes yes yes yes yes yes outputs text image yes yes yes yes yes yes yes table types inputs outputs related tasks recurrent attentive models recurrent attentive models applied visual question answering vqa fuse language image features stacked attention network proposed identifies image regions relevant question via multiple attention layers progressively filter noises pinpoint regions relevant answer image generation sequential variational framework draw shown substantial improvement standard variational vae similar ideas also explored machine reading comprehension models take multiple iterations infer answer based given query document novel neural network architecture called reasonet proposed reading comprehension reasonet performs inference number steps determined termination gate according difficulty problem reasonet trained using policy gradient methods segmentation language expressions task image segmentation first proposed given image natural language description system identify regions image correspond visual entities described text authors proposed approach composed three main components convolutional network encode source images lstm network encode natural language descriptions fully convolutional classification upsampling network segmentation one key differences approach way integrating image text features region image extracted spatial features concatenated textual features inspired alignment model approach spatial feature aligned different textual features based attention models approach yields superior segmentation results benchmark dataset conditional gans image generation generative adversarial networks gans widely used powerful tool image generation conditional gans often employed constraints generated image needs satisfy example deep convolutional conditional gans used synthesize images based textual descriptions proposed use conditional gans translation different tasks lbie takes image text input presenting additional challenge fusing features source image textual description framework overview proposed modeling framework shown based neural networks generic image segmentation colorization tasks framework composed convolutional image encoder lstm text encoder fusion network generates fusion feature map integrating image text features deconvolutional network generates outputs target image upsampling fusion feature map optional convolutional discriminator used training colorization models image encoder image encoder convolutional neural network cnn given source image size cnn encoder produces spatial feature map position feature map containing feature vector channels language encoder language encoder recurrent long memory lstm network given natural language expression length first embed word vector word embedding matrix use lstm produce word contextual vector encodes contextual information word order dependencies resulting language feature map recurrent attentive fusion module fusion network fuses text information image feature map outputs fusion feature map position image region containing editing feature vector fusion network devised mimic human image editing process region source image fusion network reads language feature map repeatedly attention different parts time enough editing information collected generate target image region number steps varies region region internal state internal state time step denoted sti spatial feature map poition image region containing vector representation editing information state initial state spatial feature map source image sequence internal states modeled convolutional gated recurrent units described attention attention time step denoted spatial feature map generated based current internal state language feature map attention attention implemented follows exp sti update current internal state infusing attention feature map implemented follows relu convolutional operator note intermediate output fusion feature map time step termination gates termination gates one image region termination gate generates binary random variable according current internal state image region sti fusion process image region stops editing feature vector image region set terminate gates true fusion process entire image completed fustion network outputs fusion feature map define categorical distribution ftg sti ftg ski probability ith feature map stopping time inference algorithm describes stochastic inference process fusion network state sequence hidden dynamic chained attention recurrent fashion fusion network outputs image region editing feature vector step controlled ith termination gate varies region region algorithm stochastic inference fusion network require spatial feature map image require language feature map expression ensure fusion feature map function usion initialize tmax attention sample set end end set end end end function image decoder image decoder deconvolutional network takes input fusion feature map produced fusion module unsamples produce editing map size target image number classes segmentation channels colorization discriminator discriminator takes generated image outputs probability image realistic structure discriminator follows uses convolutional neural network extract features image model discriminator serves function judging whether image looks natural whether compatible language descriptions latter constraint left loss described later loss training denote loss expectation taken categorical variables generated termination gates loss output sample space exponential size intractable sum entire sample space denote density naive approach approximation subsample loss update parameters via gradient monte carlo estimate loss log log subset sampled distributon experiments found monte carlo estimate suffers high variance resolve issue employ reparameterization trick replaces every sampled cat another random variable generated distribution exp log zit exp log temperature annealed via fixed schedule auxiliary random variables samples drawn gumbel independent parameters log log uti uti unif define loss rewritten update approximated taking gradient monte carlo estimates loss obtained sampling use two different losses segmentation colorization respectively segmentation segmentation assume unique answer pixel whether referred stage segmentation response map size produces log probability class pixel use softmax loss training softmax colorization colorization goal generate realistic images constraint natural language expressions input scene representations introduce mixture gan loss loss optimization discriminator parametrized introduced constructing gan loss response map predicted color channels combined grayscale source image produce generated color image generator loss gan loss taking input loss channels target image response map log discriminator trained first generating sample via algorithm combined grayscale source image produce optimize following loss log log generator loss discriminator loss optimized alternatively training stage experiments conducted three experiments validate performance proposed framework new dataset cosal introduced test capability understanding descriptions associating inferred textual features visual features framework also yielded performance benchmark dataset referit image segmentation third experiment carried flowers dataset colorization task experiments cosal dataset synthetic dataset cosal colorizing shapes artificial language introduced studying correlation images descriptions image dataset consists nine shapes paired textual description image goal task defined given blackwhite image corresponding description train model automatically colorize nine shapes following textual description figure shows example task requires sophisticated coreference resolution inference logical reasoning dataset created follows first divide image regions region contains shape randomly sampled set shapes squares fat rectangles tall rectangles circles fat ellipses tall ellipses diamonds etc shape filled one color choices chosen random position size shape generated uniform random variables illustrated figure difficulty task increases number color choices experiments specify start point descriptive sentences image divided two categories direct descriptions relational descriptions former prescribes color certain shape diamond red latter depicts one shape conditional another shape left diamond blue understand direct descriptions model needs associate specified shape textual features relational description adds another degree difficulty calls advanced inference capability reasoning ratio direct descriptions relational descriptions varies among different images colors shapes image uniquely determined description experiment randomly generated images corresponding descriptions training purpose images descriptions testing metric task use average iou nine shapes background evaluation metric specifically region compute iou ratio total intersection area total union area predicted colors ground truth colors also compute iou background white image iou classes shapes background computed entire test set averaged model implementation convolutional network implemented image feature extractor layer kernel stride output dimension relu used nonlinearity layer layer kernel size inserted every two layers sentence textual description encoded bidirectional lstms share parameters lstms units fusion network attention model units gru cells use units termination gate uses linear map top hidden state gru cell two convolutional layers kernel size output dimension put top fused features classifier upsampling layer implemented top deconvolutional network kernel size stride upsample classifier original resolution upsampling layer initialized bilinear transforms maximum termination steps vary model reduced simply concatenating features extracted convolutional network last vector lstm results table shows average iou two models respectively results show model attention achieves better performance relational descriptions dataset direct descriptions two models achieve similar performance demonstrates framework capability interpreting multiple sentences associating source image attention yes yes number direct descriptions table average iou two models without attention attention performance varies among datasets different ratios direct relational descriptions figure illustrates model interprets nine sentences inference step sentences red attended first step yellow green attended next two consecutive steps one observation experiment model first extracted information direct descriptions focused relational descriptions additional reasoning experiments referit dataset referit dataset composed photographs real world scenes along natural language descriptions distinct objects photographs dataset contains different object categories including animals people buildings objects background elements grass sky training development datasets include images figure right ground truth image left illustration sentences attended time step red yellow green represent first second third time step respectively model precision precision precision precision precision iou scrc bbox grounder bbox etc model table results previous models model referit dataset metric following use two metrics evaluation overall overall iou predicted ground truth region averaged entire test set precision threshold percentage test data whose per image iou prediction ground truth threshold thresholds set model implementation architecture used image encoder images size textual descriptions encoded lstm units fusion network attention model uses units gru cells units top classifier upsampling layer similar implementation section maximum number inference steps relu used top convolutional layer applied parameters network results table shows experimental results model baseline methods referit dataset see framework yields better iou precision hypothesis outperformance resulted unique attention mechanism used fusion network efficiently associate individual descriptive sentences different regions source image much discrepancy two models probably due fact textual descriptions dataset simple examples semantic labels provided supplementary materials figure first row original images second row results translation model without text input third row results model taking textual descriptions account figure first row original images second third fourth rows results generated framework arbitrary text input petals flower experiments flower dataset dataset experiment use flowers dataset contains images flower categories image five textual descriptions following split dataset classes training classes testing given grayscale image flower description shapes colors flower goal colorize image following description model implementation convolutional network similar used encoding images textual descriptions encoded lstm units fusion network attention model uses units gru cells units image encoder composed deconvolutional layers followed convolutional layers upsample fusion feature map target image space maximum length spatial rnn discriminator composed five layers convolutional networks stride output dimension discriminator score average final output relu used nonlinearity following convolutional layer except last one uses sigmoid function results due lack available models task compare framework previous model developed translation baseline colorizes images without injection texts colorization results images test set shown figure see without text input baseline approach often colorizes images color dataset images painted blue white framework generate flowers similar original colors specified texts figure also provides examples images generated arbitrary text input using trained model conclusion future work paper introduce problem image editing lbie propose generic modeling framework two lbie image segmentation colorization heart proposed framework fusion module uses recurrent attentive models dynamically decide region image whether continue fusion process models demonstrated superior empirical results three datasets including referit dataset image segmentation flower dataset colorization proposed cosal dataset evaluating performance lbie system future extend framework image editing subtasks another interesting area build image editing system allows users edit images interactive natural way references akata reed walter lee schiele evaluation output embeddings image classification proceedings ieee conference computer vision pattern recognition pages antol agrawal mitchell batra lawrence zitnick parikh vqa 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simple lower bounds approximate isoperimetric inequalities jan yitong abstract prove log lower bound complexity deterministic las vegas randomized algorithms solving approximate ann problem hamming space model cells using table size lower bound matches highest known lower bounds static data structure problems lower bound proved general framework relates complexity ann isoperimetric inequalities underlying metric space tighter connection ann lower bounds isoperimetric inequalities established stronger richness lemma proved techniques introduction nearest neighbor search nns problem fundamental problem computer science problem database points metric space dist preprocessed data structure query time given query point metric space asked find point database closest according metric paper consider decision approximate version nns approximate nearneighbor ann problem algorithm asked distinguish two cases point databases query point radius points database away query point approximation ratio complexity nearest neighbor search extensively studied model classic model data structures model database encoded table consisting memory cells upon query algorithm answers query making adaptive table complexity problem measured tradeoff time cost terms number answer query space cost terms sizes table cells substantial body work complexity nns various metric space widely believed nns suffers curse dimensionality problem may become intractable solve dimension metric space becomes high consider important example hamming space log sufficiently state key lab novel software technology nanjing university china yinyt work supported nsfc grants large constant conjecture nns metric remains hard solve either approximation randomization allowed individually series pioneering works technique asymmetric communication complexity known richness lemma lower bounds form log stands number cells table proved deterministic approximate due liu randomized exact due barkol rabani lower bound highest possible lower bound one prove communication model fundamental barrier overcome elegant technique introduced seminal work thorup lower bounds deterministic ann randomized exact improved log represents number bits cell recently previous work applying technique thorup certificates data structures lower bound deterministic ann improved log last lower bound behaves differently polynomial space poly space polylog linear space particular bound becomes space cost strictly linear entropy database randomization approximation allowed complexity nns substantially reduced tables log log log log log tight bound proved randomized approximate nns hamming space consider decision version randomized ann solved table polynomial size tables size technique called introduced panigrahy prove log log lower bounds randomized ann later extended general asymmetric metrics among lower bounds randomized ann lower bounds panigrahy proved explicitly complexity significance complexity nns discussed papers recent breakthrough upper bounds also attributes solving problem random database retrospectively randomized exact lower bounds due density version richness lemma also hold random inputs lower bounds hold monte carlo randomized algorithms fixed complexity leaves open important case complexity deterministic las vegas randomized algorithms ann number may vary different inputs contributions study complexity deterministic las vegas randomized algorithms approximate ann problem number answer query may vary different pairs average taken respect distribution input queries databases ann hamming space hard distribution inputs natural every point database sampled uniformly independently hamming space query point also point sampled uniformly independently according earlier lower bounds recent lsh algorthm input distribution seems capture hardest case nearest neighbor search also central obstacle overcome efficient algorithms simple proof show following lower bound complexity ann hamming space natural input distribution theorem log deterministic las vegas randomized algorithm solving problem hamming space model using table size must expected complexity log expectation taken uniform independent input database query random bits algorithm lower bound matches highest known lower bounds static data structure problems lower bound known polynomial evaluation also deterministic ann due previous work also prove lower bound ann lower bound matches highest known lower bound problem fact prove lower bounds unified framework relates cellprobe complexity ann isoperimetric inequalities regarding expansion property metric space inspired notions metric expansion defined define following notion expansion metric space let dist metric space point denoted set points within distance consider distribution say weakly independent distribution point measure constant say distribution point set denotes set points within distance point consider database every point sampled independently query sampled independently denote input distribution prove following lower bound theorem metric space dist assume followings weakly independent distribution distribution deterministic las vegas randomized algorithm solving problem dist model cells using table size must expected complexity log log log log log input distribution key step prove theorem stronger version richness lemma prove section proof stronger richness lemma uses idea called introduced panigrahy later refined larsen new richness lemma well connection techniques richness lemma techniques interests preliminary let dist metric space let problem defined follows database points preprocessed stored data structure upon query accessing data structure want distinguish following two cases point database dist points database dist cases answer arbitrary abstractly given universe queries universe databases data structure problem function maps every pair query database answer example query universe metric space database universe set tuples points maps query database boolean answer neighbor database points database neighbor arbitrary otherwise note due technical reason usually use indicate case given data structure problem code alphabet encodes every database table cells cell storing word bits use denote set indices cells use denote content cell table write tuple contents cells upon query algorithm adaptive retrieves contents cells table called outputs answer last adaptive means algorithm actually decision tree round address cell probe next determined query well contents cells probed previous rounds together pair code decision tree called scheme randomized schemes algorithm takes sequence random bits internal random coin paper consider deterministic las vegas randomized algorithms therefore algorithm guaranteed output correct answer terminates scheme fixed size table well length cell fixed two parameters together give space complexity number may vary pair inputs may random variable algorithm randomized given distribution complexity scheme given expected number answer sampled expectation taken input distribution internal random bits algorithm richness lemma complexity richness lemma rectangle method introduced classic tool proving cellprobe lower bounds data structure problem natural communication problem scheme interpreted communication protocol algorithm table communications given distribution data structure problem distribution combinatorial rectangle monochromatic richness lemma states problem dense enough rich easy solve communication contains large monochromatic specifically problem solved alice sending bits bob sending bits total contains monochromatic size uniform measure model cells tables size complexity means monochromatic size log log lower bounds proved refuting large specific data structure problems prove following richness lemma complexity lemma let distributions respectively let product distribution deterministic randomized las vegas scheme solving table cells containing bits expected input distribution monochromatic compared classic richness lemma new lemma following advantages holds complexity gives stronger result even restricted complexity newly introduced parameter confused overhead caused complexity argument rather strengthens result even lower bounds gives bound classic richness lemma lemma claims existence family rectangles parameterized therefore prove lower bound enough refute one rectangle family see gives power prove highest lower bounds even worst case known static data structure problems proof lemma uses argument called introduced panigrahy approximate nearest neighbor search later refined larsen polynomial evaluation proof greatly influenced larsen approach rest section dedicated proof lemma proof richness lemma lemma fixing random bits sufficient consider deterministic algorithms high level idea proof simple fix table procedure called procedure chooses subset many cells resolve maximum amount positive queries associates database string call certificate represent contents cells due nature cellprobing algorithm certificate fixed set queries resolve fixed also observe density problem good databases amount positive queries resolved certificate constructed procedure least queries hand since many certificates therefore least good databases least databases associated pick popular certificate positive queries resolves together good databases associated form large monochromatic proceed formal parts proof given database let denote set positive queries use denote distribution induced let pxy denote set cells probed algorithm resolve query database fix database let subset cells say query resolved resolved probing cells table storing database pxy denote pxy set positive queries resolved database assume two databases indistinguishable meaning tables storing respectively cell contents due determinism algorithm resolve set positive queries databases procedure fix database suppose procedure following procedure chooses unique measure positive queries resolved maximized one procedure chooses arbitrary one use denote set cells chosen procedure also denote set positive queries resolved chosen set cells database procedure chooses informative set cells size resolve maximum amount positive queries use denote contents along addresses cells chosen procedure database call certificate chosen procedure let two databases simple observation two databases certificate chosen procedure respective sets positive queries resolved certificate going well proposition databases let denote number resolve query database assumption lemma inputs sampled product distribution claim many good columns databases high density low average costs claim collection ygood substantial amount good databases ygood every ygood followings true amount positive queries large average complexity among positive queries bounded deterministic means chosen set function proof claim proved series averaging principles first consider ydense set databases least positive queries averaging principle ydense since ydense ydense distribution induced ydense construct ygood ydense set ydense average complexity bounded markov inequality ygood hence ygood note ygood rest consider good databases fix claim every good database ygood procedure always picks subset many cells resolve substantial amount positive queries claim every ygood holds proof fix good database ygood need prove exists claims follows immediately resolve positive queries construct hypergraph vertex set pxy positive queries database associated hyperedge pxy precisely set cells probed algorithm resolve query database also define measure hyperedges total measure positive queries associated formally every pxy since probability distribution hyperedges moreover recalling average size hyperedges probabilistic method whose proof full paper must exist size induced construction positive queries associated hyperedges induced pxy pxy precisely positive queries pxy therefore pxy recall every ygood since claim proved recall certificate constructed procedure certificate let denote set database every possible assignment good databases ygood certificate due determinism procedure classifies ygood many disjointed subclasses recall ygood averaging principle following proposition natural denoted proposition exists certificate hand fixed since databases proposition must abuse notation write let satisfies proposition due claim proposition note every set positive queries database thus monochromatic finishes proof lemma rectangles conjunction problems many natural data structure problems expressed conjunction relations query point database points consider data structure problem let database tuple points function given data structure problem defined conjunction subproblems many natural data structure problems defined way example membership query finite domain function indicates whether two points unequal metric space distance dist function defined dist dist function value arbitrary cases partial match pmd alphabet function defined otherwise show refuting large rectangles function give lower bounds conjunction problem let distributions respectively let product distribution let function data structure problem defined conjunction lemma defined assume deterministic randomized las vegas scheme solving table cells cell containing bits expected input distribution followings true density distribution constant contain monochromatic measure least log log log distribution proof union bound density distribution lemma assumption existing scheme parameters altogether imply monochromatic constants depending let largest set columns form formally clearly monochromatic must definition conjunction must hold none means hence recall monochromatic due assumption lemma either therefore either always choose log log satisfy less immediately lower bound log log otherwise due monochromatic must hold gives log therefore gives lower bound log isoperimetry ann lower bounds given metric space distance dist say two points dist otherwise point denoted set points given point set define set points point natural notion metric expansion introduced definition metric expansion let metric space probability distribution fix radius define min expansion distribution defined largest introduce refined definition metric expansion using two parameters definition let metric space probability distribution distributions metric expansion defined actually special case expansion metric space means notion allows describe extremal expanding situation metric space expanding stop measure rather way close measure generality may support higher lower bounds approximate given radius approximation ratio recall near neighbor problem defined conjunction function arbitrary cases observe actually version near neighbor following proposition gives intrinsic connection expansion metric space size monochromatic rectangle relation proposition distribution function defined contain monochromatic measure distribution proof since definition monochromatic must hold therefore either proposition together lemma immediately gives following corollary reduces lower bounds problems isoperimetric inequalities corollary let distribution metric space let assume deterministic randomized las vegas scheme solving table cells cell containing bits expected input distribution followings true constant distribution log log log remark lower bound proved following form swt corollary unless unrealistically large comparable corollary always gives first lower bound log metric space strictly improves lower bound example expanding would give lower bound logdsw particular space linear becomes lower bound ann hamming space let hamming space hamming distance dist recall represents around case hamming ball radius centered set set points within distance point denote hamming ball radius pthe volume zero vector obviously following isoperimetric inequality harper well known lemma harper theorem let hamming space let every words hamming balls worst vertex expansion following upper bound volume hamming ball well known boolean entropy function consider hamming problem hard distribution problem uniform independent distribution database database point sampled uniformly independently query point sampled uniformly independently theorem let log solved deterministic las vegas randomized scheme table cells cell containing uniform independent database query bits expected log proof choose satisfy going show let uniform distribution distribution lower bounds follows directly corollary first chernoff bound point thus trivially hand log sufficiently large holds let consider harper theorem means words expanding distribution apparently small enough hence lower bound ann norm let metric space distance dist let distribution defined first define distribution defined following isoperimetric inequality proved lemma lemma holds consider problem defined metric space distance hard distribution problem database database point sampled independently according query point sampled independently according following lower bound proved fix assume log log define log choose log theorem metric space defined solved deterministic las vegas randomized scheme table cells cell containing bits expected input distribution proof followings true log claim distribution see expansion true let lemma set means due corollary either nsw second bound always higher ranges first bound gives references amirali abdullah suresh venkatasubramanian directed isoperimetric inequality application bregman near neighbor lower bounds stoc alexandr andoni dorian croitoru mihai hardness nearest neighbor focs alexandr andoni piotr indyk mihai optimality dimensionality reduction method focs alexandr andoni ilya razenshteyn optimal hashing approximate near neighbors stoc omer barkol yuval rabani tighter lower bounds nearest neighbor search related problems cell probe model journal computer system sciences conference version stoc allan borodin rafail ostrovsky yuval rabani lower bounds high dimensional nearest neighbor search related problems discrete computational geometry pages conference version stoc amit chakrabarti bernard chazelle benjamin gum alexey lvov lower bound complexity approximate searching hamming cube discrete computational geometry pages conference version stoc amit chakrabarti oded regev optimal randomised cell probe lower bound approximate nearest neighbour searching siam journal computing conference version focs harper optimal numberings isoperimetric problems graphs journal combinatorial theory piotr indyk nearest neighbors spaces handbook discrete computational geometry pages jayram subhash khot ravi kumar yuval rabani lower bounds partial match problem journal computer system sciences conference version stoc michael kapralov rina panigrahy nns lower bounds via metric expansion emd icalp kasper green larsen higher cell probe lower bounds evaluating polynomials focs ding liu strong lower bound approximate nearest neighbor searching information processing letters peter bro miltersen noam nisan shmuel safra avi wigderson data structures asymmetric communication complexity journal computer system sciences conference version stoc rina panigrahy kunal talwar udi wieder geometric approach lower bounds approximate search partial match focs rina panigrahy kunal talwar udi wieder lower bounds near neighbor search via metric expansion focs mihai mikkel thorup higher lower bounds rich problems siam journal computing conference version focs alan siegel universal classes fast high performance hash functions tradeoff applications focs yaoyu wang yitong yin certificates data structures icalp yitong yin simple lower bounds approximate isoperimetric inequalities arxiv preprint
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oct asymptotic theory maximum likelihood estimates multivariate generalised linear models efstathia bura sabrina duarteb liliana forzanib ezequiel smuclerc mariela suedc institute statistics mathematical methods economics wien vienna austria department statistics george washington university washington facultad unl researcher conicet santa argentina instituto uba conicet fcen buenos aires argentina abstract regression dimensionality reduction method many applications asymptotic theory reduced rank estimators parameter matrices multivariate linear models studied extensively contrast theoretical results available multivariate generalised linear models develop theory concave criterion functions maximised parameters spaces neither convex closed results used derive consistency asymptotic distribution maximum likelihood estimators multivariate generalised linear models response predictor vectors joint distribution illustrate results real data classification problem binary covariates keywords exponential family rank restriction parameter spaces corresponding author email introduction multivariate multiple linear regression model response pdimensional predictor vector postulates matrix error term var based random sample satisfying model ordinary least squares estimates parameter matrix minimizing squared error loss function bxi obtain bols xit xit regression introduces rank constraint minimized subject constraint rank min solution rrr bols first bols see reinsel velu singular vectors regression attracted attention regularization method introducing shrinkage penalty moreover used dimensionality reduction method constructs latent factors predictor space explain variance responses anderson obtained test hypothesis rank given number derived associated asymptotic theory assumption normality assumption joint normality izenman obtained asymptotic distribution estimated regression coefficient matrix drew connections regression principal component analysis correlation analysis specifically principal component analysis coincides regression izenman monograph reinsel velu contains comprehensive survey theory history reduced rank regression including time series many applications despite application potential regression received limited attention generalised linear models yee hastie first introduce regression class multivariate generalised linear models covers wide range data types response predictor vectors including categorical data multivariate vector generalised linear models topic yee book accompanied associated packages vgam vgamdata yee hastie proposed alternating estimation algorithm shown result maximum likelihood estimate parameter matrix multivariate generalised linear models bura duarte forzani asymptotic theory restricted rank maximum likelihood estimates parameter matrix multivariate glms developed yet general maximum likelihood estimator concave sense maximizes empirical mean concave criterion function asymptotic theory defined concave function received much attention huber haberman niemiro among classical references recently hjort pollard presented unified framework statistical theory convex criterion functions minimized open convex sets euclidean space geyer studied restricted closed subset euclidean space rank restriction reduced rank regression imposes constraints studied result neither convex closed parameter spaces paper develop theory concave criterion functions maximized parameters spaces neither convex closed apply results obtain asymptotic theory reduced rank regression estimators generalised linear models specifically derive asymptotic distribution properties maximum likelihood estimators multivariate generalised linear models response predictor vectors joint distribution asymptotic theory develop covers regression linear models special case show improvement inference asymptotic theory offers via analysing data set yee hastie analysed throughout function denotes row vector stands column vector derivatives denotes symmetric matrix second order derivatives vector valued function denotes matrix let random vector taking values measurable space distributed according law interested estimating finite dimensional parameter using independent identically distributed copies sequel use denote mean let subset euclidean space known function assume parameter interest maximizer map defined one estimate maximizing empirical version optimization criterion specifically given known function define throughout denotes empirical mean operator hereafter used interchangeably criterion function depending two appropriate given setting assume unique maximizer deterministic function defined criterion function defined maximizing maximum criterion function attained supremum finite value almost maximizes criterion function sense satisfies sup small used instead definition estimator satisfies called weak criterion function called strong proposition appendix lists conditions existence uniqueness strong consistency defined concave parameter space convex regularity conditions stated theorem van der vaart asymptotic expansion distribution consistent strong see definition given nonsingular symmetric second derivative matrix restricted consider optimization image function necessarily injective specifically restrict optimization problem set requiring condition exists open set map even unrestricted problem defined exists priori guarantee supremum attained considering restricted problem nevertheless lemma establishes existence restricted strong regardless whether original real maximiser weak strong proofs provided appendix lemma assume exists criterion function condition holds exists strong criterion function proposition next establishes existence consistency weak restricted sequence concave function assumptions unrestricted problem stated proposition appendix proposition assume condition holds assumptions proposition appendix exists strong restricted problem morevoer strong restricted problem converges probability derive next asymptotic distribution constrained estimator well defined lemma even uniquely determined unique asymptotic distribution one could use taylor series expansion derive asymptotic results building idea condition introduces parametrization neighborhood allows applying standard tools order obtain asymptotic distribution restricted condition given exist open set open set rqs twice continuously differentiable setting condition prove sbn strong criterion function apply theorem van der vaart obtain asymptotic behavior sbn combined taylor expansion yield linear expansion finally requiring condition relates parametrizations suffices derive asymptotic distribution terms condition consider condition condition condition ensures well defined regardless fact may contain multiple moreover also agrees consequently orthogonal projection onto respect inner product defined definite positive matrix satisfies denotes generalised inverse matrix gradient necessarily full rank contrast gradient proposition assume conditions hold assume also unrestricted problem satisfies regularity conditions appendix strong restricted problem converges probability satisfies influence function unrestricted estimator defined nonsingular symmetric second derivative matrix defined according moreover asymptotically normal mean zero asymptotic variance avar side remark conjecture estimators maximizers criterion function restrictions satisfy conditions true also holds important since asymptotic distribution restricted estimators derived directly asymptotic distribution unrestricted one asymptotic theory maximum likelihood estimator reduced rank multivariate generalised linear models section show maximum likelihood estimators multivariate generalised linear models restricted strong conditional using results section obtain existence consistency asymptotic distribution maximum likelihood estimators reducedrank multivariate generalised linear models exponential family let random vector assume distribution belongs kparameter canonical exponential family pdf pms exp vector known functions nonnegative known function vector natural parameters taking values exp integral replaced sum discrete set natural parameter space assumed open convex strictly convex function defined moreover assume convex infinitely differentiable particular every matrix second derivatives since strictly convex every multivariate generalised linear models let random vector multivariate response vector predictors multivariate generalised linear model postulates conditional distribution given belongs fixed exponential family hypothesizes natural parameters henceforth call emphasise dependence depends linearly vector predictors thus pdf pms given exp depends linearly frequently subset natural parameters depends complement normal linear model constant variance example accommodate structure partition vector indexing model components assume natural parameter space exponential family open convex subset assume let vect denotes generic vector true parameter suppose independent identically distributed copies satisfying true parameter vector available given realisation conditional factor depend parameter interest let definition maximum likelihood estimator mle exits parameter indexing model subject theorem next establishes existence consistency asymptotic normality mle theorem assume satisfies model subject true parameter regularity conditions appendix maximum likelihood estimate exists unique converges probability moreover asymptotically normal covariance matrix defined partial reduced rank multivariate generalised linear models number natural parameters number predictors large precision estimation interpretation results adversely affected way address assume parameters live lower dimensional space assume vector predictors partitioned parameter corresponding rank min way natural parameters related predictors via set matrices rank min following yee hastie refer exponential conditional model subject restrictions imposed partial reduced rank multivariate generalised linear model multivariate generalised linear model special case partial reduced rank multivariate generalised linear model obtain asymptotic distribution reduced model show conditions satisfied defined next maintain consistency notation introduced section vectorise matrix involved parametrisation model reformulate accordingly parameter space vectorised object understanding use symbol indicate matrix space component product space identified image operator vec rmn sequel keep notation simple possible concatenate column vectors without transposing write moreover write understanding stands vec problem belongs entire parameter belongs however restricted problem assume true parameter belongs let consider without loss generality assume set matrices whose first rows linearly independent therefore rkfirst trivial since first rows linearly independent consider rkfirst finally let map proposition conditions satisfied defined respectively rank restriction existence maximum likelihood estimate guaranteed sense defined replaced however using lemma work strong sequence criterion function theorem states main contribution theorem let denote true parameter value assume satisfies model subject defined exists strong maximising sequence criterion function defined moreover weak sequence converges probability strong sequence asymptotically normal covariance matrix avar defined remark since projection eigenvalues using partial multivariate generalised linear models sults efficiency gain application marital status workforce study yee hastie analyse data questionnaire collected large new zealand workforce observational study conducted homogeneity analysis restricted subset european males missing values variables used yee hastie interested exploring whether certain lifestyle psychological variables associated marital status especially response variable marital status levels single separated divorced widower married living partner reference group data regressors collected binary respectively coded presence negative healthwise goal investigate unhealthy variables related adjusting age level education variables described table variable name description marital marital status divorced living partner age years log years education secondary high school last months largest number drinks one day current smoker usually wear hat shirt suntan lotion outside summer suffer nerves would call nervous person feelings easily hurt would call tense highly strung feel miserable reason often feel worry awful things might happen worrier mood often binge smokenow sun nerves nervous hurt tense miserable fedup worry worrier mood table variables used workforce study categorical response taking values probabilities expressed multinomial vector fit generalised linear model presented paper otherwise pdf written exp exp natural parameter related pdf identity log let vector predictor variables consider dependence made explicit notation yee hastie fit multinomial regression model log log log intercept coefficient matrix parameters estimate multinomial linear model fitted data level binge significant log smokehow tense log significant log next fitted partial reduced rank multivariate generalised linear model two continuous variables subject restriction represent continuous variables binary predictors aic criterion estimates rank one see yee hastie using asymptotic results current paper duarte developed test based bura yang also estimates dimension therefore notation rank restriction results drastic reduction total number parameters reduction estimation burden also reflected tight confidence intervals compared unrestricted model seen tables table yee hastie consequence variables nervous hurt significant unrestricted generalised linear model significant reduced furthermore variables binge smokenow nervous tense significant responses significant coefficients positive correspond variables binge smokenow nervous tense hurt widowers since positive value binary variables indicates poor lifestyle negative psychological characteristics analysis concludes men features chance single divorced widowed higher chance married adjusting age education also coefficients corresponding response log twice large log suggesting effect predictors differs group computations performed using package vgam developed yee variable log log log intercept glm glm glm table estimators standard errors generalised linear model first two rows reduced generalised linear model last two rows indicates significant level variable log log log binge glm smokenow glm sun glm nerves glm nervous glm hurt glm table mles standard errors parentheses full rank generalised linear model first two rows partial reduced rank generalised linear model last two rows indicates significant level variable log log log tense glm miserable glm fedup glm worry glm worrier glm mood glm table mles standard errors parentheses full rank generalised linear model first two rows partial reduced rank generalised linear model last two rows indicates significant level discussion exception work yee vector generalised linear models vglms yee wild yee hastie yee regression almost exclusively restricted data response variable continuous estimation multinomial logit models studied yee hastie distribution results estimators obtained paper fill gap developing asymptotic theory restricted rank maximum likelihood estimates parameter matrix multivariate glms illustrate potential impact results refer real data analysis example section order assess significance predictors yee hastie calculate standard errors coefficient matrix factors independently infer significance components matrix components matrix separately asymptotic distribution either estimator obtained assuming fixed known way first analyse check predictors significant examine influence response standard errors unclear product standard errors measures relates significance product components coefficient matrix moreover practical approach readily extended using results theorem obtain errors component coefficient matrices simultaneously assess statistical significance predictor response using approach yee hastie predictor found significant across responses example yee hastie find predictor hurt significant three groups single widower hand assess significance combination thus find hurt significant widower single men groups see table proofs consistency long established see instance theorem van der vaart functions defined typically proof consistency mestimators assumes parameter interest point maximum ascertained assumptions proposition assumption proposition yields uniform consistency estimator property needed order establish consistency proposition assume strictly concave function convex open subset euclidean space function well defined unique maximizer compact set sup exists unique criterion function moreover proof compact subset collection measurable functions assumption integrable envelope moreover fixed map continuous since concave defined open set stated example van der vaart conditions guarantee class lim sup need prove exist unique maximizer converges maximizer first consider deterministic case ignoring moment sequence random functions since belongs open set exists closed ball contained uniform convergence guarantees lim sup sup sup let since concave continuous attains maximum compact set denote note strictly smaller zero boundary ball shown therefore conclude local maximum let satisfy convexity implies exists satisfies therefore implying therefore maximum global strict concavity guarantees global maximum unique thus unique solution optimization problem repeating argument prove convergence sequence turning stochastic case uniform convergence set assumed guarantees deterministic result applied element obtains result proof lemma let sequence unconstrained maximization problem since lim sup lim sup define exists sup sup let since strong criterion function proof proposition proposition established existence unique maximizer criterion function invoke lemma guarantee existence strong criterion function let strong criterion function start deterministic case sup defined sequence real numbers proof proposition define obtain small enough sup sup large enough condition therefore sup since large enough combining obtains sup deduce prove sup sup let prove choose take distance maximizer defined proposition assumed distance smaller thus sup turn yields convergence probability equivalent existence almost everywhere convergent subsequence therefore applying deterministic result set probability one exists ank converges zero obtain result regularity conditions proposition theorem condition open subset euclidean space measurable function differentiable almost every derivative condition exists measurable function neighborhood condition map admits second order taylor expansion point maximum nonsingular symmetric second derivative matrix regularity conditions van der vaart proved theorem book strong sequence criterion function probability moreover asymptotically normal mean zero avar result invoked following proofs proof proposition assume sequence converges probability assumed open bicontinuity guarantees converges probability note sup sup sup except term omitted last three inequalities therefore strong maximizing sequence criterion function next verify conditions satisfied specifically condition holds since measurable function differentiable almost every fact differentiable also differentiable moreover derivative function neighborhood continuity neighborhood denotes maximum neighborhood first inequality holds condition valid unconstrained problem second inequality follows since continuously differentiable thus lipschitz condition satisfied condition observe function twice continuously differentiable satisfy required regularity properties respectively moreover since second derivative matrix second derivative matrix matrix nonsingular symmetric full rank nonsingular symmetric apply theorem van der vaart obtain first order taylor series expansion around defined respectively obtains expansion proof theorem write vec matrix form vec vect vector parameters model note value defined notation allows simplify expression function replace regularity conditions required derive consistency asymptotic distribution mle compact sup sup sup prove existence uniqueness consistency mle present model next show assumptions stated proposition satisfied strict convexity implies fixed strictly concave function concavity preserved expectation thus concave identifiability condition satisfied exponential family section allows applying lemma van der vaart conclude taking expectation respect conclude moreover contradicts hypothesis finally integrability follows conditions required van der vaart theorem derive asymptotic distribution easily verifiable integrability assumptions stated second derivative matrix finally observe deduce according general formula asymptotic variance asymptotic variance given lemmas corollary required prove proposition written let lemma assume rkfirst exists proof let matrix comprised first rows linearly independent rank rank hence invertible take expression thus since rank corollary let rkfirst written defined satisfies lemma open set proof show complement closed consider rank strictly less assume converges note tnt also equal zero hence rank lemma open sets proof homeomorphism equivalent respectively products open sets lemma lemma one one bicontinuous proof suffices note rkfirst satisfies required properties let rfirst therefore inverse thus one one bicontinuous lemma open open set equivalently complement rkfirst proof suffices show rkfirst sequence first rows linearly independent closed set let converges write obtain comprises first rows therefore first rows linearly independent proof proposition verify conditions satisfied condition lemma set defined open belongs condition since first rows linearly independent applying lemma obtains open set consider lemma open set bicontinuous lemma furthermore given function twice continuously differentiable jacobian full rank fact latter order full column rank see also direct implication theorem condition consider associated shown lemma since since invertible order span next since diag ird applying theorem obtains rank rank therefore proof theorem theorem verified conditions theorem satisfied maximum likelihood estimator model satisfying asymptotic variance mle estimator given also showed conditions satisfied proof proposition result follows proposition since avar references anderson estimating linear restrictions regression coefficients multivariate normal distributions annals mathematical statistics bura duarte forzani sufficient reductions regressions exponential family inverse predictors journal american statistical association bura yang dimension estimation sufficient dimension reduction unifying approach multivariate cook sufficient dimension reduction via inverse regression minimum discrepancy approach journal american statistical association duarte modelos lineales generalizados rango reducido suficiente dimensiones tesis doctoral universidad nacional del litoral argentina geyer asymptotics constrained annals statistics haberman concavity estimation annals statistics hjort pollard asymptotics minimisers convex processes huber behavior maximum likelihood estimates nonstandard conditions proceedings fifth berkeley symposium mathematical statistics probability vol izenman regression multivariate linear model journal multivariate analysis izenman modern multivariate statistical techniques regression classification manifold learning new york springer puntanen styan isotalo matrix tricks linear statistical models personal top twenty springer science business media reinsel velu multivariate reduced rank regression lecture notes statistics new york springer niemiro asymptotics defined convex minimization annals statistics van der vaart asymptotic statistics cambridge university press yee vector generalized linear additive models new york usa springer yee vgam vector generalized linear additive models package version url https yee hastie vector generalized linear models statistical modelling yee wild vector generalized additive models journal royal statistical society series
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scalability gpu explore model checker nathan cassee thomas neele anton eindhoven university technology eindhoven netherlands use graphics processors gpus promising approach speed model checking extent becomes feasible instantly verify software systems development gpu explore model checker runs computations gpu years extended various techniques possibilities improve performance continuously investigated paper discuss hash table tool works heart functionality propose alteration hash table isolated experiments seems promising analyse effect integrated tool furthermore investigate current scalability gpu explore experimenting input models varying sizes running tool one latest gpus nvidia introduction model checking technique systematically determine whether concurrent system adheres desirable functional properties numerous examples successfully applied however fact computationally demanding means yet commonly used procedure software engineering accelerating computations graphics processing units gpus one promising way model check system design mere seconds minutes opposed many hours gpu explore model checker performs computations gpu initially consisted state space exploration engine extended perform deadlock safety checking checking liveness properties also investigated positive results liveness checking yet integrated official release tool finally order reduce memory requirements gpu explore partial order reduction successfully integrated since first results achieved gpu explore considerable progress made instance original version running nvidia able explore state space model approximately minutes many improvements gpu explore algorithms reported gpu hardware cuda compiler reduced seconds levels become much feasible interactively check debug large models furthermore gpu developments continue many options still investigated performance important tool gpu explore however far scalability tool yet thoroughly investigated instance currently access nvidia titan gpu equipped global memory models using gratefully acknowledge support nvidia corporation donation geforce titan used research timo kehrer alice miller eds workshop graphs models gam eptcs cassee neele wijs cassee neele wijs far global memory suffices models used analysis gpu explore require state space exploration current paper report results obtained scaling models utilise addition also experimentally compared running gpu explore titan gpu maxwell architecture released gpu explore running titan gpu pascal architecture released year later provides insights regarding effect recent hardware developments tool finally analyse scalability critical part tool namely hash table structure used model checking keep track progress made system state space even small improvement hash table design may result drastic improvement performance gpu explore recently identified conducting isolated experiments still potential improvement current paper particularly investigate whether changing size buckets data structures act containers states stored positive effect running time structure paper follows section discuss related work next overview inner working gpu explore presented section hash table proposed alterations discussed section present results obtained experimentation section finally conclusions pointers future work given section related work literature several different designs parallel hash tables found first hash table gpus proposed alcantara based cuckoo hashing secondly laarman designed hash table shared memory systems implementation later used basis hash table underlying lts min model checker hash tables gpu proposed moazeni sarrafzadeh bordawekar misra chaudhuri cuckoo hashing implemented alcantara publicly available part cudpp library unfortunately best knowledge implementations available hash table designs besides gpu explore several gpu model checking tools bartocci developed extension spin model checker performs exploration gpu achieved significant large models gpu extension parallel model checking tool divine called developed barnat model checking process offload cycle detection procedure gpu tool even benefit use multiple gpus achieves significant model checking properties valid edelkamp sulewski address issues arising limited amount global memory available gpu implement hybrid approach tool called using gpu next state computation keeping hash table main memory accessed multiple threads running central processing unit cpu keep part state space global memory store rest disk record disk queried process call delayed duplicate detection even though disk access causes overhead manage achieve tools http scalability gpu explore model checker consumers producer rec gen work send work work rec par using send rec trans send rec trans end par figure example lts network one producer two consumers right communication ltss specified using exp syntax files containing specification producer consumer respectively edelkamp also applied gpu programming probabilistic model checking solve systems linear equations gpu order accelerate value iteration procedure gpus well suited purpose enable times traditional cpu implementation developed tool called gpurc performs full exploration process gpu similar gpu explore implementation applies dynamic parallelism relatively new feature cuda allows launching new kernels within running kernel tool shows good compared traditional tools although added benefit dynamic parallelism minimal finally gpus also successfully applied accelerate computations related model checking instance use gpu construct state space decomposition minimisation investigated probabilistic model checking implemented gpu accelerated parameter synthesis parameterized continous time markov chains gpus gpu explore gpu explore model checker practically runs entirely gpu general progress checked host side thread running cpu written cuda extension offered vidia cuda compute unified device architecture provides interface write applications vidia gpus gpu explore takes network labelled transition systems ltss input construct synchronous product ltss using many threads exploration optionally checking presence deadlocks violations safety properties negation safety property added automaton network lts directed graph nodes represent states edges transitions states transition action label representing event leading one state another example network shown figure initial states indicated detached incoming arrows one producer generates work sends one two consumers happens means synchronisation send rec actions actions executed independently process ltss combined using relevant synchronisation rules defined right figure using syntax open tool state space network consists states transitions general approach gpu explore perform state space exploration discussed cassee neele wijs threads exploration threads exploration shared mem cache shared mem cache cache texture cache global memory network input global hash table figure schematic overview gpu hardware architecture gpu explore tion leaving many details relevant understanding current work interested reader referred cuda program host launches cuda functions called kernels executed many times parallel specified number gpu threads usually threads run kernel using different parts input data although gpus allow multiple different kernels executed simultaneously gpu explore use feature thread executed streaming processor threads grouped blocks predefined size block assigned streaming multiprocessor consists fixed number sps see figure thread number registers allow fast access threads block together share memory exchange data located shared memory finally blocks share data using global memory gpu relatively large slow since global memory used exchange data host kernel gtx itan used experiments global memory sms sps sps total writing gpu applications challenging due execution model gpus single instruction multiple threads threads partitioned groups called warps threads warp run sharing program counter always execute program instruction hence thread divergence phenomenon threads forced execute different instructions due constructions access physically distant parts global memory negatively affects performance model checking tends introduce divergences frequently requires combining behaviour processes network accessing storing state vectors system state space global memory gpu explore mitigated combining relevant network information much possible integers storing textures allow read access use dedicated cache speed random accesses furthermore global memory hash table used store state vectors figure currently used hash table designed optimise accesses entire warps space partitioned buckets consisting integers precisely enough one warp fetch bucket one combined memory access state vectors hashed buckets placed within bucket available slot bucket full another hash function used find new bucket block accesses global hash table collect vectors still require exploration state vector process states group threads assigned construct successors using parallelism since access global memory slow block uses dedicated scalability gpu explore model checker state cache figure serves store collect newly produced state vectors subsequently moved global hash table batches cache duplicates detected gpu explore hash table states discovered search exploration phase gpu explore inserted global memory hash table hash table used keep track open closed sets maintained breadth first search based exploration state space since many accesses reads writes hash table performed exploration performance critical model checker order allow efficient parallel access hash table prevent corruption state vectors insertion atomic operation even state vector spans multiple bit integers given requirements considered several hash table implementations one proposed alcantara uses cuckoo hashing cuckoo hashing key hashed bucket hash table case collision key already bucket evicted rehashed using another hash function different bucket continue last evicted key hashed empty bucket hash functions exhausted chain becomes long hash table considered one originally designed gpu explore refer hashing mechanism gpu explore hashing experimentally compared two hash tables comparisons concluded cuckoo hashing average performs better meet demands needed gpu explore however based performance evaluation possible performance increase identified gpu explore hashing section discusses proposed modification implementation gpu explore hashing figure division threads warp bucket size gpu explore gpu explore hash table consists two parts storage used store discovered vectors set hash constants used hash function determine position vector hash table memory used store vectors divided buckets size integers eachjbucket addik tionally split two equally sized half buckets therefore single bucket store vectors reason writing vectors buckets group threads many cases atomic writes scheduled uninterrupted sequence results vectors consisting multiple integers written without write operations corrupting noted gpu explore hash table uses closed hashing vectors stored hash table opposed pointers vectors cassee neele wijs inserting warp threads inserts single vector bucket way threads divided bucket observed figure figure visualizes single bucket gpu explore hash table vector length thread warp assigned one integer bucket one integer vector assignment done left right per half bucket example first threads first vector group assigned first slot bucket first thread vector group assigned first integer vector assigning every thread single part vector bucket thread relatively simple task executed parallel insertion algorithm first hashes vector insert determine bucket belonging vector thread checks place bucket compares integer position corresponding integer vector comparing insertion algorithm uses cuda warp instructions quickly exchange data threads warp determine whether single vector group threads found vector vector found insertion algorithm terminates vector found algorithm continues figure example inserting vector gpu explore hash table figure shows example vector inserted gpu explore hash table first two slots already occupied first two slots already occupied first six threads compare element vector element bucket icons arrows indicate result comparisons seen one thread match however elements slot match insertion algorithm report find vector found bucket insertion algorithm selects first free slot bucket case third slot selection procedure done efficiently using cuda warp instructions next associated threads attempt insert vector selected slot using compare swap operation insertion fails another warp claimed slot already another vector algorithm takes next free slot bucket bucket free slots next set hash constants used next bucket probed repeated hash constants used insertion succeeds buckets insertion algorithm reports failure exploration stops gpu explore hash table initialized using eight hash functions exploration step blocks found new vectors use insertion algorithm insert new vectors found global hash table hash table therefore vital part gpu explore buckets length integers chosen fact warps cuda consist threads way every integer bucket assigned single thread besides design choice also allows coalesced memory access warp accesses continuous block integers operation executed single memory fetch operation uncoalesced accesses hand done individually coalescing accesses available bandwidth global memory used efficiently scalability gpu explore model checker configurable bucket size current implementation hash table makes possible gpu explore achieve considerable increase performance model checkers suffers one disadvantage namely initially scanning bucket threads vector length active time threads inactive await result atomic insertion active group insertion fails still free slot hash table another group threads becomes active attempt atomic insertion remaining threads await result operation figure division threads warp buckets size therefore buckets lot free slots insertions fail majority threads warp inactive furthermore vector size generally exceed four integers means attempting atomically insert vector majority threads warp inactive ergo one possible improvement gpu explore hash table reduce bucket size fewer slots per bucket therefore fewer threads needed insert single vector single insertion still uses one thread per integer bucket turn vectors inserted parallel figure shows division threads warp looks like buckets size instead used observed warp insert four elements parallel diagram group threads inserts different vector accesses different buckets global memory logical consequence improvement scanning bucket fewer threads inactive vector inserted free slot suppose vector size certain model new bucket size inserting vector using gpu explore hash table threads inactive hand four buckets size simultaneously accessed threads inactive four vectors simultaneously processed opposed one however threads active time smaller buckets also lead thread divergence within warp first course accessing different buckets simultaneously likely leads uncoalesced memory accesses furthermore also possible insertion procedure one group needs work another warp instance consider first group warp fails find vector designated bucket also write bucket since latter full case group needs fetch another bucket time another group warp may find vector designated bucket therefore able stop insertion procedure situation two groups diverge second group wait first group finished inserting means use smaller buckets advantageous performance increase smaller buckets outweighs performance penalty divergence paper address whether true practical explorations suggested performance increase experimentally validated comparing implementation hash table varying bucket size original gpu explore hash table cassee neele wijs cbs runtime duplication sequence figure results inserting sequence randomly generated integers hash tables gpu explore gpu explore configurable buckets cbs bucketed version bucket size four integers used isolation part gpu explore results comparison presented discussed section experiments section discuss experiments evaluate scalability gpu explore section report experiments evaluate effect bucket size runtimes gpu explore next two sections goal determine whether gpu explore scales well varying size model performance hardware respectively experiments use nvidia titan maxwell installed machine running inux int number blocks set threads per block hash table allocated global memory benchmarks selected varied set models cadp toolset toolset beem database models suffix modified obtain larger state space varying bucket size test different bucket sizes two types experiments performed first hash table varying bucket sizes tested isolation time taken insert elements measured second gpu explore modified uses hash table modifiable bucket size bucket size set compile time performance version gpu explore compared original gpu explore exploration several input models input data performance evaluation hash tables isolation set sequences scalability gpu explore model checker randomly generated integers sequence consisting vectors length integer sequences vary often element occurs sequence duplication means every unique element occurs sequence duplication means unique element sequence occurs times therefore sequence duplication unique elements note experiment tries replicate state space exploration many duplicate insertions performed high many transitions state space lead state experiment replicates performance hash table models vary average states sequence measured total time required insert elements results comparison depicted figure refer standard gpu explore hash table hash table configurable bucket size cbs experiment cbs bucket size four integers compared cbs slower sequences element occurs times however sequences higher duplication degree cbs starts outperform sequences duplication less time spent atomically inserting elements since elements already hash table cbs performs three times better see later amount duplication relevant model checking addition testing performance hash tables isolation performance cbs compared standard gpu explore hashing part gpu explore hash table underlying gpu explore implemented wijs neele modified allow configuration bucket size compile time compared time required state space exploration implementation gpu explore configurable bucket size original implementation gpu explore four bucket sizes compared gpu explore namely bucket sizes integers relative performance bucket sizes recorded performance gpu explore baseline model experiment run five times average running time five runs taken result comparisons illustrated figure observed total exploration time gpu explore configurable bucket size models larger runtime gpuexplore show small performance increase bucket size instances however new hash table causes slow three reasons promising performance shown previous experiments reflected first increased complexity hash table negatively affects register pressure number registers thread requires execute kernel register usage kernel increases compiler may temporarily place register contents global memory effect observed hash table tested isolation case far fewer registers per thread required increase register pressure also reason gpu explore configurable bucket size set slower gpu explore static bucket size furthermore smaller bucket sizes result thread divergence uncoalesced memory accesses reading hash table therefore available memory bandwidth used less efficiently leading drop performance apparently increased potential parallel insertions vectors overcome drawback lastly exploring gpu explore discovers duplicates states several incoming transitions average models used experiments exceptions higher around however figure concluded hash table isolation starts outperform static hash table element duplicated times partly explains performance seen figure runtime relative static bucket size cassee neele wijs wafer des bucket size figure relative runtime gpu explore variable bucket size runtime gpu explore fixed bucket size integers used reference normalized varying model size addition experimentally comparing effect different bucket sizes also investigated gpu explore behaves exploring state spaces different size performed experiment two different models first version gas station model number pumps fixed two varied number customers two twelve none instances requires state vector longer two integers model simple implementation network one token continuously passed forward nodes node two transitions communicate neighbours three internal transitions varied number nodes network executed tool five times instance computed average runtime results listed table smallest instances performance gpu explore measured lot worse compared larger instances two main reasons first relative overhead suffered initialization work scanning higher second parallelism offered gpu fully exploited size one search layer small occupy blocks work gas station model peak performance achieved instance ten customers million states larger instances performance decreases slightly due increasing occupancy hash table leads hash collisions therefore time lost rehashing results token ring model show another interesting scalability aspect performance drop instances nodes nodes caused fact instance nodes smallest state vector exceeds bits length longer state vectors lead memory accesses throughout generation algorithm scalability gpu explore model checker table performance gpu explore gas station token ring model varying amount processes token ring gas station states time states time due pascal architecture titan used benchmarks based maxwell architecture launched since nvidia released several gpus aspects improved architecture revised cuda cores gpu global memory available investigate well gpu explore scales faster hardware performed several experiments titan maxwell architecture txm titan pascal architecture txp latter released one used installed das cluster node running ent inux txm experiments performed blocks txp gpu explore set use blocks improvements hardware allows gpu explore launch blocks evaluate compared cpu approach also conducted experiments enerator tool latest version adp toolbox performed nodes das cluster equipped ntel aswell ghz cpu memory ent inux results listed table reported runtimes averages run corresponding tool ten times larger models see times running gpu explore txp compared running txm average indicates gpu explore scales well higher memory bandwidth larger amount cuda cores comparing gpu explore txp cpu analysis average take state spaces account consisting least million states average speedup considering model checking linear achieved roughly amounts using cpu cores respectively combination cassee neele wijs table performance comparison enerator adp gpu explore running titan maxwell architecture txm pascal architecture txp runtime seconds model acs odp transit wafer des states txm txp average observation frequently times even times achieved clearly demonstrates effectiveness using gpus model checking conclusion future work paper reported number scalability experiments conducted gpu model checker gpu explore earlier work identified potential improve hash table however experiments varied bucket size gpu explore provided insight specific input models bucket size set small becomes noticeable context entire computation gpu explore additional register use per thread introduced uncoalesced memory accesses thread divergence make worthwile make bucket size configurable may different applications experiments hash table isolation point hashing made efficient way besides also conducted experiments models different sizes scaled gas station model token ring model obtained encouraging results second model gpu explore generate million states per second first model peak gpu explore able generate million states per second exploring state space million states three minutes believe impressive numbers demonstrate potential gpu model checking finally reported experiments conducted new gpu hardware titan scalability gpu explore model checker pascal architecture provides benchmark set models average titan maxwell architecture also compared runtimes gpu explore running pascal titan cpu enerator tool adp toolbox measured average entire benchmark set models models yielding state space least million states often times observed one case even times future work future work consider various possible extensions tool first ability write explored state spaces disk open possibility postprocess analyse state spaces could done directly application bisimulation reduction gpu second work making tool user friendly currently providing input model quite cumbersome since gpu explore requires user express system behaviour low level description formalism networks ltss specifying systems would much convenient modelling language would supported investigate modelling languages would suitable integration current tool finally also consider application gpu explore conduct computations similar model checking performance analysis requires incorporate time input models instance including special actions represent passage time references dan alcantara vasily volkov shubhabrata sengupta michael mitzenmacher john owens nina amenta building efficient hash table gpu gpu computing gems jade edition morgan kaufmann publishers christel baier katoen principles model checking mit press henri bal dick epema cees laat rob van nieuwpoort john romein frank seinstra cees snoek harry wijshoff distributed system computer science research infrastructure long term ieee computer barnat petr bauch brim milan designing fast ltl model checking algorithms gpus jpdc ezio bartocci richard defrancisco scott smolka towards spin model checker spin acm new york usa rajesh bordawekar evaluation parallel hashing techniques presentation gtc last checked february http dragan stefan edelkamp damian sulewski anton wijs parallel probabilistic model checking general purpose graphics processors sttt dragan stefan edelkamp damian sulewski anton wijs extension prism general purpose graphics processing units pdmc ieee nathan cassee anton wijs analysing performance gpu hash tables state space exploration gam eptcs open publishing association cassee neele wijs milan petr nicola paoletti brim marta kwiatkowska precise parameter synthesis stochastic systems tacas lncs springer sjoerd cranen jan friso groote jeroen keiren frank stappers erik vink wieger wesselink tim willemse overview toolset recent advances tacas lncs springer stefan edelkamp damian sulewski efficient model checking general purpose graphics processors spin lncs springer stefan edelkamp damian sulewski external memory search delayed duplicate detection gpu mochart lncs springer hubert garavel lang radu mateescu wendelin serwe cadp toolbox construction analysis distributed processes sttt david heimbold david luckham debugging ada tasking programs ieee software alfons laarman scalable model checking thesis university twente lang refined interfaces compositional verification forte lncs springer prabhakar misra mainak chaudhuri performance evaluation concurrent data structures gpus icpads maryam moazeni majid sarrafzadeh hash table graphics processors saahpc thomas neele anton wijs dragan jaco van pol partial order reduction gpu model checking atva lncs springer john nickolls ian buck michael garland kevin skadron scalable parallel programming cuda queue rasmus pagh flemming friche rodler cuckoo hashing esa lncs springer radek beem benchmarks explicit model checkers spin lncs steven van der vegt alfons laarman parallel compact hash table memics lncs springer anton wijs achieving discrete relative timing untimed process algebra iceccs ieee anton wijs gpu accelerated strong branching bisimilarity checking tacas lncs springer anton wijs model checking properties application gpus cav part lncs springer anton wijs dragan gpuexplore state space exploration using gpus tacas lncs anton wijs dragan model checking safety properties using gpus sttt anton wijs wan fokkink combining performance functional analysis iceccs ieee scalability gpu explore model checker anton wijs katoen dragan graph decomposition strongly connected maximal end components cav lncs springer anton wijs katoen dragan efficient gpu algorithms parallel decomposition graphs strongly connected maximal end components formal methods system design anton wijs thomas neele dragan gpuexplore unleashing gpu model checking lncs springer zhimin yang liu yun liang jun sun gpu accelerated counterexample generation ltl model checking icfem lncs springer zhimin yang liu jun sun jianqi shi shengchao qin gpu accelerated reachability checking iceccs
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generate pseudorandom permutations groups feistel revisited hector oct abstract recent results alagic russell given evidence cipher may secure quantum adversaries quantum queries considered groups prompts question whether classical schemes may generalized arbitrary groups whether classical results still apply generalized schemes paper generalize cipher feistel cipher show even mansour original notions secrecy obtained group variant cipher generalize result kilian rogaway cipher pseudorandom super pseudorandomness also group case using slide attack match bound found generalizing feistel cipher arbitrary groups resolve open problem patel ramzan sundaram showing feistel cipher arbitrary group super pseudorandom finally generalize result gentry ramzan showing cipher implemented using feistel cipher public permutation last result also consider case group generalize bound introduction even mansour introduced proved security des inspired block cipher scheme call scheme given public permutation strings two different random secret keys message could enciphered article based work done master thesis university copenhagen details thesis contact aehogo obvious decryption using inverse public permutation scheme minimal sense needed xor key permutation otherwise remaining key could easily found improvement dunkelman keller shamir showed need single key scheme would still retain indistinguishability random pseudorandom another consideration block ciphers feistel cipher construction luby rackoff showed build pseudorandom permutations pseudorandom functions eventually kuwakado morii showed scheme feistel scheme could broken quantum adversaries quantum queries rather discard beautiful constructions entirely alagic russell considered whether would possible define scheme abelian groups order retain security quantum adversaries quantum queries showed security reduction hidden shift problem certain groups result inspires ask whether feistel schemes generalized groups whether get pseudorandomness model prior work extension simplification scheme dunkelman keller shamir attacked construction using variants slide attacks order show security bound optimal considered variants scheme addition involution permutation also kilian rogaway inspired desx define construction scheme special case referred kuwakado morii able break scheme feistel scheme strings using simon algorithm able query oracle superposition states kaplan using kuwakado morii results showed break many classical cipher schemes turn incited alagic russell work hidden shift problem posit feistel cipher construction groups bit strings might secure quantum adversaries quantum queries many feistel cipher variants exist different relaxations round functions see example latter also considered feistel ciphers groups vaudenay also considered feistel ciphers groups order protect ciphers differential analysis attacks called decorrelation removed schemes considered greater degree abstraction black rogaway consider ciphers arbitrary domains general question existence quantum pseudorandom permutations see summary results work random oracle model consider groups family finite groups consider pseudorandom permutations given informally following definition informal keyed permutation group pseudorandom permutation prp indistinguishable random permutation probabilistic distinguishers access polynomially many queries super pseudorandom permutation sprp permutation distinguisher given access inverse well define group scheme encryption scheme encryption algorithm plaintext uniformly random key define two problems group scheme existential forgery efp cracking efp adversary must eventually output pair satisfies correctness adversary given ciphertext asked find corresponding plaintext holds group scheme probability adversary succeeds efp polynomially bounded theorem informal uniformly random permutation chosen uniformly random probabilistic adversary success probability solving efp negligible specifically bounded amount queries respectively basic reduction latter inference result also get theorem informal super pseudorandom permutation chosen uniformly random probabilistic adversary success probability solving efp negligible corollary informal super pseudorandom permutation chosen uniformly random probabilistic adversary success probability solving negligible bound theorem find theorem informal probabilistic adversary limited polynomially many queries polynomially many inverse queries group scheme group super pseudorandom permutation apply slide attack find attack matches bound given considering group feistel cipher whose encryption algorithm consists multiple uses round function pseudorandom function show feistel cipher pseudorandom super pseudorandom regardless underlying group note feistel cipher super pseudorandom proven finally consider group scheme instantiated using feistel cipher uses encryption algorithm modelled random functions plaintext uniformly random key show one main results theorem informal probabilistic adversary queries inverse random oracles queries queries success probability distinguishing random oracle bounded may also rewrite main theorem following theorem informal adversary total queries success probability distinguishing random oracle bounded note main result due however consider group version add details proof sketches outline paper section state assumptions paper section give definitions hold paper general leaving specialized definitions various sections section introduce generalized scheme arbitrary groups stating proving results section define generalized feistel cipher arbitrary groups prove small results section consider implementation generalized scheme using generalized feistel cipher public permutation section give concluding remarks general definitions following work random oracle model may assume existence random permutation oracle group elements let family finite groups group pair set operation satisfying group axioms also assume group polynomial poly need concept pseudorandom also called indistinguishable random several forms notation write element chosen uniformly random set following consider positive integer security parameter specified unary per convention assume exists uniquely specified group size definition let efficient keyed function pseudorandom function prf probabilistic distinguishers limited polynomially many queries exists negligible function negl afm fgm negl fgm set functions pseudorandom function say pseudorandom function definition let efficient keyed permutation pseudorandom permutation prp probabilistic distinguishers limited polynomially many queries exists negligible function negl negl set permutations definition let efficient keyed permutation said super pseudorandom permutation sprp probabilistic distinguishers limited polynomially many queries inverse exists negligible function negl negl set permutations super pseudorandom permutation said super pseudorandom permutation first remark results section initially proven project prior start thesis worked complement thesis thus chosen include parts inclusion accounts brevity certain results begin defining scheme arbitrary groups refer group scheme definition define group scheme triple key generation algorithm encryption algorithm decryption algorithm key generation algorithm takes input security parameter fixes outputs group outputs key encryption algorithm takes input key plaintext outputs public permutation decryption algorithm takes input key ciphertext outputs inverse public permutation definition satisfies correctness two forms security group scheme subsection prove classical results new scheme considering even mansour two notions security existential forgery problem cracking problem cracking problem stronger two definition existential forgery problem efp consider following game group key generated adversary gets security parameter unary group receives oracle access oracles eventually outputs pair queried say succeeds cracking problem consider following game group key generated adversary gets security parameter unary group presented receives oracle access oracles decryption oracle outputs queries outputs plaintext say succeeds success probability probability uniformly random chosen encryption outputs even mansour show efp security infers security limiting factors prohibiting problems inference result employed groups fact nothing disallowing use proof efp security efp security scheme noted therefore omit indeed redefining notions proof take account working necessarily abelian group able prove group scheme satisfies efp notion security specifically following theorem assume let key probabilistic adversary success probability solving efp bounded succ number number success probability negligible even mansour inference result get corollary corollary assume let key probabilistic ppt adversary success probability solving cracking problem negligible even mansour also note results may extended instances permutation pseudorandom permutation simple reduction hence get following two results theorem assume pseudorandom permutation let key probabilistic adversary polynomially many queries oracles success probability solving existential forgery problem negligible corollary assume pseudorandom permutation let key probabilistic ppt adversary success probability solving cracking problem negligible pseudorandomness property group scheme although notions security strong interested pseudorandomness property group scheme offers kilian rogaway show scheme satisfies pseudorandom permutation property encryption oracle permutation oracles scheme indistinguishable random adversary polynomially many queries oracles note show pseudorandomness property state discussion section proof may adapted include decryption oracle scheme satisfies super pseudorandom permutation property done analysis decryption oracle arbitrary group concur however also able generalize proof construction entirely remarkable key usually different group inverse hence able use proof adjustments games analysis proof given appendix posterity completeness present result following theorem theorem assume let key probabilistic adversary limited polynomially many oracle queries adversarial advantage bounded def adv aek number number success probability negligible stated simply theorem probabilistic adversary limited polynomially many queries group scheme group super pseudorandom permutation removing decryption oracle get following corollary corollary probabilistic adversary limited polynomially many queries group scheme group pseudorandom permutation remark see group group scheme reduces scheme given proof given proves security scheme proof given proves pseudorandomness equivalently claims proven multiple round group scheme sprp security depends last round also sprp slide attack would like show security bound found optimal slightly alter simple optimal attack cipher constructed original version works abelian groups adjustments also present another slide attack modular addition desx construction consider group cipher group binary operation publicly available permutation oracle define following values hereby consider attack follows arbitrary values arbitrary values query store values hash table sorted first coordinate exists match step check guess seen birthday probability must exist slid pair satisfying property exists random pair holds probability expect collisions hash table including collision slid pair correct key found data complexity attack queries queries hence attack bound matches lower bound given theorem theorem therefore found scheme optimal considering approximation probability birthday problem collision randomly chosen elements set elements feistel consider feistel cipher arbitrary groups call group feistel cipher following complement treat group feistel cipher construction great detail main accomplishment section settling open problem posed definitions define feistel cipher group series round functions elements definition given efficiently computable necessarily invertible function called round function define group feistel cipher case multiple rounds index round functions denote group feistel cipher concurrently denote input round output respectively denote left right parts output note round output may invert round setting computing get holds rounds regardless invertibility round functions get feistel cipher invertible let pseudorandom function define keyed permutation def fkr sometimes index keys omit key index entirely results completeness show preliminary results group feistel ciphers considered first note pseudorandom permutation distinguisher need compare also pseudorandom permutation consider pseudorandom function pick distinguisher sets identity element queries oracle receives second query distinguisher lets receives may find inverse elements acquires may compute random would occur negligibly many times may query permutationoracle polynomially many times retrieved many times queries able distinguish random permutation probability one would expect group feistel cipher see figure indeed pseudorandom permutation theorem pseudorandom function pseudorandom permutation proof proposition generalized proof given katz lindell analogous result xor difficulties therefore omit among considerations showed feistel cipher abelian groups super pseudorandom left open problem proof groups present proof proposition group feistel cipher super pseudorandom proof proof using following procedure choose two pairs tikz figure adapted group scheme group feistel cipher figure encryption schemes query encryption oracle get query decryption oracle guess oracle else guess random algorithm succeeds probability random permutation algorithm succeeds negligibly often super pseudorandomness group feistel cipher refer reader paper show strong result using certain hash functions round functions following corollary corollary let group characteristic let pseudorandom functions adversary polynomially many queries family permutations consisting permutations form indistinguishable random super pseudorandom permutations sprps implementing group scheme shown scheme feistel cipher generalizable arbitrary groups might consider implement one given gentry ramzan considered exactly scheme however paper sketches proofs refer another edition paper full details unable find copy place specify exist generalize result decided fill details generalizing proof section consider generalized version gentry ramzan construction namely group scheme instantiated group feistel cipher public permutation key consisting two subkeys chosen independently uniformly random round functions modelled random function oracles available parties including adversary consider operation group operation otherwise discern group operation elements following shall follow proof closely however make quite modifications mostly due nature generalization note consider scheme opposed version see figure main theorem section following theorem let modelled random oracles let subkeys chosen independently uniformly random let let probabilistic adversary queries queries queries definitions begin proof need several definitions identical definitions rewording definition let denote permutation oracle either oracles respectively get transcripts set queries set queries set queries sets hxqc yqc discern two types oracle queries cipher queries oracle queries respectively queries bounds computational complexity adversary may assume deterministic proof theorem hence may consider algorithm given set queries determine next query definition query hxi upper equality case indexes defined final output definition let tuple transcripts length respectively say possible every hxi let define two useful ways may answer queries already defined definition let process cipher query oracles use uses replaced another independent random function definition let denote process answers oracle queries using answers cipher query follows queries exists pair return queries exists pair return otherwise return uniformly chosen element latter definition may consistent function permutation formalize exactly event definition let possible inconsistent exist either containing transcript called inconsistent note assume never repeats part query answer determined previous queries every possible atranscript consistent note let denote transcripts seen cipher queries answered respectively oracle queries noting case function replaced another random function also note using notation likewise lemmas let begin finding results aid proving main theorem first compare distributions using result afterwards shall consider distributions equal lastly shall consider distributions equal combining results allow prove main theorem remark whenever write mean subkeys chosen independently uniformly random proof proposition follows argument proposition lemma proof let possible consistent consistent simply difference cipher queries consistent overlap pairs hence need consider choose elements many possible elements without replacement let consider probability inconsistent inconsistent either given happens probability queried would return corresponding queried would return uniformly random element likewise queried inverse hence inconsistent thereby get consistent consistent inconsistent inconsistent inconsistent distribution independent consistency let focus distributions show identical unless input cipher query equal oracle input order first define event badg definition every specific key define badg set possible consistent satisfying least one following yil lemma let possible consistent atranscript badg proof know badg one occur hence using union bound badg occurs occurs occurs occurs lemma let possible consistent badg proof want show query answers subtranscripts games equally distributed condition neither events occur game fix key recall adversary query oracle determine answer previous queries games query query answer equally distributed games underlying random function games game query uniformly random answer random function likewise game query uniformly random answer random function consider permutation oracle consider pair games input first round function game output always uniformly random element newly selected game already queried output round function corresponding oracle answer else uniformly random element newly selected former event game never occurs event never occurs distributions equal games access random function second third round function outputs equal distributions games input fourth round function game output always uniformly random element newly selected unless case output equal output first round function game already queried input first round function output round function corresponding oracle answer output equal output first round function else uniformly random element newly selected former event game never occurs event never occurs distributions equal ask query determine answer based previous queries see inverse permutation oracle using yields analogous distributions thus distributions two games must equal let show distributions identical unless value input two separate occasions also define key bad altered pertains current oracles definition every specific key function define bad set possible consistent atranscripts satisfying least one following events xli yir yil yjr yjl xli yjl xli yir yil lemma let possible consistent atranscript bad proof bad satisfies using uniform independently chosen may achieve upper bound individual event probabilities use union bound many ways picking also many ways picking many ways picking probability two elements chosen equal may bound event accordingly achieve using union bound bad occurs occurs lemma let possible consistent bad following proof based proof refers first part argument need generalization argument also include proof since possible hxi therefore respective answers gives respectively never repeats part query definition answer independent uniform element modelled random function oracles oracle outputs independent uniform elements hence second part proof fix bad seek compute since possible note define xli yir yil yil yir xli yil second equality latter equivalent yil observe similarly also hence bad implies inputs distinct implies also modelled random function independent thus assumed chosen bad bad proof theorem complete proof theorem combine lemmas following probability estimation proof theorem let set possible consistent atranscripts following ease notation sake reader let badg denoted badg bad bad furthermore abbreviate inconsistency incon let consider cases incon badg badg badg badg incon badg badg last estimate used lemma consideration maximal amount possible inconsistent pairs time incon incon bad bad bad bad incon incon bad bad last estimate used lemma proof lemma let use temporary estimate badg badg bad bad last estimate also used lemma may assume wlog badg badg badg likewise badg badg badg lemma respectively lemma get following continued estimate using triangle inequality every combination query elements badg bad max badg max bad denote total amount queries may quickly estimate reword main theorem theorem let modelled random oracles let let let adversary total queries proof given theorem get using get final estimate reordering conclusion generalized even mansour scheme well feistel cipher work arbitrary groups proved classical results pertain group versions based work hope opens avenues proving classical schemes may made quantum secure generalizing certain groups work suggest generalizing classical schemes using underlying group structures hidden shift reductions author would like thank thesis advisor gorjan alagic topic enlightening questions answers well encouragements along way author would also like thank department mathematical sciences university copenhagen lending facilities writing process references gorjan alagic alexander russell cryptography based hidden shifts eurocrypt john black phillip rogaway ciphers arbitrary finite domains pages springer berlin heidelberg orr dunkelman nathan keller adi shamir minimalism cryptography scheme revisited eurocrypt yan zong ding michael rabin everlasting security stacs annual symposium theoretical aspects computer science antibes juan les pins france march proceedings pages shimon even yishay mansour construction cipher single pseudorandom permutation cryptology craig gentry zulfikar ramzan eliminating random permutation oracles cipher asiacrypt jeremy jean tik cryptographers http jonathan katz yehuda lindell introduction modern cryptography crc press edition marc kaplan gaetan leurent anthony leverrier maria breaking symmetric cryptosystems using quantum period finding arxiv hidenori kuwakado masakatu morii quantum distinguisher feistel cipher random permutation isit pages ieee hidenori kuwakado masakatu morii security cipher isita pages ieee joe kilian phillip rogaway protect des exhaustive key search analysis desx cryptology michael luby charles rackoff construct pseudorandom permutations pseudorandom functions siam moni naor omer reingold construction pseudorandom permutations revisited journal cryptology preliminary version proc stoc sarvar patel zulfikar ramzan ganapathy sundaram ciphers xor exclusive selected areas cryptography annual international workshop sac john newfoundland canada august revised papers pages serge vaudenay provable security block ciphers decorrelation pages springer berlin heidelberg berlin heidelberg mark zhandry note prps corr super pseudorandomness group scheme following assume adversary unbounded computationally may make polynomially many queries oracles act black boxes truly random permutation intend play pseudorandom random permutation game given encryption oracle related decryption oracle randomly chosen equal probability following two options random key chosen uniformly used encrypt random permutation chosen used encrypt adversary wins game distinguish chosen probability significantly better explicitly wish prove following group scheme theorem assume let key probabilistic adversary limited polynomially many oracle queries adversarial advantage bounded def adv aek total number total number success probability negligible proof may assume deterministic essence unbounded computationally affords possibility derandomizing strategy searching possible random choices picking effective choices computed effectiveness choice example see may also assume never queries pair sets respectively let define two main games could play oracle interactions see next page explicit game descriptions note steps italics impact response queries simply continue answer queries note key turns bad say key bad sets exist either good bad keys otherwise game consider random game corresponds latter probability definition game see letting denote probability playing game simply giving uniformly random answers queries notation let game initially let empty flag unset choose answer query follows game initially let empty flag unset choose answer query follows query choose set flag bad define thereby also return query choose redefine set flag bad else set flag bad goto step define thereby also return query choose set flag bad define thereby also return query choose redefine set flag bad else set flag bad goto step define thereby also return query choose set flag bad define thereby also return query choose redefine set flag bad else set flag bad goto step define thereby also return query choose set flag bad define thereby also return query choose redefine set flag bad else set flag bad goto step define thereby also return game consider experiment corresponds game played prior probability define probability define game outlined note parts italics impact response queries however time key becomes bad choose new random value repeatedly response key longer bad reply value intuitively game behaves like game except game checks consistency want win collision see letting denote probability playing game proof given appendix defined games way outcomes differ event key turns bad thus circumstance causes difference instructions carried games also cause games set flag bad let bad denote event flag gets set bad case flag set bad two following lemmas follow previous statement lemma bad bad lemma using two lemmas able prove lemma lemma adv bad using lemmas adv bad bad bad bad let define yet another game game game initially let empty flag unset answer query follows query choose define thereby also return query choose define thereby also return query choose define thereby also return query choose define thereby also return queries answered choose exists becomes bad set flag bad game runs game except choose key queries answered checks badness flag checking whether key become bad shown flag set bad game flag set bad game consideration cases see appendix hence get following lemma lemma bad bad using lemma bound bad order bound adv adversary queries elements elements key chosen uniformly random probability choosing bad key adv bad restating theorem get theorem probabilistic adversary limited polynomially many queries generalized scheme group super pseudorandom permutation proof probability game recall definition see write denote final transcripts drop index understood begin defining game game initially let empty choose answer query follows query return else choose define return query return else choose define return query return else choose define return query return else choose define return notice difference game game defining latter defined values oracles beforehand former defines goes still adversary tell difference betweeni playing game game defining thus wish show adversary may distinguish playing game playing game even negligibly showing adversary may distinguish outputs given two games games begin choosing uniformly random key show value games identical hereby assume key fixed arbitrary value remainder proof considering definitions game game see two games define differently former defining latter defines computes show game also answers queries referring although directly given partial functions game functions defined values including query define partial function following defined def defined undefined otherwise using definition defining value implicitly defines value first question whether whether clashes values differences values differing defined lemma let partial functions arising game partial function proof proof induction number define steps game steps steps becomes defined initial case induction proof trivial empty values may clash suppose step define possibility becomes occur new value clashes prior defined value defined clashes arise defined step value clash argument similar become defined well although case defined step forces new uniformly random value chosen clash occurs analogously clashes arise hence must may also consider game sense define value game implicitly define value game wish show oracle game expressed terms correspond exactly game case query beginning game first note game never defines unless queried alternately queried however never repeats query guess answer never queries part already defined pair may assume undefined queried therefore see concurrently queried defined defined let consider two cases defined undefined case defined game returns setting leaves unchanged value remains unlike next case case undefined game repeatedly chooses uniformly undefined set follows uniformly distributed seen showing argument follows definition case setting also sets contrast prior case defined consider query game case defined returned unchanged case undefined choose set returned thus behaviour game game identical queries briefer arguments following cases arguments similar case query assume element pair queried like case see also case defined returned leaving unchanged games case undefined chosen uniformly cases corresponding sets query pairs thus behaviour game game identical queries case query instead assume element pair queried case using definition game definition see defined defined also holds defined assumption beginning case hence defined indeed games secure value case undefined chosen uniformly defined cases thus behaviour game game identical queries case query assume element pair queried case using definition game definition well case assumption see defined defined hence defined indeed games secure value case undefined chosen uniformly defined cases thus behaviour game game identical oracle queries proof probability game game match recall definition see write denote final transcripts also introduce following definition definition say two overlap say pairs identical likewise two pairs overlap definition oracles must identical therefore wlog may assume queries oracles let prove lemma lemma bad bad proof need prove game flag set bad game flag set bad want show exists either becomes bad consider cases flag set bad cases use analogous argument following defined overlapping pairs identical assume exists becomes bad either need check four oracle queries flag game set bad needs consideration cases assume assume
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bounded holomorphic functions hyperbolic numbers candidates hyperbolic activation functions jun eckhard hitzer university fukui liouville theorem states bounded holomorphic complex functions necessarily constant holomorphic functions fulfill socalled conditions conditions mean complex independent direction holomorphic functions ideal activation functions complex neural networks liouville theorem makes useless yet recently use hyperbolic numbers lead construction hyperbolic number neural networks describe conditions hyperbolic numbers show exists new interesting type bounded holomorphic functions hyperbolic numbers constant give examples functions therefore substantially expand available candidates holomorphic activation functions hyperbolic number neural networks keywords hyperbolic numbers liouville theorem conditions bounded holomorphic functions introduction sake mathematical clarity first carefully review notion holomorphic functions two number systems complex hyperbolic numbers liouville theorem states bounded holomorphic complex functions necessarily constant holomorphic functions functions fulfill socalled conditions conditions mean complex contrary complex case hyperbolic logistic function bounded due absence singularities thus general terms seems suitable activation function concretely following facts however might disadvantage real imaginary part different squashing values component functions significantly differ zero around independent direction respect incremental ratio defines derivative taken holomorphic functions would ideal activation functions complex neural networks liouville theorem means careful measures need taken order avoid poles function becomes infinite yet recently use hyperbolic numbers boundaries hyperplanes real unipotent output buchholz argued lead construction hyperbolic number neural networks describe generalized cauchyriemann conditions hyperbolic numbers show exist bounded holomorphic functions hyperbolic numbers constant give new example function therefore excellent candidates holomorphic activation functions hyperbolic number neural networks shown hyperbolic number neural networks allow control angle decision complex numbers isomorphic clifford geometric algebra generated single vector negative square algebraic basis isomorphism realized mapping hyperbolic numbers isomorphic clifford geometric algebra generated single vector positive square algebraic basis isomorphism hyperoblic numbers realized mapping complex variable functions follow treatment given assume complex function given absolute convergent note slightly correct two formulas buchholz think necessary delete buchholz original etc power series real functions real variables since obtained algebraic way complex number arbitrary functions must satisfy certain conditions several equivalent ways obtain conditions following riemann state function function complex variable derivative independent direction complex plane respect incremental ratio taken requirement leads two partial differential equations named cauchy riemann relate one method obtaining equations following consider expression function derivative respect shall zero first perform bijective substitution based computing derivative help chain rule need derivatives hyperbolic numbers hyperbolic numbers also known numbers form commutative algebra canonical hyperbolic system numbers defined hyperbolic conjugate defined requiring real imaginary part vanish obtain conditions functions complex variable fulfill conditions functions functions follows fulfill laplace equation similarly using chain rule obtain called harmonic functions important many fields science notably fields electromagnetism astronomy fluid dynamics used accurately describe behavior electric gravitational fluid potentials study heat conduction laplace equation heat equation liouville theorem states bounded holomorphic function fulfills conditions constant therefore complex neural networks meaningful use holomorphic functions activation functions used special measures need taken avoid poles complex plane instead separate componentwise split real scalar functions real part imaginary part usually adopted therefore standard split activation function complex domain given laplace equation simple example elliptic partial differential equation general theory solutions laplace equation known potential theory solutions laplace equation taking hyperbolic conjugate corresponds isomorphic algebra taking main involution grade involution maps hyperbolic invariant corresponding lorentz invariant physics modulus defined positive definite hyperbolic numbers fundamentally different complex numbers complex numbers quaternions division algebras every element unique inverse hyperbolic numbers always inverse instead idempotents divisors zero define following idempotent basis fulfills inverse basis transformation simply setting get corresponding coordinate transformation well inverse coordinate transformation figure hyperbolic number plane horizontal vertical showing hyperbolas modulus green straight lines modulus red divisors zero hyperbolas modulus blue hyperbolic conjugate becomes due idempotent basis idempotent basis using hyperbolic invariant becomes multiplicative following consider product quotient two hyperbolic numbers expressed idempotent basis due repeat product zero even though factors numbers along axis therefore called divisors zero divisors zero inverse hyperbolic plane diagonal lines divisors zero pairs hyperbolas constant modulus shown fig hyperbolic number functions assume hyperbolic number function given absolute convergent power series real functions real variables example hyperbolic number function exponential function ehy cosh sinh possible divide moreover product hyperbolic number axis times hyperbolic number axis cosh sinh since obtained algebraic way hyperbolic number arbitrary functions must satisfy certain conditions several equivalent ways obtain conditions function function hyperbolic variable derivative independent direction hyperbolic plane respect incremental ratio taken requirement leads two partial differential equations called generalized gcr conditions relate obtain gcr conditions consider expression function derivative respect shall zero first perform bijective substitution based computing derivative help chain rule need derivatives requiring real vanish obtain gcr conditions functions hyperbolic variable fulfill gcr conditions functions functions functions called hyperbolic holomorphic functions follows fulfill wave equation cosh lim cosh lim function representative turn real neural node activation function holomorphic hyperbolic activation function via note another holomorphic hyperbolic activation function studied namely sinh sinh especially similarly wave equation important linear partial differential equation description waves occur physics sound waves light waves water waves arises fields like acoustics electromagnetics fluid dynamics wave equation prototype hyperbolic partial differential equation let compute partial derivatives exponential function gcr conditions therefore clearly fulfilled means hyperbolic function holomorphic since exponential function range product also values range therefore function values components function values repeatedly applied chain rule differentiation similarly obtain function pictured fig let verify function fulfills gcr conditions using chain rule obtain clearly see partial derivatives fulfill gcr conditions exponential function expected definition exponential function therefore manifestly holomorphic hyperpolic function bounded case holomorphic hyperbolic functions gcr conditions imply liouville type theorem like holomorphic complex functions easily demonstrated counter example artanh quadrant fig including negative left artanh quadrant fig including positive top artanh figure function horizontal axis left corner paper plane vertical axis figure produced compare quote given introduction split activation function used clearly holomorphic real part depends depends thus gcr conditions fulfilled geometric interpretation multiplication hyperbolic numbers order geometrically interpret product two complex numbers proves useful introduce polar coordinates complex plane similarly geometric interpretation product two hyperbolic numbers first introduce hyperbolic polar coordinates radial coordinate hyperbolic polar coordinate transformation given artanh quadrant hyperbolic plane fig limitted diagonal idempotent lines including positive right quadrant fig including negative bottom product constant hyperbolic number assuming hay artanh hyperbolic number assuming hyperbolic polar coordinates geometric interpretation scaling modulus hyperbolic rotation movement along hyperbola physics einstein special relativistic hyperbolic rotation corresponds lorentz transformation one inertial frame constant velocity tanh another inertial frame constant velocity tanh neural networks based hyperbolic numbers dimensionally extended spacetime therefore ideal compute electromagnetic signals including satellite transmission conclusion compared complex numbers hyperbolic numbers well complex functions hyperbolic functions saw according liouville theorem bounded complex holomorphic functions necessarily constant bounded hyperbolic holomorphic functions exist one function already beeng studied studied promising example hyperbolic holomorphic function detail distinct notions idempotents divisors zero special hyperbolic numbers introduced introducing hyperbolic polar coordinates geometric interpretation hyperbolic number multiplication given hyerbolic neural networks offer compared complex neural networks therefore advantage suitable bounded hyperbolic holomorphic activation functions would certainly interest study convergence accuracy decision boundaries hyperbolic neural networks activation function similar acknowledgment want acknowledge god beginning word god word god god beginning things made without nothing made made life life light mankind want thank dear family well nitta kuroe references guerlebeck holomorphic functions plane space birkhauser chp buchholz sommer hyperbolic multilayer perceptron proceedings international joint conference neural networks como italy vol nitta buchholz decision boundaries hyperbolic neurons proceedings international joint conference neural networks ijcnn june buchholz phd thesis theory neural computation clifford algebras university kiel catoni mathematics minkowski birkhauser laplace equation wikipedia accessed august http laplace greek term logos note greek philosopher named heraclitus first used term logos around designate divine reason plan coordinates changing universe wave equation wikipedia accessed august http equation online function grapher http number wikipedia accessed august http notes collaboration ishi doran lasenby geometric algebra physicists cambridge university press cambridge hitzer relativistic physics application geometric algebra adhav proceedings international conference relativity university amravati india january bible new international version niv gospel according john chapter verses http strong bible lexicon entry logos available online blue letter bible http
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oscillations social force model tobias kretz ptv group karlsruhe germany jul july abstract social force model one prominent models pedestrian dynamics naturally much discussion criticism spawned around concerns existence oscillations movement pedestrians contribution investigating circumstances parameter choices model variants oscillations occur prevented shown oscillations excluded model parameters fulfill certain relations fact parameter choices oscillations occur exploited verify specific computer implementation model introduction social force model pedestrian dynamics model aims describing movement pedestrians predominant purpose simulating pedestrian movement computers force pedestrian pedestrian typically form function grows increasing distance pedestrians depend velocities one pedestrians function suppresses forces pedestrian located outside current walking direction pedestrian social force model first introduced variant later called elliptical specification second variant circular specification proposed third variant elliptical specification difference three variants lies mainly way velocities two interacting pedestrians considered computation force variant considers velocity pedestrian exerts force variant consider velocities distance pedestrians variant considers relative velocity pedestrians pedestrian exerts force pedestrian force acts analytical considerations paper mainly simplest variant considered nevertheless also discussed results change qualitatively variants applied oscillations paper unrealistic artifacts trajectory pedestrian approaching another pedestrian destination wall understood occurrence oscillations sense contribution discussed number contributions often claimed oscillations avoided social force model made small paper shown correct exact conditions value model parameter oscillations occur derived remainder paper first single pedestrian approaching destination investigated pedestrian approaching another pedestrian standing still finally two pedestrians approaching case model reduced one dimension noise term set zero first section approaching destination problem shown severe certain conditions oscillations prevented continue infinitely long time argued case relevant simple pragmatic solutions second third case yield restrictions choice parameters produce realistic behavior pedestrian approaching destination section interested discuss equations motion solution single pedestrian approaching destination coordinate point required come standstill assume beginning pedestrian walking desired speed straight towards destination coordinate tangential component walking velocity describe pedestrian walking positive coordinate negative direction towards destination since variants social force model differ force pedestrians driving force term results section hold three variants assume desired velocity always externally given always pointing pedestrians current position towards assumption simplest one questioned obvious oscillations around intention investigate quantitative details oscillations general equation motion pedestrian reads external parameter typically values seconds require pedestrian reach also stand still arrival condition pedestrian speed larger considering walking direction smaller walk destination left negative side coordinates desired velocity points direction positive coordinates time following moment pedestrian solved integration constants need determined initial conditions choose moment pedestrian however later usage want set general remember particular case two conditions determine values integration constants compute time tturn pedestrian stops turns around position happens tturn tturn initial case simplifies tturn tturn half way actual question fast pedestrian returns passed speed long loop takes choose moment pedestrian standing still tturn time treturned pedestrian returns therefore understood time tturn absolute point time begin determining value integration constants equation follows equations treturned following equation treturned treturned treturned substitutions treturned another substitution transforming last equation obviously one solution general solution lambert function definition inverse relation yey resubstituting reasons convenience resubstitute treturned interval two branches denoted within interval two branches holds case would treturned thus branch gives backwards time solution interested already computed therefore continue branch although inverse relation yey remainder write case numerical value solution treturned using equation equation get speed treturned time pedestrian returns dependence speed time pedestrian last treturned used defining equation function properties index determines recurrence offspring example absolute time pedestrian first time offspring denotes absolute time pedestrian returns first time offspring case properties describe passage offspring loop durance turning position index enumerates loops means two properties index equation means passage offspring depends like time takes loop depends like rewriting equation terms writing turn around distance nth loop table shows results first passages process begins speed figures visualize data typical values takes passages amplitude gets smaller takes seconds oscillation period seconds resolve computer implementation would time step maximum half value seconds simulation steps per second turn case infinitely many oscillations proof given sum diverges pedestrian oscillate infinitely long around destination coordinate first sight unsatisfactory result regard social force model oscillations unrealistic however ask realistic assumptions initial conditions concerns particular desired velocity demanded beginning pedestrian come stop nevertheless set desired velocity time one particular value simplistic real persons plan ahead adjust desired speed desire stop certain position table numerical results first passages resp oscillations value understood oscillation period oscillation total time oscillation values dimensionless numerical values lambert function computed using wolfram alpha figure left value beginning oscillation right maximum amplitude oscillation figure left period oscillation right total time nth passage offspring time nth oscillation pedestrian reaches offspring first time speed figure shows evolution amplitude time adapt desired velocity beforehand achieve ask modify dynamically account precisely ask distance pedestrian start braking social force model sets desired speed opposing current speed long take comes stand still equation motion reads following initial conditions defined moment pedestrian starts brake moment stops distance stand still braking begin looking solving results gives finite positive thus sufficient set desired speed zero pedestrian wants stop would imply infinitely large braking distance infinitely long time come stand still assume pedestrian braking maximum desired speed minimum time braking minimum braking distance come standstill initial speed different pragmatic solution problem planning applications usually requirement set speed pedestrian arrives destination furthermore usual destination point given cross section crossed pedestrian move specific area restriction given speed arrival problem disappears third objection also real person comply request stand still center exactly given coordinate real people always sway little bit try stand still one require simulated pedestrian oscillations around destination point found functional form surely match swaying real people stand still amplitude ones model quickly fall real people real swaying required reproduced models implicitly required limit precision set movement order magnitude acceptable pedestrian approaching standing pedestrian theory want investigate situation pedestrian approaching another pedestrian one standing fixed position one dimension empirical investigation see example since pedestrian exerting force moving variant elliptical specification social force model produce result variant circular specification one investigating different variant elliptical specification speed pedestrian force acts modifies strength force assume pedestrian standing still facing positive direction pedestrian approaching pedestrian positive coordinate speed directed towards negative well desired speed constant time assumed pedestrian time far away pedestrian walking desired walking speed equation motion pedestrian variants social force model radius pedestrian assume simplicity pedestrians radius parameters social force model pedestrian come standstill distance pedestrians bump one another must larger therefore first condition parameters social force model yield realistic results assume oscillations occur generally also occur close equilibrium thus oscillations pedestrian already close oscillations expanding exponential function equation taylor series around gives equation equation damped harmonic oscillator known three solutions damped pedestrian approaches equilibrium point oscillating around critically damped pedestrian approaches equilibrium point quick possible without oscillations damped pedestrian approaches equilibrium point slower would possible without oscillations three cases realized depends relation parameters damped critically damped damped thus addition equation second condition parameters variants social force model chance yield realistic results implications parameters found literature literature one find values parameters gained empirical observations laboratory experiments four parameters given one use relations test realistic results scenario discussed expected four parameters values given two equations allow set limits table shows compilation data relate relations seen four parameters taken experiment relation easily holds relation clearly violated whereas parameters calibrated given work relations require pedestrians walk quite slowly calibrated given needs relatively large values relations hold indication circular specification alone parameter space yields realistic results small source bmax violated violated table bold face marks value literature normal face computed equations one parameter calculated another one range given range utilized parameter space derived parameter large possible assumption input calibration simplified nomad model discussed said publication identical circular specification social force model except radius pedestrians stated explicitly exponent stated explicitly said contribution distance nomad model means body surface body surface parameter meanings identical means center center distance value parameter would change factor even equation could violated elliptical specification comparison elliptical specification rigid analysis much difficult circular specification since elliptical specification depend distance two pedestrians also relative velocity still one estimate effect added consideration relative velocity occurrence oscillations elliptical specification social force model force pedestrian pedestrian defined cos cos gives position pedestrians model parameters pedestrians positions derived properties time dependent properties constant one dimension pedestrians facing simplifies cos assumed otherwise therefore force either zero arithmetic geometric mean current projected distance resp directly seen large distances equation reduces approximately circular specification depends distance pedestrian approaching another pedestrian positive values positive negative therefore inequality arithmetic geometric means holds obviously long force equation larger compared circular specification consequently pedestrian overshoot certain parameter choices circular specification case pedestrian overshoots equilibrium point turns around would desirable force larger smaller circular specification however since case becomes positive holds therefore equation exponential factor gives smaller value circular specification yet fraction factor value may large values parameter outweigh damping effect modification exponential function three indications also phase pedestrian movement general oscillations suppressed elliptical specification first large values parameter already circular specification oscillations equation tells means isolated view way back problem pronounced system may actually even evolve point exists second one expand equation series small values moment pedestrian turns around therefore circular elliptical specification yield identical force starting move backward positive equation tells elliptical specification yields smaller forces difference circular specification leveled third admittedly arguable additional assumptions one expand small also reduce complexity equation regard various approximate forms equation derived way one analytically solvable contains parameter omitting indices changed leads less strict requirewhere comparison equation factor ment avoiding oscillations equation force zero pedestrian starts approach finally note case sufficiently large distance typical pedestrian walking speed since equation diverges positive values pedestrian slow eventually rest turn around departing pedestrian thus always simulation initiated may yield dynamics fit line argumentation compare extrinsically prepared garden eden states dealt example model extension one simply want exclude construction simulations realistic applications makes sense choose parameters oscillations occur however verify computer implementation social force model interesting use parameters around critically damped conditions check oscillations occur according expectations model specific validation work amend validation tests like rimea test cases usually formulated model independently subsection model specific verification process carried exemplarily utilizing ptv viswalk figure simulation scenario figure shows setting implemented simulation model walking area modeled wide pedestrian facing positive direction stopped red traffic signal second pedestrian set model moving negative direction distance expected difference table stand still distance dependence parameter column expected shows values according equation parameter settings since soc mean controls contribution elliptical specification set zero remainder refers soc iso software pedestrian left set zero well minimize effects fact nature simulation stochastic noise set zero value noise parameter set zero pedestrians unnecessarily blur results distance expected difference dist exp diff table stand still distance dependence parameter column expected shows values according equation difference theoretical expectation simulation cases first investigate second pedestrian comes rest according equation depends values parameters soc iso soc iso latter set keeping soc iso constant increasing value steps second pedestrian pass first one first time value overlap visually distance central points pedestrians comes close table figure show values stand still distance dependence values parameter parameters kept constant theoretical expectation met well cases table figure show values stand still distance dependence values parameter soc iso parameters kept constant theoretical expectation met well cases stand still distances table unrealistically high smallest values parameter chosen parameters way scan realistic parameter values parameters one well demonstrate certain cases small values oscillations others large values none figures show time evolution position approaching pedestrian various values parameter figure left visualization data table expectation regression curve would right visualization data table expectation regression curve would figure left position approaching pedestrian time various values parameter right system critically damped increasing oscillations get smaller vanish neglecting damped case damping exponential function approximately expectated time distance two reversal points damped cases table shows comparison actual expected various values generate damped behavior two pedestrians approaching two pedestrians approach one dimension may come stand still equilibrium distance might jointly equilibrium distance move one two possible directions figure left position approaching pedestrian time damped critical cases regard value right zoom region oscillations parameters identical pedestrians one rewrite coupled equations motion pedestrians one equation movement center mass one equation relative motion latter one approximated oscillator equation compared case one additional factor friction term implying system critically damped leads conclusion case two mutually approaching pedestrians sets stronger restriction parameter choice case one pedestrian approaching another standing least oscillations avoided variant social force model case suppresses oscillations even one two pedestrians standing still number reverses simulation approximate expectation table comparison simulated approximately expected time reversals conclusions could shown social force model proposed elliptical specification circular specification special case around equilibrium reduces equations damped harmonic oscillator implies indeed one careful choose parameters pedestrians trajectories yield unrealistic results however time means parts parameter space small oscillations exactly oscillations look parameter values found literature circular specification shows parameters deemed yield realistic results certain calibration scenarios may produce oscillating behavior unless desired walking speed set rather small values equations social force model elliptical specification easily treated analytically still discussion equations could argued compared circular specification oscillations clearly suppressed elliptical specification therefore appears superior two preceding variants also reasons discussed paper found already paper different reasons therefore good idea either simply use elliptical specification combine one earlier variants simply adding forces reduce risk oscillations phenomenon oscillations used verify specific computer implementation social force model possible show reproduces expected results comparison seen attempt falsify either two theoretical expectations software implementation attempt failed potential issues could found method applied verify implementation social force model variants generally example model specific verification tests carried contribution examples phenomenon oscillations bears potential formulate tests acknowledgments thank mohcine chraibi normen rochau useful discussions references references helbing molnar social force model pedestrian dynamics physical review arxiv helbing farkas vicsek simulating dynamical features escape panic nature arxiv johansson helbing shukla specification social force pedestrian model evolutionary adjustment video tracking data advances complex systems steffen seyfried repulsive force continous space models pedestrian movement arxiv preprint eprint chraibi seyfried schadschneider generalized model pedestrian dynamics physical review chraibi kemloh schadschneider seyfried models pedestrian dynamics networks heterogeneous media treml avoiding numerical pitfalls social force models physical review corless gonnet hare jeffrey knuth lambertw function advances computational mathematics corless jeffrey knuth sequence series lambert function proceedings international symposium symbolic algebraic computation acm monir numerical evaluation lambert function application generation generalized gaussian noise exponent ieee transactions signal processing publisher wolfram alpha llc retrieved april https winter human balance posture control standing walking gait posture gorrini shimura bandini ohtsuka nishinari experimental investigation pedestrian personal space transportation research record journal transportation research board werner helbing social force pedestrian model applied real life scenarios pedestrian evacuation dynamics seer rudloff matyus validating social force based models comprehensive real world motion data transportation research procedia hoogendoorn daamen microscopic calibration validation pedestrian models models using experimental data traffic granular flow springer schadschneider schreckenberg garden eden states traffic models journal physics mathematical general arxiv brunner kirchberger lebeda oswald kraft thoss seyfried hartnack wader spennes kretz rimea richtlinie mikroskopische initiatoren des schwendimann waldau gattermann moroge schreckenberg eprint http german ptv ptv vissim user manual ptv group karlsruhe germany publisher wolfram alpha llc retrieved january https exp publisher wolfram alpha llc retrieved january https exp proof pedestrian heading destination point never comes rest section show diverges first step proof begin equation holds therefore holds follows equation follows also see figure plot visualization evolution allows see first step easily second step verify physically obvious mathematically needs shown step effect check potential errors done since holds figure plot visualization evolution basically created results according equation third step proof equations read follows thus show behavior also fourth step large would approach value smaller sum would converge approaches may diverge follows compute limit large expand series convenience write knowing approach increasing computing right side limit towards derivatives one write compare figure considering terms second order using approximations equation gives large approaches thus like harmonic series ratio subsequent terms approaches note equation implies therefore sum depends trivially sum especially convergence behavior last step prove also therefore beyond harmonic series lower estimate implying harmonic series also sum diverges assume written placed two elements harmonic series requires obviously case lower estimate right side positive term replaced minimum value negative term upper limit thus seen table hold also therefore harmonic series lower estimate series thus series diverges series figure plot comparing exp approximation
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robust computer algebra theorem proving oracle dec gopal sarma school medicine emory university atlanta usa nick hay vicarious fpc san francisco usa nnickhay keywords oracle safety cas theorem proving math oracles received june context superintelligent systems term oracle two meanings one refers modular systems queried tasks another usage referring class systems may useful addressing value alignment control problems superintelligent system answers questions aim manuscript survey contemporary research problems related oracles align research goals safety examine existing question answering systems argue high degree architectural heterogeneity makes poor candidates rigorous analysis oracles hand identify computer algebra systems cass primitive examples oracles mathematics argue efforts integrate computer algebra systems theorem provers systems largely developed independent one another provide concrete set problems related notion provable safety emerged safety community review approaches interfacing cass theorem provers describe architectural deficiencies identified cass suggest possible lines research practical software projects scientists interested safety introduction recently significant public attention drawn consequences achieving humanlevel artificial intelligence small communities analyzing impact related technologies decades forecasts made many recent breakthroughs dramatically accelerated pace research areas diverse robotics computer vision autonomous vehicles name researchers industrialists view advances artificial intelligence potential overwhelmingly beneficial humanity medicine transportation fundamental scientific research areas actively transformed advances artificial intelligence hand issues privacy surveillance access inequality economics policy also utmost importance distinct specific technical challenges posed research problems context forecasting one set issues stands apart namely consequences artificial intelligence whose capacities vastly exceed human beings researchers argued superintelligence poses distinct problems modest systems described particular emerging discipline safety focused issues related potential consequences goal structures systems significant capacity exert influence world vantage point fundamental concern deviations values superintelligent agent could significantly detrimental consequences one strategy advocated addressing safety concerns related superintelligence oracle system answers questions words oracle directly influence world capacity except via user system oracle directly take physical action except answering questions posed system operator argued may provide way bypass immediate need solving value alignment problem would powerful resource enabling safe design autonomous deliberative superintelligent agents weaker notion term oracle call oracle refers modular component larger system queried tasks article view computer algebra systems primitive oracles mathematical computation likely become quite powerful time horizons many expect superintelligent systems developed assumption math oracles prove useful development systems addressing architectural problems cass integration interactive theorem provers provides concrete set research problems align issues safety addition systems may also useful proving functional correctness aspects architecture section briefly discuss unique challenges allocating resources safety research section briefly summarize motivation developing oracles context safety give overview safety risks control strategies identified superintelligent oracle ais section analyze contemporary question answering systems argue sarma hay contrast computer algebra systems current systems poor candidates rigorous analysis oracles section review differences theorem provers computer algebra systems efforts integrating two known architectural problems cass close list additional research projects related mathematical computation may interest scientists conducting research safety metascience safety research resource allocation standpoint safety poses unique set challenges areas academic research operate long potentially uncertain time horizons say academia engage longterm research research quantum gravity example approaching nearly century worth effort theoretical physics however key difference fundamental research sciences humanities safety possibility negative consequences indeed significant ones key technological breakthroughs taking place without corresponding advances frameworks safety issues controversial largely due disagreement achieving subsequent consequences specifically notion intelligence explosion whereby intelligence software systems dramatically increases due capacity model source code yet receive adequate scientific scrutiny analysis affirm importance safety research also agree cautioned proceeding speculative lines thinking lack precision perspective article possible fruitfully discuss issues related safety maintaining connection practical research problems extent goal similar spirit widely discussed manuscript concrete problems safety however aim bit bold authors robust computer algebra theorem proving oracle concrete problems state outset analysis set aside questions related superintelligence goal explicitly tackle superintelligence related safety concerns believe areas contemporary research overlap novel ideas concepts arisen among researchers purely focused analyzing consequences systems whose capacities vastly exceed human beings clear claim strategy searching research objectives align aims superintelligence theory sufficient cover full spectrum issues identified safety researchers doubt prospect superintelligence raises entirely new issues context contemporary research however considering young field believe perspective adopted article moderate stance take field critical growth phase new culture created article focuses one area safety landscape oracle identify set concrete software projects relate abstract conceptual ideas safety bridge gap practical contemporary challenges longer term concerns uncertain time horizon addition providing concrete problems researchers engineers tackle hope discussion useful introduction concept oracle newcomers subject state outset within context oracle analysis limited scope systems perform mathematical computation oracles general nonetheless considering little effort directed superintelligence control problem confident fruit addressing general issues awaiting discovery brief overview oracle described oracle system answers questions although term traditionally used denote fully general systems also imagine oracles whose capabilities limited latter systems refer oracles safety researchers argued superintelligent oracles potential help address safety concerns autonomous deliberative superintelligent systems oracle ability influence world user system consequently oracle may position assist researchers solving difficult technical conceptual problems related design independent deliberative agents yet superintelligent oracle ais pose distinct set safety concerns problems identified example presume superintelligent system limited ability interact world may nonetheless potential negatively impact world manipulating user system armstrong bostrom sandberg write vulnerable oai social engineering advanced oai literally offer world genuine altruism defence offer world free poverty hunger war disease death well personal trinkets could want could guess weaknesses individual find right arguments convince granting power liberty moral profitable thing able play rivalries institutional weaknesses ethical dilemmas virtuosity human could given potential dangers human manipulation oai researchers searched control strategies mitigate risks examples control methods proposed physical physically containing oai system limiting eliminating external like previous security measure precluding oai physically manipulating world direct manner epistemic capability oais output could restricted even beyond question answering example restricting answer yes questions sarma hay virtual oai could run simulated world even system capabilities expanded behavior could observed modeled resetting oai questionthis would prevent oai engaging social engineering manipulating answers users questions although capacities oracles limited nonetheless pose safety risks architectural deficiencies oracles might exploited larger system manipulate human user could give answers difficult verify allow oai execute complex intricate plans unbeknownst user therefore flaws oracles inherently risky used solely domain applicability may well dangerous part larger system general capabilities though control strategy narrowest sense creating robust oracles important objective designing safe oais furthermore ensuring robustness subsystems might mitigate need stronger control strategies oai would fewer weaknesses exploit without saying arguments presented highly schematic dependent specific technologies knowledge limited work translating analyses superintelligent oracle ais concrete language modern artificial intelligence goal manuscript spirit anchor schematic philosophical arguments practical contemporary research narrow focus mathematical domain remainder article use term oracle limited sense subsystem particular oracles performing mathematical computations hope analysis presented intrinsic value developing robust math oracles well provide intuition context identifying concrete problems relevant developing safe superintelligent oracle systems contemporary systems qualify oracles obvious class contemporary systems would seem qualify oracles question answering systems qass stated basic criterion characterizing oracles fundamental mode interaction answering questions posed user queries part larger system contemporary qass largely aimed using natural language processing techniques answer questions pertaining useful facts world places movies historical figures important point make qass highly variable nature underlying technology instance ibm original watson system competed jeopardy developed prior recent advances deep learning fundamentally transformed areas ranging computer vision speech recognition natural language processing particular task system nonetheless able perform level beyond accomplished human participants introduction info panes popular search engines hand based recent machine learning technology indeed advances also power latest iterations watson system end spectrum wolfram alpha also question answering system architecturally centered around large curated repository structured data rather datasets unstructured natural language systems currently useful humans navigating world planning social outings arriving quick useful answers ordinary questions clear remain useful quite capacity many years standalone components superintelligent systems although underlying techniques deep learning nlp fundamental interest right fact systems qass seems artifact utility robust computer algebra theorem proving oracle consumers another important observation contemporary qass much underlying architecture replaced taking advantage structured data example wolfram alpha demonstrates nlp machine learning based systems underlying technology used part larger pipelines turn unstructured data textual sources structured data fact simply underscores contemporary qass particularly appealing model systems analyze oracle safety computer algebra oracles mathematical computation question answering systems described rely natural language processing varying degrees addition domain applicability tended towards ordinary knowledge useful wide array consumers another type question answering system computer algebra system cas computer algebra traditionally referred systems computing specific results specific mathematical equations example computing derivatives integrals group theoretic quantities etc sense think computer algebra emphasize argument contemporary qass good candidates analysis oracle ais argument traditional formulation oracle tool safety fully expect significant breakthroughs made advancing theory practice techniques safety hope manuscript provide motivation pursue research rather point viewing contemporary systems lens superintelligence seems little reason believe current qass remain sufficiently architecturally stable used standalone components systems many years hand certainly important problems examine evaluating broader impact qass bias nlp systems overgeneralization privacy name issues overlap set problems identified examples concrete problems safety addition beginning see conferences devoted contemporary ethical issues raised machine learning see example workshop ethics natural language processing set algorithms performing applied mathematician theoretical physicist might work paper pencil indeed early work computer algebra came quantum field first computer algebra systems veltman schoonschip performing field theoretic computations led theory electroweak unification computer algebra systems grown popularity functionality expanded substantially cover wide range standard computations mathematics theoretical physics including differentiation integration matrix operations manipulation symbolic expressions symbolic substitution algebraic equation solving limit computation many others computer algebra systems typically run read evaluate print loop repl research education context popularity also grown result notebook model pioneered mathematica system allowing computations cass closely mimic sequential paper pencil work mathematicians theoretical physicists assessing utility cass important note little reason believe computer algebra subsumed branches research machine learning indeed recent research demonstrated applications machine learning computer algebra theorem proving discuss detail via algorithm selection former case proof assistance latter certainly visible machine learning computer algebra theorem proving much active deep areas research also likely profit advances fields artificial intelligence opposed replaced time horizons likely see artificial intelligence beyond expect systems become quite powerful possess capabilities may useful construction general systems therefore worth examining systems perspective safety sarma hay briefly clarifying nomenclature proceeding want explicitly describe issues relating nomenclature arisen discussion thus far state choices terminology given phrase oracle become common usage safety community continue use phrase first word capitalized well acronym oai clarification needed may also use full phrase superintelligent oracle without capitalization modest use cases word oracle either refer oracles state domain knowledge oracle applicable least abstract consider extending terminology domains physics oracles cell biology oracles ethics oracles therefore remainder article concerned safety robustness issues design math robust computer algebra integrated theorem proving today consider standard feature much closer interaction proof assistance computer algebra software several areas benefit including specification interfaces among components certification results domains applicability justification optimizations direction use efficient algebra proofs stephen watt future computer algebra systems threshold described computer algebra systems thought question answering systems subset mathematics related set systems interactive proof assistants interactive theorem provers itps itps also systems mathematics different mathematical context computations one wishes construct proof general kind statement words rather computing specific answers specific questions itps used show candidate mathematical structures software systems possess certain properties sense distinction theorem proving computer algebra viewed historical anomaly perspective philosophical logical efforts early century led mechanization mathematics distinction computing nth laguerre polynomial constructing proof induction might viewed rather artificial although benefit hindsight see two types tasks quite different practice role itps research world different cass whereas cass allow researchers perform difficult computations would impossible paper pencil constructing proofs using itps often difficult even rigorous methods pure mathematics broad terms overhead using itps formalize theorems arises fact proofs systems must proceed strictly set formalized axioms system verify computation consequently itps related systems automatic theorem provers largely used verifying properties software systems require assurance hardware verification mistakes lead costly recalls quotation suggests many academic researchers view integration interactive proof assistants computer algebra systems desirable numerous efforts years exploring possible avenues achieving objective complete list given integrating theorem proving computer algebra would opening wealth potentially interoperable algorithms date remained largely unintegrated cite one example authors developed framework exchange information maple computer algebra system isabelle interactive theorem prover show robust computer algebra theorem proving oracle simple problem involving proof elementary polynomial identity could solved combined system neither system alone see fig cite example demonstrate simply stated elementary problem solved existing environments either computer algebra proof assistance computer algebra system capacity structural induction theorem provers generally rather weak expression simplifiers numerous examples one academic literature another key difference cass itps architectural soundness respective systems discuss computer algebra systems architectural deficiencies practical issue vast majority use cases pose problems integration theorem provers nature designed architecturally sound context superintelligent systems architectural problems cass potential points weakness could exploited malicious purposes simply lead unintended detrimental consequences therefore use phrase robust computer algebra refer cass lack problems identified research literature section combine discussion robust computer algebra integration interactive theorem provers spectrum approaches address issues varying degrees taxonomy approaches many possible avenues tackle integration theorem provers computer algebra systems give broad categories characterizing integration theorem provers built top computer algebra systems include analytica theorema redlog logical extensions axiom system classification first described kaliszyk wiedijk paper arguing architecture list fourth category given frameworks mathematical exchange two systems category includes mathml openmath omscs mathscheme logic broker bridges information exchange solutions pairs systems category include bridges combining pvs hol isabelle maple nuprl weyl omega isabelle summit recently lean mathematica example given bridging isabelle maple example approach category embedding computer algebra system inside proof assistant approach taken kaliszyk wiedijk holcas system system expressions precise semantics proof assistant proves correctness simplification made computer algebra system one primary aspect integration differentiates approaches degree trust theorem prover places computer algebra system computer algebra systems give false impression monolithic systems globally semantics reality large collections algorithms neatly packaged unified interface consequently often corner cases lack precise semantics lead erroneous solutions consider following example system incorrectly gives solution even though given polynomial indeterminate value however expression treated fraction polynomials first simplified solve operation applied words unclear semantics solver module simplifier leads incorrect result another simple example following integral making substitution gives indeterminate result clear inspection solution integral simply belongs class problems induct sarma hay circles represent object nodes whose labels reported table rectangles figure example polynomial identity proven integrating maple computer algebra system represent link nodes contain labels complex series steps corwith isabelle maple simplifier used expanding powerful complement responding final ofphase ofwhich proof arethe folded triangular rest theorem provingtoarchitecture isabelle allows setup within proofthe induction node link nodes labelled simplify identify points systems cooperate solution problem namely maple invoked exai safety research pand polynomial power note restconsider folded node hides away several agenda development safe superintelliadditional maple calls meant perform evaluations disequalitites figure example incorrect solution simple polynomial equation computer algebra system gent systems outlined require formal guarantees correctness point sequence computations computer algebra used current systems would unable provide necessary framework constructing proof figure problem arising symbolic integration due evaluation substitution known specialization problem namely expression evaluation variable substitution commute seen theorem proving benefit tremendously wealth algorithms expression simplification mathematical knowledge computer algebra potential cost compromising reliability combined system possible application current research qualitatively certified computations taxonomy approaches bridging theorem provers computer algebra described key distinction degree trust theorem prover places computer algebra system instance approaches build theorem provers top computer algebra systems address architectural issues cass integrative sound extreme building computer algebra system top theorem prover allows degree trust par theorem prover however approach distinct disadvantage computer algebra systems represent many hundred worth effort intermediate approaches involving common languages symbolic exchange adhoc bridges bring light important notion spectrum provable safety namely robust computer algebra theorem proving oracle ability assign probabilities correctness computations authors present algorithm assigning probabilities statement formal language might ask strategies might look like similar goal mind significantly weaker interfaces theorem provers computer algebra systems provide concrete example ask question along lines fundamentally interface computer algebra system weaker link decrease confidence final result much instance example given figure revise confidence result knowing polynomial simplification conducted within computer algebra system worth asking simple answers question require major theoretical advances made instance might imagine curating information computer algebra experts known weaknesses use information simply give qualitative degree confidence given result example repository formal proofs generated using integrated systems steps proof require computer algebra flagged also assigned qualitative measure uncertainty relationship highly informal method giving qualitative certification computations formal algorithm developed compared existing techniques software industry ensuring correctness one hand unit testing theoretically trivial yet quite powerful practice something along lines automated checklists software complexities modern software would impossible handle without extensive software testing frameworks hand formal verification provide substantially stronger guarantees yet major undertaking correctness proofs often significantly demanding construct software consequently discussed section formal verification much less frequently used industry typically exceptional circumstances high guarantees correctness required hardware verification integrated systems computer algebra theorem proving give rise quite interesting perhaps ironic opportunity pursue simple strategies giving qualitative estimates correctness computation logical failures error propagation examples described demonstrate errors initial calculations may well propagate give rise results systems capable performing mathematical computation become increasingly sophisticated embedded part design workflows science engineering beyond see today could imagine errors quite costly difficult debug case superintelligent system concerning scenarios would systematic errors computer algebra could exploited adversarial purposes led unintentional accidents large scale issue error propagation another example concrete context pursuing simple strategies assigning qualitative measures certainty computations performed integrated theorem proving computer algebra systems instance may less inclined trust result computer algebra system invoked early computation opposed later curated data computer algebra experts reliability failure modes various algorithms might also chain together informal estimates arrive single global qualitative estimate multiple systems developed independently based fundamentally different architectures might also significantly confident result could verified two separate systems additional topics related ideas merit investigation broader context mathematical computation integrating smt solvers interactive theorem provers satisfiability sarma hay modulo theories smt solvers important element automated reasoning efforts analogous described bridge smt solvers interactive theorem provers identifying important widely used algorithms computer algebra computer algebra systems grown become massive collections algorithms extending domains well outside realm mathematics purely mathematical capacities cass prove useful future systems would valuable rank order algorithms popularity importance one approach would basic textual analysis source code github stackexchange would also allow targeted efforts directly address issues soundness core algorithms expression simplification integration context holcas system described example would valuable rough estimates number required implement minimal cas widely used functionality top theorem prover proof checkers integrated systems proof checkers important tools landscape formal verification theorem proving indeed often much less computationally expensive verify correctness proof generate scratch availability proof checkers widely used interactive theorem provers one reason confident correctness formal proofs described strategies integrating computer algebra theorem provers potentially result combined system less trustworthy theorem prover alone therefore availability proof checkers combined systems would valuable resource verifying proof correctness certain mathematical domains potentially provide avenue surmounting need directly make cas architecturally robust development integrated proof checkers likely substantial undertaking require novel architectures integrating core cas itp systems distinct described however largely unexplored topic merits investigation analyzing scaling properties algorithms computer algebra theorem proving function hardware resources premise analysis presented cass integrated theorem proving likely remain sufficiently architecturally stable useful several decade construction systems hand argued earlier much less clear true visible consumeroriented question answering systems make arguments rigorous would valuable develop quantitative predictions capabilities existing algorithms computer algebra theorem proving provided substantially expanded hardware resources instance might examine problems mathematics theoretical physics solutions cass intractable current resources may feasible future hardware cognitive science computer algebra role computer algebra played theoretical physics mathematics influenced thinking process researchers computer algebra simply convenience shifted way problems solved fundamentally enabled new problems solved would completely intractable otherwise robust computer algebra theorem proving oracle cognitive science mathematical thought substantial topic overlaps many established areas research however systematic review research mathematics theoretical physics since advent computer algebra role mathematical thought process underexplored topic would interesting avenue pursue understanding role cass itps integrated systems may come play superintelligence particularly case neuromorphic systems modeled human cognition questions also relate understanding scaling properties cas theorem proving algorithms well cataloguing widely used algorithms computer algebra conclusion aim article examine preexisting research objectives computer science related disciplines align problems relevant safety thereby providing concrete practical context problems otherwise longer time horizon research particular focused notion oracle used safety community observed word oracle two meanings context superintelligent systems one usage refers subsystem larger system queried tasks superintelligent systems restricted answer questions examined contemporary question answering systems qass argued due architectural heterogeneity systems readily lend rigorous analysis safety perspective hand identified computer algebra systems cass concrete primitive examples oracles examined architectural deficiencies cass identified theorem proving community argued integration interactive theorem provers itps cass objective area research respective communities several decades provides set research problems practical software projects related development powerful robust math oracles multidecade time horizon independent role oracles systems may also prove useful tools safety researchers proving functional correctness components architecture natural choices systems use would interfaces wolfram language widely used computer algebra system one hol family theorem provers coq substantial repositories formalized proofs modern itp lean rather representing bold profound new agenda view projects concrete achievable goals may pave way substantial research directions topics discussed long rich academic history number projects appropriate students anywhere undergraduates phd students beyond good undergraduate research projects would probably start basic data science catalogue core computer algebra algorithms usage popularity would useful estimate certified implementations algorithms would entail whether formally verified implementations along lines kaliszyk wiedijk holcas system cas built top theorem prover also useful would systematic study role computer algebra played mathematics theoretical physics would interesting overlap cognitive psychology three projects together would make approachable undergraduate thesis beginning project graduate student solid phd thesis devoted topic oracle might involve tackling approaches oracles stemming reinforcement learning well advanced theorem proving cas related topics investigating development hybrid architecture would allow student worked projects several years would develop unique skill set spanning philosophy machine learning theorem proving computer algebra context superintelligent oracle ais may possess ability manipulate human user differentiate addressing architectural algorithmic deficiencies subsystems versus general control methods containment strategies given strong mathematical capabilities likely useful construction general systems designing robust cass oracle important counterpart general control strategies system fewer loopholes exploit controlling oais poses distinct set challenges concrete mathematical analysis infancy nonetheless considering little attention given superintelligence control problem general optimistic potential translate analyses oais arisen safety community mathematical software frameworks modern artificial intelligence acknowledgements would like thank stuart armstrong david kristoffersson marcello herreshoff miles brundage eric drexler cristian calude several anonymous reviewers insightful discussions feedback manuscript would also like thank guest editors informatica ryan carey matthijs maas nell watson roman yampolskiy organizing special issue references bostrom superintelligence paths dangers strategies oxford university press shanahan technological singularity mit press sarma hay artificial intelligence future life institute russell dewey tegmark research priorities robust beneficial artificial intelligence magazine vol armstrong sandberg bostrom thinking inside box controlling using oracle minds machines vol fallenstein taylor christiano reflective oracles foundation game theory artificial intelligence logic rationality interaction springer armstrong value policy networks oracle preparation armstrong good safe uses oracles arxiv bostrom future progress artificial intelligence survey expert opinion fundamental issues artificial intelligence springer grace salvatier dafoe zhang evans exceed human performance evidence experts arxiv may rovelli quantum gravity scholarpedia vol russell fear supersmart robots scientific american vol eden moor soraker steinhart singularity hypotheses scientific philosophical assessment springer verlag chalmers singularity philosophical analysis journal consciousness studies vol amodei olah steinhardt christiano schulman concrete problems safety arxiv june tegmark open letter research priorities robust beneficial armstrong orseau safely interruptible submitted robust computer algebra theorem proving oracle ferrucci brown fan gondek kalyanpur lally murdock nyberg prager building watson overview deepqa project magazine vol knight ibm pushes deep learning watson upgrade mit technology review wolfram jeopardy ibm wolfram alpha stephen wolfram blog conference computer aided verification springer fix fifteen years formal property verification intel years model checking springer kern greenstreet formal verification hardware design survey acm transactions design automation electronic systems vol weinzierl computer algebra particle physics arxiv high energy physics phenomenology kropf introduction formal hardware verification springer science business media huang machine learning computer algebra tech university cambridge computer laboratory ballarin computer algebra theorem proving phd thesis university cambridge computer laboratory irving szegedy alemi chollet urban sequence models premise selection advances neural information processing systems kaliszyk wiedijk certified computer algebra top interactive theorem prover towards mechanized mathematical assistants springer komendantskaya heras grov machine learning proof general interfacing interfaces arxiv watt future computer algebra systems threshold proceedings bundy hutter jones moore meets formal software development dagstuhl seminar dagstuhl reports vol beeson mechanization mathematics alan turing life legacy great thinker springer klein elphinstone heiser andronick cock derrin elkaduwe engelhardt kolanski norrish formal verification kernel proceedings acm sigops symposium operating systems principles acm kaivola ghughal narasimhan telfer whittemore pandav taylor frolov reeber replacing testing formal verification intel coretm processor execution engine validation international windsteiger theorema system mathematical theory exploration international congress mathematical software springer bertoli calmet giunchiglia homann specification integration theorem provers computer algebra systems international conference artificial intelligence symbolic computation springer clarke zhao theorem prover mathematica international conference automated deduction springer dolzmann sturm redlog computer algebra meets computer logic acm sigsam bulletin vol jenks sutor axiom scientific computation system springer poll thompson adding axioms axiom tech computing laboratory university kent miner importance mathml mathematics communication notices ams vol buswell caprotti carlisle dewar gaetano kohlhase open math standard tech open math society calmet 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development ibm proceedings international conference software engineering ieee erdogmus effectiveness approach programming ieee transactions software engineering vol sarma jacobs watts ghayoomie larson gerkin unit testing model validation biological simulation vol keller matter trust skeptical communication coq external provers phd thesis polytechnique armand faure keller werner modular integration solvers coq proof witnesses international conference certified programs proofs springer harrison towards hol light international joint conference automated reasoning springer pollack believe machinechecked proof twenty five years constructive type theory vol hardy hadamard psychology invention mathematical field dehaene number sense mind creates mathematics oxford university press drijvers gravemeijer computer algebra instrument examples algebraic schemes didactical challenge symbolic calculators springer drijvers learning mathematics computer algebra environment obstacles opportunities zentralblatt didaktik der mathematik vol lakoff mathematics comes embodied mind brings mathematics basic books wolfram elementary introduction wolfram language wolfram media paulson foundation generic theorem prover journal automated reasoning vol paulson isabelle generic theorem prover vol springer science business media bertot interactive theorem proving program development coq art calculus inductive constructions springer science business media moura kong avigad van doorn von raumer lean theorem prover international conference automated deduction springer
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jun hyperbolic actions bounded cohomology subgroups part finite lamination subgroups michael handel lee mosher march abstract second part two part work prove every finitely generated subgroup either virtually abelian second bounded cohomology contains embedding part focus finite lamination subgroups meaning set attracting laminations elements finite construction hyperbolic actions subgroups general theory part applicable introduction second part two part work main theorem alternative second bounded cohomology finitely generated subgroups outer automorphism group free group finite rank theorem every finitely generated subgroup either virtually abelian embedded copy uncountable dimension background see part paper theorem reduced theorem main result paper theorem explains produce useful hyperbolic actions certain class finite lamination subgroups fully state theorem shall first define properties arise hypotheses consider group action hyperbolic space recall loxodromic action gromov boundary dynamics first author supported national science foundation grant grants program years second author supported national science foundation grant unique pair recall also nonelementary exist independent loxodromic elements meaning sets disjoint given loxodromic element say satisfies wwpd see proposition part ordered pair discrete subset space distinct ordered pairs let ian denote finite index normal subgroup consisting outer automorphisms whose induced action trivial subgroup ian virtually abelian restrictions definition proper free factor whose conjugacy class fixed element natural restriction homomorphism virtually abelian image explained part property ian virtually abelian restrictions plays role theory analogous role played irreducible subgroups mapping class groups theory bestvina fujiwara decomposition theory bestvina feighn handel associates finite set attracting laminations associated subgroup set finite finite lamination subgroup otherwise infinite lamination subgroup part proved two theorems application theorem namely theorem part infinite lamination subgroups virtually abelian restrictions actions free splitting complex wwpd elements theorem part general result determining second bounded cohomology group possessing hyperbolic action sufficiently rich collection wwpd elements also part combining theorem theorem reduced proof theorem following result main theorem present paper theorem finitely generated finite lamination subgroup ian abelian virtually abelian restrictions exists finite index normal subgroup action hyperbolic space following hold every element acts either elliptically loxodromically action nonelementary every loxodromic element commutator subgroup satisfies wwpd respect action remarks wpd versus wwpd lectures topic stated stronger version theorem saying group image isom loxodromic elements commutator subgroup satisfy wpd requires stronger hypothesis saying roughly virtually abelian restrictions holds free factors broader class subgroups also makes proof application theorem considerably intricate interests keeping paper growing ever longer settled version theorem presented methods proof theorem first step theorem reduce statement subgroups automorphism groups free groups stated theorem key step reduction proved proposition section construction automorphic lift satisfying hypotheses theorem exists free factor rank conjugacy class natural homomorphism lifts homomorphism aut whose image aut virtually abelian section shall show minimizing rank free factor reduce theorem following statement identified rewritten recall canonical isomorphism inn associates inner automorphism aut denote theorem consider subgroup kernel inn giving following image commutative diagram short exact sequences inn aut finitely generated virtually abelian suppose abelian set proper nontrivial free factor fixed action subgroups exists finite index normal subgroup action hyperbolic space following properties hold every element acts either elliptically loxodromically action nonelementary every loxodromic element satisfies wwpd respect action proof theorem begins section assuming ian may replacing intersection ian note conclusion theorem holds holds finite index subgroup consider maximal proper free factor system proof breaks cases depending number minimum integer represented subgraph marked graph complement edges case number equals handled section using action simplicial tree naturally associated free factor system case number takes majority paper section end full introduction case see section brief one uses dynamics strata produce certain hyperbolic suspension space applying combination theorem prove hyperbolicity construction including flaring properties needed apply combination theorem found sections construction action based abelian subgroup methods found sections pieces put together case theorem proved section prerequisites theory assume reader familiar certain basic concepts already reviewed part paper particular section marked graphs topological representatives free factor systems relative train track maps attracting laminations section properties ian kernel section number free factor system elements subgroups fully irreducible relative free factor system needed paper conduct reviews basic concepts contents introduction lifting automorphism group definition automorphic lifts sufficient condition abelian constructing automorphic lifts proof proposition proof theorem implies theorem automorphic extensions free splitting actions case analysis theorem proof case introduction case theorem outline case motivation suspension actions combination theorems flaring top stratum path functions lpf flaring path functions lpf negative flaring positive flaring properties proof lemma special flaring condition appendix graph homotopy principle flaring hyperbolicity free splitting role flaring piecewise riemannian metric path functions constructing coning nielsen axes geometry dynamics construction proof hyperbolicity abelian subgroups background review cts principal automorphisms rotationless outer automorphisms rotationless abelian subgroups disintegration subgroups almost invariant subgraphs admissible families paths disintegration subgroup coordinate homomorphism disintegration group train track semigroup action homotopy semigroup action semigroup actions dynamics group action suspension action suspension action proof theorem case lifting automorphism group section first thing study structure finitely generated finite lamination subgroups ian abelian virtually abelian restrictions motivating question study fixes conjugacy class proper nontrivial free factor natural restriction map lifted homomorphism aut sections devoted constructions automorphic lifts using construction section prove implication theorem theorem section consider free splitting study natural subgroup aut free splitting action extends study used section prove one two major cases theorem definition automorphic lifts recall fact group subgroup normalizer free factor letting stab stabilizer conjugacy class natural restriction homomorphism stab denoted choosing aut representing preserving taking outer automophism class aut throughout paper use theorem virtually abelian subgroups ian abelian sometimes write virtually abelian reminder one may freely include ignore adverb virtually front adjective abelian context subgroup iaf finite rank free group one freedom example following definition see corollary definition virtually abelian restrictions subgroup ian virtually abelian restrictions proper free factor stab restriction homomorphism virtually abelian image definition let ian finitely generated finite lamination subgroup virtually abelian virtually abelian restrictions automorphic lift homomorphism aut proper image virtually free factor stab group abelian following triangle commutes aut diagram horizontal arrow natural restriction homomorphism stab domain restricted vertical arrow natural quotient aut inn figure notation associated automorphic lift aut image virtually abelian quotient virtually abelian group kernel free rank possibly infinite horizontal rows exact homomorphism emphasize role sometimes refer automorb image phic lift rel adopting notation theorem set image kernel inn image thus obtaining commutative diagram shown figure note two properties follow definition abelian free group rank first holds proper virtually abelian restrictions see definition second consequence first combined virtually abelian otherwise free group defining requirement virtually solvable aut injects solvable abelian subgroups virtually abelian put conditions rank could even infinite referring elements thought ambiguously element free factor corresponding element inner automorphism group inn ambiguity cause little trouble using canonical isomorphism inn given rank automorphic lift aut defined rank note rank otherwise aut finite case subgroups virtually abelian completes definition first two main results section proposition ian finitely generated finite lamination subgroup virtually abelian virtually abelian restrictions exists automorphic lift aut proof found section preceded lemma section one recovers proposition simple fact every finite lamination subgroup virtually abelian otherwise intersection ian would automorphic lift aut proper free factor would follow rank course fact simple proof expressed terms isomorphism aut leave reader sufficient condition abelian section prove lemma gives sufficient condition finitely generated finite lamination subgroup ian abelian negation condition becomes property must hold abelian lemma let ian finitely generated finite lamination subgroup maximal proper free factor system restriction abelian number abelian proof theorem exists fully irreducible rel relative train track theory unique lamination pair carried nonattracting subgroup system one two forms either ana ana maximal infinite cyclic group carried see definition every nonperiodic line carried evidently carried carried ana applying theorem follows weakly attracted either iteration iteration follows two laminations unique elements carried existed attracting lamination supported generic lines would weakly attracted either iteration iteration either case set laminations would form infinite set contradicting finite lamination group since finite element finite orbit action since ian follows lemma element fixed particular stab corollary exists homomorphism stab property holds stab inequality consider following homomorphism range abelian claim kernel also abelian claim completes proof lemma every solvable subgroup virtually abelian every virtually abelian subgroup ian abelian prove claim proposition every subgroup kernel consisting entirely upg elements abelian need check element kernel upg corollary every element ian upg need check kernel equivalently suppose contrary exists dual repelling lamination since trivial component neither laminations supported since contrait follows shown diction constructing automorphic lifts proof proposition let ian finitely generated finite lamination subgroup virtually abelian virtually abelian restrictions choose maximal proper free factor system restricted group denoted abelian since ian follows component fixed lemma group abelian applying lemma extension extension may therefore choose marked graph pair representing single edge may choose components roses endpoint rose vertex component number components equals number components number either one two cover cases separately case two components say construct commutative diagram follows aut aut aut aut subgroup aut preserves homomorphism induced restricting aut aut aut evidently injective since automorphism determined restrictions complementary free factors homomorphisms denoted top bottom arrows diagram induced canonical homomorphisms automorphism groups outer automorphism groups homomorphism composition stab latter map natural restriction homomorphism must construct may choose marked graph pair representing free factor system following properties two rose components subgraph ranks equal rank rank respectively edge oriented decomposes oriented common initial point denoted respective terminal vertices rose vertices isomorphism restricts isomorphisms given let homotopy equivalence represents preserves restricts locally injective path corollary possibly trivial paths closed paths fact plus sign occurs ian homeomorphism isotopic identity restricts identity moves point may also assume define aut aut automorphism induced use isomorphism define note unique lift aut preserves two lifts differ inner automorphism preserves implying malnormality trivial follows uniqueness homomorphisms hence commutativity diagram straightforward construction homomorphism injective lift inclusion since injective commutativity diagram also injective least one two maps aut automorphic lift image aut virtually least one corresponding images would virtually abelian virtually abelian abelian injective would virtually abelian contradiction case single component say hbi proof case similar case main differences place direct sum use fiber sum marked graph pair representing connected subgraph shall construct following commutative diagram aut diagram use following notations conjugate restricted inner automorphism adjoint isomorphism adb aut aut adb canonical epimorphism aut epimorphism qab aut also fiber sum qab following subgroup aut aut aut aut qab define homomorphism qab define aut aut subgroup preserves homomorphism jointly induced two restriction homomorphisms rab aut aut aut two compositions qab rab aut evidently note injective aut restricts identity follows commutes since rank hence trivial construct homomorphism may case choose marked graph pair representing rose whose rank equals rank common initial point terminal endpoints equal rose vertex isomorphism restricts follows applying corollary previous case find aut represents represented homotopy equivalence preserves fixes takes possibly trivial paths based map homomorphism unique element aut represents inner automorphism preserves malnormality malnormality trivial follows injective since injective follows injective denote obviously compositions qab hence induced homomorphism topologically represented completes construction diagram commutativity evident previous case done show least one two homomorphisms aut aut image virtually abelian virtually abelian image contained virtually abelian subgroup injectivity implies virtually abelian contradiction proof theorem implies theorem let ian finitely generated finite lamination subgroup virtually abelian virtually abelian restrictions applying proposition exists free factor stab exists automorphic lift aut may assume rank minimal amongst choices adopt notation figure section matching notation theorem choosing isomorphism rank image aut hypotheses theorem immediate finitely generated virtually abelian definition automorphic lifts abelian also image must check one remaining hypothesis theorem namely assuming proper nontrivial free factor preserved action consider restriction homomorphism contradiction preserved aut claim composition aut automorphic lift since rank rank claim proved contradicts assumption aut automorphic lift minimal rank canonical isomorphism inn denoted restricts isomorphism inn subgroup preserves since malnormal follows thus restricts injection inn also group free group rank would virtually solvable hence trivial cyclic virtually abelian contradiction since image map aut preserves contains follows image virtually abelian since class since malnormal follows preserves class tracing definitions one easily sees composed homomorphism aut equal composition stabout latter map natural restriction homomorphism thus aut automorphic lift completing proof claim applying conclusions theorem using free group hyperbolic action aut obtain finite index normal subgroup satisfying conclusions theorem subgroup finite index normal subgroup composition action theorem element acts elliptically loxodromically holds element theorem theorem action nonelementary holds action theorem since image abelian follows image homomorphism commutator subgroup contained kernel hence image contained theorem loxodromic element wwpd element respect action corollary says wwpd preserved pullback follows every loxodromic element satisfies wwpd respect action theorem automorphic extensions free splitting actions hypothesis theorem interest transferred context finite rank free group perhaps identified isomorphically free factor aut following higher rank free group subgroup irreducibility property proper nontrivial free factor preserved hyperbolic spaces prove theorem one needs actions groups section focus natural situation produces actions trees free splittings recall free splitting minimal simplicial action simplicial tree stabilizer edge trivial two free splittings simplicially equivalent exists simplicial isomorphism sometimes use notation simplicial equivalence class free splitting formally action given homomorphism isom denote briefly formal notation isom refers group simplicial equivalently group using geodesic metric given barycentric coordinates simplices note element isom determined restriction vertex set fact determined restriction subset vertices valence two free splittings equivalent simplicial isomorphism equivalence class free splitting denoted group acts naturally set equivalence classes free splittings given free splitting set conjugacy classes nontrivial vertex stabilizers free splitting free factor system called vertex group system denoted function assigns free splitting vertex group system induces function set simplicial equivalence classes free splittings set free factor systems every free splitting realized marked graph pair sense quotient universal cover component total lift collapsed point one may also assume component noncontractible case marked graph pair topologically represents vertex group system twisted equivariance functional notation given two free splittings automorphism aut map said equivariant special case simply called equivariance twisted equivariant map behaves well respect stabilizers shown following simple fact lemma free splittings aut equivariant map stab stab addition map simplicial isomorphism inclusion item equality stab stab furthermore acts loxodromically acts loxodromically case axes satisfy remark one could approach proof first working equivariant case reducing twisted case equivariant case conjugating action using instead give direct proof proof prove stab stab twisted equivariance stab stab stab prove consider inverse simplicial automorphism implication second line may inverted applying sides equation second line rest suffices prove direction direction proved using equivariant assuming loxodromic axis consider line calculating exactly one shows equation stab stab holds may assume generator infinite cyclic group stab stab since stabilizer line infinite cyclic group follows loxodromic axis equal free splittings number recall number free factor system see example section minimal number edges amongst marked graph pairs representative tight relationship free splittings single edge orbit free factor systems number precise fact section function restricted free splittings one edge orbit free factor systems number result bijection hence stab stab whenever correspond bijection bijection fact number components equals number vertex orbits equals single component rank free factorization rank quotient graph groups circle one edge one vertex hand two components rank rank chosen conjugacy classes free factorization quotient graph groups arc one edge two vertices stabilizer free splitting automorphic extension consider free splitting stabilizer subgroup stab let aut preimage stab standard projection homostab morphism aut short exact sequence stab inn stab setup shall define natural way action stab extends given free splitting action proceed follows aut composed action assuming words representative element addition stab stab follows actions equivalent meaning exists simplicial automorphism isom equation rewritten action notation simply says satisfies equivariance suppose conversely aut exists equivariant isom since conjugates action action follows proves equivalence stab stab following lemma also contains uniqueness information regarding conjugating maps lemma free splitting following equivalent stab aut representing exists equivariant isomorphism aut representing exists isomorphism furthermore satisfies equivalent conditions stab representing equivariant isomorphism uniquely determined denoted corresponding inner automorphism two maps equal remark item note defined inn stab uniqueness statement special case general uniqueness statement make use later lemma two free splittings aut exists one simplicial isomorphism particular taking exists one equivariant simplicial isomorphism proof suppose equivariant simplicial isomorphism exists follows equivariance acts loxodromically axis acts loxodromically axis lemma map induces set axes loxodromic elements therefore uniquely determined follows restriction set vertices valence uniquely determined may expressed form certain choice hence since uniquely determined restriction vertices valence follows uniquely determined amongst simplicial isomorphisms proof lemma uniqueness clause immediate consequence lemma item follows uniqueness clause map clearly satisfies twisted equivariant isom defined consider function stab given lemma defines action stab sides action equation clearly satisfy stab twisted equivariance hence equation holds application uniqueness clause lemma following lemma summarizes discussion together evident geng rewrites twisted equivariance property eralization subgroups stab using action notation instead functional notation lemma associated free splitting unique isometric assigns stab unique simplicial isomoraction stab phism stated lemma denoted action notation satisfying following twisted equivariance action notation aut action stab generally restriction subgroup stab unique isometric action satisfies twisted equivariance remark twisted equivariance equation action dot used ambiguously action action aut meaning particular action dot clear context furthermore contexts two meanings overlap always agree example lemma says action dots always respect standard isomorphism inn given next lemma key step proof loxodromic wwpd portions theorem see remarks statement brief given free splitting certain subgroup lemma gives criterion verifying restriction free splitting action certain group nonelementary understand statement lemma reader may note applying lemma entire setup first paragraph stab lemma satisfied free splitting subgroup lemma let free splitting vertex set let aut subgroup subset stab nontrivial let inn let action normal subgroup map equivariant subgroup stabfn fixed whole group free group rank action nonelementary remarks lemma applied section prove oneg stab edge case theorem first case group assumptions first paragraph lemma hold automatically lemma lemma also applied section prove case theorem place lemma applied case group aut contained stab act nonetheless kind enough give action satisfying assumptions first paragraph lemma proof action trivial edge stabilizers restriction free splitting action clearly also point trivial stabilizer follows nontrivial either elliptic fixed point set fix single point loxodromic pair stabj stabj stabj infinite cyclic group claim point fixed every element otherwise point unique stabilizer stabfn unique nontrivial vertex stabilizer fixed action bijection also fixes lemma follows fixes also nontrivial vertex stabilizer namely stabilizer uniqueness therefore since contradicted hypothesis lemma thus proving holds claim proof action nonelementary follows standard argument loxodromic otherwise applying claim follows nontrivial elliptic elements fix fix case loxodromic see example proposition since rank exists stabj follows independent loxodromic elements case analysis theorem proof case adopting notation theorem may assume ian choose maximal proper free factor system fully irreducible relative extension meaning free factor system strict nesting invariant finite index subgroup equivalently since ian invariant proof theorem breaks two cases case number equals case number proof theorem case assuming extension using stab applying fact exists free splitting one edge orbit whose vertex stabilizer system forms free factor system equality stabilizer subgroups stab stab follows stab applying lemma consider resulting action show restricted action satisfies conclusions stab since every isometry tree either theorem using elliptic loxodromic conclusion theorem holds proving conclusion theorem wish apply lemma must check hypotheses assumptions first action stab paragraph lemma hold applying lemma subgroup aut preserve also hypothesis theorem subgroup proper nontrivial free factor particular preserve nontrivial vertex stabilizer free splitting finally using hypotheses virtually abelian follows theorem saying abelian rank otherwise group aut would solvable hence virtually abelian contradiction conclusion lemma therefore holds saying restricted action nonelemenb therefore nonelementary verifies conclusion tary action theorem conclusion theorem says loxodromic element wwpd immediate consequence next element action lemma also used case theorem lemma group action simplicial tree normal subgroup restricted action trivial edge stabilizers loxodromic element wwpd element action proof work simplicial metric assigns length edge given oriented line let stabilizers actions denoted stabg stabj stabg consider loxodromic let denote axis oriented translates positive direction endpoints since trivial edge stabilizers group stabj stabj stabj infinite cyclic prove satisfies wwpd respect action suffices prove ordered pair isolated point orbit proposition may assume generator infinite cyclic group stabj already true without affecting wwpd property may replace generator letting denote integer valued length fundamental domain action follows edge path length fundamental domain stabj choose subsegment length corresponding neighborhood consisting endpoint pairs oriented lines containing oriented subsegment consider follows axis contains hence length also map takes restricted orientation subsegment restricted orientation subsegment let edges oriented segment parameterized since translation number follows stabj since trivial edge stabilizers follows stabj shown preserves orientation follows shows isolated orbit unique element intersection completes proof theorem case introduction case theorem proof theorem case take rest paper section give broad outline methods proof followed motivation coming constructions geometric group theory outline case make heavy use theory abelian subgroups developed feighn handel brief outline main features theory need disintegration groups see section element uniformly bounded power rotationless meaning roughly various natural finite permutations induced element trivial rotationless particularly nice relative train track representative called associated abelian subgroup contains called disintegration subgroup idea disintegration group first disintegrate decompose pieces one piece stratum equal identity elsewhere one pieces form generators group choosing list exponents one per stratum composing associated powers pieces however order composition continuous homotopy equivalence abelian exponents list chosen independently instead two constraints imposed strata partitioned collection almost invariant subgraphs exponent must constant certain linear relations required amongst strata wrap around common twist path following key theorem disintegration groups lets study abelian subgroup finite index working entirely appropriate disintegration group disintegration theorem theorem every rotationless abelian subgroup exists every representing intersection finite index proof theorem case detailed proof carried section based material whose development soon commence sketch finite index argument may assume abelian group rotationless abelian let maximal proper free factor system case means number disintegration theorem obtain apply conclusions theorem representative proper core filtration element representing may assume replacing finite index subgroup construction disintegration group core filtration element properly contained would represent free factor system hence contradicting maximality thus maximal proper core filtration element since number top stratum stratum maximality every stratum strictly either edge terminal endpoint attached zero stratum enveloped hard work proof breaks two major phases first sections construction hyperbolicity hyperbolic metric space needed verifying conclusions theorem constructed terms see section motivation construction first describe simpler case height indivisible nielsen path starting lift universal cover let total lift collapse point obtain induces component obtaining simplicial tree map map let mapping cylinder obtained identifying construction complex geometric meaning unique inversion height indivisible nielsen path closed obtained forming circuit see line lifting projects line called geometric axis projections lifts fundamental domains axis mapping cylinder defined portion mapping cylinder corresponding geometric axis quasiflat contradicting hyperbolicity thus take mapping cylinder instead constructed mapping cylinder coning geometric axis using one cone point per geometric axis attaching arcs connect cone point endpoints fundamental domains axis section use combination theorem prove hyperbolicity theorem generalization combination theorem requires verify flaring hypothesis section study relative flaring properties specifically flares relative lower filtration element geometric case flares relative nielsen path section study flaring properties induced map geometric case flaring properties induced map obtained coning geometric axis required flaring hypothesis verified section allowing application theorem deduce hyperbolicity major phase proof sections use theory disintegration groups obtain isometric acb appropriate wwpd elements tion one uses number homomorphism lamination attracting lamination case stretch factor one would like think stratum eigenvalue yet make sense yet appropriate topological representative define subgroup kernel lift sequence inclusions sequence inclusions aut hard work section use theory disintegration groups construct element natural action semigroup acts stretching edge uniform factor equal resulting map equivariant obtain action twisted equivariant restricted subgroup one described isometries allows identify action lemma section suspend semigroup action geometric case graph obtained coning geometric restriction obtain required axes obtaining isometric action action finally section put pieces together verify conclusions theorem basis wwpd conclusions case theorem lemma already played role case motivation suspension actions combination theorems may motivated construction hyperbolic suspension action looking familiar examples somewhat unfamiliar way example mapping torus hyperbolization thurston consider homeomorphism closed oriented hyperbolic surface genus associated deck action map uniquely determines outer automorphism associated extension group aut inverse image infinite cyclic group natural homomorphism aut choice aut representing naturally determines semidirect product structure deck transformation action extends naturally action whereby projects action denoted lift whose induced action circle infinity agrees induced action equivalently unique equivariant lift although deck action isometries extended action isometries however way suspend action obtaining isometric action follows suspension space mapping cylinder obtained quotient identifications one may check shall section different context action generated letting act deck group extending naturally rest letting according formula one may also check hyperbolic metrics slices extend path metric uniquely action isometries returning land familiar group may identified fundamental group mapping torus surface homeomorphism thurston hyperbolization theorem applied suspension space may identified universal cover equipped deck transformation action group example mapping torus hyperbolization consider irreducible train track representative nongeometric fully irreducible outer automorphism natural extension group aut semidirect product structure determined choice aut representing unlike previous situation started surface homeomorphism started topological representative extend action deck action extend action semigroup inverse image aut semigroup mapping unique equivariant associated map isometries semigroup action lift although deck action isometries isometric action way suspend semigroup action defined exactly mapping cyclinder namely quotient space appropriate metric done theorem apply properties train track map prove flaring hypothesis allowing one apply combination theorem conclude gromov hyperbolic thus proving hyperbolic group flaring methods proof case use combination theorem sarder generalization bestvina feighn combination theorem common feature examples shared construction paper metric bundle lipschitz projection map minimum distance equals point contained image geodesic section map studying large scale geometry important study quasigeodesic sections flaring properties context convenient translate concepts dynamical systems parlance quasigeodesic section turns suspension semiflow whose true orbits geodesic sections see closing paragraphs section combination theorems predecessor share key hypotheses regarding asymptotic flaring behavior see definition remark though combination theorems hold much general settings certain kinds metric bundles general metric base spaces section study flaring context relative train track map used section extend contexts building suspension space metric bundle allowing application combination theorem flaring top stratum assume reader familiar basic concepts laid section first part paper particular terminology notation top stratum occurring heading properties cts brief review order fix notations follows rest paper see section detailed citations drawn primarily present need cts described needed section give thorough review cts notations fix relative train track representative associated filtration satisfying following top stratum transition matrix top eigenvalue attracting lamination corresponding denoted dual repelling lamination exists reversal one indivisible periodic nielsen path height meaning contained case inverse nielsen paths following hold decomposes paths endpoints vertices unique illegal turn least one endpoint disjoint assume oriented initial point terminal point initial terminal directions distinct fixed directions stratum geometric exists closed path case matrix one obtains eigenlength function lpf defined paths endpoints vertices following properties edge lpf general lpf lpf lpf edges lpf lpf exists lpf lpf lpf path endpoints vertices circuit crossing edge sufficiently large path splitting terms set edges paths endpoints follows three cases consider ageometric case exist parageometric case exists closed geometric case exists closed occasionally need consider cases separately part attempt smoothly handle three cases simultaneously using following conventions ageometric case exist notation simply ignored exception see notational convention following corollary parageometric case exists closed notations ignored completes notations main result section proposition uniform flaring property top stratum result generalizes theorem lemma covers special case nongeometric train track map exists closed two features distinguish present general situation special case first flaring property formulated relative penultimate filtration element second height indivisible nielsen path exists flaring property formulated relative taken together two features require careful isolation annihilation metric effects paths path task particularly tricky geometric case exists closed situation must also annihilate effects iterates path functions lpf throughout paper following standard terminology literature relative train track maps finite path graph definition locally injective continuous function often restrict requiring endpoints vertices case concatenation edges without backtracking general concatenation edges locally injectivity may fail hence backtracking may occur called edge path generalizing eigenlength function lpf notations path function graph simply function assigns finite path endpoints vertices number subject two properties symmetric meaning assigns length zero trivial path require nontrivial paths also generally require version triangle inequality see lemma preceding discussion regarding coarse triangle inequalities path functions interest henceforth section adopt adopt notations big path functions lpf defined terms auxiliary little path functions lpf omitting certain terms function lpf already defined notations consider path edge path decomposition define number terms contained define lpf summing lpf respectively terms occur subpath note edge little eigenvector equation lpf lpf implies big eigenvector equation lpf lpf edges see holds sides zero follows fact hence subpath following quasicomparability result obvious consequence fact eigenvector transition matrix positive entries lemma constant edges lpf lpf next lemma says part subpaths never overlap lemma isolation path endpoints vertices unique decomposition following properties term edge copy every subpath copy term decomposition note assume finite statement leaving proof lemma end section give applications path endpoints vertices let denote number subpaths lemma decomposition corollary formulas lpf finite path endpoints vertices letting lemma decomposition general exist similarly lpf lpf lpf general lpf exist notational convention exist motivated corollary extend meaning notations lpf case exist defining lpf corollary finite path endpoints vertices unique decomposition following properties exist exists iterate iteration exponent equals closed contains subpath exists least one subpaths contains edge addition closed subpath contains edge note context corollary subpaths nondegenerate subpaths allowed degenerate proof everything immediate consequence lemma except perhaps item follows fact least one endpoint disjoint notations following immediate consequence lemma corollary lemma applied furthermore part corollary formulas lpf finite path endpoints vertices using corollary decomposition lpf lpf lpf furthermore letting lpf lpf latter inequality corollary gives freedom switch back forth lpf using combinatorial situations lpf geometric situations much rest paper coarse triangle inequalities lpf path functions lpf satisfy version triangle inequality two paths endpoints vertices terminal vertex equals initial vertex lpf lpf lpf denotes operation straightens edge path rel endpoints obtain path lpf best get coarse triangle inequalities lemma exists constant finite paths endpoints vertices terminal point coincides initial point lpf lpf lpf proof consider lemma decompositions may write following hold concatenation terms lemma decomposition either degenerate initial subpath single term decomposition either degenerate terminal subpath single term lemma decomposition concatenation terms decomposition follows similarly lpf lpf lpf properties next lemma describes properties relative train track map respect path functions lpf lemma exist constants finite path endpoints vertices following hold lpf lpf lpf lpf proof found satisfying applying corollary find satisfying maximizing obtain satisfying exists let endpoint set single point geometric two points otherwise exist let point fixed elementary homotopy theory see lemma section map pairs homotopy inverse category topological pairs homotoping rel vertices may assume takes vertices vertices edges possibly degenerate paths exists using applying sides obtain prove following general fact homotopy equivalence pairs takes vertices vertices takes edges paths assuming exists exist path endpoints vertices obviously suffices first inequality taking also suffices second inequality taking case prove consider corollary decomposition applying corollary also inductive application lemma combined fact follows maximum number edges crossed edge exist done exists corollary least one follows therefore proof lemma lemma equivalent saying two distinct subpaths overlap edge lifting universal cover subpaths equivalent saying path projects either least one edge common first case assume project project reversal may assume former consider decomposition legal turn illegal contained induced decompositions intersection contains height illegal turn assume interchanging may assume projecting see projection implies initial directions respectively equal terminal initial directions fixed distinct initial directions also fixed distinct using eigenlength function lpf notations lpf lpf lpf lpf lpf lpf thus path tightened form large subpaths cancel concatenation tightens contradicts two terms concatenation turn taken concatenation point also seen remaining case orientations projections agree case either initial terminal subpath inverse impossible flaring path functions lpf section continue adopt notations path function define relation paths endpoints vertices relation means exist paths endpoints vertices note relation symmetric need emphasize dependence relation write formally definition path function said satisfy flaring condition respect exist integers sequence paths endpoints vertices flaring inequality max holds following two properties hold threshold property property remark two properties translations present context two requirements original treatment flaring bestvina feighn threshold property corresponds requirement large girth pseudoorbit property corresponds requirement proposition general flaring path functions lpf satisfy flaring condition respect constants much work prove following version proposition special situation definition lpf note implies choice determines whole sequence lemma special flaring exist positive integers finite path endpoints vertices max proof special flaring condition takes sections similar spirit theorem based two special cases negative flaring result lemma section positive flaring result lemma section two special cases united section using uniform splitting property given lemma order prove special flaring condition embarking apply lemma proof proposition applying corollary easy see satisfies flaring condition lpf satisfies flaring condition thus suffices assume special flaring condition holds use prove general flaring condition idea proof standard exponential growth given proposition swamps constant error represented relation denote shorthand hard work carefully keep track various constants notations fixing consider relation given formally fix integer whose value determined application lemma arranged set equal constant lemma consider sequence paths endpoints vertices choose vertex say integer may assume endpoints paths uniformly bounded length endpoints replacing reduces flaring property stated flaring property uniform changes constants choose several paths endpoints vertices denoted follows first using exist hence next anticipating application lemma choose let hence finally choose hence particular trivial paths also require chosen record following identities see choices possible work lift universal first choose lift induction lifts cover determined equation using initial terminal vertices contain vertices amongst choose initial terminal vertices respectively follows define define path initial vertex initial vertex define path terminal vertex terminal vertex projecting obtain paths identities follow evident fact paths may concatenated form path two paths initial endpoint two paths terminal endpoint initial endpoint similarly need new upper bounds quantities represent difference bounds expressed terms known upper bound derived applying applying lemmas inductively starting applying induction step new upper bounds used derive expression threshold constant large differences become insignificant new upper bounds form expression represents certain polynomial integer coefficients variables substitute proofs inequalities almost identical carry proof latter start induction since path degenerates point inducting forward direction interval using inducting backward direction interval using summarize proved follows similarly let max use setting value threshold constant apply lemma special flaring condition constant obtain integers setting threshold requirement follows max max additional threshold requirement max max thus proved general flaring condition given using max negative flaring continue adopt notations context lemma special flaring next lemma establishes flaring negative direction path define maximum paths subpath leaf realized definition may always assume generic leaf every finite subpath every leaf subpath every generic leaf notice already subpath leaf lemma exists exists integer following holds finite subpath generic leaf realized endpoints vertices finite path intuitively least nongeometric case see proof proposition page point lemma leaf illegal turns uniform density respect path function turns die exponential rate iteration proof proof height illegal turn path said play illegal turn subpath otherwise turn play generic leaf realized birecurrent periodic weakly attracted iteration follows split concatenation paths copies applying lemma follows illegal turn play addition quasiperiodic respect edges lemma integer exists integer finite subpaths contains subpath copy follows exists integer subpath least three height illegal turns play arguing contradiction lemma fails using value given exist positive integers finite paths endpoints vertices paths subpath derive contradictions separate cases case upper bound decompose either partial edges trivial path endpoints vertices lemma exists positive integer depending path splitting terms either single edge copy subpath therefore height illegal turn interior play paths height illegal turn play since obtained tightening two illegal turns play contradiction choice case upper bound consider line weak limit subsequence crosses least one edge apply weak attraction results follows two options either closure contains weakly attracted lemma geometric height closed indivisible nielsen path exists third option namely iterate lemma since contains subpath leaf crosses edges neither shows closure contain weakly attracted would exist hence sufficiently large contains subpath crosses edges lemma choose splits endpoints middle edge subpath hence number edges crossed goes infinity contradicting shown first two options leads contradiction choice concludes proof nongeometric remains show geometric third option also avoided careful choice hence desired contradiction achieved show exists weak limit subsequence contains least one edge iterate closed path may done setting application lemma simple give direct proof lift universal cover write edge eij first last terms allowed partial edges path let equal twice number edges given say eij well covered full edges periodic line eij subpath since projects contains intersection distinct periodic lines contain two fundamental domains eij well covered lines unique exists moreover eij well covered inductively move follows forward backward past well covered edges lift either encountering edge well covered encountering initial terminal subsegments uniform length passing subsequence one following therefore satisfied exists sequence integers eik well covered number edges crossed bounded independently subcase holds existence weak limit crosses edge iterate follows immediately subcase holds obtained tightening since number illegal turns height first last terms uniformly bounded number edges cancelled tightening uniformly bounded follows contains subpath contradiction fact leaf positive flaring properties section continue adopt notations context lemma special flaring item lemma establishes flaring positive direction lemma also includes several useful properties used section tie together negative positive flaring results prove lemma lemma applications make use path map properties laid section particularly lemma lemma roughly speaking path defined finite path follows finite path contains straightened image contains subpath obtained deleting initial terminal subpaths longer bounded cancellation constant path longest subpath survives deletion operation choices lemma following conditions hold path endpoints vertices decomposition subpaths endpoints vertices similarly lpf lpf place respectively positive integers exist finite paths endpoints vertices exist constants subpaths leaves exists positive integer constant path splits concatenation paths copies expansion factor remark notation disambiguated requiring exponent binds tightly hence items applied follows note makes statements weaker intepreted subpath proof prove proof general follows easy induction statement lpf proved exact manner decompositions copy contribute contribute contribute case decompositions concatenating decompositions given lemma produces decomposition given lemma completes proof next prove fix integer path obtained removing initial terminal segments cross number edges bounded independently bounded cancellation constant fold lows bounded independently suffices find depending blu applying lemma obtain finite paths endpoints vertices follows induction first inequality follows immediately corollary subpath leaf prepare proving second inequality given positive integer let set paths contain subpath form since endpoint contained fact maximal subpath form neither initial terminal adjacent contributes applying corollary follows fixed amongst paths ratio positive lower bound independent proving key observation exists positive integer subpath contained equivalent saying weak limit lines equivalent saying leaf follows lemma fact item therefore follows combining observation proving focus primarily analogue lpf connecting version stated applying corollary assumption splitting lpf lpf shall show replace expense replacing square root requiring lpf exceed threshold constant precise claim exists lpf lpf lpf suffices prove corollary follows lpf using claim obtain lpf lpf two applications corollary obtain lpf lpf prove claim case evident suppose induction lpf lpf since subpath since latter splits terms edge copy path follows may deconcatenated form splits terms exactly either trivial exists proper subpaths follows lpf lpf lpf applying item follows lpf lpf lpf lpf using splitting obtain lpf lpf lpf path obtained truncating initial terminal segments longer bounded cancellation constant since finite number paths lpf finite upper bound applying item follows lpf lpf lpf apply lemma since subpath follows subpath lemma subpath lemma thus paths hence item lpf lpf lpf lpf complete induction show appropriate threshold constant quantity right lpf equivalently lpf lpf lpf lpf since suffices show lpf lpf taking threshold constant induction complete proof lemma special flaring condition proof complete proof general flaring condition stated proposition also complete shown section special flaring condition fails exists sequence exist paths endpoints vertices max assuming argue contradiction consider integer satisfying conclusions lemma lemma integer subpath crosses edges let max choose integer satisfying conclusion lemma uniform splitting lemma respect constant conclusion says finite path endpoints vertices path splits terms copy context shall refer uniform splitting exponent let maximal collection subpaths endpoints vertices disjoint interiors subpaths cross edges complementary subpaths satisfy endpoints vertices well first claim lim passing subsequence may assume choose subpaths disjoint interiors since since may apply negative flaring lemma obtain subpaths endpoints vertices also subpaths lim ratios positive upper lower bounds independent upper bound follows corollary lower bound comes lemma therefore obtain lim using limit using follows sufficiently large first inequality follows applying lemma subdivision paths complementary subpaths contradicts verifying first claim second claim constant constraints placed claim applied lim see let number subpaths let number subpaths let lemma applied obtain follows multiplying inequality using follows multiplying inequality follows proves second claim follows applying lemma use constants involved statement definition choice uniform splitting exponent since follows splits terms either copy consider constants lemma constraining may combine lemma obtain iii sufficiently large values construction paths occur order subscript subpaths disjoint interiors applying lemma lemma occur order subpaths using follows paths path disjoint interiors applying lemma follows putting together see subpath paths occur order subpaths path disjoint interiors subpaths number together complementary subpaths one decomposition paths ignoring complementary subpaths lemma therefore implies last inequality follows sufficiently large applying inequality conclude sufficiently large therefore already constrained put one constraint applying lemma follows max follows allows apply positive flaring lemma conclusion letting long long sufficiently large combining sufficiently large obtain combining iii obtain second inequality holds sufficiently large gives final contradiction completes proof lemma appendix graph homotopy principle lemma section used earlier proof lemma used later construction homotopy semigroup action section elementary result homotopy theory precision state result language category theory give complete proof define category subcategory standard homotopy category pairs follows objects pairs finite graph finite subset morphism denoted homotopy class rel homotopy equivalence restricts bijection define fundamental group functor category category indexed groups follows pair associate indexed family groups morphism associate indexed family group isomorphisms category functor axioms implicit discussion easily checked let autgp denote group automorphisms category let autgp aut call pure automorphism group category denote finite index subgroup consisting autgp identity lemma graph homotopy principle category groupoid every morphism inverse morphism meaning homotopic identity rel homotopic identity rel fundamental group functor faithful pair morphisms restricted maps equal induced isomorphisms equal two morphisms inverses restrictions inverses isomorphisms inverses fundamental group functor restricts injective homomorphism defined pure automorphism group autgp aut proof proved follow immediately prove direction obvious direction suffices prove special case fixes induces identity homotopic identity rel proof know freely homotopic identity map idg space induces identity fundamental group choose homotopy idg claim closed path trivial applying claim alter homotopy follows using homotopy extension property may homotope map keeping stationary stationary stationary outside arbitrarily small neighborhood arrange note homotopy stationary means restricted map constant independently obtain homotopy rel identity done subject claim prove claim consider closed path based representing arbitrary element obtain path homotopy path concatenated path follows since follows center hence trivial completing proof prove start homotopy inverse may assume maps inverses homotopy extension property may homotope stationary outside small neighborhood track homotopy point moves back since fixes point homotopic identity induced map inner automorphism represented closed curve based element form closed curve based let morphism obtained identity homotopy stationary outside small neighborhood track homotopy closed curve applying homotopy extension property letting may apply conclude morphism isomorphism inverse remark note proof depends heavily fact center trivial proof breaks instance replaced torus fact analogue lemma graph replaced torus subset torus false hand analogue space whose fundamental group trivial center true flaring hyperbolicity throughout section continue notations regarding outer automorphism relative train track representative penultimate filtration element top stratum main result section construction carried section gromov hyperbolic space used later sections proving case theorem construction based results found sections particular proposition flaring result proposition expressed context certain natural free splitting statement proposition found section preliminary work carried sections free splitting role flaring begin description free splitting associated marked graph subgraph together description features associated height nielsen paths explain intuition behind flaring result proposition let denote free factor system corresponding subgraph form noncontractible components conjugacy class image injection injection determined inner automorphism appropriate choices base points paths definition free splitting let denote free splitting corresponding subgraph means starting associated universal covering map tree deck action obtained collapsing point component total lift denote collapse map since let induces via action evidently action free splitting minimal action simplicial tree trivial edge stabilizers note set conjugacy classes nontrivial vertex stabilizers action precisely free factor system indeed stabilizer vertex equals nontrivial component stabilizer covering noncontractible component case stabilizer conjugate projecting fixing definition lifting standard isomorphism deck transformation group universal covere group recall covering space theory ing map bijective correspondence automorphisms lifts aut represent bijection given following relation equivariance equivariance hence induces since preserves lifts preserves implies corresponding map twisted equivariance property property automorphism understood important discussion often drop notations writing simply instead next section impose additional metric constraint see heading stretch properties definition nielsen paths nielsen set paths geometric parageometric cases exist unique inverse pair nielsen lift nielsen path paths height nielsen path geometric case projection nielsen path line closed distinct initial terminal directions nielsen line projects iterate nielsen line projection nielsen line nielsen set collection subsets defined follows ageometric case geometric case set nielsen lines parageometric case set nielsen paths furthermore basepoint set denoted defined follows geometric case nielsen line unique decomposition concatenation nielsen paths defined set concatenation points parageometric case nielsen path endpoint set note geometric case basepoint sets distinct nielsen lines disjoint follows two facts lying endpoint exactly base point first point contained nielsen line second two nielsen paths contained corollary lift contained locally injective complement projection map endpoints vertices projections consider finite path integer parageometric case closed must equal say superscript representing exponent note paths precisely paths form exists give intuitive explanation proposition flaring result expressed context free splitting path function vanishes paths lifted path function projected path function vanishes paths turn function pairs vertices induces choose lift map map wish intepret metric case proposition expresses flaring metric respect action attempt carry process using either little path functions lpf indeed obtain metrics dpf vertices however case geometric desired flaring condition fails nielsen line iteration produces sequence nielsen lines ftk furthermore map ftk takes base lattice base lattice preserving path distance since base lattice nielsen line infinite diameter invariant path metric demonstrates failure flaring instead carry process starting either big path functions lpf obtaining metrics dpf base lattices nielsen lines uniformly bounded diameter may hope avert failure flaring however dpf actually metrics satisfy coarse version triangle inequality lemma order work actual path metrics kind fuzzy coarse metrics use relative hyperbolicity construction geometric case cone base lattice nielsen axis embedding naturally graph equipped path metric closely related dpf extending naturally ageometric case simply set dpf parageometric case consistency construct coning endpoint pair nielsen path cases extends naturally thus statement proposition unify cases expressing flaring respect along way state prove two important properties almost trivial ageometric parageometric cases significant content geometric case proposition gives quasicomparability coarse metric dpf true metric proposition proves gromov hyperbolicity metric piecewise riemannian path functions metric tree finite path initial terminal endpoints determined endpoints denoted path functions lpf lpf vanish paths lift path function via universal covering map vanishes paths hence projects via path function notations lpf lpf path path endpoints vertices letting functions projection projection follows definitions lpf lpf remark point lpf vertices map map hence paths either degenerate contained lpf lpf lpf associated path functions lpf lpf equivariant functions dpf lpf pairs vertices given dpf lpf dpf lpf furthermore edge lpf lpf immediate consequence fact varies finite set edges finite set positive numbers lpf positive minimum record fact later use lemma exists vertices dpf lpf using inequality lpf may construct piecewise riemannian metricr tree denoted vertices dpf follows function dpf metric vertex set next translate context several results path functions section proposition vertices unique decomposition path following properties hold exist exists precisely maximal paths contains subpath exists least one subpaths nondegenerate geometric case nondegenerate dpf lpf lpf furthermore given path finite singly infinite whose endpoints vertices assuming exists unique decomposition alternating concatenation maximal paths paths contain subpath two consecutive least one nondegenerate nondegenerate geometric case proof items translations corollary item translation corollary proofs immediate results combined definitions furthermore clause quick consequence following observations hold nested pair finite subpaths first every subpath subpath also every maximal subpath whose dpf distances greater lpf maximal subpath remark item consider paths geometric case nondegenerate used improve constants applications underlying nondegeneracy fact base point closed nielsen path disjoint parageometric case hand one paths may degenerate happens orientation reversal nielsen path initial vertex terminal vertex necessarily disjoint fact underlies item projects case degenerate lifts path path form nondegenerate closed path based constructing coning nielsen axes embed tree graph denoted extend action action follows index nielsen set letting basepoint set definition cone adding new vertex attaching unique edge points called cone points edges called cone edges since simplicial action takes nielsen paths nielsen paths hence induces basepoint preserving permutation nielsen set action extends uniquely permuting cone points cone edges let set vertices whose stabilizer subgroup stab infinite lemma formula stab defines injection set nontrivial subgroups subgroup equal stab conjugate fundamental group noncontractible component geometric stratum conjugate infinite cyclic subgroup proof fact may extracted theorem direct proof easy sketch partition cone points induces equivariant hence stabilizer preserving collapse map nontrivial stabilizer bijection set components using covering space theory stabilizers latter components precisely subgroups conjugate fundamental group noncontractible disjoint component furthermore since distinct components intersections stabilizers trivial stabilizers subtrees unequal nontrivial completes proof nongeometric geometric equivariant hence stabilizer preserving bijecej nielsen line mapping tions definition precisely lines cover collapse map closed nielsen path element represented denoted root free unique turn notations precisely infinite cyclic covering space theory stabilizers lines subgroups conjugacy class group two different lines distinct stabilizers proof completed noting conjugate fundamental group noncontractible component circuit contained notations piecewise riemannian metric piecewise riemannian metric extends piecewise riemannian metric denoted follows nongeometric case nothing geometric case group acts freely transitively set cone edges extend single cone edge extend cone edges equivariantly obtain note length independent parageometric case group acts freely set cone edges two orbits cone edges corresponding two endpoints extend single cone edge orbits extend equivariantly two obtain also require length orbits cone edges define cone height length cone edge metric path metric define metric vertices minimizing path length equals minimum continuous paths graph endpoints infimum evidently minimized embedded edge path endpoints note since tree embedded edge paths need determined endpoints denote path bypasses path let called bypass note path graph bypass path cone point midpoint furthermore bypass completely determined endpoints thus correspondence set paths set bypasses extending map following definition project obtaining choosing aut representing lift equivariant maps respect inclusion extend equivariant map follows noted convenience often suppress notation maps action induces action nielsen set extend set cone points setting furthermore map restricts bijection basepoint sets extend endpoint map uniquely isometry cone edges edge following equation follows big eigenlength equation section eigenlength equation follows equivariantly homotoping relative endpoints edges may arrange stretches edge uniform factor since extension isometry edges follows conditions constitute additional constraints maps record stretch properties maps stretch edge constant factor path map permutes cone edges taking cone edge isometrically image stretch properties first step lemma follow recall basic definitions given constants map metric spaces embedding addition point distance point domain subinterval say sometimes conflate constants setting using terminology like etc sometimes ignore altogether use terminology like etc lemma map proof let aut representative corresponding satisfies equivariance equation see definition map lipschitz stretch properties noted complete proof suffices show lipschitz map coarse inverses meaning composed maps moves point uniformly bounded distance construct taking advantage twisted equivariance combined fact action finitely many vertex edge orbits kind twisted equivariant version lemma consider vertex set partition points whose trivial infinite respectively using equivariance follows vertex subgroup inclusion stab stab follows furthermore subgroup inclusion equation stab stab consequence lemma combined fact restricts homotopy equivalence union noncontractible components fact follows restricted map bijection set define restriction equal inverse restriction map automatically equivariant define restriction follows choose one representative orbit action choose arbitrarily extend equivariance defined equivariant map extend edge stretch distance constant factor hence obtain equivariant map since finitely many orbits edges lipschitz since equivariant equivariant follows two compositions equivariant ordinary untwisted sense compositions therefore moves point uniformly bounded distance hence coarse inverses geometry dynamics section prove several propositions regarding including proposition interpretation flaring proofs follow statement propositions proposition quasicomparibility inclusion vertex set vertex set coarse metric dpf metric meaning exist constants vertices dpf proposition hyperbolicity graph metric gromov hyperbolic furthermore geodesics stable following sense exists vertices path close geodesic endpoints state earlier flaring result proposition setting given given sequence vertices defined interval integers say sequence proposition flaring exist integers pair defined integer interval max proof proposition quasicomparibility ageometric case exist dpf done henceforth assume exists letting denote cone height follows bypass length given vertices using proposition obtain decomposition path accompanying formula dpf dpf lpf lpf subpath element nielsen set endpoints distinct points corresponding bypass defined thus obtain path length calculation follows length lpf lpf applying proposition constant applying proposition follows least one lpf lpf hence lpf lpf therefore length lpf lpf dpf lpf lpf proves first inequality using turn opposite inequality starting shall normalize choice cone height lpf bypass length lpf proving proposition normalized fashion implies proposition value normalized version equivalent version given vertices choose embedded edge path endpoints satisfying two optimization conditions minimal subject minimal unique decomposition alternating concatenation subpaths bypasses obtain formula lpf lpf lpf claim otherwise decompose nielsen path case nondegenerate nondegenerate nondegenerate construct path replacing rwith corresponding bypass choice normalization contradiction cases lead path exhibiting contradiction described follows case degenerate replace subpath unique bypass endpoints case degenerate replace subpath unique bypass endpoints last remaining case degenerate let corresponding bypass consider concatenated edge path denoted obtained replacing bypass corresponding subpath straightening concatenation produces path may inductively apply coarse triangle inequality lpf given lemma conclusion dpf lpf lpf lpf lpf lpf lpf lpf since subpath lpf lpf since lpf lpf therefore dpf lpf lpf alpf lpf lpf lpf max completes proof proposition proof proposition hyperbolicity exist ageometric case since tree done parageometric case similarly easy also subsumed general proof exist purpose apply result kapovich rafi proposition let graph obtained attaching edge unordered pair element nielsen set put simplicial metric assigning length edge map extending inclusion defined take attached edge map evidently vertices corresponding bypass vertices commutes inclusions conclusions proposition namely hyperbolicity stability geodesics therefore follow demonstrate conclusions place conclusions identical conclusions proposition applied inclusion map graph hyperbolic arc uniformly hausdorff close geodesic endpoints need verify inclusion map satisfies hypotheses proposition respect simplicial path metrics assign length edge one hypothesis proposition hyperbolic holds path metrics trees another hypothesis inclusion lipschitz graph map means takes edge bounded length edge path immediate since inclusion isometry onto image remaining hypotheses proposition numbered first two trivially satisfied using inclusion map vertices vertices surjective use instead last hypothesis says exists integer vertices connected edge diameter path uniformly bounded case needs attention single edge connected edge happens subpath element nielsen set concatenation sequence nielsen paths concatenation points form consecutive sequence form pair connected edge since vertex along contained one nielsen paths hence distance one follows diameter bounded proof proposition flaring denote constants proposition shorthand fix consider specified pair projecting orbits defined choose vertices ver respectively denote path let projection also denote unique paths let projections assumption applying proposition follows lpf lpf lpf let proposition path function lpf satisfies flaring condition definition respect using obtain constants specify also let max applying proposition follows lpf flaring condition lpf therefore applies conclusion lpf max lpf lpf lpf max lpf lpf applying proposition max max max max completes proof construction continue fix choice aut representing consider corresponding equivariant map definition suspension space formally depend choice see remarks end section regarding dependence constructions follow definition suspension space slices fibers semiflow define suspension space namely quotient modulo gluing identification let denote equivalence class continuous projection map given closed connected subset denote refer slice special case singleton refer fiber fiber may regarded copy sense obtain homeomorphism given special case integer action induced action follows action using equivariance note homeomorphism equivariant respect actions generally integer homeomorphism equivariant semiflow partially defined formula uniquely extends requiring semiflow equation hold particular define first hitting map letting flow forward along flow segment thus obtaining formula completes definition definition piecewise riemannian metric geodesic metric define piecewise riemannian metric recall piecewise riemannian metric see beginning section edge integer define riemannian metric depending two cases case use metric note stretches length constant factor hence gluing map takes isometrically onto case equivalently geometric parageometric case cone edge use metric note also cone edge maps isometrically gluing map takes isometrically onto metrics rectangles glue isometrically along common boundaries follows first two edges common endpoint restrictions metric metric equal fixing letting vary allows glue rectangles get piecewise riemannian metric next noted gluing map maps isometrically onto letting vary allows glue get desired piecewise riemannian metric minimizing path lengths using piecewise riemannian metric obtain geodesic metric may similarly define geodesic metric slice minimizing path length respect restricted piecewise riemannian metric special case fiber letting edge image edge metrics related stretches metric factor whereas preserves metric particular map bilipschitz homeomorphism therefore isometry integer homeomorphism general completes definition use end proof theorem verifying wwpd conclusions shall need following metric property lemma two fibers proof lemma follows noting edge projection function property tangent vector using riemannian metric right hand side inequality following metric property used later study inclusion map fibers distort distance lemma integer exist constants integer inclusion map proof construct cycle equivariant lipschitz maps follows inclusion map equivariant maps first hitting maps lipschitz map equivariant coarse inverse equivariant map construction consider commutative diagram diagram map equivariant map untwisted equivariant top bottom maps instances equivariant isometry whose inverse twisted equivariant lemma map equivariant lipschitz coarse inverse follows equivariant lipschitz coarse inverse whose lipschitz coarse inverse constants depend may therefore fill diagram map makes diagram commute untwisted equivariant lipschitz coarse inverse lipschitz coarse inverse constants depending going round cycle obtain two equivariant move points uniformly bounded distances depending maps therefore lipschitz coarse inverses hence constants depending remark definition suspension space depends ostensibly choice representative aut fact different choices produces suspension spaces isometric easily checked using fact distinct choices differ inner automorphism proof hyperbolicity prove hyperbolic metric space applying combination theorem descendant combination theorem hypotheses theorem consist opening hypothesis four numbered hypotheses must check last four flaring condition prove application proposition opening hypothesis theorem metric bundle hypothesis says metric bundle respect map must therefore verify map satisfies definition metric bundle given definition first must lipschitz map fiber must geodesic metric space respect path metric induced already verified also item definition translated setting requires interval exists path uniformly bounded length passes endpoints respectively obtain path choose take path defined whose length equals remaining verification needed metric bundle set inclusions uniformly proper meaning inclusions uniformly lipschitz fact construction exists single nondecreasing gauge function inclusions following property define consider connected geodesic path length projection length letting integers image implying image implies turn applying lemma hence may assume nondecreasing functions hence nondecreasing functions using done hypotheses theorem base fiber hyperbolicity hypotheses require base space hyperbolic evident fibers hyperbolic uniform hyperbolicity constant proposition gives constant since fiber homeomorphic follows hyperbolic uniform hyperbolicity constant depending hypothesis theorem barycenters hypothesis says barycenter maps uniformly coarsely surjective varies review means section around heading barycenter map given geodesic metric space gromov boundary consider triple space barycenter map coarsely map follows exists constants depending two points exists embedded line endpoints use notation quasigeodesic line lemma exist constants depending triple exists point comes within distance lines point distance constants chosen point called barycenter map taking triple barycenter called barycenter map say barycenter maps uniformly coarsely surjective means exists coboundedness constant image barycenter map comes within distance point hyperbolic space action fundamental domain bounded diameter diam coboundedness constant barycenter map hence uniform coboundedness constant barycenter maps since fibers comes equipped homeomorphism barycenter maps uniform coboundedness constant hypothesis theorem aka definition slight restatement hypothesis specialized present context flaring exist integers two sections projection map meaning identity interval following implication holds max statement tautologically implies flaring hypothesis given definition difference latter statement quantifier order starts exist integers remainder statement unchanged simple geometric estimation argument yields converse implication need proving flaring first reduce discretized version taking place solely fibers integer values follows flaring exist integers integer two maps sections projection map following implication holds max show flaring implies flaring choose consider integer pair sections projection map let let semiflow restricts homeomorphisms defined integer property distance follows functions sections constant depends applying flaring exist integers implication holds using maps homeomorphisms take hypothesis implies hypothesis conclusion implies conclusion remains verify flaring applying proposition restate convenience form matching flaring flaring exist integers pair defined integers following implication holds max proof flaring flaring commutative diagram first hitting map note diagram bottom equivariant top equivariant left equivariant isometry right equivariant isometry choose consider integers consider pair maps sections follows recall synonymous hence denote follows sequences defined since isometry hypothesis implies hypothesis similarly since isometries conclusion implies conclusion using therefore proved flaring flaring completes verification hypotheses combination theorem space therefore hyperbolic abelian subgroups section reviews background material needed rest paper section contains basic material regarding automorphisms outer automorphisms see also comprehensive overview section reviews elements theory abelian subgroups developed focussing disintegration subgroups background review cts notations reviewed features associated filtration assumption top stratum studying disintegration groups shall need defining properties derived properties cts section reader may ignore assumption top stratum shall refer material specific cts general material certain compilation results multiple sources one may also consult comprehensive overview general properties strata section stratum either irreducible stratum meaning transition matrix irreducible matrix zero stratum meaning zero matrix irreducible stratum satisfies one following stratum remark matrix perronfrobenius matrix eigenvalue neg stratum section single edge orientation either trivial closed path distinct initial terminal directions neg stratum fixed stratum trivial linear stratum nielsen path equivalently periodic nielsen path superlinear stratum otherwise properties strata stratum also referred linear edge linear edges following features twist path twist coefficient section linear edge widi unique closed nielsen path root free meaning iterate shorter closed path equivalently base point element group represented root free say twist path integer twist coefficient another linear edge twist path determine conjugacy class inversion neg nielsen paths definition neg edge indivisible nielsen path contained linear edge every nielsen path form wik exceptional paths definition paths form linear edges twist path twist coefficients sign properties strata properties stated notations respect top stratum arbitrary somewhat different emphasis lines section recall spaces lines canonical homeomorphism point subsets paths topology induced hausdorff topology compact subsets topology basis element finite path consisting paths subpath homeomorphism natural bijection induced universal covering map refer homeomorphism saying line realized corresponding line attracting laminations section remark associated attracting lamination set lines whose realization property finite subpath edge exists subpath distinct strata corresponding laminations distinct set laminations independent choice representative nielsen paths corollary fact inversion exists one indivisible periodic nielsen path contained initial terminal directions distinct least one endpoint contained geometricity fact geometric stratum exists closed path properties zero strata definition definition definition zero stratum following properties envelopment exist indices irreducible stratum stratum component noncontractible zero stratum contractible component say zero strata enveloped denote hrz union enveloped zero strata filtration element union noncontractible components hrz taken paths paths exists edge irreducible stratum exists maximal subpath path say specifically path stratum envelopes every edge path follows endpoints vertices contained properties complete splittings section definition splitting path concatenation expression characteristic property short completely split relative train track map following complete splitting edge unique splitting complete meaning term either edge irreducible stratum indivisible nielsen path neg indivisible nielsen path exceptional path taken path zero stratum principal automorphisms rotationless outer automorphisms consider automorphism aut induced boundary homeomorphism fixed subgroup denoted fix denote denoted per fix respectively sets periodic points fixed points period say attractor consider per collection neighborhood attractor neighborhood basis also repeller fix denote sets attracting repelling nonrepelling within per points respectively pern per fixn fix associated inner automorphism use following special notations fix bic bic bic following equivalences collected lemma based results fact aut nontrivial invariant stab commutes fix fix principal automorphisms definition say principal automorphism fixn furthermore fixn fixn neither equal nontrivial equal set endpoints lift generic leaf attracting lamination outer automorphism representing let aut denote set principal automorphisms representing let set aut representing either principal equivalently symmetry equation useful situations one trying prove certain property symmetric inversion one may reduce case principal replacing inverses case already principal use reduction argument little comment many places rotationless outer automorphisms definition remark concept forward rotationless outer automorphisms defined proved forward rotationless outer automorphisms precisely outer automorphisms representatives use stricter property rotationless defined symmetric inversion better adapted study abelian subgroups definition incorrectly stated definition given replace one changes required arguments use written using current correct definition say rotationless two conditions hold first per fix forward rotationless condition replaced weaker pern fixn second integer map bijection recall remark injective always true bijective holds surjective holds expansion factor homomorphisms section consider attracting lamination action set lines consider subgroup stab stabilizes expansion factor homomorphism unique surjective homomorphism stab stab furthermore exists stab relative train track representative stratum corresponding eigenvalue transition matrix equals twistors aka axes lemma preceding page recall two elements said unoriented conjugate conjugate ordinary conjugacy class denoted unoriented conjugacy class denoted marked graph nontrivial conjugacy classes correspond bijectively circuits orientation preserving homeomorphisms domain note root free oriented circuit representing factor nontrivial covering map unoriented conjugate oriented circuits representing differ arbitrary homeomorphism consider nontrivial unoriented root free conjugacy class two representatives aut following equivalent fix stab fix furthermore equivalent conditions hold integer exists pair equivalent conditions hold said twistor number distinct elements set stab called multiplicity twistor properties independent choice representing number twistors multiplicity twistor finite follows first representing twistor equivalent neg linear edge twists around meaning twist path either represents furthermore multiplicity one number linear edges twist around lemma rotationless abelian subgroups principal sets rotationless abelian subgroups definition corollary abelian subgroup rotationless elements rotationless equivalently rotationless generating set every abelian subgroup contains rotationless subgroup index bounded uniformly constant depending namely subgroup powers integer power every element rotationless lemma assuming rotationless subset elements principal set represented aut determines uniquely amongst representatives fix principal set fixed map defines homomorphism aut section canonical map aut subgroup also set fix unique principal set maximal inclusion subject property contains set defines section aut remark comparison homomorphisms rotationless abelian subgroups definition lemma consider rotationless abelian subgroup consider also two principal sets define lifts aut let maximal principal sets containing respectively also lifts defined suppose aut suppose also hence follows set fixed distinct automorphisms representing element nontrivial root free situation associated comparison homomorphism characterized equation number distinct comparison homomorphisms finite coordinate homomorphism lemma definition lemma every abelian subgroup finite lamination group set finite addition rotationless stab thus obtain finite collection expansion factor homomorphisms defined namely choosing enumeration expansion homomorphisms comparison homomrophisms obtain coordinate homomorphism injective disintegration subgroups section fix rotationless representative corresponding filtration need notations section using structure see section build definition disintegration subgroup associated certain abelian subgroup constructed section explicit combinatorial procedure review using also obtain description coordinate homomorphism see section noting depends choice definitions section depend describe structures associated collection linear edges augmenting structures described heading properties strata section def two linear edges common twist path said linearly equivalent abbreviated associated linequivalence class denoted linw recall distinct elements linw distinct twist coefficients lemma def given twist path distinct linear edges linw path form called path exceptional path twist coefficients sign every completely split path concatenation terms complete splitting either sign exceptional path hence single term terms consist initial terms complete splitting terminal two overlap edge uniquely defined obtained complete splitting conglomerating single term consider twist path linw associated family set paths also associated linear family set linw every path belongs unique family every neg linear edge path belongs unique linear family almost invariant subgraphs intuition behind almost invariant subgraphs partition collection strata determined following requirement one perturbs letting restrictions strata replaced iterates varying integer exponents one wishes perturbation resulting outer automorphisms commute outer automorphism represented exponents constant edges contained almost invariant subgraph def define equivalence relation irreducible strata follows given path short path edge taken connecting path zero stratum enveloped define relation amongst irreducible strata denoted meaning exists short path term edge note let directed graph whose vertex set set irreducible strata directed edge relation let denote connected components graph define almost invariant subgraph union strata comprising vertices together zero stratum enveloped one note set almost invariant subgraphs partitions set nonfixed strata admissible families defn lemma defn integers define edge otherwise fixed homotopy equivalence representing outer automorphism denoted construction preserves given filtration notation used denote one would like arrange among things edges need hold general hold certain relation amongst linear edges terms satisfied given almost invariant subgraph linear edges say triple related exists stratum short path twist path integer linw term say admissible related triples letting twist coefficients respectively letting almost invariant subgraphs containing respectively following equation holds notation consider almost invariant graph pair linear edges triple related associated family defined set paths form let denote set families associated related triples linear edges say pair linear edges exists almost invariant subgraph related equivalently linear family letting unique twist path linw exists family set equivalence relation linear edges generated called written note relation amongst linear edges refinement relation note exceptional path occurs term complete splitting iterate short path family element linear edges paths notation corollary almost invariant subgraph use terminology refer subpaths form elements set defined notation namely completely split paths term one following nielsen path short path stratum contained example edge hiz stratum path family furthermore admissible almost invariant subgraph path following iterate also path disintegration subgroup recall definition disintegration subgroup also recall admissible semigroup use aid understanding properties subgroup defn cor lem cor subgroup generated set elements admissible subgroup abelian rotationless dependence rather suppressed definition notation used see example admissible semigroup corollary proof let denote set admissible forms let subgroup generated map defined injective semigroup homomorphism extends isomorphism every element written difference two elements every written form record simple consequence definitions later use fact function assigns map satisfies following homomorphic property edge proof fixed edge sides equal otherwise edge almost invariant graph hence hence path hence sides equal repeat theorem cited section disintegration theorem theorem every rotationless abelian subgroup exists every representing intersection finite index remark statement disintegration theorem made explicit fact theorem holds choice representing already implicit notation used disintegration groups definition representative allowed coordinate homomorphism disintegration group disintegration group rotationless abelian injective coordinate homomorphism defined end section whose coordinate functions expansion factor homomorphisms comparison homomorphisms review one may use structure pick subset coordinate functions consisting expansion factor homomorphisms subset comparison homomorphisms scaling expansion factor homomorphisms obtain homomorphism denoted still injective lemmas preceding paragraphs let either neg linear homomorphism coordinate functions denoted defined two cases depending whether neg linear let almost invariant graph containing case neg linear twist path twist coefficient set homomorphism whose value admissible given case letting eigenvalue transition matrix homomorphism unique homomorphism admissible letting attracting lamination associated stratum note scaled copy meaning also define comparison homomorphisms one linequivalent pair linear edges using following formula following properties hold particular first two relate functions coordinate functions stratum correspond expansion factor coordinate functions correspondence scaled copy explained set consisting coordinate functions neg linear stratum together functions pairs neg linear strata exactly set comparison homomorphism coordinate functions homomorphism injective admissible train track semigroup action throughout section continue adopt notations regarding rotationless representative whose top stratum associated attracting lamination consider disintegration group reviewed section recall element stabilizes shall focus subsemigroup consisting value top coordinate homomorphism meaning asymptotic stretch factor either stretch attracting lamination using theory disintegration groups construct topological representative whose action edges agrees action appropriate iterate namely lifting projecting tree topological representatives universal cover extending coned graph obtain semigroup actions detailed properties described section important feature action isometric action fact action lengths uniform amount depending appropriate value coordinate homomorphism see section hand restricting semigroup actions subgroup obtain true isometric actions properties studied section actions constructed section basis construction section isometric group action hyperbolic suspension complex use throughout sections section establish notations notations given rotationless representative associated filtration top stratum associated objects notations described notations following additional objects notations section properties zero strata huz union zero strata enveloped set zero strata form empty highest irreducible stratum union noncontractible components contractible components zero strata definitions denotes free splitting corresponding marked graph pair associated nielsen set basepoint set also denotes action obtained coning basepoint set cone point edges definition associated automorphism aut representing unique equivariant maps follows lift induced respect collapse map extension permutes cone points cone edges maps satisfy stretch properties recorded section disintegration group denoted full aut called extended disintegration group denoted associated top stratum following objects almost invariant subgraph containing huz coordinate homomorphism associated scaled copy expansion factor homomorphism obtained lifted coordinate homomorphism projection map kernels homomorphisms denoted ker ker thus commutative diagram short exact rows ker inn inn aut letting subsemigroups follows inclusions homotopy semigroup action section prepare construction semigroup action work downstairs construct homotopy semigroup action means clear construction intuitive idea associate topological representative although action equation hold exactly hold homotopy relative values edges hnz determined appropriate iterates values determined homotopy graph homotopy principle section letting vert denote set vertices define subsets vert follows first vert vert huz latter equation holding edge huz endpoints definition zero strata next huz note huz subgraphs edges common also frontier huz huz finally denotes set points fixed inclusion follows remark summary construction homotopy semigroup action define topological representative following hold idg denotes identity group element idg identity map edge huz huz identity particular particular fixes point subset height indivisible nielsen path exists fixes endpoint define homotopy following hold edge homotopy straightens edge path relative endpoints form path cancelling edges without ever cancelling edges stationary preserves recall homotopy preserves subset means stationary means constant first construct maps homotopies along way proving various requirements constructing construction applying graph homotopy principle lemma recall section abelian semigroup admissible recall also injection defined topological representative injection semigroup homomorphism whose image generates since restricted map fta homotopy equivalence fixes point lemma map pairs fta homotopy equivalence category pairs hence homotopy class rel element fta autgp pure automorphism group graph point category lemma fact maps fta ftb homotopic rel map autgp defined semigroup homomorphism since commutative group generated subsemigroup simple semigroup argument shows extends uniquely group phism autgp choose representative already choose fta identity identity notice straightening carried applied need operator definition later use denote homotopy inverse fta represents autgp may define follows huz identity clearly identity verifying construction satisfies item note first exists oriented edge initial vertex initial vertex path equals addition fix shows restricts completing proof item also follows continuous proof homotopy equivalence topological representative delayed end item encompassed following generalization also used item endpoints see holds general endpoints concatenation three special types suffices check types type edges huz type endpoints set type height indivisible nielsen path type special case handled already item type first note fta equation follows section general choose note group autgp equation fta may calculate fta fta ftb ftb second equations second third lines follow section type let height indivisible nielsen path may write alternating concatenation nontrivial paths form path huz nielsen path endpoints lemma definition claim nielsen path prove claim holds case left hand side equals using autgp claim holds general case holds applying claim complete proof also completes construction proof except delay proof homotopy equivalence topological representative constructing given turn construction motopy let first using homomorphism translates exists homotopy rel denoted second huz second equation holds applying together fact section putting together may define edge huz restriction homotopy rel endpoints restriction equal items evident construction item follows definition relative train track map tells edge integer path path homotopy straightenes produce cancels edges topological representatives since generates may choose hence since three homotopy since homotopy equivalences follows also since topological representatives respectively follows topological representative semigroup actions deriving various properties turn construction action construction extend action define map carrying associated following steps first let image homomorphism aut consider map part homotopy semigroup action constructed section let unique equivariant lift let equivariant map induced collapsing component point straightening edge stretches length constant factor record several properties semigroup action twisted equivariance map equivariant semigroup action property property follows fact edge consequence section item applied edges vertex action property takes vertices vertices restricts bijection set vertices nontrivial stabilizer proof denote vertex sets let denote subset stabfn nontrivial section items map takes set since topological representative follows restricts homotopy equivalence amongst noncontractible components letting restricts total lifts follows restricts bijection amongst components set components nontrivial stabilizer quotient map components corresponds bijectively vertex set follows restricts self map restricts bijection proving property stretch property maps edge edge path stretching length uniform factor follows definition piecewise riemannian metric section eigenlength equation section section item edge restriction train track property edge path injective follows property together property applied statement property recall section subgroup stab aut recall particularly lemma stab containing inn unique section says subgroup stab action extending given action free group inn element action satisfies twisted equivariance property stab hence stab restricted action identical furthermore restriction given lemma unique isometric action action satisfies equivariance projected image map proof consider restricts identity edge section item map therefore permutes edges mapping isometrically image follows map isometry since satisfies equivariance property follows stab lemma since applying stab arbitrary proved stab lemma also proved map restriction given lemma namely unique action action stab assigning unique equivariant isometry exists invariance nielsen data restricts order preserving bijection basepoint sets particular induces bijection nielsen paths definition follows immediately section item finally immediate consequence property semigroup action extends uniquely extension denoted semigroup action permutes cone points permutes cone edges isometries particular letting dynamics group action isometries simplicial tree equipped geodesic metric satisfy dichotomy isometry either elliptic loxodromic dichotomy hold isometries gromov hyperbolic spaces general third category isometries namely parabolics section prove lemma says part dichotomy hold action element lemma dynamics action acts loxodromically acts either loxodromically elliptically precisely acts loxodromically axis nielsen line definition case furthermore uniform uniformly positive stable translation length loxodromic elements acting following sense exist constants loxodromically integer remarks lemma terminology uniform uniformly positive stable translation length refers corollary stable translation length lim uniform positive lower bound independent choice loxodromic rate positive lower bound approached element uniform property applied section purposes characterizing described section loxodromic elements action lemma proof could already presented almost back section restricted action inn methods available section additional things needed prove lemma like restriction satisfies twisted equivariance larger action preserves nielsen paths associated objects section proof throughout proof apply section regarding action nielsen paths elements nielsen set basepoint sets cone points cone edges particular action preserves nielsen paths takes maximal path maximal path acts elliptically fixes vertex consider geometric case acts loxodromically axis nielsen line fixes corresponding cone point either case acts elliptically suppose acts loxodromically axis nielsen line always happens nongeometric case terminology proof applies cases vertex integer consider proposition decomposition path meaning maximal subpaths alternating subpath since intersection contains concatenation fundamental domains action follows contains must contain distinct distinct applying fact takes maximal paths maximal paths consequence collection contains least distinct edges applying proposition obtain dpf min lpf edge applying proposition immediately follows acts loxodromically estimate vertices similar estimate holds arbitrary points replacing upper bound distance arbitrary point vertex set suspension action section complete proof case theorem throughout section adopt notations concerning rotationless representative top stratum disintegration group free splitting associated pair particular section used coned free splitting construct suspension space proving gromov hyperbolic using flaring properties action section studied extended aut subsemigroup constructed disintegration group train track semigroup action section shall show suspend semigroup action completely general construction applies isometric action disintegration group top stratum section put pieces together complete proof suspension action recall notations short exact sequence representing choose automorphism follows determines splitting short exact sequence hence semidirect expressed terms inner automorphism product structure may determined namely thus element group operation written form unique notation defined recall definition construction using chosen automorphism representing corresponding equivariant map denoted abbreviated form suspension space namely quotient identifications recall also various notations associated introduced definitions suffices carry define action element following steps first define group action next define action element define action element composition finally verify two sides semidirect product relation act identically act equation let note formula well defined using properties section group action evidently acts isometry using semigroup action properties action equation satisfied completes definition isometric action agrees note restriction inn action action given definition defined general special case equation holds lemma extension general case follows easily next let act shifting downward evidently isometry two sides semidirect product equation act required way preserves horizontal level set notice since action since action vertical translation equal meaning follows integer vertical translation length action record later use fiber lemma proof theorem case first set notation based formulation case section outline proof section apply results conclusions section obtain hyperbolic action theorem proved order aut image setup proof subgroup quotient homomorphism aut kernel finitely generated satisfying hypotheses theorem abelian virtually abelian proper nontrivial free factor conclusion theorem existence certain kind hyperbolic free replace action finite index normal subgroup conclusion proved using finite index subgroup hyperbolic action finite index normal subgroup conclusion follows restriction action conjugates one need observe intersection conclusions theorem action imply conclusions restricted finite index subgroup may therefore assume rotationless abelian subgroup using existence integer constant power element inn inn aut inn associated objects commutative diagram short figure group exact rows rotationless replace abelian group finite index subgroup powers maximal proper free factor system number applying disintegration theorem obtain representative disintegration group subgroup finite index choose penultimate filtration element represents free factor system since number top stratum replacing may thus assume adopt notations regarding free splitting associated marked graph pair action obtained coning elements nielsen set disintegration group associated using follows setting extended disintegration group kernel kernel may augment commutative diagram notations obtain commutative diagram short exact rows shown figure notations also subgroups subsemigroups semigroup action described section associating let map pick let suspension space constructed definition let isometric action constructed section make heavy use identification integer sections turn proof three conclusions theorem action acts either elliptically proof conclusion every element acts either loxodromically show generally every element elliptically loxodromically section general element form lemma loxodromic suppose preserves consider particular restriction restriction elliptic fixes point hence fixes point elliptic loxodromic axis nielsen line fixes corresponding cone point elliptic act loxodromically whose remains consider axes nielsen lines applying lemma acts loxodromically integer constants independent consider function assigns integer minimum translation distance vertices action inf prove acts loxodromically apply classification isometries gromov hyperbolic spaces says every isometry elliptic parabolic loxodromic elliptic parabolic function bounded thus suffices prove consider also integer consider equivariant map given isometry metric metric map conjugates action action applying inequality applying twisted equivariant isometric conjugacy vertex seen earlier uniformly proper maps uniform gauge function see section diverge constant arbitrarily large values holds vertex implying contradicting sufficiently large completes proof conclusion furthermore proved following useful later following equivalent acts loxodromically acts loxodromically acts loxodromically axis nielsen line nonelementary shall focus proof conclusion action restricted actions subgroup proving first nonelementary action latter follows whole nonelementary action shall apply lemma must check hypotheses lemma let set vertices nontrivial stabilizer action restricts shown section semigroup action property element semigroup action acts equivariant bijection follows immediately semigroup action extends uniquely group action obtain action satisfying hypotheses property restricting first paragraph lemma hypothesis theorem proper nontrivial particular subgroup stabfn free factor fixed finally free group rank otherwise since fixed solvable subgroup aut abelian would follow hence virtually abelian contradicting hypothesis virtually abelian verified hypotheses lemma conclusion action nonelementary since action nonelementary minimal subtree line contains finite path contained nielsen line furthermore contained axis loxodromic element whose axis therefore nielsen line applying lemma action loxodromic choosing stabfn letting follows also acts loxodromically axis also choice intersection either empty finite since neither lines nielsen axis ray line infinite dpf diameter goes infinitely far away line follows independent loxodromic elements proving nonelementary finally using previous paragraph whose axes nielsen lines showed proof conclusion act loxodromically furthermore since lines infinite hausdorff distance follows also infinite hausdorff distance shown item section proves independent loxodromics hence action nonelementary proof conclusion element acting loxodromically wwpd element acting loxodromically shown earlier consider proof condition equivalent acting loxodromically acting loxodromically axis nielsen line shall show first wwpd element action let stab denote subgroup action finally action stabilizes may say axis stab meaning common axis elements stab act loxodromically shown secthe action extends action tion since normal since restriction free splitting action inn may apply lemma conclusion wwpd element action express conclusion let equivalently set axes conjugate subgroups stab explicitly letting bijectively indexed set left coset representatives stab group may index bijectively using geodesic metric wwpd property element action says lengths segments metric uniformly bounded constant independent obtain definition applying corollary obtain proving diamu diamu denotes next use show wwpd element action purpose may work entirely using coarse metric set say means first recall lemma satisfies metric axioms except triangle inequality replaced coarse triangle inequality respect constant subtree let lemma corollary dpf coarse metrics quasicomparable proposition inclusion dpf coarse metric metric also path uniform quasigeodesic metric particular lines uniform quasigeodesics prove wwpd element suffices prove exists diamu inequality actually shall need prove complete train ideas wwpd element action reason implies wwpd element using uniformly quasigeodesic exists constant sequence intersection lik contains subpath nested union equals hence diamu prove letting point minimizes assume let similarly defined similar distance bounds let projection meaning unique point let similarly defined since since concatenation follows lemma combining coarse triangle inequality obtain similarly two points least one denoted contained similarly least one denoted follows coarse triangle inequality diamu completes proof wwpd element action let finally show wwpd element action set uniform subgroups conjugate note fails wwpd sequence lines exists constant sequences lik defined nested sequence subpaths whose union equals lines uniform quasigeodesics hyperbolic space exist constants lines points minimum distance subsegments throwing away finitely many terms sequence may assume applying lemma immediately follows line sequence set thus exists integer set contains infinitely many lines sequence passing subsequence may assume lines semiflow flowing let line corresponding finite hausdorff distance choose sequences distances uniformly bounded passing subsequence may assume subsegments nested union distances also uniformly bounded hence uniform properness inclusion distances uniformly bounded follows exists lines uniformly close subsegments whose length hence exists constant diamdj contradicts isometry set lines corresponds set lines references bestvina bromberg fujiwara constructing group actions applications mapping class groups publ math inst hautes sci bestvina feighn combination theorem negatively curved groups diff geom bestvina fujiwara bounded cohomology subgroups mapping class groups geom topol electronic bestvina feighn handel laminations trees irreducible automorphisms free groups geom funct anal tits alternative dynamics exponentiallygrowing ann math solvable subgroups virtually abelian geometriae dedicata culler morgan group actions proc london math soc feighn handel abelian subgroups geometry topology recognition theorem groups geom dyn gaboriau jaeger levitt lustig index counting fixed points automorphisms free groups duke math gromov hyperbolic groups essays group theory gersten msri publications vol springer handel mosher relative free splitting free factor complexes loxodromic outer automorphisms preparation lipschitz retraction distortion subgroups geom topol hyperbolic actions bounded cohomology subgroups part infinite lamination subgroups subgroup decomposition part geometric models memoirs ams appear subgroup decomposition part relative kolchin theorem memoirs ams appear subgroup decomposition part iii weak attraction theory memoirs ams appear subgroup decomposition part relatively irreducible subgroups memoirs ams appear virtually abelian subgroups ian abelian kapovich rafi hyperbolicity free splitting free factor complexes groups geom dyn sardar combination theorem metric bundles geom funct anal
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probabilistic demand side management approach consumption admission control lorant kovacs rajmund drenyovszki andras olah janos levendovszky kalman tornai istvan pinter new generation electricity network called smart grid recently conceived vision cleaner efficient cheaper electricity system one major challenges electricity network generation consumption balanced every moment paper introduces new concept controlling demand side means automatically electric appliances make sure demand match available supplies based statistical characterization need new approach instead using hard limits estimate tail probability demand distribution control system using principles results statistical resource management keywords smart grid demand side management admission control introduction main issue electricity networks keeping almost perfect balance electricity generation consumption time balance demand supply crucial since oversupply means waste energy undersupply causes performance degradation grid parameters phase voltage level unfortunately control supply side almost impossible large time constants fossil nuclear plants possibility applying cost ineffective auxiliary generators additionally smart grids percentage renewable resources increased gives rise uncertainty generation side hence best way keep balance manage demand side demand side management dsm means new kind challenge system operators control power grid local scale possible installing intelligent measurement devices smart meters however new perspective households controlled intelligent devices residential sector accounts total energy consumption contains time shiftable appliances high number amount consumption involved direct control eliminate error daily prediction based generation actual demand spread electric vehicles could mean additional opportunity average cars parked europe time hence batteries electric vehicles serve extra storage capacity power grid paper propose new approach shorttime demand side management introduced method takes account probabilistic nature load aid consumption admission control algorithm shiftable appliances reshapes probability density function pdf aggregate consumption related work influencing demand side context smart grid electricity networks usually referred demand side management dsm demand response demand response mechanism managing customer consumption response supply side conditions demand side management covers activities programs undertaken service providers influence amount timing electricity use many solutions proposed dsm like direct control smart appliances pricing load scheduling good references found different dsm approaches direct control system operators remove extreme values electricity consumption peak shaving encourage additional energy use periods lowest system demand valley filling load control demand response strategy presented simulating summer period airconditioning units conducted two control algorithms takes account users comfort via heuristic consumer utility metric uses binary policies fairness maintained two scheduling algorithms priority based round robin results show significant energy cost savings achieved proposed algorithms minimize operating cost residential microgrid milp model proposed decision variables used model demand also supply electrical thermal energy covers solar energy distributed generators energy storages loads model predictive control scheme proposed iteratively produce control sequence microgrid case study reveals performance minimum cost control comparison benchmark control policies results show savings annual operating cost dsm model three layers introduced model consists admission controller load balancer demand response load forecaster modules whenever user turns appliance request sent admission controller capacity available peak hours accepts request initiates operation appliance appliance operation exceeds capacity request rejected forwarded load balancer task load balancer solve optimization problem assign future timeslot appliance start later approach residential energy consumption scheduling proposed introduces pricing mechanism based convex increasing cost function authors present distributed algorithm optimization problem proposed techniques literature consider load curves however papers deal randomness load side exception uncertainty considered well authors propose milp optimization model performs scheduling adopted dsm model forecasts load curve user previous knowledge energy usage additionally real time pricing inclining block rates combined model effective pricing optimization information appliances revealed time schedule appliances updated accordingly simulation results show reducing total ratio energy expenses users rest paper organized follows problem formulation system model described section concept proposed consumption admission control algorithm introduced section results presented discussion section finally paper concluded section figure applied model problem formulation system model large number appliances tolerate delay executing program washing machine later time additionally near future spread electric vehicles mean huge amount elastic demand power system result time shiftable demands system hand devices show stochastic consumption behaviour neither start use time operation known priori hence statistical approach efficiently solve control task foundation approach new consumption admission control algorithm modify pdf consumption unit household street city etc temporary shiftable appliances system operators perspective pdf close dirac delta function meaning constant load ideal however reach optimum realistic goal keep probability density function narrow possible mass pdf lies lower upper limit sake even realistic model allow tail probabilities smaller predefined probability paper assume service provider calculates communicates parameters governing behaviour customer using parameters subscriber smart meter appliances local level resulting fully distributed solution problem parameters coming service provider govern cac algorithm follows capacity upper limit every time slot allowed exceeded small probability paper concentrate upper limit oversonsumption probability however plan extend approach lower limit underconsumption probality future work tail probability referred quality service qos parameter satisfy overload grid keep certain limit hence stability grid parameters frequency voltage level etc task admission control appliances way probability distribution function pdf aggregate consumption satisfies prescription service provider cooperative attitude subscriber motivated rewards underlying model depicted figure model subscribers assumed smart meter following properties communicate service provider smart appliances register consumption statistics appliances smart traditional ones temporarily appliances model stochastic deterministic shiftable appliances taken consideration devices executing fixed program seen deterministic defined categories examples appliances listed table table device categories used model shiftable nonshiftable stochastic electric heating air conditioner refrigerator lighting vacuum cleaner deterministic washing machine dishwasher circulation pump figure measures used model depicted admission control algorithm uses discrete time slots denoted figure status appliances system parameters capacity limits qos supposed unchanged new consumption requests supposed handled instantaneously time slots deterministic component consumption well stochastic one figure illustration original modified pdf aggregate consumption free parameters govern algorithm stochastic part described estimated calculated probability distribution function maximum cmax minimum cmin capacity limits changed every time instant service provider csys builds natural upper limit lines fuses cmax number appliance classes number appliances class total number enabled appliances appliance class means set appliances statistical descriptors considering deterministic stochastic appliances model write inequality figure measures used model constant value probability expressed consumption admission control algorithm decision enable disable appliance system carried consumption admission control cac algorithm mentioned section aim algorithm sharpen shape pdf aggregate consumption customer resulting near constant load time domain smart meter calculates aggregate pdf individual appliances individual pdf communicated smart appliances measured case traditional ones concept originally applied call admission control atm communication networks paper following mathematical model used let denote random variable consumption jth appliance aggregate consumption random variable number enabled appliances case new incoming consumption demand cac checks whether inequality holds enabled plus incoming appliance denotes probability event probability limit overconsumption therefore cac keeps upper tail probability aggregate consumption limit probability overconsumption calculated based probability density function aggregate consumption pdf aggregate consumption calculated analytically convolution individual pdfs appliances lower limit checked manner upper limit probability underload higher goal expressed estimation probability overconsumption convolution operation time consuming case high number appliances classes suggested estimate probability terms inequalities large deviation theory ldt bounds markov chebisev bennett hoeffding chernoff upper bounds estimation overconsumption derived calculation following upper bound bounding method tail probability independence random variables expected value expressed variance widely known markov inequality needs expected value give upper bound probability random variable greater equal positive constant case inevitably advantage markov inequality simplicity tight upper bound chebysev inequality also simple also tight upper bound hoeffding inequality exponentially decreasing upper bound results tighter estimation even far expected value compared markov chebysev inequalities also based expectation random variables independent additionally variables upper lower bounds hoeffding inequality expressed clear increase cmax upper bound decreases exponential rate bennett inequality gives exponentially decreasing upper bound like hoeffding assumes bounded input random variables formulated following form bennett inequality needs additional statistical information compared hoeffding standard deviation appliances maximum value chernoff inequality also exponentially decreasing upper bound aggregate consumption random variable containing consumption enabled shiftable appliances plus incoming one another approach estimating aggregate pdf based central limit theorem clt denotes cdf standard normal cdf must emphasize clt upper bound tail probability speed convergence main question regarding estimations based central limit theorem absolute error clt estimation decreasing towards tails relative error increasing results discussion order clear picture performance consumption admission control algorithm simulation environment established matlab investigated following aspects cac algorithm relation qos empirical probability overconsumption case different ldt bounds model complexity load time series load shape modification made cac number enabled appliances case different ldt bounds clt throughout simulations used stationary load time series explore statistical behaviour cac algorithm clear real benefit new algorithm comes fore nonstationary environment day longer consumption period relation qos empirical probability overconsumption cmax lge esxj called logarithmic momentum generating functions parameter satisfies possibly tightest bound new demand shiftable appliance appears enabling disabling calculated using one section present investigation regarding relation predefined qos empirical probability overconsumption ratio predefined qos empirical probability overconsumption denoted using upper bound tail probability leads underestimation number appliances enabled results lower bound results point view service provider means guaranteed qos causes spare capacities following assumptions made simulations load appliances modelled bernoulli iid series time instants one appliance class appliances statistical descriptors number appliances class consumption demand temporarily disabled appliances deleted aim investigation measure performance different tail probability estimation methods plugged cac case different probability state appliances figure depict results case respectively results figure show empirical probability almost meet qos tail probability exactly calculated analytical aggregate pdf see small deviation case small probabilities due difficulty measuring rare events case monte carlo simulations pon cmax mean analytic clt chernoff bennett hoeffding chebisev model complexity load time series cac algorithm needs appliance level statistical information therefore load time series simulations generated approach aggregate time series built appliance level consumption time series used different appliancelevel models simulations bernoulli iid first order markovian higher order markovian tree cases models used bernoulli iid realistic consumption model aim prove cac concept requires measuring probability state maximum value consumption realistic widely used model first order markovian model model described transition probability matrix realistic model among three approaches applied distributions holding times states separately leads generally higher order markovian hom model benefit hom models capability model long range dependence samples usual property real load time series figure examples iid hom time series seen cases models fitted measured data coming redd dataset dataset contains appliance level power data homes several weeks sampling time seconds figure different bounds using chernoff bennett inequalities cac algorithm sets one order magnitude lower ratio regardless results acceptable decrease number accepted appliances details see section applying hoeffding bound leads results highly depend pon value applying chebisev markov bound lead poor results regardless values performance clt based cac close analytic calculation note clt upper bound tail probability consequence values occur pon cmax mean analytic clt chernoff figure iid bernoulli top hom middle model original measurement refrigerator bottom bennett hoeffding chebisev figure different bounds cac algorithm descripted equations assumes iid appliance load time series important question complex time series models affect cac figure demonstrates slight performance degradation even hom model pieces microwave ovens probability simulation length time instances pon cmax mean original time series modified time series analytic clt chernoff bennett chebisev markov hoeffding load however case small probabilities clt results higher ratio chernoff bennett remains almost range like figure load shape modification made cac however basic mathematical idea cac limit underconsumption probability direct objective dsm methods expressed load shape modification time domain instance valley filling peak clipping cac algorithm stated forms pdf aggregate consumption towards function equivalent constant load time domain section demonstrate effectiveness cac algorithm regarding load shaping assumptions consumption demand temporarily disabled appliances deleted selection appliances temporarily disabled based random selection guarantees fairness one appliance class appliances shiftable stochastic type figure original aggregate consumption time series modified one seen figure one see form algorithm yield almost load shaping hypothesis treatment consumption demand temporarily disabled appliances referred scheduling strategy plays key role algorithm perform load curve modification prove changed scheduling strategy cac strategy figure load shape modification ability cac algorithm scheduler figure alternative method handle disabled appliances case assumptions scheduler shifts consumption temporarily disabled appliance one time instant guarantees sum consumed energy remains modification load curve selection appliances temporarily disabled based random selection guarantees fairness one appliance class appliances schiftable stochastic type original time series modified time series load figure microwave oven hom model time instances time instances figure load shape modification scheduling clear cac scheduler able modify load shape red curve figure closer constant load factor increased widely used measure efficiency electric energy usage calculated based results planning investigate sophisticated scheduling methods future work number enabled appliances case different ldt bounds clt cac algorithm scheduler disables shiftable appliances aggregate consumption exceeds cmax upper limit higher probability allowed task step determine number appliances enabled appliance class qos must satisfied figure depicts number enabled appliances case different ldt bounds clt assuming one appliance class modelled bernoulli iid model appliances pon cmax mean analytic clt chernoff bennett hoeffding chebisev markov number enabled appliances figure number enabled appliances one class figure shows decision curve slightly nonlinear convex parameters figure highly nonlinear even state two decision regions generally linearly separable figure number enabled appliances number enabled appliances monotonously increasing function case one appliance class figure estimation probability overconsumption applying ldt bounds lead lower number enabled appliances compared analytically calculated value values applying ldt bounds stated causes spare capacities system clt bound lead values higher means breach contract exact percentages enabled appliances analytically calculated value reference collected table table percentage enabled appliances analytic clt chernoff bennett hoeffding chebisev markov reference reference qos guaranteed yes yes yes yes yes case two appliance classes cac algorithm decide enable different combinations appliances figure green colour indicates allowable set appliances red colour indicates combinations qos satisfied assumptions two appliance classes modelled bernoulli iid model appliances classes tail probability exactly calculated analytical aggregate pdf figure number enabled appliances one class separator curve depends different ldt bounds clt applied cac next two figures figure depict investigations regarding number enabled appliances case different tail probability estimation methods assumptions two appliance classes modelled bernoulli iid model appliances classes reference tail probability exactly calculated analytical aggregate pdf first experiment figure values second experiment figure difference performance degradation smaller first case difference values ratio large case higher ratio figure separator curves lie far causing severe performance degradation num apps satisfy qos cmax mean analytic clt pon chernoff bennett hoeffding pon figure number enabled appliances two classes latter case figure exact separator fact reflected estimation methods num apps satisfy qos cmax mean analytic clt chernoff bennett pon pon figure number enabled appliances two classes table number enabled appliances seen certain parameters table percentage enabled appliances analytic reference reference reference clt chernoff bennett table percentage enabled appliances analytic clt reference reference reference chernoff bennett performance decrease caused different ldt bounds smallest case chernoff bound highly sensitive ratio case utilization loss caused chernoff bound case clt near performance experiments number enabled appliances higher reference causes breach contract regarding qos criterion time computational complexity clt substantially lower chernoff bound analytical convolution result recommend using analytical computation possible case lack time importance satisfying qos chernoff bound comes fore clt lowest computational need quite good performance guarantee qos criterion conclusions future work paper new statistical approach proposed managing balance demand available supplies smart grids smart meter subscriber performs task shiftable appliances based two parameters obtained supplier upper capacity limit allowable probability overconsumption qos smart meter influences probability distribution function aggregate consumption order keep tail probabilities given threshold new approach takes uncertainty consumption account furthermore work fully distributed manner since calculations performed smart meter conducted several simulations evaluate performance cac result introduced consumption admission control method promising candidate demand side management smart grid environment acknowledgement research supported european union hungarian republic project design optimization modernization efficient operation energy supply utilization systems using renewable energy sources icts references lukas swan ismet ugursal modeling energy consumption residential sector review modeling techniques renewable sustainable energy reviews volume issue october pages issn pasaoglu fiorello martino scarcella alemanno zubaryeva thiel driving parking patterns european car drivers mobility survey european commission jrc institute energy transport petten netherlands http khan javaid arif saud qasim khan peak load scheduling smart grid communication environment advanced information networking applications aina ieee international conference may ullah mahmood razzaq ilahi khan javaid survey different residential energy consumption controlling techniques autonomous dsm future smart grid communications basic appl sci fang misra satyajayant xue guoliang yang dejun smart grid new improved power grid survey communications surveys tutorials ieee fourth quarter koutitas control flexible smart devices smart grid smart grid ieee transactions phillip oliver kriett matteo salani optimal control residential microgrid energy volume issue june pages issn costanzo zhu guchuan anjos savard system architecture autonomous demand side load management smart buildings ieee transactions smart grid vol wong jatskevich schober autonomous management based energy consumption scheduling future smart grid ieee transactions smart grid samadi wong schober tackling load uncertainty challenges energy consumption scheduling smart grid ieee transactions smart grid vol june levendovszky van der meulen tail distribution estimation call admission control atm networks proccedings ifip third workshop performance modelling evaluation atm networks ilkley west yorkshire july lin bai probability inequalities springerverlag berlin heidelberg hoeffding probability inequalities sums bounded random variables journal american statistical association vol bennett probability inequalities sum independent random variables journal american statistical association vol chernoff measure asymptotic efficiency tests hypothesis based sum observations annals mathematical statistics vol dasgupta anirban asymptotic theory statistics probability springer ardakanian keshav rosenberg markovian models home electricity consumption proc acm sigcomm work green netw greennets zico kolter matthew johnson redd public data set energy disaggregation research proceedings sustkdd workshop data mining applications sustainability authors addresses lorant kovacs phd kecskemet college faculty mechanical engineering automation gamf izsaki kecskemet hungary rajmund drenyovszki msc kecskemet college faculty mechanical engineering automation gamf izsaki kecskemet hungary andras olah phd catholic university faculty information technology utca budapest hungary janos levendovszky phd budapest university technology economics deptartment telecommunications egry budapest hungary levendov kalman tornai phd catholic university faculty information technology utca budapest hungary istvan pinter phd kecskemet college faculty mechanical engineering automation gamf izsaki kecskemet hungary
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logical methods computer science vol submitted published resource usage analysis naoki kobayashi kohei suenaga lucian wischik aoba sendai miyagi japan address koba aoba sendai miyagi japan address kohei microsoft way redmond usa address lwischik abstract propose resource usage analysis extended resource primitives goal resource usage analysis statically check program accesses resources files memory valid manner type system extension previous behavioral type systems guarantee safety property invalid access performed well property necessary accesses close operation file eventually performed unless program diverges sound type inference algorithm type system also developed free programmer burden writing complex type annotations based algorithm implemented prototype resource usage analyzer authors knowledge first resource usage analysis deals expressive concurrent language like introduction computer programs access many external resources files library functions device drivers etc resources often associated certain access protocols example opened file eventually closed file closed access allowed aim resource usage analysis statically check programs conform access protocols although number approaches including type systems model checking proposed far resource usage analysis similar analyses focused analysis sequential programs treat concurrent programs especially involving dynamic channels resources present paper propose method resource usage analysis concurrent languages dealing concurrency especially important concurrent acm subject classification key words phrases type system verification concurrent programs resource usage analysis preliminary version appeared international conference verification model checking abstract interpretation vmcai charleston usa january logical methods computer science kobayashi suenaga wischik creative commons kobayashi suenaga wischik programs hard debug also actual programs accessing resources often concurrent use extended resource primitives target language analysis applied wide range concurrency primitives including dynamically creating passing channels uniform manner main new difficulty dealing concurrent programs control structures complex concurrent programs sequential programs example consider following process read close read reads sends signal channel parallel close waits signal channel closes synchronization channel closed read capture kind causal dependency communications resource access use ccs processes extra type information called behavioral types example process given behavioral type using behavioral types introduced construct type system resource usage analysis manner similar previous behavioral type systems type judgment form usual type environment behavioral type approximating behavior free channels resources example process typed behavioral types also used augment channel types judgment given chanh res behavioral type simply single input command characteristic feature kind type system behavior input continuation accounted output input channel argument type res specifies resource sent along channel read first closed using type environment output process shri typed res shri behavioral type output followed continuation continuation obtained substituting argument type way types propagate information resources channels passed thorough channels accessed important property type system types express abstract behavior processes certain properties processes verified verifying corresponding properties types using example model checking techniques latter properties behavioral types amenable automatic verification techniques like model checking former ones types channel mobility also types typically represent behavior part entire process technical contributions present work summarized follows formalization type systems resource usage analysis proof soundness augmented previous behavioral types hiding renaming constructors adapted problem resource usage analysis resource usage analysis processes used types also previous work type systems igarashi kobayashi however used fragment without hiding renaming chaki used fragment without renaming present paper uses hiding renaming inclusion hiding renaming important accuracy automatic inference see remark realization fully automatic verification making analysis precise igarashi kobayashi gave abstract type system without giving concrete type inference algorithm chaki requires type annotations full automation enabled combination number small ideas like inclusion hiding renaming type constructors approximation ccslike type petri net reduce problem checking conformance inferred types resource usage specification verification usual safety property invalid resource access occur also extended safety call partial liveness necessary resource accesses closing file eventually performed unless whole process diverges partial liveness guaranteed chaki type system noteworthy point type system guaranteeing partial liveness parameterized mechanism guarantees deadlockfreedom sense kobayashi definition type system combined mechanism model checking abstract interpretation another type system whatever verify yoshida graph type system implementation prototype resource usage analyzer based proposed method implementation tested http rest paper structured follows section introduces extension primitives creating accessing resources section introduces type system resource usage analysis guarantees processes never perform invalid resource access section gives type inference algorithm type system section extends type system guarantee necessary resource accesses closing opened files eventually performed unless program diverges section describes prototype resource usage analyzer implemented based present work section discusses related work section concludes processes section introduces syntax operational semantics target language syntax definition processes set processes defined following syntax processes else true false values kobayashi suenaga wischik range countably infinite set var variables ranges set labels called access labels called trace set denotes set sequences access labels prefixes like bind tighter parallel composition access label specifies kind access operation typical access labels going use paper initialization read write close process accesses resource behaves like often write init read write close acci accr accw accc process creates new resource bound name accessed according behaves like specifies set acceptable sequences operations allowed new resource example creates resource first initialized read written arbitrary number times closed prefix closure write empty sequence often abbreviate sequence write xhe often omit trailing write xhe respectively xhe bound free variables defined customary manner also binds identify processes assume implicity applied bound variables always different free variables operational semantics formally define operational semantics process calculus operational semantics almost standard reduction semantics except trace sets represent resources accessed future may change reduction definition structural preorder least reflexive transitive relation closed rules figure stands remark previous behavioural type systems structural relation asymmetric standard symmetric structural relation used type preservation property would hold necessarily imply type system introduced next section definition set reduction labels ranged var define target target target definition let set sequences access labels defined definition reduction relation least relation closed rules figure write write reflexive transitive closure notice invalid access resource occurs program accesses specification resource specification set resource usage analysis free free figure structural preorder xhe target target true else false else figure reduction relation hand indicates resource correctly used far indicates one correctly completely used definition process contain form kobayashi suenaga wischik give type system guaranteeing process reduced process words process never performs invalid access section example following process first creates resource first initialized read arbitrary number times closed spawns four processes synchronize channels accessed valid order init initialize send signals read wait signal read signal read wait signal read signal close wait close example following program prototypical recursive functions replicated service listens channel either terminates recursion sending message back reply channel recursively invokes reply private channel example recursive step read following program use integer decide whether recurse though language integers operations primitives trivial extend language type system primitives rhi else shn read init close program corresponds following program init parbegin read read parend close example consider following producer accp chi producer consumer acc phi producer buf producer hbuf consumer hbuf xhi first two processes producer consumer define behavior producers consumers producer repeatedly waits receive signal performs put buffer accp sends signal consumer repeatedly waits receive signal performs get buffer accp sends signal third process creates new buffer put get applied alternately creates two channels used synchronization runs infinitely many producers consumers remark treat resources primitives paper could alternatively express resource tuple channels corresponds access operation example resource example expressed tuple consisting example taken ealier version modified resource usage analysis three channels init read close could directly reuse previous type systems infer properties discussed paper different precision treating resources primitives however simplifies type systems introduced later sections clarifies essence expressed resource tuple channels would need primitives simultaneous creation multiple channels need care whether communications resource access channels succeed hand resource access primitives simplifies particular extended type system discussed section type system section introduces type system prevents invalid access resources type system section guarantee liveness property necessary accesses eventually made extensions guarantee property discussed section types first introduce syntax types use two categories types value types behavioral types latter describes process accesses resources communicates channels mentioned section use ccs processes behavioral types definition types sets value types behavioral types defined bool res chanh communication labels behavioral type ccs process describes kind communication resource access process may perform describes process performs communication resource access types describe processes first perform action behave according actions respectively input output access operation invisible action describes process performs communications resource access according possibly parallel describes process behaves according either describes process behaves like arbitrary number times possibly parallel abbreviated xia denotes simultaneous renaming describes process behaves like hidden channel example describes process performs output invisible action type describes process behaves like recursive process fined type describes process behaves like except actions whose targets replaced invisible action describes process behaves like except actions whose targets replaced formal semantics behavioral types defined later using labeled transition semantics value types bool type booleans res type resources type chanh abbreviated chanh describes channels carrying tuples consisting values types type approximates replication semantics section differentiated section predicate disabled kobayashi suenaga wischik receiver channel may use elements tuple communications resource access example chanh res res describes channels carrying pair resources party receives actual pair first read close sometimes omit write chanh chanh empty also write chanhi note treated constructor rather operator performing actual substitution write latter throughout paper xia slightly different relabeling standard ccs allows communication relabeling ccs difference calls introduction special transition label section definition set free variables written defined xia defined xia bind variables respectively identify behavioral types renaming bound variables rest paper require every channel type chanh must satisfy example chanh res valid type chanh res semantics behavioral types give labeled transition relation behavioral types transition labels distinct reduction labels definition label indicates potential react presence substitution identifies also extend target function transition labels target target target transition relation behavioral types least relation closed rules figure write reflexive transitive closure also write constraint removed assume free variables codom never clash bound variables judgment form given later particular need implicit assumption figure resource usage analysis xia target target target target target figure transition semantics behavioral types remark confused hiding operator ccs replaces actions example make transition set tracesx defined set possible access sequences described definition traces tracesx note tracesx hence trace set definition kobayashi suenaga wischik define subtyping relation intuitively means process behaving according also viewed process behaving according put another way means simulates define closed types containing free type variables definition subtyping subtyping relation closed behavioral types largest relation implies often write write remark note subtyping relation defined converse one used igarashi kobayashi generic type system due two different dual views behavioral types think behavioral types describing behavior processes hand igarashi kobayashi think behavioral types describing assumption environment kind process accepted environment difference write behavioral types lefthand side write choice instead remark depending property type system guarantee finer subtyping relation may need chosen example definition allows may want disallow relation want infer property like simultaneous writes following properties satisfied proofs see appendix lemma precongruence closed behavioral type constructor tracesx tracesx typing consider two kinds judgments values processes mapping finite set variables value types type environment describes types variables describes possible behaviors example chanh bool implies may send booleans along channel twice judgment chanh chanh bool means may perform input may send boolean received value note judgment type approximation behavior free channels may less specified must example chanh holds chanh invariant perform invalid access neither write dom domain write empty type environment write distinct type environment dom dom write type environment dom dom dom define value judgment relation least relation closed true bool false bool resource usage analysis write abbreviation definition type judgment relation least relation closed rules given figure explain key rules rule first premise implies continuation output process behaves like second premise chanh implies tuple values sent may used input process according therefore whole behavior output process described stands vik vik true false example stands note previous behavioral type systems resource access communications made receiver counted behavior output process see remark rule first premise implies continuation input process behaves like following previous behavioral type systems split two parts first part describes behavior received values taken account channel type second part describes resource access communications performed values taken account behavioral type input process condition requires access communication behavior conforms channel arguments behavior premise implies behaves like behaves like require channel unlike previous behavioral type systems interested resource access behavior communication behavior used accurately inferring resource access behavior check process behavior conforms resource usage specification rule allows type process replaced approximation remark weakening derived appendix lemma needed rule following example shows information usage resources input process propagated output process example let consider yhx read let chanh res following judgment holds output input processes yhx read used subtyping relations using obtain res kobayashi suenaga wischik chanh xhe chanh bool else chanh res res tracesx figure typing rules using get res since tracesx obtain using example recall example rhi else shn read init close let let chanh int res chanhi int res chanhi resource usage analysis long tracesx obtain see section algorithm establishes tracesx remark type example demonstrates recursion hiding renaming used together general order type recursive process form shyi need find type satisfies moreover type inference section must find least thanks type constructors recursion hiding renaming always expressed recall lemma remark reader may wonder rules asymmetric sense information continuation receiver process transferred sender process vice versa design choice comes observation channel resource exchanged sender receiver general statically known sender put information behavior channel resource type sender example consider process xhyi since receiver scope put information used output type sender xhyi still useful possible recover symmetry treatment senders receivers extent see section previous paper following theorem states process performs invalid access resource theorem type soundness safety suppose safe safe proof make use following lemma proof see appendix proof theorem focus single reduction step lemma know judgements preserved reduction must show safety also preserved induction derivation reduction interesting case since rules alter case given tracesx lemma assume safe hence induction hypothesis conditions tracesx get tracesx safe type inference algorithm section discusses algorithm takes closed process input checks whether holds similar type systems algorithm consists following steps extract constraints type variables based version typing rules reduce constraints trace inclusion constraints form tracesxn kobayashi suenaga wischik chanh xhe chanh bool else chanh res tracesx res figure syntax directed typing rules decide whether constraints satisfied algorithm step sound complete give overview step first two steps almost previous work step extracting constraints typing rules presented section transformed typing rules shown figure figure type environment obtained merging bindings defined every dom type equality syntactic equality means two sets typing rules equivalent following sense derivable exists holds derivable conversely derivable based rules obtain algorithm figure takes process outputs triple consisting type environment behavioral type set constraints figure defined given dom dom dom dom dom dom dom dom triple output satisfies following properties holds substitution resource usage analysis exists substitution may contain variables representing unknown behavioral types value types set constraints substitution replaces closed behavioral types value types intuitively triple expresses set type judgments first property says triple contains valid judgments second property says every valid judgment subsumed triple give formal proof properties usual proved induction structure step reducing constraints given closed process produces triple set constraints consists unification constraints value types behavioral types occurring variables constraints form ischan means channel type subtype constraints behavioral types form constraints form tracesx remove first two kinds constraints unification constraints value types ischan applying standard unification algorithm thus obtain following constraints tracesxm assume different since replaced lemma also assume contains type variables constraint since otherwise always add tautology subtype constraint also replaced using lemma therefore constraints reduced lemma tracesxm least solutions subtype constraints thus reduced type checking validity trace inclusion constraints form tracesx example recall example applying algorithm first part step obtain following constraints tracesx applying second part step obtain tracesx kobayashi suenaga wischik ptv fresh ptv bool true false xhe let ptv chanh fresh let chanh fresh let else let ptv bool let dom ischan else let let res let res tracesx figure type inference algorithm step constraint solving present approximation algorithm checking trace inclusion constraint tracesx trace set regular language actually extend algorithm deal case deterministic petri net language see remark first describe algorithm example example reduced typability process equivalent constraint tracesx resource usage analysis removed since step approximate behavior petri net part similar translation usage expressions petri nets kobayashi previous work since behavioral types expressive recursion hiding renaming however need approximate behavior behavioral type unlike previous work case infinite make tractable make sound approximation pushing top level eliminate pictured treat clarity also use version petri nets labeled transitions rectangles places net dots labeled etc transitions net write number tokens node behavior together corresponds initial marking say nodes restricted names constitute basis note tracesx tracesx ptraces ptraces set traces petri net thus ptraces sufficient condition tracesx key point still infinite states reachable states expressed form parallel composition copies could express linear combination finitely many processes petri net kobayashi suenaga wischik step construct deterministic minimized automaton accepts language initial marking step construct another petri net simulates behavior simultaneously problem tracesx ptraces equivalent reachability problem example initial marking transitions ptraces following unsafe state unreachable explain behavior able make transition specification automaton able step use approximation algorithm decide reachability problem manner similar kobayashi analyzer typical steps described detail see section step step construction first introduce notion basis basis analogous vector space state expressed linear combination elements basis definition pair basis following conditions satisfied nat exist nat finitely many note basis whenever exist let write index one tuple index picks one among therefore basis behavior simulated labeled petri net given use syntax represent elements petri net rather standard tuple notation marking state tokens place written transition consumes marking produces expressed label transition set places pbn resource usage analysis initial marking pbn set transitions consists pbj pbn index pbj pbn index pbj pbn index index pair transitions omit basis write let write ptraces set means construction ptraces tracesx construction outlined applied basis found heuristic algorithm basis found approximate moving example replaced respectively let approximation easy prove sound approximation sense tracesx tracesx compute basis follows see appendix details since appear inside recursion first eliminate constructors let resulting expression contain let set behavioral types subexpressions behavioral types obtained expanding recursive types contain unnecessary unfolding finite set basis therefore construct petri net construction ptraces tracesx tracesx ptraces sufficient condition tracesx steps construction reduction tracesx reachability problem let pna tna sets places transitions respectively let minimized deterministic accepts let set states transition function definition composition written defined follows set places pna set transitions tna tna initial state initial state initial state since states minimized automaton accepting states kobayashi suenaga wischik ptraces reduced reachability problems theorem ptraces marking satisfies following conditions reachable undefined thus reduce ptraces finite set reachability problems hence ptraces decidable corollary ptraces every transition rule form undefined marking reachable remark actually extend algorithm checking tracesx deal case belongs class deterministic petri net languages precisely class languages deterministic petri nets language deterministic petri net complement petri net language therefore construct petri net accepts intersection language ptraces reduced emptiness problem petri net decidable due decidability reachability problem useful resource usage specifications regular languages deterministic petri net language example consider resource point program execution number times operation pop performed less number times push performed specification expressible deterministic petri net language extensions type system given far guarantees invalid resource access performed necessary access performed eventually example type system guarantee file eventually closed discuss extensions type system guarantee properties interested type systems satisfy either partial liveness stronger liveness property partial liveness contain resource access must performed liveness fair reduction sequence eventually performs necessary resource access reduction sequence fair input output action infinitely enabled eventually succeed without fairness assumption process satisfy liveness property presence divergent process restrictive standard term actually partial liveness viewed safety property bad state reachable necessary accesses yet performed system make move resource usage analysis idea take resource type system previous sections combine existing system annotates communications eventually succeed specifically existing system might guarantees annotated communications eventually succeed unless process diverges combination would guarantee partial liveness existing system could lockfreedom guarantees annotated communications eventually succeed even presence divergence assuming strongly fair scheduler combination would guarantee full liveness formally state resource access must performed extend trace sets definition extended trace set set sequences access labels possibly ending special label closed prefix operation intuitively special label means resource access need performed example trace set means close operation needs performed means close operation need performed state partial liveness property formally write possibly empty sequence definition process partially live whenever type system partial liveness property extend syntax processes allow input output prefix annotated information whether communication guaranteed succeed definition extended processes set extended processes given attributes attribute indicates annotated input output operation appears operation succeed unless whole process diverges give guarantee often omit tag assume exists type system guaranteeing process sense definition indeed type systems moreover static analysis tool typical automatically infer annotations definition active written active additionally writp ten well annotated active exists example note since although output never succeeds appear introduce type system guarantees partial liveness extend behavioral types extending input output attribute indicate whether action guaranteed succeed kobayashi suenaga wischik disabled disabled disabled disabled disabled disabled disabled disabled disabled disabled disabled disabled disabled disabled disabled disabled disabled disabled disabled disabled disabled disabled disabled disabled xia disabled disabled disabled figure definition disabled example process type implies process may send values twice first send guaranteed succeed sent value received process guarantee second send transition semantics behavioral types unchanged attribute ignored revise definitions subtype relation traces using following predicate disabled intuitively means describes process may get blocked without accessing resources definition disabled least binary relation extended behavioral types sets variables closed rules figure definition set etracesx extended traces disabled disabled means may last access attached sequence definition etracesx definition largest relation closed behavioral types satisfies following properties exists disabled implies disabled set variables note definition implies etracesx etracesx typing rules section except rules shown figure changes attributes attached tracesx replaced etracesx resource usage analysis chanh chanh res etracesx figure typing rules partial liveness important invariant maintained typing rules type process annotated process annotated example derive chanhi following theorem states soundness extended type system theorem well annotated partially live proof make use three lemmas first two show typing preserved reduction third means type process properly captures possibility process blocked subject reduction exists proof see appendix well annotated well annotated proof trivial definition well annotated disabled well annotated bool codom implies disabled proof see appendix ready prove theorem suppose well annotated show inversion typing rules get tain res tracesx sequence channel types variables bound also implies well annotated thus disabled get well annotated disabled implies disabled etracesx required example annotated version example else read init close kobayashi suenaga wischik suppose chanh int res chanhi long etracesx obtain type inference type inference algorithm extended type system almost algorithm basic type system discussed section changes constraint generaltion algorithm attribute annotations input ouptut processes propagated types example case output processes becomes let ptv chanh fresh constraint tracesx replaced etracesx second change forces adjust reduction constraint reachability problem petri nets recall step algorithm section first need use eptraces defined corresponds etracesx instead ptraces reduction definition eptraces set pdisabled initial marking pdisabled means disabled holds behavioral type expressed second construction automaton needs adjusted accepts extended traces example automaton used explanation step section replaced one accepts changes validity constraint etracesx reduced reachability problem petri net composition petri net automaton defined manner definition theorem eptraces marking satisfies following conditions reachable pdisabled undefined resource usage analysis implementation implemented prototype resource usage analyzer based extended type system described section tested examples given present paper implementation tested http analyzer takes program input uses typical annotate input output action attribute whether action guaranteed succeed automatically recall syntax extended processes section annotated program analyzed based algorithm described section followings design decisions made current implementation restrict resource usage specification regular languages although future may extend based remark step algorithm checking etracesx blindly approximate pushing future might utilize existing model checker handle case already finite step solving reachability problems petri nets approximate number tokens place element finite set approximation reduces petri nets finite state machines use bdd compute approximation reachable states figure shows part successful run analyzer first process second line input program runs server returns new initialized resource write output input actions resource access specification expressed number newr refers specification second process runs infinitely many client processes sends request new resource receiving reads closes third process line version replicated service example boolean passed first argument instead integer current system adapted handle integers affect analysis since system ignores value simply inspects branches conditional note program creates infinitely many resources infinitely many states first output annotated version input program produced typical output input attribute recall section remaining part shows trace inclusion constraint constructed petri net final line reports verification succeeded implies safety property section partial liveness property section satisfied related work resource usage analysis similar analyses recently studied extensively variety methods type systems model checking proposed however deal concurrent languages knowledge none deal partial liveness property total liveness property discussed section nguyen rathke propose system kind resource usage analysis functional languages extended threads monitors language neither resources monitors created dynamically hand target language type system applied programs may create infinitely many resources due existence primitives dynamic creation resources recall example figure also programs kobayashi suenaga wischik input new create create acc init new create false false close else acc read output result analysis new create create newr acc new create false false close constraints etrace acc init acc read acc close included initial marking places transitions close error found figure sample run analyzer use wide range communication synchronization primitives type systems deal concurrency certain degree section associating resource unique capability access resource type system control resource access order ensuring uniqueness capability keeping track access currently allowed capability approach however resource accesses completely serialized programmers care appropriately passing capabilities threads type systems also require rather complex type annotations igarashi kobayashi type system resource usage analysis extended deal threads introducing following typing rule spawn resource usage analysis describes resources used according used interleaving manner however obvious accurately capture information possible synchronizations model checking technologies course applicable concurrent languages suffer state explosion problem especially expressive concurrent languages like resources communication channels dynamically created passed around appropriate abstraction must devised effectively performing resource usage analysis model checking actually typebased analysis considered kind abstract model checking behavioral types extracted first two steps type inference algorithm abstract concurrent programs captures access behavior resource conformance abstract program respect resource usage specification checked model checking problem would interesting study relationship abstraction behavioral type abstraction techniques concurrent programs used model checking community perspective advantage approach type describes behavior much smaller state space whole program particular infinitely many resources dynamically created whole program infinite states often case behavioral types still finite indeed example figure limitation current analysis programs abstracted one way hand usual abstract model checking techniques refine abstraction step step verification succeeds technically closest type system igarashi kobayashi chaki rajamani rehof type systems developed checking communication behavior process viewing set channels resource possible use type systems directly resource usage analysis summarize similarities differences type systems type system present paper whether types supplied programmer inferred automatically types inferred automatically igarashi kobayashi generic type type system present paper type channel must annotated chaki type system annotated type contains information values channels particular sent along channel used senders receivers information used make type checking process compositional purpose resource usage analysis discussed think burden programmers declare channels going used since primary concern resources accessed channels ideal would allow user specify types infer others like purpose need develop algorithm check conformance inferred type declared type seems generally harder decide trace inclusion constraint tracesx expect able develop sound algorithm properly restricting language declared types languages used behavioral types three type systems use fragment ccs language types check dependency communications types igarashi kobayashi generic type system however lacks hiding type system applied obtain precise information resource usage fact analysis would fail even program example chaki type system use hiding lacks renaming constructor without kobayashi suenaga wischik renaming constructor general type necessarily exist hinders automatic type inference recall remark algorithms checking conformance inferred types respect specifications igarashi kobayashi generic type system check conformance inferred types respect specifications left open suggested could solved model checking problem chaki type system conformance expressed checking global behavior checking conformance declared types respect inferred types type checker piper conditions verified using spin restricted process corresponding conformance check work check trace inclusion constraints tracesx algorithm based reduction petri nets works even infinite states guaranteed properties igarashi kobayashi generic type extended type system present paper guarantee certain property necessary communications resource accesses eventually performed unless whole process diverges chaki type system type system section present paper guaranteed properties depend choice language behavioral types subtyping relation latter type systems ordinary simulation relation used process type describes possible behavior process behavior like certain resource access eventually performed rajamani recently introduced elaborate notion simulation relation called even conformance relation however type system still guarantee lack process hand relying external analysis check extension section keeps typing rules subtyping relation simple achieving guarantee necessary resource accesses eventually performed unless whole process diverges kobayashi type systems implementation form basis extended type systems partial total liveness properties discussed section used producing programs conversely behavioral types introduced paper used refine type systems yoshida honda also studied type systems guarantee certain properties type systems also used checking whether programs sense section section utilized existing analysis enhance result resource usage analysis type systems concurrent languages may also useful example type system atomicity used infer atomicity sequence actions source program using atomicity information may able reduce state space behavioral types check trace inclusion relation etracesx efficiently conclusion formalized type system resource usage analysis proved soundness also developed sound incomplete last phase deciding resource usage analysis trace inclusion relation tracesx algorithm order liberate programmers burden writing complex type annotations also implemented prototype resource usage analyzer based algorithm remains much future work necessary assess effectiveness analysis including design type system algorithm deciding trace inclusion relation tracesx detail refine analysis necessary also necessary make analyzer devising method generating comprehensive explanation verification result currently analyzer gives answer extensions type system deal typical synchronization primitives like internal choice also left future work references ball cook levin rajamani slam static driver verifier technology transfer formal methods inside microsoft integrated formal methods volume pages ball rajamani slam project debugging system software via static analysis proceedings acm symposium principles programming languages pages chaki rajamani rehof types models model checking programs proceedings acm symposium principles programming languages pages cormac flanagan type effect system atomicity proceedings acm sigplan conference programming language design implementation pages deline enforcing protocols software proceedings acm sigplan conference programming language design implementation pages deline adoption focus practical linear types imperative programming proceedings acm sigplan conference programming language design implementation foster terauchi aiken type qualifiers proceedings acm sigplan conference programming language design implementation pages fournet hoare rajamani rehof conformance cav volume lecture notes computer science pages honda yoshida uniform type structure secure information flow proceedings acm symposium principles programming languages pages igarashi kobayashi generic type system theoretical computer science igarashi kobayashi resource usage analysis acm transactions programming languages systems preliminary summary appeared proceedings popl kobayashi typical static analyzer tool available http kobayashi partially typed process calculus acm transactions programming languages systems kobayashi type system processes information computation kobayashi information flow analysis acta informatica kobayashi saito sumii process calculus proceedings volume lecture notes computer science pages august marriott stuckey sulzmann resource usage verification proceedings first asian symposium programming languages systems aplas volume lecture notes computer science pages kobayashi suenaga wischik mayr algorithm general petri net reachability problem siam journal computing milner communication concurrency prentice hall nguyen rathke typed static analysis concurrent resource access control draft pelz closure properties deterministic petri nets stacs annual symposium theoretical aspects computer science volume lecture notes computer science pages peterson petri net theory modeling systems rajamani rehof models contract conformance first international symposium leveraging applications formal methods skalka smith history effects verification proceedings first asian symposium programming languages systems aplas volume lecture notes computer science pages yoshida graph types monadic mobile processes volume lecture notes computer science pages yoshida liveness guarantee presence nontermination nondeterminism technical report msc technical report university leicester april resource usage analysis appendix appendix properties subtyping relation section states proves properties subtyping relation used proof type soundness theorems particular proofs lemmas appendices type inference algorithm described section particular transforming constraints behavioral types actually two subtyping relations basic one definition extended one definition since proofs almost state prove properties basic extended ones simultaneously places additional condition check extended case places marked extended case discussing basic case attributes attached actions ignored also omit even extended case important lemma simulation relation subtyping relation reflexive transitive let relation behavioral types whenever implies disabled implies disabled condition required extended case proof part trivial definition show part suppose simulation show simulation whenever implies extended case disabled implies disabled suppose case trivial definition check case suppose show suppose also since simulation exists since transitive implies extended case show suppose disabled since simulation disabled implies disabled lemma structural congruence proof proofs standard kobayashi suenaga wischik next show precongruence first show basic type constructors lemma precongruence simple cases proof follow fact following relations xia show closed arbitrary type constructors ftv set free bound behavioral type variables lemma precongruence general cases ftv proof let prove induction derivation also prove disabled implies disabled induction structure extended case words hence lemma start case analysis last rule used required condition follows immediately consider case case case also construction required case show left case assumed make induction hypothesis note free gives exists remains prove condition resource usage analysis get used lemma part case show left case assumed make induction hypothesis gives remains prove exist hence cases cases follow immediately induction hypothesis case must derived induction hypothesis exists using get required case make assumed clash two substitutions swap around giving induction hypothesis hence required kobayashi suenaga wischik case xibx must derived induction hypothesis get remains prove let exists xic xic used fact preserved lemma part cases similar use fact preserved lemma extended case addition need show disabled implies disabled follows straightforward induction structure lemma substitution xia xia xia xib xia xib xia xia target xia xia proof parts straightforward although part case labels xib prove part construct relation simulations interesting case infer xib xib gives hence hence required part construct xia clash prove simulations focus two cases resource usage analysis suppose xia inferred target must infer requires target prove follows assumed clash also target target let target suppose either cases required inferred target suppose xia must infer requires target prove follows let target suppose either contradiction hence required lemma exclusion projection target target target target proof straightforward lemma simulation tracesx tracesx proof proofs largely standard part follows immediately definitions subtyping traces part suppose let ftv lemma suffices prove simulation suppose show exists induction derivation kobayashi suenaga wischik case analysis last rule used show main cases cases similar straightforward case derived thus case derived induction hypothesis exists thus remains show get case derived induction hypothesis exist thus get required case case derived assumed without loss generality free thus induction hypothesis exists using obtain required case equal exists required extended case also need prove disabled implies disabled proved induction derivation disabled show case disabled derived using last rule figure cases follow immediately induction hypothesis two cases consider resource usage analysis case disabled must deduced disabled induction hypothesis disabled assumption disabled required note case case let disabled must derived disabled note induction hypothesis get disabled disabled using last rule figure get disabled required appendix proof subject reduction property section prove subject reduction property used proofs theorems appendix prove basic extended cases simultaneously lemma weakening dom dom proof part straightforward part proved straightforward induction derivation lemma judgement substitution values processes xia proof part either case result follows case remark types never free names part induction derivation cases follow straightforwardly lemma consider four particular cases case inferred induction hypothesis lemma tion get xia xia hence required xia case inferred res tracesz assume clash lemma get xia giving tracesz tracesz xia hence tracesz xia lemma get res induction hypothesis res xia two together give xia kobayashi suenaga wischik process push substitution definition substitution operator behavior xia use lemma push hence required extended case replace traces etraces reasoning inferred case zhwi chanh part implies chanh induction hypothesis get three give process push substitution definition substitution operator behavior push using several parts lemma case inferred chanh use three deductions first part get chanh second assumption lemma get substitution right disappears assume clash substitution left pushed inside lemma together give third induction hypothesis get three give previous case push substitution process behavior get required lemma proof part induction derivation cases use lemma case uses lemma interesting case judgement must inferred res tracesx lemma infer res resource usage analysis assume lemmas get lemma get tracesx tracesx tracesx gives finally gives required extended case replace traces etraces reasoning part induction derivation show main cases cases straightforward case given xhe must deduced xhe xhe must deduced chanh chanh respectively must show pick existence guaranteed follows definition subtyping relation definition remains prove start judgment lemma hence kobayashi suenaga wischik therefore required result follows show follows lemma lemma lemma assumption definition case given must derived res show let behavioral type satisfies guaranteed exist follows case given must derived res tracesx show exists induction hypothesis exists satisfies res using get definition traces tracesx get tracesx using get remains show exists follows latter relation follows rule extended case replace traces etraces reasoning case follows immediately part induction hypothesis appendix proofs lemma theorem section gives proof lemma disabled used proof theorem lemma disabled well annotated bool codom implies disabled imply definition proof first note well annotated well annotated sufficient show iii resource usage analysis bool codom imply disabled prove induction derivation case analysis last rule case case disabled case case since disabled case case since disabled case case note implies implies induction hypothesis get disabled disabled implies disabled case case imply induction hypothesis get disabled also implies disabled required case case happen condition must form true else false else contradicts case case chanh imply induction hypothesis get disabled definition disabled get disabled case case happen since must form contradicts case similar case case must derived induction hypothesis get disabled condition disabled appendix computing basis behavioral type section appendix section let behavioral type form contain obtained pushing described section show compute basis kobayashi suenaga wischik constructor eliminated running algorithm elimupf elimup dom elimupf elimupf elimupf elimupf elimupf elimupf elimupf elimupf elimupf xia elimupf elimup elimupf elimupf elimupf elimupf target otherwise keeps recursive definitions cache avoiding repeated computation contain always terminates since range finite set powerset constructor removed manner eliminate renaming constructor using following algorithm elimrenf elimren dom elimrenf elimrenf elimrenf elimrenf elimrenf elimrenf elimrenf elimrenf elimrenf xia elimrenf elimren applying algorithms obtain equivalent type contain elements atoms defined modulo recursive types appear transitions atoms forms basis definition let behavioral type contain set atoms atoms least set satisfies following conditions atoms atoms atoms atoms atoms atoms atoms atoms atoms atoms atoms atoms
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nalysis variance andom bjects paromita department statistics university california davis davis usa oct october abstract mean variance provide way obtaining mean variance general metric space valued random variables used statistical analysis data objects lie abstract spaces devoid algebraic structure operations examples spaces include covariance matrices graph laplacians networks univariate probability distribution functions derive central limit theorem variance mild regularity conditions utilizing empirical process theory also provide consistent estimator asymptotic variance results lead test compare populations based variance general metric space valued data objects emphasis comparing means variances examine finite sample performance inference procedure simulation studies several special cases include probability distributions graph laplacians leads tests compare populations networks proposed methodology good finite sample performance simulations different kinds random objects illustrate proposed methods data mortality profiles various countries resting state functional magnetic resonance imaging data key words functional data analysis mean variance central limit theorem two sample test samples probability distributions wasserstein metric samples networks graph laplacians fmri corresponding research author email pdubey supported national science foundation grants introduction increasing abundance complex data multiple disciplines need statisticians develop techniques suitable analysis data settings data objects take values metric space devoid vector space structure common occur example pairwise distances observed data objects available natural assume observed samples random objects drawn distribution metric space standard problem testing important case concerning two sample case basic interest statistics becomes challenging data objects lie general metric spaces case euclidean data classical sample comparison problem gaussian data corresponds classical analysis variance anova comparing means primary interest uses comparisons summary variation measures within groups general metric space valued random variables provided direct generalization mean implies corresponding generalization variance may used quantify spread distribution metric space valued random variables random objects around mean mean resides object space thus amenable algebraic operations implies central limit theorem directly applied obtain limit distributions means contrast variance always scalar makes tractable key result paper central limit theorem empirical variance data objects general metric spaces weak assumptions result interest demonstrate applied derive test based groupwise variances testing null hypothesis equal population distributions random objects emphasis testing equality means variances recent years study nonparametric tests equality two distributions euclidean data expanded cover data increasingly encountered frameworks feature large complex data tests suitable random objects metric spaces typically based pairwise distances major approaches based nearest neighbors graphs energy statistics kernels nearest neighbor based tests henze henze penrose schilling count number neighbor comparisons observations neighbors belong sample choice tuning parameter impacts resulting inference two sample tests friedman rafsky pairwise distances pooled sample used form graph minimum spanning tree test statistic number subtrees formed removal edges defining nodes belong different samples recently modified versions test proposed chen friedman chen also based similarity graph pooled observations weighted edge count test order deal problem unequal sample sizes since difficult form within sample edges samples smaller sizes weights proportional reciprocal sample sizes assigned within sample edges related approach minimum distance pairing rosenbaum dataset split disjoint pairs minimizing total distance within pairs test statistic number cross matches class statistics popularly known energy statistics initially proposed case multivariate two sample test baringhaus franz rizzo based fact independent random vectors distribution distribution finite expectations inequality holds equality two distributions compared framework extended case metric space valued random variables lyons metric spaces strong negative type leading distinction measures finite first moments respect underlying metric extension energy statistics approach spaces admitting manifold stratification patrangenaru ellingson explored recently guo patrangenaru recent review rizzo kernel based two sample testing gretton aims finding maximum mean discrepancy mmd largest difference expectations functions unit ball reproducing kernel hilbert space rkhs corresponds distances embeddings distributions rkhs distribution free tests obtained based large deviation bounds empirical estimates also asymptotic distribution mmd energy statistics approach linked kernel method two sample testing distance based energy statistic computed using negative type equivalent mmd statistic obtained special type kernel induced sejdinovic every positive definite kernel mmd statistic corresponds energy statistic obtained derived kernel used compute mmd connections kernel methods energy statistics wasserstein distance distributions compared explored ramdas commonly used kernels kernel framework like gaussian kernel might sensitive choice tuning parameter needed scale kernel determining good scaling factor challenging goal inference complex data confirm simulations problematic adopting kernel method obtain inference empirically tests good power performance either location type alternatives scale type alternatives usually simultaneously major challenge tests choice required tuning parameters often major impact resulting inference challenges associated existing inference procedures motivate proposed test simple based test statistic mimics statistics tests classical anova based replacing within sums squares corresponding variances separate combined samples easy compute sample comparisons mean based testing corresponding large sample theory including laws large numbers central limit theorems empirical means explored previously data objects lie special metric spaces smooth riemannian manifolds bhattacharya patrangenaru bhattacharya bhattacharya topologically stratified spaces certain restrictions like phylogenetic trees kendall barden bhattacharya lin virtually results depend local linear tangent similar approximations specific manifold spaces apply random objects general metric spaces require local euclidean approximation central limit theorem means recently applied space graph laplacians ginestet interest obtain inference networks required choosing high dimension approximating space thus leading problems small sample high dimensional data ensuing complications inference since mean based testing exploits local euclidean approximation property underlying space extend data general metric spaces space univariate probability density functions serve one examples illustrate proposed approach another drawback methods lose power data high dimensional large covariance matrices inverses need estimated network nodes mean graph laplacians covariance matrix dimension address issue strong assumptions sparsity invoked estimation large covariance matrices inverses order implement corresponding tests provides additional motivation approach simple dimension data enters indirectly properties metric obtain asymptotic properties draw empirical process theory similar spirit recent study regression relationships general metric space valued random objects responses real valued predictors petersen goal paper develop simple straightforward extension anova one basic tools inference statistics case metric space valued random objects starting point totally bounded metric space equipped metric show consistency sample mean derived using results petersen concerning regression estimators mild assumptions derive central limit theorem variance mild assumptions provide consistent estimator asymptotic variance method applicable wide class objects including correlation matrices univariate probability distributions manifolds also space graph laplacians making use central limit theorem derive new test random objects study asymptotic distribution test statistic null hypothesis equality population distributions well power function customary test heteroscedasticity population groups prior classical anova evaluate assumption equal variances across populations compared one popular test purpose proposed levene test statistic form usual anova applied could principle monotonic function absolute deviations observations group centers proposed test unifies levene test classical anova testing inequality population means data objects general metric spaces therefore aims location scale type alternatives instead location alternatives objective classical anova test statistic sum two components one component proportional squared difference pooled sample variance weighted average groupwise variances weights proportional sample sizes groups special case euclidean data part statistic proportional squared usual anova useful detect differences means populations key auxiliary result statistic converges zero rate null hypothesis equality means population distributions bounded away zero otherwise component test statistic accounts differences variances population groups euclidean setting simplifies generalization levene test applied squared absolute deviations observations group means assumptions central limit theorem hold asymptotic distribution component test statistic number populations compared paper organized follows basic set defined section theory regarding asymptotic behavior variance provided section test introduced section followed empirical studies study power performance section data applications section include samples univariate probability distributions employ wasserstein metric samples graph laplacians networks use frobenius metric section illustrate proposed tests human mortality records countries time period analyze evolution age death distributions countries years section analyze resting state fmri dataset compare probability distribution positive correlations fmri signals posterior cingulate area brain alzheimer disease patients similarly aged normal subjects finally section analyze brain connectivity networks patients dementia derived fmri signals certain brain regions buckner compare connectivity networks demented patients three different age groups preliminaries mean generalization centroids metric spaces special case euclidean data includes arithmetic mean median geometric mean different choices distance functions variance corresponding generalized measure dispersion around mean formally following totally bounded metric space metric probability measure random objects following random variables take values population mean argmin random sample random variables distribution corresponding sample mean argmin sample mean obtained minimizing sum functions data objects population variance quantifies spread random variable around mean corresponding sample based estimator note asymptotic consistency sample mean follows results petersen following assumption objects exist unique infd assumption instrumental establish weak convergence empirical process population process turn implies consistency consistency implies consistency since diam diam sup finite since totally bounded central limit theorem clt hold empirical variance need assumption entropy integral space wellner van der vaart log metric centered covering number using open balls radius specifically clt assume results power proposed test section need additional assumption entropy integral whole space entropy integral finite log random objects satisfy assumptions include space univariate probability distributions finite second moments equipped wasserstein metric spaces correlation matrices graph laplacians fixed dimensions equipped frobenius metric two univariate distributions finite variances distance also known earth movers distance closely related optimal transport villani quantile functions corresponding respectively two matrices dimension consider frobenius metric trace better characterize space graph laplacians denote weighted undirected graph set vertices set edges given adjacency matrix equality zero holds graph laplacian defined diagonal matrix degrees vertices assumption graphs simple self loops multi edges one one correspondence space graphs graph laplacians therefore graph laplacians used characterize space networks ginestet following results imply spaces described provide examples spaces satisfy assumptions inference methods therefore apply compare samples univariate distributions correlation matrices networks proposition space satisfies assumptions set consists univariate probability distributions finite second moments wasserstein metric proposition space satisfies assumptions set consists graph laplacians connected undirected simple graphs fixed dimension correlation matrices fixed dimension frobenius metric proofs supplementary materials central limit theorem variance following proposition lays foundations proving central limit theorem empirical variance proposition suppose assumptions hold proposition makes possible deal sum dependent random variables replacing sum random variables crucial step derivation central limit theorem since totally bounded population variance var always finite central limit theorem variance follows theorem assumptions proposition var intuitive sample based estimator estimator following result shows proposition assumptions proposition cov cov var var therefore estimator proposition variant proposition follows simple application delta method quantity finite boundedness combining theorem proposition slutsky theorem leads simple application delta method gives asymptotic distribution standard deviation square root variance since consistent estimators one use construct asymptotic confidence intervals variance standard deviation depend quality large sample approximations bootstrap provides alternative often better finite sample properties weak assumptions bickel freedman beran fairly general assumptions resampling methods like bootstrapping permutation tests work whenever central limit theorem holds janssen pauls basic criterion bootstrap confidence sets correct coverage probability asymptotically convergence bootstrap distribution root case monte carlo approximations bootstrap distribution root provide approximate quantiles construction confidence sets empirical measure generated puts mass underlying measure measurable set indicator function set weak law large numbers given sample size drawn replacement bootstrap proximation root sample based estimators variance asymptotic variance obtained bootstrap sample mean bootstrap sample exists unique almost surely conditionally applying central limit theod rem theorem conditionally since continuous distribution function real line sup distribution function conditional standard normal distribution function sup establishes asymptotic consistency bootstrap distribution therefore bootstrapping viable option construction confidence intervals variance comparing populations random objects assume sample random data objects belong different groups size wish test null hypothesis population distributions groups identical versus alternative least one groups different population distribution compared others consider sample means random objects computed data falling group corresponding sample variances variance estimates argmin well pooled sample mean corresponding pooled sample argmin well weights base inference procedures auxiliary statistics almost surely equal numerator classical euclidean anova specifically corresponds weighted variance group means weights proportional group sizes correspond group variance classical anova setting hence regarded generalization classical anova general setting metric space valued data euclidean setting simple algebra shows special case also proportional weighted average squared pairwise distances group means analogous anova numerator expected small null hypothesis equality population distributions indeed case following proposition demonstrates proposition suppose exist unique almost surely let null hypothesis equality population distributions assumptions groups nfn inference classical anova requires gaussianity equality population variances hence targets differences group means capture differences population distributions assumptions obviously restrictive employ statistics account differences among population variances euclidean case turns slightly modified version traditional levene test substituting squared distances observations group means instead distances following result provides asymptotic distribution null hypothesis proposition assumptions proposition null hypothesis nun construct test statistic combining way distribution null hypothesis given proposition gaining power alternatives ensuring asymptotic mean diverges departing null hypothesis nun constructing scale estimated standard deviation equal suitably scaled respect variability second term scaled terms order null hypothesis consistency proposition imply theorem null hypothesis assumptions proposition level test accordingly reject null hypothesis equality population distributions test statistic turns bigger quantile distribution rejection region defines test study consistency proposed test consider contiguous alternatives capture departures null hypothesis equal population distributions terms differences means variances begin defining following population quantities argmin denotes expectation probability distribution jth population defined proposition random objects distributed according jth population distribution euclidean setting analogous pooled population mean pooled population variance respectively general case interpreted generalizations quantities euclidean case proportional weighted sum differences groupwise population means pooled population mean simple algebra seen property still holds general metric case proposition states mild assumptions existence uniqueness pooled groupwise means statistics consistent estimators population quantities assumptions zero equality population means positive otherwise population quantity proportional weighted average pairwise differences groupwise variances nonnegative zero population groupwise variances equal proposition suppose exist unique sample based estimators almost surely assume infd also infd let defined proposition population quantity nonnegative zero iff population means equal study power performance proposed test consider sequences alternatives non negative sequences either strictly greater case either reflects contiguous alternatives interest asymptotic behavior power function inf following result require assumption examples considered satisfy assumptions shown propositions following result provides sufficient conditions consistency proposed test family contiguous alternatives asymptotic power alternatives choice level theorem assumptions proposition assumption sequences contiguous alternatives either power function satisfies nan nbn asymptotic tests may work well common situations group sample sizes modest arguments similar provided end section resampling methods like bootstrapping permutation tests using proposed test statistic adopted obtain accurate level tests empirical experiments presented next section found sample sizes groups large asymptotic test stable accurate small sample sizes using bootstrap permutation based critical values test statistic instead asymptotic critical values leads accurate inference simulation studies order gauge performance proposed test performed simulation experiments various settings random objects consider include samples univariate distributions equipped metric samples graph laplacians scale free networks model albert frobenius metric samples multivariate data usual euclidean metric case considered two groups equal size constructed empirical power functions proposed test departures null hypothesis equality population distributions two groups level empirical power computed finding proportion rejections monte carlo runs comparing performance proposed test existing tests used bootstrap version proposed test critical value test obtained bootstrap distribution monte carlo simulations performance bootstrap version found correct level sample sizes asymptotic version test always produce correct level test small sample sizes also investigated finite sample power asymptotic test increasing sample sizes comparing power functions group sizes simulations explored location differences also differences shape scale population distributions compared proposed test graph based test chen friedman energy test based pairwise distances rizzo kernel based test gretton graph based test constructed similarity graph pooled observations two groups constructing minimal spanning tree graph pooled pairwise distance matrix following suggestion chen friedman union kth msts kth mst spanning tree connecting observations minimizes sum distances across edges subject constraint spanning tree contain edge mst computing statistic energy test rizzo used pairwise distance matrix obtained specified metric space random objects kernel based method chose gaussian kernel kernel width median pairwise distances suggested gretton multivariate euclidean setting two sample case additionally compared proposed test hotelling test power power power functions power functions figure empirical power function probability distributions group group left empirical power function probability distributions right solid red curve corresponds bootstrapped version proposed test test statistic dashed blue curve graph based test chen friedman black curve energy test rizzo dotted green curve corresponds kernel test gretton level tests indicated line parallel sample sizes groups fixed power power power functions power functions figure empirical power function probability distributions group group left empirical power function probability distributions right proposed test different sample sizes tests level indicated line parallel solid curve corresponds sample sizes dashed curve dotted curve first type random objects study random samples univariate probability distributions datum distribution random distance two probability distributions choose metric first scenario group generate distributed group compute empirical power function tests second scenario drawn randomly group group empirical power evaluated first scenario emphasizes location differences populations second emphasizes scale differences results presented figure find first scenario mean differences proposed test graph based test perform similarly outperformed energy test perform better kernel based test second scenario scale differences proposed test outperforms tests figure indicates proposed test consistent large sample sizes scenarios figure indicates insufficient sample sizes bootstrap version proposed test stable rejection regions overall reliable asymptotic version test sample sizes increase asymptotic test becomes reliable yields results similar bootstrap test power power power power functions power functions power functions figure empirical power function probability distributions group group leftmost panel corresponds group sample sizes middle panel corresponds rightmost panel solid curve corresponds asymptotic version proposed test dashed curve corresponds bootstrap version level test indicated line parallel next consider samples graph laplacians scale free networks model frobenius metric popular networks power law degree butions commonly used networks related world wide web social networks brain connectivity networks scale free networks fraction nodes network connections nodes large values approximately typically range specifically used algorithm generate samples scale free networks nodes one might encounter brain networks group set group selected fixed interval studying empirical power function left panel figure indicates scenario proposed test better power behavior graph based test kernel based test kernel based test automatic scaling parameter choice gretton published software low power graph based test high false positive rate right panel figure shows empirical evidence proposed test also consistent scenario sample size increases figure especially small samples bootstrapping test statistic leads correct empirical level test increasing sample size asymptotic bootstrap versions test perform similarly power power power functions power functions figure empirical power functions networks model parameter left panel solid red curve corresponds bootstrapped version proposed test test statistic blue dashed curve graph based test chen friedman black curve energy test rizzo green dotted curve kernel test gretton sample sizes fixed right panel solid power function corresponds proposed asymptotic test dashed power function test dotted power function test level tests indicated line parallel power power power power functions power functions power functions figure empirical power function networks model model parameter leftmost panel corresponds group sample sizes middle panel rightmost panel solid curve corresponds proposed asymptotic test dashed curve bootstrap version proposed test level level indicated line parallel power power power functions power functions figure empirical power function vectors component distributed independently beta group beta varying group left empirical power function degrees freedom vectors distributed independently truncated group truncated multivariate varying degrees freedom group component vectors truncated lie right sample sizes level solid red curves correspond bootstrapped version proposed test test statistic dashed blue curves graph based test chen friedman dotted green curves corresponds kernel test gretton black curves energy test rizzo magenta curves hotelling test multivariate setting considered vectors distributed truncated multivariate normal distributions group components truncated lie group chose indicating degrees freedom degrees freedom increases shape distribution becomes similar group obtained empirical power functions second scenario took five components vectors distributed independently beta group group five components assumed distributed independently beta empirical power obtained function figure illustrates cases proposed test overwhelmingly outperforms comparison tests data illustrations mortality data human mortality database provides data form yearly lifetables differentiated countries gender presently includes yearly mortality data countries available converted density country gender calendar year first converting available lifetables histograms applying local least squares smoothing used hades package http bandwidth random objects consider resulting densities considering time period countries database records available time period obtained densities age death age interval densities obtained quantile functions compute wasserstein distance metric choose distribution space standard deviations computed function calendar year shown along pointwise bootstrap confidence bands figure separately males females one finds small peak variance mortality males followed larger peak female standard deviation time male standard deviation time figure yearly standard deviations solid line along pointwise bootstrap confidence limits dashed lines females top males bottom female male standard deviation standard deviation time female group male group standard deviation standard deviation time time time figure yearwise standard deviations solid line along pointwise bootstrap confidence limits dashed lines females group females group males group males group counter clockwise starting top left females later peak also quite prominent peaks might possibly attributed major political upheaval central eastern europe period since several countries dataset belong regions countries dataset experienced turmoil associated end communist role belarus bulgaria czech republic estonia hungary latvia poland lithuania russia slovakia ukraine females group group density age males group group density age figure mean densities years groups red blue females top males bottom check whether indeed countries responsible variance peak around split dataset two groups group consisting eastern european countries group countries repeated earlier analysis figure shows group variance distributions indeed decreasing trend years males females variance shows distinct females years males years figure testing differences population distributions groups years females top males bottom proposed test fluctuations males females group figure illustrates wasserstein mean densities countries various calendar years seems clear difference mean densities two groups males females implementing bootstrap version proposed test compare distributions groups obtaining carried bootstrap version proposed test due relatively small sample sizes figure illustrates obtained year null hypothesis populations identical rejected males females level years analyzing connectivities using fmri data alzheimer disease alzheimers disease irreversible progressive brain disorder slowly destroys memory thinking skills eventually leading severe dementia found associations abnormalities functional integration brain regions recent studies sui indicated selectively targets regions highconnectivity hubs brain posterior midline particular posterior pcp described buckner nexus hub high cortical connectivity functional connectivity region could potential biomarker hub seed voxel identified voxel signal highest correlation signals nearby voxels quantify connectivity following petersen analyze distribution correlations signal seed voxel pcp hub signals voxels within cube voxels centered seed voxel subjects analysis consisted cognitively normal elderly patients demented elderly patients diagnosed removal outliers underwent fmri scan davis imaging research center preprocessing recorded bold signals implemented adopting standard procedures correction head motion correction normalization addition linear detrending account signal drift filtering include frequencies signals subject recorded interval seconds measurements available second intervals study actually normal subjects since disease known progress age fair comparison selected matching ages demented patients check age matching worked age distributions normal elderly subjects analysis patients compared wilcoxon rank sum test null hypothesis equal age distributions two groups yielded subject target density function positive correlations within pcp hub density estimated observed correlations kernel density estimator utilizing standard gaussian kernel bandwidth negative correlations commonly ignored connectivity analyses densities estimated resulting sample densities sample across subjects figure shows wasserstein mean probability distribution functions compare two populations distributions applied asymptotic bootstrap version proposed test wasserstein mean probability distribution functions probability distribution function demented subjects normal subjects correlation figure wasserstein mean probability distribution functions positive correlations pcp hub normal subjects blue demented patients red samples density functions yielded bootstrap indicating significant differences exist terms connectivity patients normal subjects comparing brain networks alzheimer patients recent advances neurological studies revealed brain hubs regions high connectivity brain interconnect functional integration specialized roles studying interconnections hubs reveal important insights brain diseases like disorders cognition associated disrupted connectivity cortical hubs discussed buckner one question interest whether interconnections change aging patients dementia consider connections cortical hubs listed table buckner order analyze cognitively impaired patients considered demented subjects discussed preceding subsection subject connectivity matrix obtained whose entries correlations average fmri signals described subsection cubes around seed voxels hubs connectivity matrices thresholded discussed buckner obtain adjacency matrices networks hubs nodes presence edge indicates correlation greater graph laplacians formed adjacency matrices cognitively impaired patients split three groups based ages subjects assigned groups based whether aged age left panel figure shows difference average graph laplacians subjects group subjects group right panel difference average graph laplacians subjects group subjects group since group sample sizes small applied bootstrap version proposed test see differences significant populations graph laplacians three different groups cognitively impaired patients null hypothesis equality population distributions graph laplacians rejected bootstrap conclusion test indicates evidence support inter hub connections show changes age patients alzheimer disease hubs hubs hubs hubs figure left panel shows difference average graph laplacians demented patients aged demented patients aged right panel shows difference average graph laplacians demented patients aged demented patients aged references albert emergence scaling random networks science barden owen central limit theorems means space phylogenetic trees electronic journal probability baringhaus franz new multivariate test journal multivariate analysis beran impact bootstrap statistical algorithms theory statistical science bhattacharya bhattacharya nonparametric inference manifolds applications shape spaces cambridge university press bhattacharya lin omnibus clts means nonparametric inference spaces proceedings american mathematical society bhattacharya patrangenaru large sample theory intrinsic extrinsic sample means manifolds annals statistics large sample theory intrinsic extrinsic sample means manifolds annals statistics bickel freedman asymptotic theory bootstrap annals statistics buckner sepulcre talukdar krienen liu hedden sperling johnson cortical hubs revealed intrinsic functional connectivity mapping assessment stability relation alzheimer disease journal neuroscience chen chen weighted test multivariate object data journal american statistical association chen friedman new test multivariate object data journal american statistical association les nature quelconque dans espace friedman rafsky multivariate generalizations smirnov tests annals statistics ginestet balachandran rosenberg kolaczyk hypothesis testing network data functional neuroimaging annals applied statistics gretton borgwardt rasch smola kernel test journal machine learning research guo patrangenaru testing equality two distributions high dimensional object spaces arxiv preprint henze multivariate test based number nearest neighbor type coincidences annals statistics henze penrose multivariate runs test annals statistics janssen pauls bootstrap permutation tests work annals statistics kendall limit theorems empirical means independent distributed random variables brazilian journal probability statistics levene robust tests equality variances contributions probability statistics lyons distance covariance metric spaces annals probability patrangenaru ellingson nonparametric statistics manifolds applications object data analysis crc press petersen functional data analysis density functions transformation hilbert space annals statistics regression random objects euclidean predictors annals statistics appear arxiv preprint ramdas trillos cuturi wasserstein testing related families nonparametric tests entropy rosenbaum exact test comparing two multivariate distributions based adjacency journal royal statistical society series statistical methodology schilling multivariate tests based nearest neighbors journal american statistical association sejdinovic sriperumbudur gretton fukumizu equivalence statistics hypothesis testing annals statistics sui zhu cui sui zhang liu duan zhang wang zhang jiang functional connectivity hubs could serve potential biomarker alzheimers disease reproducible study current alzheimer research szarek metric entropy homogeneous spaces quantum probability rizzo testing equal distributions high dimension interstat energy data annual review statistics application villani topics optimal transportation american mathematical society wellner van der vaart weak convergence empirical processes applications statistics springer series statistics upplementary aterials main proofs proof proposition let space quantile functions corresponding space univariate distribution functions finite second moments random object taking values let denote corresponding quantile function convexity space properties distance population sample thereby means exist unique given qyi proving proving observe quantile functions part bigger class comprising monotone functions metric wasserstein metric corresponds metric space quantile functions metric entropy space upper bounded log constant per theorem wellner van der vaart therefore thus establishing entropy integral whole space log establishes proof proposition space graph laplacians graphs considered proposition convex ginestet space correlation matrices properties frobenius distance imply exist unique convexity proving hold space graph laplacians space correlation matrices bounded subsets euclidean space matrices euclidean metric equivalent frobenius metric matrices metric entropy space seen bounded log log constant due simple volume comparison argument szarek log log implying establishing using similar upper bounds entropy integral whole space log log establishing proof proposition define first step control behavior uniformly small define functions function class envelope function let integral theorems wellner van der vaart imply small sup therefore sup want show exists small inf sup since sup sup inf inf inf using inf assumption using markov inequality bound equation small expression bounded sup sup using consistency mean possible choose completes proof proof theorem first term proposition second term converges distribution applying central limit theorem random variables theorem follows directly slutsky theorem proof proposition observe proposition hence decomposed two components equation proposition converges distribution applying central limit theorem random vectors cov cov var var observing differentiable function gradient function simple application delta method establishes asymptotic normality totic variance given gradient function evaluated proof proposition null hypothesis groupwise means equal assumptions nfn follows applying proposition observations also individual groups noting slutsky theorem completes proof proof proposition null hypothesis let using proposition since find denominator simple algebraic manipulation shows numerator nun null hypothesis applying theorem individual groups find continuing see nun symmetric idempotent matrix orthogonal projection space orthogonal column space rank trace equals property orthogonal projector matrices applying continuous mapping theorem slutsky theorem nun limiting distribution quadratic form normal random variables therefore distributed distribution degrees freedom equal rank proof proposition proving consistency pooled mean arguments essentially proof lemma petersen consider weak law large numbers applied individual groups using lim diam find asymptotically equicontinuous probability sup allows use theorem wellner van der vaart conclude converges weakly applying wellner van der vaart converges zero probability assumptions corollary wellner van der vaart implies proving consistency enough prove consistency consistency groupwise variances follows earlier results observe diam implies first term second term also weak law large numbers applied individual groups whence converges probability clearly nonnegative individual group equality zero holding therefore always nonnegative zero proof theorem proof relies following auxiliary result uniform consistency estimators assumption boundedness entropy integral space lemma assumptions theorem holds denote respectively respectively lim lim lim statements supremum taken respect underlying true probability measure class possible probability measures generate random observations proof lemma found following subsection titled additional proofs similar notation proof proposition define statistic represented replicating steps proof proposition find converges distribution asymptotically moreover consequence lemma continuity uniformly consistent estimator sets defined uniform consistency implies sup sup writing quantile distribution represent limiting power function sup implies sequence hypotheses lim lim inf lim since lim sup lim converges distribution random variable find nan nbn lim completing proof additional proofs proof lemma since proof similar ignore index proof observe respectively equal observing inf inf inf sup sup replicating steps proof proposition one obtains given log finite entropy integral envelope function class indexes empirical process markov inequality next observe sup sup similar arguments see jdiam finite entropy integral envelope function class indexes empirical process markov inequality equations sup completing proof since proof similar ignore index proof proving uniform consistency enough prove uniform consistency rest follows lemma continuity similarly proof find observe analogy proof implies next observe sup sup similar arguments proof imply jdiam finite entropy integral envelope function class indexes empirical process markov inequality equations sup completes proof note since inf inf inf sup using similar arguments proofs individual groups control behavior defining function classes obtain constants sup sup markov inequality next observe nkn sup similar arguments markov ity constants equations sup completes proof
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sep dehn functions subgroups artin groups noel brady ignat soroko abstract show positive integer exist artin groups containing subgroups whose monodromy automorphisms grow consequence produce examples artin groups containing finitely presented subgroups whose dehn functions grow contents introduction preliminaries growth bounding dehn function bieri double cube complexes special cube complexes morse theory cube complexes groups log definition structure structure constructing special cover permutation representation cover exploring hyperplane pathologies proof main theorems growth upper bounds growth lower bounds growth lower bounds growth open questions references introduction intense interest subgroups artin groups raags recent years due largely work ian agol dani wise others virtual fibering question topology date september mathematics subject classification primary noel brady ignat soroko agol showed compact oriented irreducible subgroup raag virtually fibers haglund wise showed fundamental groups special cubical complexes subgroups raags building machinery agol went solve virtual fibering conjecture fundamental result curved cubical complexes hyperbolic fundamental groups virtually special paper consider two questions subgroups raags first question asks groups virtually embed raags hagen wise show hyperbolic groups virtually embed raags also known groups virtually embed raags hyperbolic examples exponentially growing monodromy automorphisms groups exponential linear monodromy automorphisms gersten gives explicit example group virtually embed raag group considered gersten cat group prompts following open question every cat group virtually embed raag family hydra groups considered provides test case question monodromy automorphisms grow polynomially arbitrary degree proved hydra groups virtually embed raags topic ongoing research author construct analogues hydra groups base subgroup replaced complicated raag cat polynomially growing monodromy automorphisms arbitrary degree virtually special theorem positive even integer exist virtually special groups growth function pnq growth function pnq since finite index subgroups groups since special groups embed raags obtain following corollary corollary positive integer exist artin group containing subgroup whose monodromy automorphism growth function second question asks kinds functions arise dehn functions finitely presented subgroups raags recall dehn functions capture isoperimetric behavior cayley complexes groups lot known dehn functions arbitrary finitely presented groups see example group hyperbolic dehn function linear cat groups particular raags dehn functions either quadratic linear examples cat groups contain finitely presented subgroups whose dehn functions form dense set restricting case ambient groups raags gets harder find examples subgroups wide variety dehn functions examples finitely presented kernels raags polynomial dehn functions degree shown dehn function kernels raags quartic bridson provides example raag containing finitely presented group exponential dehn function second result shows finitely presented subgroups raags whose dehn functions polynomial arbitrary degree dehn functions subgroups artin groups theorem positive integer exists artin group contains finitely presented subgroup dehn function extent currently known isoperimetric behavior subgroups raags would interesting one could produce examples types dehn functions paper organized follows section introduce growth functions automorphisms prove folklore result proposition growth automorphism free group invariant taking powers automorphism passing subgroup finite index rely gilbert levitt growth theorem whose proof uses machinery also provide example due yves cornulier demonstrates arbitrary groups invariance growth function passing subgroup finite index hold section devoted providing estimates dehn function bieri double group terms growth monodromy automorphism use bridson lower bound proposition adapt proof upper bound given setting case groups needed construction proposition using estimates show later section polynomially growing monodromy automorphisms involved construction upper lower bounds dehn function bieri double actually coincide monodromy automorphism polynomial growth order dehn function bieri double grows like section recollect relevant definitions related morse theory groups special cubical complexes used sections section introduce groups play central role construction define groups log notation graphical tool encode conjugation relations prove group cat exhibit explicit formulas monodromy automorphism proposition defer section proof automorphism growth goal section exhibit finite special cover presentation complex group arbitrary even construction done several stages first arbitrary engineer certain right action set cardinality may thought torus action defines finite cover cellularly embeds proposition subcomplex product graphs free three four since hyperplane pathologies definition special cube complex proposition eliminate fourth hyperplane pathology observe even values complex terminology follows special square complex exist another finite cover proposition section bring pieces together prove theorems even values groups virtually special monodromy automorphism growing obtain growth functions odd degree observe presentation group noel brady ignat soroko combinatorial subcomplex obtained deleting hyperplane corresponding last generator thus pullback finite special square complex covering makes virtually special group monodromy automorphism growing prove theorem look bieri double special finite index subgroups prove lower upper bounds dehn function coincide order bieri double naturally embeds raag whose underlying graph join underlying graph raag containing empty graph two vertices section provide computation growth function monodromy automorphism abelianization finally section list two open questions related study paper preliminaries growth follows consider functions following equivalence relations definition two functions said equivalent means exist constants pnq agpnq definition two functions said equivalent means exist constants pnq agpbn extend equivalence relations functions assuming constant interval remark notice implies implies however relation strictly finer latter identifies single exponential functions whereas relation use relations dealing growth functions automorphisms relations discussing dehn functions groups done traditionally term definition essential proving equivalence dehn functions let free group finite rank finite generating set let associated word metric automorphism define pnq max paq apa equal following properties used sequel proposition let free group finite rank finite generating set let automorphism finite generating set iii finite index subgroup invariant finite generating set dehn functions subgroups artin groups proof let since sets generate exist words let constants denote maximal lengths respectively obviously one pai pai pbj pbi fix arbitrary assume without loss generality values pbj pail pail max pai pnq therefore pnq pbj pnq symmetry pnq pnq proves proving parts iii state following remarkable result gilbert levitt levitt growth theorem cor let free group finite rank free generating set given autpf exist integer constants word length pgq satisfies pgq consider sequence growth parameters levitt growth theorem corresponding generators exist constants nmi pai nmi order parameters lexicographically clearly nmj nmi pick maximal respect order constants arbitrary nmi means side less equal side large enough prove one direction notice dmi pai pbi dmi nmi aii pdnq hence exist constant cbig pnq max pai pnq cbig dmi thus opposite direction pai nmi abii pdnq pdnq noel brady ignat soroko taking maximum get arbitrary pnq max pai cbig pnq cbig maxi cbig proves hence prove iii one direction notice first finite index see hence one writing word setting obtain arbitrary pbj pbj max pai pnq pnq pbj pnq opposite direction notice exist integer every api let sequence growth parameters generators let maximal consider new generating set arbitrary pai nmi abii abii similar meaning fourth inequality holds since general automorphism autpf one indeed one write cyclically reduced whereas taking maximum get arbitrary pnq max pai cbig pnq cbig maxi cbig means since part proves part iii proved remark proof property proposition works automorphisms arbitrary finitely generated groups remark property iii proposition hold arbitrary finitely generated groups grateful yves cornulier providing following example let infinite dihedral group inner automorphism linear growth restriction index subgroup xby trivial see observe implies hence thus inb paq looking cayley graph shows element distance word minimal length representing element indeed grows linearly view item proposition suppress dependence generating set adopt notation pnq pnq arbitrary generating set dehn functions subgroups artin groups abelianization fab consider induced automorphism fab fab denote generating set fab corresponding let denote viewed subset define pnq max following true lemma pnq pnq proof let fab natural homomorphism pai hence length shortest word generators uki element bigger pai former equal hence pai taking maximum get required inequality embedding may consider vector space basis uki let linear operator given basis uki matrix paij qki let denote maxi consider two norms endpck one operator norm respect sup sup another one supremum norm space sup max lemma max sup proof max max max sup following fact see cor lemma exist constants sup corollary growth function grsup sup equivalent growth function grop following results proved proof lemma equivalence class function grop depends conjugacy class glpk noel brady ignat soroko view corollary lemma need consider growth jordan normal forms matrices lemma suppose matrix jordan normal form eigenvalues equal grsup maximal pnq size jordan blocks combining get corollary let automorphism free group abelianization eigenvalues equal size largest jordan block jordan normal form pnq pnq proof indeed pnq pnq lemma lemma sup lemma lemma lemma bounding dehn function bieri double section outline known upper lower bounds dehn function bieri double group lower bound established lemma see proposition argument upper bound see proposition follows outline theorem latter paper argument given setting groups adapt reasoning setting definition bieri double let group bieri double group one denotes free group generating set autpf paqq homomorphism given generators paq definition dehn function let group given finite presentation word lying normal closure free group paq define areapwq min paq paq dehn function function defined pnq maxtareapwq denotes length word generators viewed equivalence dehn functions independent choice presentation see denote pnq pnq dehn functions subgroups artin groups proposition lemma proposition let automorphism denote word length respect fixed generating set dehn function pnq bieri double one max pbq pnq bpf proposition let automorphism free group assume pnq pnq dehn function pnq bieri double one pnq remark proved using pnq pnq see however reader unfamiliar machinery make exposition independent result instead follows apply proposition automorphisms whose growth functions pnq pnq computed section shown equivalent order prove proposition need preliminary results combings groups start definitions let group finite generating set associated word metric definition combing normal form set words letters denote length word free monoid definition let unbounded given eventually constant paths define min maxtda ptqqu tpn definition given combing asynchronous width function defined pnq max definition finitely generated group said asynchronously combable exists combing constant pnq definition length combing function given lpnq max relation combings dehn functions manifested following result proposition lemma let group finite set semigroup generators exists combing whose asynchronous width bounded constant whose length bounded function lpnq dehn function pnq presentation satisfies pnq nlpnq connection groups bieri doubles following result useful noel brady ignat soroko theorem theorem asynchronously combable every split extension asynchronously combable remark proof result follows groups combings whose asynchronous width bounded constants combing split extension whose length bounded constant taken product concatenation combings meaning traverse path first path note product combings opposite order may bounded asynchronous width example groups shows let bieri double free group set free generators goal obtain upper bound length lpnq combing paq terms growth automorphism treat combing free group unique reduced word fixed system generators represents given element group prove following proposition adapting reasoning groups theorem case groups proposition let pnq increasing function bounding growth paqq length lpnq combing paq group paq satisfies lpnq pnq proof take arbitrary write paq let dayts length shortest word generators payts representing element would like show dts considering natural homomorphism paq one observes hence dts therefore suffices show denote shortest word generators written paq denote dehn functions subgroups artin groups one easily checks hence denote homomorphism defined generators paq pgq therefore pwi pwi hand observation moreover since paqq paq get pwi finally get element estimate pwi rby rby shows finishes proof proposition ready prove upper bound dehn function bieri double proof proposition noted free group asynchronously combable constant also therefore theorem asynchronously combable hence proposition pnq nlpnq due proposition lpnq therefore pnq noel brady ignat soroko cube complexes special cube complexes article haglund wise established fundamental groups special cube complexes admit embeddings artin groups gives natural class subgroups artin groups suggests construct examples within class summarize relevant definitions results special cube complexes section definition cube complex copy viewed metric space euclidean metric suppress call cubes face metric subspace obtained restricting coordinates either cube complex complex obtained gluing possibly varied dimensions together along faces via isometries cubes cube complex cube complex called square complex cube complex simple link every vertex simplicial complex simplicial complex flag collection pairwise adjacent vertices spans cube complex curved link vertex flag simplicial complex definition hyperplane midcube subset obtained restricting one coordinates hyperplane cube complex connected component new cube complex formed follows cubes midcubes restriction midcube defines attaching map edge dual hyperplane midpoint vertex definition parallelism walls two oriented edges cube complex called elementary parallel square containing attaching map sends two opposite edges orientation respectively define parallelism oriented edges equivalence relation generated elementary parallelism oriented wall parallelism class oriented edges note every hyperplane defines pair oriented walls consisting edges dual describe four pathologies interaction hyperplanes cube complex forbidden special cube complexes definition hyperplane contains one midcube cube definition hyperplane exists combinatorial map complexes mapping identically recall cellular map complexes combinatorial restriction open cell homeomorphism onto image hyperplane called definition hyperplane two edges dual belong common square share common vertex addition consistent choice orientation edges dual makes common vertex origin terminus hyperplane called directly dehn functions subgroups artin groups definition two distinct hyperplanes interosculating intersect edges dual dual belong square share common vertex definition special cube complex curved cube complex called special hyperplanes selfosculations definition virtually special group group called special exists special cube complex whose fundamental group isomorphic group virtually special exists special cube complex finite index subgroup isomorphic fundamental group definition artin group let simplicial graph artin group raag associated finitely presented group given presentation xai rai pai definition salvetti complex given artin group salvetti complex associated curved cube complex defined follows let circle endowed structure complex single single let let saj torus product structure every full subgraph verticespkq aik define torus cartesian product plexes observe identified combinatorial subcomplex salvetti complex associated full subgraph thus single edge pai tributes square attaching map general full subgraph contributes cell cardpverticespkqq theorem cube complex special admits local isometry salvetti complex artin group since local isometries cat spaces one gets following corollary fundamental group special cube complex isomorphic subgroup artin group morse theory cube complexes use following definitions definition morse function map defined cube complex morse function every cell characteristic map composition extends affine map constant dim image discrete noel brady ignat soroko definition morse function morse function cube complex cellular map property lifts morse function universal covers definition ascending descending links suppose cube complex morse function corresponding morse function let note link naturally isomorphic link lift say cell contributes ascending respectively descending link achieves minimum respectively maximum value ascending respectively descending link defined subset link lkpv naturally identified ascending respectively descending link note case square complex ascending descending entire links graphs following characterization groups proven see also theorem proposition morse function whose ascending descending links trees aspherical pxq means short exact sequence pxq free group isomorphic pptqq point groups section define sequence groups study presentation complex show curved square complex groups log definition recall labeled oriented graph log consists finite directed graph labels vertices edges satisfying following vertices distinct labels edge labels chosen set vertex labels log determines finite presentation follows set generators set vertex labels set relations correspondence set edges relation form oriented edge labeled vertex vertex let let group defined log presentation figure rai clearly hnn extension stable letter natural tower inclusions dehn functions subgroups artin groups figure log description figure contribution relations link presentation complex structure one way producing structure groups verify presentation corresponding log presentation metrized curved piecewise euclidean complex let denote presentation corresponding log presentation one labeled corresponding relations construction subcomplex endow structure using regular euclidean squares using local isometric embedding attaching maps proposition presentation complex defined curved complex noel brady ignat soroko proof need check gromov link condition square reduces purely combinatorial requirement link every circuits combinatorial length less figure shows contributions relations link unique adopt following notation originates terminates contributes vertex denoted link vertex denoted link see link obtained union sequence graphs pair disjoint vertices obtained adding new pair disjoint vertices connecting one pair observe procedure preserves following property every vertices indeed shortest path old vertices length two shortest path newly added vertices least two shows step create cycles lengths two three therefore link cycles length less four corollary groups curved square complex proof indeed universal cover cat complex acts isometries properly discontinuously cocompactly structure notice relations groups form implies exists epimorphism sending every fixed generator epimorphism realized geometrically morse function defined follows consider structure consisting one one takes maps map homeomorphically onto target extends linearly extends linearly mean lifts map universal covers way depicted figure note curvature characteristic maps cells lift embeddings universal cover figure morse function preimage set dehn functions subgroups artin groups proposition morse function induces short exact sequence free group rank proof theorem suffices show ascending descending links trees ascending link formed corners formed pair originating edges labeled figure similarly descending link formed corners formed pair terminating edges labeled figure figure ascending descending links morse function figure observe ascending descending links cell indeed trees definition contributes diagonal loop furthermore bouquet diagonal loops hence graph single denoted figure proposition implies monodromy automorphism shall determine explicitly automorphism particular choice basis let diagonals shown figure note following expressions generators proposition following explicit structure noel brady ignat soroko monodromy automorphism acts follows overbar denotes inverse pam furthermore restriction proof proof proposition shown freely generated elements generator factor free choose element maps generator without loss generality may take get action monodromy automorphism generators need compute conjugations need find words generators equal figure action monodromy automorphism start triangle top forming two bottom sides would like express products free generators since descending link tree exists unique path connecting unique path connecting paths correspond paths respectively see figure thus pai dehn functions subgroups artin groups figure action monodromy automorphism similarly start triangle top forming two bottom sides unique paths descend ing link correspond words respectively see figure hence constructing special cover section construct certain permutation representation group show defines finite cover presentation embedded torus allows construct finite special cover even values permutation representation define right transitive action certain set cardinality action set denote set tuples length consisting natural inclusions identify image inclusions also denote subset consisting tuples last coordinate identifications disjoint union define right action group set suffices associate permutation equivalently right action homomorphism opposite group sympxq group permutations equals set new operation defined adopt latter approach construct homomorphism noel brady ignat soroko recall natural tower inclusions since also inclusions hpm allows define homomorphism inductively repeatedly extending homomorphisms symphm symphm follows base induction let generators act flips respective coordinates set given operator changes coordinate vice versa fixing others relations commutators rai clearly satisfied symphm since operators pairwise commute thus homomorphism symphm inductive step fixed suppose symphm already defined particular implies invariant also suppose following property holds base induction guarantees suppositions true goal extend homomorphism symphm since suffice define end set denotes composition involution interchanges coordinates leaving coordinates fixed words transfer action twisting notice definitions sets invariant relations involving generators satisfied since hold true conjugation homomorphism permutation groups relation involving last generator dehn functions subgroups artin groups translates since sends side relation acts follows last equality holds due inductive definition thus sides relation act analogously side acts side inductive definition since two sides act relation satisfied symphm proves remains proved auxiliary condition preserved inductive step implies indeed means thus since proves holds finishes inductive construction homomorphism proof thus one gets right action also denoted figure shows permutation representation noel brady ignat soroko hhh hhh hhhh hhh hhhh hhhh hhhh hhh hhh dthhhh hhhh hhh hhh dhh hhhh hhh dhh dhh hhh hhhh hhhh hhh hhhh hhhh hhh hhhh hhh hhh hhhhh dhh hhh hhhh hhhh hhhhh hhh hhhh hhhh figure case action elements arranged vertices hypercube graph marked corresponding tuples thus edge graph corresponds pair opposite edges torus defined subset represented upper corner subgraph upper half picture interchanges vertices mark edge italicized digit proposition right action defined following properties acts transitively generator acts involution differs exactly one coordinate particular fixpoints fixpoints dehn functions subgroups artin groups proof easy induction case obvious since acts coordinate flips one start obtain changing one coordinate time suppose acts transitively comprises another orbit glues one orbit thus proving transitive follow induction formulas obvious acting since act different coordinate flips suppose statement proven acting fixpoints either since otherwise fixpoint pvq would fixpoint finally say changes last coordinate whereas preserves subsets change coordinate hence composition fixpoints figure complex cover let complex following structure denote respectively denote respectively see figure denote complex product structure notice natural action preserves standard unit cubulation hence induces structure cubical complex observe homeomorphic torus follows convenient parametrize points numbers viewed identification naturally identified set introduced formed fixing edge factor say position taking product vertices positions two differ one coordinate unique directed edge first second one unique directed edge second one first one thus typical identified product form similarly arbitrary product noel brady ignat soroko choice let presentation recall consists one corresponding generators corresponding relations consider right action constructed previous section denote stabilizer point subgroup defines finite covering whose properties describe cellularly embeds proposition covering space correspondence right cosets proof szgm since action transitive proposition consists vertices identify earlier identified set consisting correspondence pairs right cosets psg sgai runs generators proposition guarantees loop actually belongs lifts uniquely determined base vertex lift base vertex fixed cyclic order relator word defines attaching map indeed since homomorphism every relator word acts identical permutation two types relators presentation based every length thus define length loops vertex loops filled loops must given nullhomotopic projected thus word values see figure sgaj sgaj sgai sgai figure typical dehn functions subgroups artin groups identify cosets szgm proposition shows every edge psg sgai changes one coordinate representing vertex therefore maps suitable let show form vertex naturally embeds embedding induced embeddings introduced denote positions endpoints following edges differ psg sgai psgai sgai psg sgaj psgaj sgaj respectively proposition fixpoints hence since square commutative conclude sets equal proposition act involutions hence since otherwise would fixpoint sgai therefore consideration actually belongs since pair parallel edges changes coordinates vertices position one position another exploring hyperplane pathologies need following description hyperplanes walls lemma hyperplanes oriented walls correspondence pairs explicitly hyperplane corresponding pair subset one following two types identification oriented wall consists fixed taking possible values oriented wall dual corresponding hyperplane proof recall structure cube complex induced standard cubulation hyperplanes subsets form modding action yields result recall oriented wall containing cube complex class oriented connected sequence elementary parallelisms via notice arbitrary product form vertex arbitrary product noel brady ignat soroko choice thus elementary parallelism via establishes equivalence since index varies independently conclude contained parallelism class three four pathologies show complex definition special cube complex proposition hyperplanes hyperplanes hyperplanes proof convenient work oriented walls dual hyperplanes since square subcomplex every wall proposition subset wall lemma walls consist form fixed arbitrary corresponding wall would conif hyperplane tain edges two different coordinate positions impossible proves corresponding wall would contain hyperplane pair edges common extremities either origins termini coincide direct origin one edge coincides terminus indirect either case tuples equal means original equal actually happening proves lies unique hyperplane prove observe hyperplane particular lemma coordinate system hyperplane following description values rest square complex union set variables since square hence point belongs square form index may less bigger suppose actually coordinates dehn functions subgroups artin groups value see set also belongs square conclude product defines combinatorial map union actually embedding identified similar reasoning applies parametrize proves every hyperplane unfortunately cube subcomplexes cartesian product three graphs hyperplanes example figure shows figure subcomplex product two segments length one segment length two figure subcomplex product three graphs interosculating hyperplanes however haglund wise proved case square complex absence first three hyperplane pathologies guarantees existence finite special cover definition simple square complex called edges divided two disjoint classes vertical horizontal attaching map square form vhv vertical horizontal edges vhproposition even integers complexes complexes hence exist finite special cover proof log definition see section groups observe even integers generators form two classes vertical horizontal relators form implies complex finite cover inherits structure vhthe complex complex indeed preimages vertical horizontal edges noel brady ignat soroko form two disjoint classes covering map phq edges contained link every vertex bipartite graph corresponding parts links vertices mapped isomorphically covering maps thus bipartite respect parts therefore links vertices boundaries form proposition hyperplanes selfh osculations hence theorem exist special cube complex finite cover proof main theorems ready prove main results theorem positive even integer exist virtually special groups growth function pnq growth function pnq proof seen proposition even integer exist special cover presentation complex group propositions section show growth function proves first part theorem second part recall restriction free subgroup first generators construction presentation complex subcomplex actually obtained deleting loop corresponding last generator also single open adjacent one sides let special cover proposition consider square subcomplex obtained deleting map onto loop labeled deleting open one sides taking connected component resulting complex claim finite special cover indeed construction lies hyperplanes twosided since subcomplexes corresponding hyperplanes special complex see complex hyperplanes observe steps deleted hyperplanes dual cells corresponding last generator change absence remaining hyperplanes therefore hyperplanes either special growth shown propositions section since corollary positive integer exist artin group containing subgroup whose monodromy automorphism growth function dehn functions subgroups artin groups proof theorem proved positive integer arbitrary even exists group pnq finite index subgroup isomorphic fundamental group special cube complex wise celebrated result see corollary exists artin group isomorphic subgroup let prove indeed commutative diagram image since finite index hence subgroup invariant free group since subgroup therefore finitely generated free group group parts iii proposition tell pnq pnq theorem positive integer exists artin group contains finitely presented subgroup dehn function proof theorem proved even integers group resp following properties pnq resp pnq contains finite index subgroup embeds artin group corollary showed resp finite index subgroup resp monodromy automorphism resp claim bieri double dehn function pnq equivalent pnq resp pnq embeds raag prove claim case subgroup monodromy automorphism notice case monodromy automorphism proved similar fashion upper bound pnq established follows propositions pnq pnq proposition implies pnq pnq pnq upper bound follows proposition lower bound bpn pbq pnq given proposition show max bpn pbq follow view pnq prove section group containing maximum definition growth functions pnq pnq achieved generator see corollary pbm noel brady ignat soroko bar element denotes image abelianization since subgroup finite index exists integer max pbm pnp nqm bpn since proves pnq prove embeds raag consider homomorphism free group rank free generators rem iii described follows let freely generated elements denote generators three copies also denote brevity xsy xty define generators follows check homomorphism easily proved using normal forms elements free amalgamated products injective thus denote artin group containing graph subgroup raag corresponding graph join empty graph two vertices remark following proof proposition given lemma see also proposition exhibit explicit sequence words bkpn realize lower bound dehn function pnq understand van kampen diagrams words look like reader referred proof lemma notation bkpn growth proposition shown free group generators convenience follows adopt notation use fact restriction first generators goal section prove growth particular pnq pnq throughout section denote word length respect system free generators tai dehn functions subgroups artin groups recall see proposition automorphism given formulas overbar denotes inverse pam upper bounds growth proposition automorphism pnq pnq need basic lemmas lemma pai proof prove statement induction observing true pai pai corollary pai lemma anm proof statement true induction anm anm corollary noel brady ignat soroko lemma proof induction case evident pan lemma proof one observes gives induction view lemma anm hence rmn rmpn npn claim pbk proof formulas get therefore pbk pbk pbk lemma gives total length pbk bounded pbk pbk denote pbk corollary lemma dehn functions subgroups artin groups inequality instead equality formula cancellations let define another function gpk follows gpk gpk gpk gpk obviously recurrent fashion easy induction shows gpk gpk gives upper bound growth pbk estimate order growth gpk let look finite differences gpk gpk gpk assume induction gpk polynomial degree conclude gpk polynomial degree since assumption true proves claim establish similar bound one could use machinery along lines prove growth growth polynomial give simple direct proof one easily checks inverse automorphism acts follows pam lemma pai pai proof define automorphism subgroup given one easily checks therefore pai pai result follows corollary noel brady ignat soroko claim pbk proof denote aim notation action written follows ptk psi first relation obvious second one follows easy induction lemma proof indeed define function follows ptk relations imply inequality equality possible cancellations reduced expression ptk obtain upper bound introduce function gpk defined recurrently follows gpk gpk gpk gpk formulas define gpk recurrently values indeed one proceeds layers numbered case given first formula case arbitrary given third one valid remaining case given second formula clearly gpk function used establish upper bound pbk pbk gpk estimate growth gpk consider finite difference gpk gpk applying recurrent relation several times get gpk gpk gpk gpk gpk gpk rgpk gpk gpk gpk dehn functions subgroups artin groups similarly gpk rgpk gpk gpk gpk since definition gpk gpk gpk gpk gpk particular gpk gpk gpk assume induction gpk polynomial function degree true since conclude gpk polynomial function degree formula similarly implies gpk polynomial degree therefore pbk gpk finishes proof claim proof proposition according corollary lemma pai according claims pbk therefore pnq pnq lower bounds growth proposition automorphism pnq pnq claim size largest jordan block jordan normal form proof sufficient prove claim direct inspection formulas see represented following matrix okm ckk identity matrix okm zero matrix dmk ckk matrices respectively given formulas ffi ffi dmk ckk known number jordan blocks matrix glpm eigenvalues given number dim kerpa rankpa noel brady ignat soroko number jordan blocks eigenvalues size least given dim kerrpa dim kerpa rankpa rankrpa easy computation shows omm dmk okm ckk ckk ffi ffi omm dmk okm ckk ffi ffi ffi ffi dmk ffi ckk note rank ckk rank ckk hence therefore number jordan blocks number jordan blocks size means one block size bigger let denote size blocks size hence proof proposition proposition follows corollary claim lower bounds growth proof theorem section needed certificate growth abelianization element basis realizes maximum definition growth functions provide corollary notation let image generator abelianization pbk proof elementary computation matrix proof claim shows represented matrix following structure okm qkk dehn functions subgroups artin groups identity matrix okm zero matrix pmk qkk matrices respectively given formulas ckn ffi ckn ffi ffi ffi ffi pmk qkk ckn pmk rows equal qkk upper triangular number diagonal sequence entries parallel main diagonal numbers ckn satisfy following identity follows matrix multiplication rule cin convention show double induction pairs cin indeed equality true pairs arbitrary since convention arbitrary since matrix representation see proof claim suppose equality cin already proved pairs fixed arbitrary pairs pair get cin needed particular coefficients vector ckn ckn polynomial since claim pbk pbk conclude pbk remark proved similar manner elements also serve certificates growth automorphisms respectively however formulas involved complicated need result construction open questions conclude paper two open questions question every cat group virtually embed raag question exist finitely presented subgroups raags whose dehn functions either polynomial noel brady ignat soroko references agol criteria virtual fibering topol agol virtual haken conjecture appendix ian agol daniel groves jason manning doc math barnard brady distortion surface groups cat groups geom dedicata bestvina brady morse theory finiteness properties groups invent birget shanskii rips sapir isomperimetric functions groups computational complexity word problem ann math brady forester snowflake geometry cat groups pages brady dehn functions curvature geometry word problem finitely generated groups basel brady bridson one gap isoperimetric spectrum geom funct bridson combings semidirect products groups geom funct bridson geometry word problem invitations geometry topology oxf grad texts oxford univ press oxford bridson subgroups artin groups mapping class groups math res bridson polynomial dehn functions length asynchronously automatic structures proc london math bridson gersten optimal isoperimetric inequality torus bundles circle quart math oxford ser bridson haefliger metric spaces curvature springer bridson pittet isoperimetric inequalities fundamental groups torus bundles circle geom dedicata dison isoperimetric function groups bull lond math soc dison riley hydra groups comment math helv gersten automorphism group free group cat group proc amer math soc hagen wise cubulating hyperbolic groups irreducible case duke math hagen wise cubulating hyperbolic groups general case geom funct anal haglund wise special cube complexes geom funct horn johnson matrix analysis cambridge university press howie asphericity ribbon disc complements trans amer math soc levitt counting growth types automorphisms free groups geom funct anal piggott detecting growth free group automorphisms action homology subgroups finite index arxiv pages sapir birget rips isoperimetric isodiametric functions groups ann math department mathematics university oklahoma norman usa address nbrady department mathematics university oklahoma norman usa address
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deciding graph isomorphism parameterized logspace maurice leibniz hannover theoretical computer science appelstr hannover germany chandoo dec abstract compute canonical representation given graph implies solving isomorphism recognition problem class accomplish split class graphs uniform ones employ generalized version argument given used show subclass helly graphs canonized logspace uniform graphs approach works logspace addition helly graphs strict subset uniform graphs thus result generalization canonization result helly graphs nonuniform case specific set ambiguous vertices arises choosing parameter cardinality obstacle solved brute force leads log space algorithm compute canonical representation therefore graphs acm subject classification graph theory keywords phrases graph isomorphism canonical representation parameterized algorithm introduction arc connected set points circle graph called graph every vertex assigned arc two vertices adjacent iff respective arcs intersect call bijective mapping representation set arcs model recognition problem graphs decide whether given graph graph canonical representation problem graphs consists computing representation given graph additional canonicity constraint whenever two graphs isomorphic models identical similarly graph interval graph every vertex assigned interval line two vertices share edge iff intervals intersect easy see every interval graph graph since every interval model model class graphs started gain attraction series papers alan tucker however still known better upper bound deciding graph isomorphism graph isomorphism general even though considerable effort done end two claimed algorithms disproven respectively subclass interval graphs algorithm isomorphism described series newer results show canonical representations interval graphs proper graphs helly graphs superset interval graphs computed logspace furthermore recognition isomorphism interval graphs two hardness results carry class graphs recognition reduction requires little additional work maurice chandoo licensed creative commons license leibniz international proceedings informatics schloss dagstuhl informatik dagstuhl publishing germany deciding graph isomorphism parameterized logspace figure graph representation main contribution extend argument used compute canonical representations hca graphs graphs split class graphs uniform ones show mentioned argument applied cases using log space parameter describes cardinality obstacle set occurs case means uniform case hence also obtain logspace algorithm uniform graphs superclass hca graphs best knowledge first algorithm decide isomorphism specifically class graphs paper structured follows section define graphs along representations recall concept normalized representations section explain mean flip trick formalize notions flip sets candidate functions idea used compute representation graphs linear time modified compute canonical helly representations hca graphs section flip trick applied uniform graphs respectively preliminaries given two sets say intersect say overlap symbols let two square matrices vertex sets say isomorphic symbols exists bijection called isomorphism two graphs adjacency matrices say consider undirected graphs without graph class subset graphs closed isomorphism define graph isomorphism problem graph class graph vertex define open neighborhood set vertices adjacent closed neighborhood subset vertices define common neighborhood also write instead vertex called universal two vertices say twins twin class equivalence class induced twin relation graph said without twins every twin class cardinality one graph define exclusive neighborhood vertices adjacent vertices none definition let function maps graphs subset subsets vertices say every pair isomorphic graphs isomorphisms holds chandoo graphs representations model set arcs circle let two points circle arc specified given part circle traversed starting going clockwise direction reached say left right endpoint write denote left right endpoint arc general arc obtained swapping endpoints covers opposite part circle say obtained flipping considering model respect intersection structure relative position endpoints matter endpoints assumed pairwise different arc covers full circle therefore model arcs described unique string follows pick arbitrary arc relabel arcs order appearance left endpoints traversing circle clockwise starting left endpoint write endpoints order appearance traversing circle clockwise starting left endpoint chosen arc every arc pick lexicographically smallest resulting string representation example smallest string model fig would result choosing following identify string representation let graph consists model bijective mapping vertices arcs called representation holds write mean arc corresponding vertex model subset let given set write denote graph graph representation say model hole exists point circle contained arc every model understood interval model straightening arcs therefore graph interval graph admits representation hole graph called helly hca graph representation maxcliques cliques holds every interval model helly property therefore every interval graph hca graph normalized representation observed intersection type two arcs one following five types disjoint contained contains jointly cover circle circle cover overlap jointly cover circle using types associate matrix every model intersection matrix square matrix entries given model define intersection matrix reflects intersection type arcs intersection matrix called interval matrix intersection matrix interval model trying construct representation graph clear whenever two vertices corresponding arcs must disjoint two vertices adjacent intersection type corresponding arcs might ambiguous would convenient intersection type every pair vertices would uniquely determined achieved associating graph intersection matrix deciding graph isomorphism parameterized logspace called neighborhood matrix defined otherwise first case applies whose condition satisfied let intersection matrix vertex set model bijective mapping say representation matrix isomorphic intersection matrix via denote set representations say normalized representation graph representation neighborhood matrix example normalized representation seen fig let denote set normalized representations lemma every graph without twins universal vertices normalized representation purpose suffices consider graphs without twins universal vertices reasons point universal vertex removed graph later added arc covers whole circle representation twin class arbitrary representative vertex chosen colored cardinality twin class twins removed lemma canonical representation problem graphs logspace reducible canonical representation problem colored graphs without twins universal vertices henceforth assume every graph without universal vertices shall consider normalized representations two vertices graph write instead example indicates arc must contained arc every normalized representation flip trick let model subset arcs flipped define resulting model consider point circle let table effects flipping arcs intersection matrix chandoo set arcs contain point flipping arcs arc contains point thus hole therefore must interval model let intersection matrices respectively observed interval matrix easily computed using input via table problem computing canonical representation colored graphs reduced canonical interval representation problem colored interval matrices solved logspace colored part mentioned explicitly easily incorporated proof adding colors leaves colored tree idea given graph compute set vertices described obtain canonical representation following argument neighborhood matrix matrix matrix must interval matrix compute canonical interval representation flip arcs back leads representation thus required set specified follows definition let graph flip set iff exists representation point circle fact argument obtain canonical representation hold required chosen interval matrix however shown equivalent definition see appendix reframe argument given follows definition candidate function let subset graphs function maps graphs subset subsets vertices call candidate function following conditions hold every exists flip set theorem candidate function graphs computed logspace canonical representation problem graphs solved logspace proof let graph decide recognition problem graphs observe flip set iff graph verify set flip set one check interval matrix trying compute interval representation representation problem let graph let subset every flip set first condition holds every representation computed previous argument return representation lexicographically smallest underlying model argmin canonical representation see indeed canonical representation consider two isomorphic graphs defined previously let isomorphism set models induced flip sets similarly define canonicity follows showing show argument direction analogous let model let flip set induces must hold since follows therefore interval matrices isomorphic meaning deciding graph isomorphism parameterized logspace since flip set well interval representations interval matrices identical underlying models due canonicity follows remain flipping resp therefore holds works logspace since representation every flip set computed logspace reduced problem computing canonical representation graphs problem computing candidate function graphs let two graph classes partition graphs candidate functions respectively candidate function graphs follows functions closed taking unions crux need able distinguish graph hence next two sections consider two classes partition graphs avoid dealing recognition two classes complete section stating candidate function used canonize hca graphs explain candidate function subclass graphs fhca fhca always returns least one flip set hca graph maxcliques hca graph flip sets due helly property exists least one maxclique every hca graph characterized common neighborhood two vertices however neither two properties hold graphs general remains argue fhca computed logspace since arguments made two candidate functions devised next sections introduce tool facilitates demonstrate fhca definition let formula graph structures free variables define function graph lemma every formula graph structures function computable logspace compute successively evaluate take union results done logspace show suffices apply structural induction full argument found appendix fhca computed formula states lemma follows fhca uniform graphs difficulty trying compute flip sets graphs general graph might different normalized representations set vertices shares common point one representation one show subset graphs namely uniform graphs issue occur therefore making easy compute flip sets graph consider arbitrary vertex looking neighbors try compute flip sets specified fig sets contain chandoo vertices contain form circle cover vertices overlap belong either depending side overlap since determine left right neighborhood matrix want express equivalence relation states two vertices overlap side two equivalence classes induced two flip sets expressed given two vertices overlap instance easy see must overlap different sides disjoint intersection type situation immediately clear distinctions required set three pairwise overlapping vertices overlap form consider possible normalized representations three vertices either jointly cover circle set overlapping intervals first case overlap pairwise overall intersection empty thus call triangle second case interval triangle interval triangle three different possible representations reflection depending three vertices placed two overlap side iff form interval triangle show easy derive information case uniform graph depend representation definition let graph set following holds holds vertex definition uniform graph graph uniform holds triangle idea behind definition captures properties must satisfy graph represented triangle definition uniform graphs guarantees never represented interval triangle show class uniform graphs property triangle predicate interval triangles invariant across normalized representations lemma let uniform graph following statements equivalent every figure uniform target flip sets deciding graph isomorphism parameterized logspace triangle triangle proof immediately follows definition uniform graphs contrasting example graph triangle depends representation consider graph obtained taking complement disjoint union three triforce circle around outer corners every edge model given precisely two triangles possible assignments vertices arcs follow automorphisms yield different representations definition let graph say least one following holds exists lemma let uniform graph following statements equivalent proof exists must hold due fact implies true must follows must since due intersect implies forms triangle therefore lemma assume exists implies therefore exists lemma follows must triangle must disjoint contradiction clear lemma state fact form property holds one representation holds representations hence name uniform graphs vertex graph defined define aforementioned equivalence relation state candidate function uniform graphs definition given graph vertex define relation holds one following applies contains contained overlap lemma let uniform graph holds iff overlap side every chandoo proof relation clear third condition holds interval triangle two follows overlap side clear overlap either form triangle cases overlap different sides theorem following mapping candidate function uniform graphs computed logspace proof show candidate function prove every uniform graph always exists flip set fact even stronger claim holds uniform graphs every set flip set let via set vertices correspond one two flip sets shown fig correctness subset vertices overlapping follows lemma show computed logspace apply lemma rewrite iff remains check entries neighborhood matrix exclusive neighborhoods expressed logic corollary canonical representation uniform graphs computed logspace theorem helly graphs strict subset uniform graphs proof first show contradiction assume exists helly graph must exist representation represented interval triangle let assume follows since must means exists follows overlap must hold either second condition follows must contained contradiction reason follows form must represented triangle contradicts helly graph see inclusion strict consider graph obtained taking triangle attaching new vertex vertex also known net graph every representation must hold represented triangle since reason helly graph deciding graph isomorphism parameterized logspace graphs definition uniform graphs follows graph exists represented interval triangle call pair witness also say via additionally witness pair call maximal exists via maximal given must always exist purpose computing candidate function class assume given graph supplied exists witness maximal justified fact iterate trying compute flip sets knowing least one conditions met additionally write ordered triple indicate let via consider vertex possible relations instance disjoint implies therefore also contain would mean universal vertex definition let graph via set normalized representations agree definition let graph via say call unambiguous condition holds definition let graph define following figure possible positions endpoints relative chandoo sets lemma graph via maximal partition proof hard see sets overlap show every vertex belongs one sets need check possible positions endpoints vertex relative consider fig know every vertex must represented must hold endpoints must one intervals exemplary depicted fig given representation overlaps number interval left endpoint situated must less equal right endpoint example right endpoint must come left graph right encodes possible placements two endpoints verified occur either remains argue intersection structure denoted assume overlaps side holds right endpoint one five intervals left endpoint must none five intervals second condition must hold contained contradiction overlaps disjoint therefore maximal since shown must circle cover entry neighborhood matrix contradicts must overlap lemma let graph via maximal vertices unambiguous proof intersection structure vertex dictates positioning endpoints relative every follows placement endpoints must hold therefore unambiguous vertex possible placements endpoints satisfy intersection structure one following four types example fig type flipping would lead type let via maximal two target flip sets want compute case immediately left endpoint immediately right endpoint must form subset deciding graph isomorphism parameterized logspace definition let graph via maximal call subset exists holds iff words set subset vertices represented normalized representation finding set construct two flip sets adding one side described challenge consists finding subset way solve parameterize input cardinality try possibilities since cardinality depend particular chosen know priori use following set superset every possible thus bounds cardinality theorem following mapping candidate function graphs computed log space taken proof show always exists flip set graph argue follows let via maximal let set must exist since one target flip sets described previously means exists describes set arcs contain point right left endpoint point right right endpoint see formula constructed true iff set variable note therefore works log space since one iterate subsets using bits apply argument lemma via requires additional log space conclusion showed canonically terms compute flip sets graphs acquire canonical representations properties uniform graphs enable easily logspace case graphs however seems sets pose obstacle trying compute flip sets simple remedy appears proposed parameterization enables use brute force changing target flip sets seem improve upon situation consequence suggest investigate space sets given restricted structure graphs could reasonable first step towards deciding isomorphism graphs polynomial time additionally shown compute flip sets graphs linear time without canonicity constraint done logspace well would mean recognition graphs acknowledgments thank anonymous reviewers helpful comments earlier drafts paper chandoo references andrew curtis min chih lin ross mcconnell yahav nussbaum francisco soulignac jeremy spinrad jayme szwarcfiter isomorphism graph classes related property discrete mathematics theoretical computer science elaine marie eschen graph recognition related problems phd thesis vanderbilt university nashville usa umi order hsu algorithms recognition isomorphism problems graphs siam june johannes sebastian kuhnert bastian laubner oleg verbitsky interval graphs canonical representations logspace siam johannes sebastian kuhnert oleg verbitsky helly graph isomorphism logspace mfcs volume lecture notes computer science pages springer berlin heidelberg johannes sebastian kuhnert oleg verbitsky solving canonical representation star system problems proper graphs logspace fsttcs volume pages schloss dagstuhl george lueker kellogg booth linear time algorithm deciding interval graph isomorphism acm april ross mcconnell recognition graphs algorithmica isomorphism test graphs phd thesis suny stony brook new york usa appendix proof lemma computed logspace successive evaluation taking union resulting sets prove use following claim verified induction see lemma formulas isomorphic graphs holds isomorphisms assignments must argue implies direction follows symmetrical argument let exist means replace previous claim rewrite set concludes lemma every formula free variables graph structures isomorphic graphs holds isomorphisms every assignment deciding graph isomorphism parameterized logspace proof show structural induction formulas base case statement clear isomorphisms inductive step consider boolean connectives quantifiers let holds isomorphisms cases similar let exists isomorphisms similar argument holds lemma given simple graph holds flip set iff interval matrix proof flip set exists point circle exactly vertices point common resulting representation flipping arcs interval representation since must hole since must isomorphic intersection matrix via follows interval matrix direction let interval matrix reach contradiction assume flip set every points circle holds exists either since interval matrix holds exists hole flipping arcs acquire representation holds every point circle exists follows flipping set arcs back hole exists contradicts interval representation
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aviation time minimization uav data collection wireless sensor networks jie gong member ieee chang senior member ieee chao jan shen member ieee xiang chen member ieee abstract paper consider scenario unmanned aerial vehicle uav collects data set sensors straight line uav either cruise hover communicating sensors objective minimize uav total aviation time starting point destination allowing sensor successfully upload certain amount data using given amount energy whole trajectory divided data collection intervals one sensor served uav data collection intervals uav navigation speed sensors transmit powers jointly optimized formulated aviation time minimization problem difficult solve first show one sensor node present sensor transmit power follows policy uav aviation speed found efficiently bisection search show general case multiple sensors aviation time minimization problem equivalently reformulated dynamic programming problem subproblem involved stage reduces handle case one sensor node numerical results present insightful behaviors uav sensors specifically observed uav optimal speed proportional given energy distance inversely proportional data upload requirement gong guangdong key laboratory information security technology school data computer science sun university guangzhou china email chang school science engineering chinese university hong kong shenzhen shenzhen china email shen state key lab rail traffic control safety beijing jiaotong university beijing china email chaoshen chen school electronics information engineering sun university guangzhou china shunde international joint research institute key lab eda research institute tsinghua university shenzhen shenzhen china email chenxiang ntroduction recently wireless communication unmanned aerial vehicles uavs considered promising technology expand network coverage enhance system throughput leveraging uavs high mobility los dominated channels one key applications data collection wireless sensor networks conventionally sensor node delivers monitored data fusion center via transmissions hence sensor node requires transmit data also relay others consequence sensors battery may drain quickly network connection may lost using uavs mobile fusion centers every sensor node directly send observed data uav addition los channel condition results higher data rate transmissions compared transmissions however uavs energy constrained due limited battery paramount shorten aviation time needed data collection mission different conventional communication techniques trajectory optimization issue wireless communications improve network connectivity uavs deployment movement optimized track network topology offline path planning uavs addressed collision avoidance fuel efficiency joint uav deployment trajectory optimization problem solved quantization theory approach joint trajectory communication power control multiple uavs studied addition uavs widely used mobile relays reference studied throughput maximization problem uav relay showed uplink power users follow staircase water filling structure joint optimization beamforming relay positions throughput maximization studied based stochastic optimization techniques round trip protocol tested evaluated experiments besides serving relays uavs also used mobile base stations bss emergent communications placement optimized space space respectively minimize required number mobile bss maximizing coverage coverage uavs mobile bss analytically studied considering interference beamwidth design addition growing research interest applying uav data collection dissemination wireless sensor networks aerial link characterization based practical protocols experiments given reference considered data collection via uplink transmission proposed mitigate interference adjusting uav heading beamforming adaptive modulation strategy adopted improve energy efficiency sensor nodes guaranteeing user fairness avoid contention due simultaneous data transmissions multiple sensor nodes frame selection scheme proposed sleep adaptation applied sensor nodes uav trajectory optimization jointly considered minimize sensors energy consumption one dimensional information dissemination problem considered sensors served cyclical tdma mode service regions sensors optimized worthwhile note existing works mentioned focus enhancing energy efficiency spectrum efficiency sensor nodes overlook fact limited aviation energy uavs one fundamental bottlenecks wireless networks matter fact dominant energy consumption uav lies propulsion control system accelerates uav maintains aviation height uav energy consumption modeled function aviation speed operation conditions climbing hovering uav trajectory optimization problem detailed propulsion energy consumption considering velocity acceleration studied however uav energy consumption model quite complex problems difficult optimally solved intuitively energy consumption uav proportional aviation time therefore energy minimization problem handled alternatively formulating aviation time minimization problem uav mission completion time minimization problem studied optimizing trajectory subject link quality constraint ground base station gbs uav cellular networks wireless sensor networks still lack research efforts consider uav sensors energy consumption well quality service sensor nodes time paper study aviation time minimization problem uav collects data set energy constrained ground sensors sensors wants upload certain amount data uav uav collect data either navigation hovering assume sensors located line uav trajectory divided nonoverlapping data collection intervals dedicated data collection one sensor node objective minimize total aviation time uav flying initial point destination jointly optimizing division intervals uav speed well sensors transmission power contributions paper summarized follows formulated uav aviation time minimization problem intrinsically difficult first consider scenario problem still difficult one sensor node present reveal insightful structures optimal solution specifically present explicit condition feasibility problem problem feasible data collection interval given show optimal power allocation sensor follows solution optimal speed efficiently obtained via bisection search data collection interval numerically found via search algorithm solving case extended solving general scenario multiple sensor nodes particular judiciously exploiting problem structure show aviation time minimization problem multiple sensors equivalently formulated dynamic programming problem stage optimal data collection interval one sensor node searched algorithm case used finding optimal uav speed sensor transmission power numerical results illustrate optimal behaviors uav sensor nodes different scenarios particular uav optimal speed proportional sensors energy budgets distance inversely proportional amount data upload randomly distributed sensors random amount data energy average minimum aviation time increases average amount data decreases average amount available energy rest paper organized follows section presents system model problem formulation section iii studies case problem solved section simulations shown section finally section concludes paper ystem odel roblem ormulation shown fig consider scenario uav flying set sensors data collection sensors located line labeled sensor needs upload information bits subject total energy budget uav flies fixed height initial point destination applies time division protocol sequentially receive uplink data sensors specifically whole aviation range divided intervals satisfying sensor node uploads data uav flies interval uav hovers location receives data sensor otherwise assume uav flies constant speed vmax receives data aviation sensor uploads data interval uav flies maximum speed vmax order minimize total aviation time paper uav process ignored analytical tractability uav uplink fig data collection uav ground sensors along line data collection modes since uav receive data either navigating hovering respectively consider data collection models two cases data collection aviation uav collects data ground node aviation time data collection time transmission distance changes flight transmit power data rate also adapt varying fading los channel model uav sensors pathloss exponent adopted model instantaneous data rate transmission interval given bandwidth reference ratio snr reference distance meter transmission power nth sensor satisfies total energy constraint besides since sensor requires upload bits uav flies data constraint notice feasibility issue data collection given amount sensor energy feasible upload bits within time duration since best channel quality experienced hovering right sensor maximum number data bits related hovering mode detailed data collection hovering uav hovers location sensor uploads data constant transmit power data rate denote transmission time hovering location transmission link static constant case thus sensor energy constraint simplified fully utilize sensor energy budget minimize aviation time transmission power maximized data constraint simplified function left hand side following property lemma function increasing function proof see appendix based lemma left hand side increasing function hence minimum satisfies equality transcendental equation effectively solved either line search bisection search uav experiences best channel condition hovering top user feasibility condition derived based follows proposition feasibility sensor data constraint feasible proof see appendix hovering mode may needed amount information bits large amount sensor energy small however may time efficient strategy comparing uav cruises collects data time therefore navigation hovering modes selected depending values incorporated problem formulation problem formulation paper aim minimize total aviation time guaranteeing sensors data successfully collected holds data collection feasible sensors problem formulated min vmax ixn ixn vmax optimization variables locations uav speeds transmission power function ievent indicator equals event true equals otherwise seen interval variables sensors coupled constraint makes difficult solve tackle problem firstly consider case show solution case extended general case iii aviation ime inimization ingle sensor ase case without loss generality set origin point horizontal axis ignore sensor index notations problem reduce min vmax vmax since hovering mode studied previous section mainly focus navigation mode problem navigation mode case given min vmax vmax vmax problem includes power allocation optimization uav speed optimization data upload interval optimization subproblems solved separately follows power allocation suppose upload interval uav speed fixed given obvious minimize aviation time sensor allocate power maximize uplink throughput left hand side thus let consider following throughput maximization problem max denote changing variable throughput maximization problem reformulated max notice uav receives data sensor otherwise uav flies maximum speed therefore condition needs specified following conclusion theorem let optimal solution problem holds satisfy max moreover optimal power allocation water level inverse channel gain corresponding optimal objective value bmax arctan proof see appendix fig illustration power allocation results theorem interpreted fig figure red solid curve represents inverse channel gain blue line represents water level area two curves represents total energy budget satisfy condition however satisfy condition shown figure must another set therefore data upload must within range theorem explicitly gives feasible region optimal data collection free space los channel model condition simplified calculating integration particular replacing term expression condition expressed following two conditions water level expressed uav speed optimization optimal power allocation maximum throughput bmax following property theorem maximum throughput bmax decreasing function proof see appendix based theorem feasibility solution guaranteed minimum speed satisfies minimum speed written function max min vmax minimum speed rewritten vmax otherwise according theorems given optimal power allocation optimization formulated min vmax vmax bmax vmax problem solved two steps firstly check feasibility problem based theorem bmax feasible second step objective function decreasing function optimal speed denoted maximum feasible speed satisfies vmax since bmax decreasing function found bisection search algorithm summary algorithm obtain optimal given problem summarized algorithm algorithm calculate given problem input output calculate according bmax set vmax bmax update reset else update reset end end set calculated according else problem given infeasible end algorithm precision parameter introduced used control precision bisection search process line examines feasibility problem inequality hold feasible solution given parameters algorithm terminates otherwise bisection search launched lines line search process continue bisection search terminates optimal solution recorded line remark interesting remark maximum throughput log bmax max constrained satisfied equality achieve maximum term right side operator max viewed energy efficiency achievable data rate per unit power average power budget therefore theorem says energy efficiency decreasing function power budget fading channels extends result awgn channel uav los channel data collection interval optimization finally consider problem determining problem written min vmax vmax optimal solution complex function efficient algorithms line search solve problem sampling points aviation range identical distance total number search pairs thus complexity search aviation ime inimization ulti sensor ase case data upload intervals sensors correlates one another particular sensor data upload interval wide one next short data upload interval deal correlation adopt approach solve aviation time minimization problem multiple sensors firstly basic concept algorithm briefly reviewed follows introduction algorithm algorithm deals decision making problems dynamic systems divided stages dynamic system expresses evolution system states influence control actions taken discrete instances time stage system form index stage system state summarizes information available decision making control action selected stage describes system state updated action taken state stage additive cost incurs objective minimize total cost finding optimal control actions given initial state formulated min notice problem optimized jointly actions stages bellman equation following proposition tells problem solved efficiently proposition algorithm prop vol every initial state minimum cost basic problem equal given last step following algorithm proceeds backward time stage stage min state space stage action space stage called bellman equation furthermore minimize right side policy optimal function termed optimal minimum cost stage problem starts stage state ends stage based proposition instead jointly optimizing stages algorithm recursively optimizes control action based optimal calculated previous step aviation time minimization apply algorithm solve aviation time minimization problem firstly objective function rewritten min vmax vmax min min vmax max minimization given pair corresponds navigation case efficiently solved algorithm uav hovers location minimization takes value solution thus according results sections define cost function vmax min max feasible elsewhere horizontal coordinates relative optimal feasible solution calculated via algorithm minimum hovering time obtained solving infeasible set cost infinity based cost function formulate aviation time minimization problem problem particular index stage system state stage end point data upload sensor denoted state space control action stage data upload interval sensor action space state update rule cost defined vmax result problem rewritten min solved recursively calculating function min vmax minimum aviation time obtain last step tmin addition optimal control actions optimal solution problem thus optimal solution original problem joint optimal speeds problem optimal power allocation remarkable computational complexity calculation functions reduced exploring property proposition concerning algorithm given optimal solution minimization problem satisfies proof see appendix according proposition reduce computational complexity calculation given launched initial point destination optimal solution given found satisfies calculation omitted optimal solutions equivalent umerical esults numerical results shown section numerical simulations set channel bandwidth khz according industry set maximum speed vmax case study optimal result case versus different values data upload requirement sensor energy constraint depicted figs respectively seen optimal transmission interval symmetric corresponds shortest average transmission distance sensor uav fig optimal solution hovering sensor receive data length data upload interval uav speed decreases data upload requirement increases decrease data upload interval increases channel gain uav sensor decrease uav speed increases transmission time uav fly maximum speed successfully receive uploaded data range minimum data upload interval depicted length decreases data upload requirement decreases time required data upload decreases fig optimal result versus opposite versus particular uav also needs hover sensor receive data length data upload interval uav speed increases amount energy increases uav fly maximum speed minimum length data upload interval decreases amount energy increases case study data collection multiple sensors studied simulation particular uav flies sensors deployed locations sensors fixed first four sensors apart one another sparsely deployed last six sensors apart one another densely deployed study impact required data energy limitation respectively location speed fig optimal solution versus case location speed fig optimal solution versus case speed location fig optimal solution sensors mbits mbits mbits mbits fig amount energy sensor set identical amount data transmitted varies set mbits mbits mbits mbits seen first four sensors sparsely located upload intervals disconnected reason transmission distance large case uav collects data sensor short range flies towards another maximum speed last six sensors amount data transmitted increases sensor sensor decreases uav speed firstly decreases increases accordingly required data uploaded successfully particularly amount data sensor extremely large uav hovers collect data maximum data rate overall aviation time minimized fig change values amount data bits mbits mbits mbits mbits firstly data bits sensors limited uav successfully receive data bits flying maximum speed addition data bits sensor reduced compared fig hovering mode necessary amount data bits still largest speed location fig optimal solution sensors mbits mbits mbits mbits aviation speed quite low set amount data sensor mbits evaluate impact energy constraint seen sufficient amount energy first four sensors uav fly maximum speed successfully receiving data last six sensors amount energy firstly decreases increases sensor sensor optimal speed also decreases first increase particular eighth sensor quite energy stringent uav hovers zero speed collect data addition transmission intervals first four sensors shift towards initial point space reserved last six sensors limited energy budget next reset amount energy simulation result shown fig found uav serves first three sensors medium aviation speed limited amount energy support maximum speed similarly energy sensor sufficient support data collection aviation hovering mode speed location fig optimal solution sensors mbits speed location fig optimal solution sensors mbits average aviation time min fig average aviation time versus average amount data random data requirement random energy random locations necessary compared figs energy constraint similar impact data requirement average performance evaluation evaluate average performance random data requirement random energy random locations amount data sensor follows uniform distribution mean value amount energy sensor follows uniform distribution mean value pair set satisfy feasibility constraint sensors uniformly distributed range results illustrated figs seen fig average amount energy sufficient average aviation time grows almost linearly increase amount energy deficient average aviation time grows exponentially increase due different relations aviation time amount data hovering mode aviation mode energy sufficient case uav collects data mainly aviation mode energy constrained case collects data mainly hovering mode fig observed amount data small average aviation time constant examined values means uav fly maximum average aviation time min fig average aviation time versus average amount energy random data requirement random energy random locations speed collect data aviation addition expected increase curves converges fixed point minimum aviation time uav flies maximum speed however figure shows convergence slow especially large values case curve firstly goes exponentially linearly slope onclusion paper solved aviation time minimization problem completing data collection mission sensor network analysis hovering mode provides feasibility condition successful data collection analysis case reveals optimal solution structures firstly optimal power allocation follows classical policy secondly maximum amount data bits successfully uploaded uav aviation decreasing function uav speed results simple bisection method find optimal aviation speed case shown division data collection intervals optimized via algorithm according numerical results observed behavior uav relies locations data amount energy amount sensors sufficient amount energy uav fly maximum speed otherwise speed proportional sensors energy budgets distance inversely proportional amount data uploaded ppendix roof emma since decreasing therefore indicates increasing addition lim lim hence ppendix roof roposition first inequality holds according lemma gap arbitrarily small tends infinity second inequality equality holds hence always feasible transmission mode bits successfully transmitted contrary left hand side always less therefore equation feasible ppendix roof heorem lagrangian function problem expressed setting get optimal power allocation expressed water level satisfied equality channel gain equivalent based found must hold continuous interval next derive necessary sufficient condition sufficiency suppose max max holds guarantee equal maximum value must larger max hand according according max equivalent therefore sufficiency proved addition maximize throughput must satisfied equality results equality condition hence obtained necessity suppose equivalently holds true let equals right hand side satisfied equality guarantees power allocation optimal energy budget fully utilized based equality max therefore hence necessity proved optimal throughput obtained substituting deducing follows arctan bmax ppendix roof heorem based achieve maximum throughput energy fully used replacing equation according first line bmax define function since decreasing function therefore first inequality holds due monotonicity second inequality holds due concavity log function based conclude increasing function since bmax decreasing function ppendix roof roposition optimal solution minimization problem min given therefore min min secondly hence min combining prove eferences zeng zhang lim wireless communications unmanned aerial vehicles opportunities challenges ieee communications magazine vol may dji inspire power beyond imagination online available http sun channel modeling channel dissertation university south carolina jul dong ota lin tang zhu data gathering wireless sensor networks journal supercomputing vol han swindlehurst liu optimization manet connectivity via smart unmanned air vehicles ieee transactions vehicular technology vol johansen path planning uavs communication constraints using splat milp journal intelligent robotic systems vol koyuncu khodabakhsh surya seferoglu deployment trajectory optimization uavs quantization theory approach zeng zhang joint trajectory communication design enabled wireless networks zeng zhang lim throughput maximization mobile relaying systems ieee transactions communications vol kalogerias petropulu mobile beamforming spatially controlled relay communications proceedings ieee international conference acoustics speech signal processing icassp cheng hsiao kung vlah maximizing throughput networks paradigm ieee wireless communications networking conference lyu zeng zhang lim placement optimization mobile base stations ieee communications letters vol mar alzenad lagum yanikomeroglu placement unmanned aerial vehicle base station maximal coverage ieee wireless communications letters vol mozaffari saad bennis debbah efficient deployment multiple unmanned aerial vehicles optimal wireless coverage ieee communications letters vol ahmed kanhere jha importance link characterization aerial wireless sensor networks ieee communications magazine vol may jiang swindlehurst optimization uav heading uplink ieee journal selected areas communications vol jun abdulla fadlullah nishiyama kato ono miura optimal data collection technique improved utility networks ieee conference computer communications infocom apr say inata liu shimamoto data gathering framework wireless sensor networks ieee sensors journal vol jul zhan zeng zhang data collection uav enabled wireless sensor network lyu zeng zhang cyclical multiple access communications tradeoff ieee wireless communications letters vol franco buttazzo coverage path planning uavs ieee international conference autonomous robot systems competitions apr zeng zhang uav communication trajectory optimization ieee transactions wireless communications vol jun zhang zeng zhang uav communication trajectory optimization connectivity constraint verdu channel capacity per unit cost ieee transactions information theory vol bertsekas dynamic programming optimal control athena scientific belmont
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weighted growth functions automatic groups nov mikael abstract growth function generating function sizes spheres around identity cayley graphs groups present novel method calculate growth functions automatic groups normal form recognizing automata recognize single normal form group element context free complexity context free grammars translated algebraic systems equations whose solutions represent generating functions corresponding symbols approach allows seamlessly introduce weightings growth function assign different even distinct weights generators underlying presentation weighting reflected growth function recover known growth functions small braid groups calculate growth functions weight generator automatic presentation braid groups according lengths braid generators introduction analytic combinatorics provides tools enumerating structures described formal grammars producing generating functions paper approach enumeration minimal length words representing group elements finitely presented automatic groups using generating functions generated formal grammars associated group automatic structure group presentation define cayley graph graph group elements vertices edge vertex generator vertex geodesic word shortest word generators inverses representing group element corresponds shortest path cayley graph radius sphere around identity set elements whose geodesic words length denote growth function generating function sequence nonnegative integers first section introduce route formal grammar generating function section demonstrate methods apply automatic group working explicit example braid group three strands date november mikael counting grammars chomsky proved contextfree language studied using generating functions article provides construction finding generating function related specific grammar starting form grammar rewriting rule translated algebraic equation terminal symbol assigned expression variables resulting generating function symbol assigned generating function rewriting assignment replaced equality concatenation multiplication disjunction addition first simple example balanced sequences grammar asb translating algebraic equation would get xys weighting symbol symbol resulting generating function count number strings number symbols result evaluating count length string equation solved straightforwardly immediately read exactly one combination string follows one unique string even length strings chomsky proved long grammar corresponding generating function rational functions anything described grammar suggests concrete approach enumeration find form grammar describing structures translate grammar system polynomial equations use basis elimination order solve system equations isolating basis elements concentrated interesting nonterminal symbol terminal variables solve rational form generating function braids automatic groups braid groups usually introduced finite presentations terms elementary braids strands braid group generators crosses strand strand give illustration weighted growth functions automatic groups figure generators braid group figure relations braid group figure inspecting effects reidemeister moves manipulations separated areas derive finite presentation figure shows relations smallest braid group relations applicable one proof word problem solvable braid groups described demonstrating automatic groups solve word problem braid groups automatic automatic group finitely presented group coupled several automata one detect whether given string generators normal form group element one detect right products normal form generator braid groups form example biautomatic groups grammars recognizing right products left products part really interests though normal form recognizer grammar normal forms mikael algebraic equations compute generating functions group sizes computed generating functions also studied extensively finitely presented groups called growth series braid groups even know grammars pick exactly one normal form group element normal form geodesic shortest possible expression specific set generators charney gives grammar braid group following transition rules state terminal construction automaton generalizes braid groups exponential growth number rules grammar improves previous constructions needed factorial growth number rules grammar produce system algebraic equations counts generator charney presentation equally solving system term order eliminate using favorite computer algebra system recovers basis weighted growth functions automatic groups first terms completely avoids one generator elimination ideal produces functional equation rewrite follows recovers growth function computed charney method going basis computations however flexible charney linear algebra approach since choose weights instance compute growth series automatic presentation weighted number elementary braid generators used word still retains strong focus automatic presentation see longer calculates geodesic shortest words presentation elementary braid generators achieve weight term translating system equations number automatic generators involved number elementary braid generators term producing system equations calculating eliminating basis produces mikael references first terms elimination order projection producing functional equation solve producing comparing small braids produces braids using two elementary generators whereas enumeration predicts reason discrepancy precisely fact geodesic measured terms elementary generators terms automatic generators hence words elementary generating set minimal representatives automatic presentation hence shows length instead conclusion methods analytical combinatorics producing generating functions directly contextfree grammars directly applicable problem computing growth functions automatic groups weighted provides insight automatic group geodesic words relate presentation different choice generators however instance braid groups automatic presentations tend sort generators moving elementary generators front inverses end word may produce geodesic simpler presentation unclear get closer growth function elementary presentation braid group references ruth charney geodesic automation growth functions artin groups finite type mathematische annalen noam chomsky marcel algebraic theory contextfree languages studies logic foundations mathematics david epstein word processing groups peters
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groups character degrees almost simple groups socle mathieu groups jan seyed hassan alavi ashraf daneshkhah ali jafari abstract let finite group denote set complex irreducible character degrees paper prove finite group almost simple group whose socle mathieu group exists abelian subgroup isomorphic study heading towards study extension huppert conjecture almost simple groups introduction let finite group let irr set complex irreducible character degrees denote set character degrees say irr complex group algebra determines character degrees multiplicities growing interest question regarding structure recovered character degree set without multiplicity character degree set use completely determine structure example groups set character degrees also share character table character degree set used distinguish solvable nilpotent groups example either recently navarro showed character degree set alone determine solvability group indeed constructed finite perfect group finite solvable group also discovered navarro rizo exists finite perfect group finite nilpotent group character degree set notice examples finite perfect groups nonabelian simple remains open whether complex group algebra determine solvability group see brauer problem however situation simple groups related groups rather different proved recently quasisimple groups uniquely determined isomorphism complex group algebras finite group quasisimple perfect nonabelian simple group late huppert posed conjecture asserts nonabelian simple groups essentially characterized set character degrees date january mathematics subject classification primary secondary key words phrases character degrees almost simple groups sporadic simple groups mathieu groups huppert conjecture corresponding author ashraf daneshkhah alavi daneshkhah jafari conjecture huppert let finite group finite nonabelian simple group sets character degrees abelian group conjecture verified alternating groups many simple groups lie type sporadic simple groups paper initiate investigation extension huppert conjecture almost simple groups whose socle sporadic simple groups group called almost simple exists nonabelian simple group aut indeed paper devoted studying finite groups character degrees almost simple groups socle one mathieu groups theorem let finite group let almost simple group whose socle one mathieu groups exists abelian group isomorphic order prove theorem establish following steps huppert introduced let almost simple group socle let group character degrees show step step chief factor isomorphic step irr step step isomorphic note prove step determine finite simple groups whose irreducible character degrees divide irreducible character degrees almost simple groups socle sporadic simple groups proposition simple groups listed table result somehow relates theorem sporadic simple groups preliminaries section present useful results prove theorem first establish definitions notation throughout paper groups finite recall group said almost simple group socle aut nonabelian simple group positive integer denotes set prime divisors group write instead irr inertia group denoted defined character irreducible character nonnegative integer called irreducible constituents set irreducible constituents denoted irr notation definitions standard could found lemma theorems suppose irr irr divides particular prime irr gallagher theorem irr irr irr groups character degrees almost simple mathieu groups lemma theorems suppose irr let irr irr particular extends irr irr irr particular irr character irreducible projective representation degree character irreducible projective representation degree lemma lemma let solvable factor group minimal respect nonabelian two cases occur prime hence exists irr irr irr irr frobenius group elementary abelian frobenius kernel prime smallest integer mod irr either divides latter case divides proper multiple divides irr irr irr proper multiple either divides divides moreover divisible nontrivial proper character degree lemma let finite group nonabelian simple group exists nontrivial irreducible character extends aut minimal normal subgroup nonabelian simple group irr extends aut irr extends proof part follows theorems prove part see lemma lemma lemma suppose every irr divides order schur multiplier lemma theorem let normal subgroup finite group let irr assume odd irr solvable degree properties almost simple groups socle sporadic section determine finite simple groups whose irreducible character degrees divide irreducible character degrees almost simple groups whose socles sporadic simple groups proposition let simple group let almost simple group whose socle sporadic simple group character degrees divides character degrees table alavi daneshkhah jafari proof suppose almost simple group socle sporadic simple group suppose also simple group whose degrees divide degrees prime divisors degrees exactly primes dividing therefore hence know possible simple groups need check degrees divide degrees could mainly done since arguments similar group give detailed proof cases frequently used referred sections suppose first isomorphic one simple groups note degrees respectively therefore isomorphic suppose isomorphic one simple groups mcl degree divisible contradiction degree divisible also contradiction therefore claimed table simple groups whose irreducible character degrees divide character degrees almost simple groups socle sporadic simple groups mcl mcl mcl continued groups character degrees almost simple mathieu groups suz suz table continued suz suz mcl mcl continued alavi daneshkhah jafari table continued mcl mcl suz mcl suz groups socle section prove theorem almost simple group whose socle theorem proved therefore need deal case convenience mention properties obtained lemma let schur multiplier group outer automorphisms degrees irreducible characters maximal subgroup whose index divides degrees one following occurs groups character degrees almost simple mathieu groups divides iii finite nonabelian simple group whose irreducible character degrees divide degrees isomorphic proof parts follows part follows proposition table part straightforward calculation proof theorem noted may assume assume finite group proof theorem follows following lemmas lemma proof assume contrary normal subgroup maximal nonabelian solvable group apply lemma one following cases prime case irreducible character degree since irreducible character degree conclude let irr lemma implies irr lemma irreducible character degree contradiction frobenius group kernel divides let lemma implies divides impossible divide every divisor let proper multiple lemma must divide contradiction lemma divisible contradiction similarly divide contradiction lemma let chief factor proof suppose nonabelian simple group positive integer since finite nonabelian simple group whose irreducible character degrees divide degrees lemma isomorphic one groups isomorphic one groups degree divisible assume isomorphic character degree irreducible constituent divides consequently character degree contradiction similarly case isomorphic factor group character degree implies character degree contradiction therefore isomorphic lemma let irr proof suppose lemma irr let maximal subgroup containing alavi daneshkhah jafari set follows lemma divides degrees must divide character degrees hence lemma one following holds suppose divide since trivial schur multiplier follows extends irr lemma irr irr turns divide degrees contradiction therefore hence index maximal subgroup containing must divide implies divides particular thus extends lemma irr leads contradiction taking suppose case equivalently moreover extends irr lemma divides degrees contradiction iii suppose extends irr thus irr irr impossible taking therefore lemma divides order schur multiplier isomorphic character degree therefore must degree divisible contradiction hence equivalently lemma subgroup trivial hence proof lemmas suppose nonabelian let normal subgroup chief factor nonabelian simple group follows lemma possesses nontrivial irreducible character irr extends lemma must irr choose largest degree since nontrivial divides degree contradiction therefore abelian since conclude lemma exists abelian group proof set since follows aut lemma conclude isomorphic case isomorphic must impossible character degree therefore isomorphic groups socle section prove theorem almost simple group whose socle note theorem proved see therefore need focus case convenience mention properties obtained groups character degrees almost simple mathieu groups lemma let schur multiplier group outer automorphisms degrees irreducible characters maximal subgroup whose index divides degrees one following occurs divides iii divides finite nonabelian simple group whose irreducible character degrees divide degrees isomorphic proof parts follows part follows proposition table part straightforward calculation proof theorem theorem true mathieu group remains assume follows assume finite group proof follows following lemmas lemma proof assume contrary normal subgroup maximal nonabelian solvable group apply lemma following two cases suppose prime irreducible character degree impossible group irreducible character prime power degree frobenius group kernel divides let lemma implies divides impossible divide every divisor let proper multiple lemma must divide contradiction lemma must divide contradiction lemma divisible contradiction similarly divide impossible therefore lemma let chief factor proof suppose nonabelian simple group positive integer since finite nonabelian simple group whose irreducible character degrees divide degrees lemma group isomorphic observe degree alavi daneshkhah jafari table triples lemma divisible implies case assume isomorphic one simple groups first column table character degree third column table irreducible constituent divides consequently character degree forth column table contradiction example isomorphic character degree divides character degree contradiction therefore lemma irr proof suppose lemma irr let maximal subgroup containing set follows lemma divides degrees must divide character degrees hence lemma one following holds suppose divide subgroup index divide extend lemma irr irr divides degrees contradiction hence degree projective representation impossible suppose thus extends follows lemma divides character degrees impossible taking iii suppose case divides follows maximal subgroup index implies therefore say abelian subgroup order let irr write since aei degree divides since index maximal subgroup least extends irr lemma irreducible character irr divides impossible could take therefore moreover know lemma degree nontrivial proper irreducible projective representation observe shows odd irr lemma group solvable contradiction groups character degrees almost simple mathieu groups table character degrees lemma degree show lemma divides order schur multiplier irreducible character degree divisible one degrees second row table contradiction therefore lemma subgroup trivial hence proof follows lemmas assume nonabelian let normal subgroup chief factor nonabelian simple group lemma nontrivial irreducible character irr extends lemma implies irr choose largest degree divides degree contradiction therefore abelian hence done lemma exists abelian group proof set since follows conclude isomorphic aut since case isomorphic conclude impossible character degree therefore isomorphic references alavi daneshkah wakefield huppert conjecture rend semin mat univ padova alavi daneshkhah wakefield huppert conjecture conway fischer families sporadic simple groups aust math bessenrodt nguyen olsson complex group algebras double covers symmetric alternating groups algebra number theory bianchi chillag lewis pacifici character degree graphs complete graphs proc amer math electronic brauer representations finite groups lectures modern mathematics vol pages wiley new york connor leemans atlas subgroup lattices finite almost simple groups arxiv june conway curtis norton parker wilson atlas finite groups oxford university press eynsham maximal subgroups ordinary characters simple groups computational assistance thackray gap group gap groups algorithms programming version huppert character theory finite groups volume gruyter expositions mathematics walter gruyter berlin huppert simple groups determined set character degrees illinois alavi daneshkhah jafari huppert simple groups determined set character degrees preprint isaacs character theory finite groups ams chelsea publishing providence corrected reprint original academic press new york answer question isaacs character degree graphs adv navarro set character degrees finite group determine solvability proc amer math navarro rizo nilpotent perfect groups set character degrees algebra nguyen wakefield projective special linear groups determined set character degrees algebra alternating sporadic simple groups determined character degrees algebr represent theory wakefield huppert conjecture monster baby monster monatsh wakefield verifying huppert conjecture comm algebra wakefield verifying huppert conjecture algebr represent theory zavarnitsine finite simple groups narrow prime spectrum arxiv alavi department mathematics faculty science sina university hamedan iran address gmail preferred daneshkhah department mathematics faculty science sina university hamedan iran address adanesh gmail preferred jafari department mathematics faculty science sina university hamedan iran address
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nov insensitive stochastic gradient twin support vector machines large scale problems zhen wanga shaob lan baia liuc dengd school mathematical sciences inner mongolia university hohhot school economics management hainan university haikou china school statistics capital university economics business beijing college science china agricultural university beijing abstract stochastic gradient descent algorithm successfully applied support vector machines called pegasos many classification problems paper stochastic gradient descent algorithm investigated twin support vector machines classification compared pegasos proposed stochastic gradient twin support vector machines sgtsvm insensitive stochastic sampling stochastic gradient descent algorithm theory prove convergence sgtsvm instead almost sure convergence pegasos uniformly sampling approximation sgtsvm twin support vector machines also given pegasos opportunity obtain approximation support vector machines addition nonlinear sgtsvm derived directly linear case experimental results artificial datasets large scale problems show stable performance sgtsvm fast learning speed keywords classification support vector machines twin support vector machines stochastic gradient descent large scale problem corresponding author email address shao preprint submitted elsevier november introduction support vector machines svm powerful tool classification already outperformed classifiers wide variety applications different svm pair parallel hyperplanes twin support vector machines twsvm pair nonparallel hyperplanes proposed developed twin bounded support vector machines tbsvm twin parametric margin support vector machines tpmsvm weighted lagrangian twin support vector machines wltsvm classifiers widely applied many practical problems training stage svm solves quadratic programming problem qpp whereas twsvm solve two smaller qpps traditional solver interior method however neither svm twsvm based solvers deal large scale problem especially millions samples order deal large scale problem many improvements proposed svm sequential minimal optimization coordinate decent method trust region newton stochastic gradient descent algorithm sgd twsvm successive overrelaxation technique algorithm dual coordinate decent method stochastic gradient descent algorithm svm pegasos attracts great attention partitions large scale problem series subproblems stochastic sampling suitable size proved pegasos almost sure convergent thus able find approximation desired solution high probability existing experiments confirm effectiveness algorithms amazing learning speed however large scale problem stochastic sampling sgd may bring difficulties svm due small subset dataset selected training fact subset suitable pegasos would weak well known svm support vectors svs small subset dataset decides final classifier stochastic sampling include svs sufficiently classifier would lose generalizations figure toy example pegasos two classes figure positive negative classes respectively include samples circle one potential svs solid blue line separating line obtained pegasos three different sampling strengthening circle sample infrequently using circle sample iii ignoring class class class class class class figure pegasos samples two classes training includes samples iterations circle sample used twice training includes samples iterations circle sample used iii training includes samples iterations circle sample excluded class class class class class class figure sgtsvm samples two classes training includes samples iterations circle sample used twice training includes samples iterations circle sample used iii training includes samples iterations circle sample excluded circle sample figure shows circle sample plays important role separating line infrequently using ignoring sample would lead misclassify compared svm significant twsvm stable sampling strongly depend special samples svs indicates sgd suitable twsvm therefore paper propose stochastic gradient twin support vector machines sgtsvm different pegasos method selects two samples different classes randomly iteration construct pair nonparallel hyperplanes due twsvm fits training samples method stable stochastic sampling thus gains well generalizations moreover characteristics inherited twsvm result sgtsvm suits many cases cross planes dataset preferential classification toy example figure shows corresponding results sgtsvm comparing figure figure clear sgtsvm performs better pegasos main contributions paper includes twsvm sgtsvm proposed easy extended classifiers prove proposed sgtsvm convergent instead almost sure convergence pegasos iii uniformly sampling proved original objective solution sgtsvm bounded optimum twsvm indicates solution sgtsvm approximation optimal solution twsvm pegasos opportunity obtain approximation optimal solution svm information please see corollaries nonlinear case sgtsvm obtained directly based original problem iteration sgtsvm includes multiplications without additional storage fastest one proposed classifiers rest paper organized follow section briefly reviews svm pegasos twsvm linear nonlinear sgtsvms together theoretical analysis elaborated section experiments arranged section finally give conclusions related works consider binary classification problem real space set training samples represented sample label organize samples class matrix samples class matrix give brief outlines related works svm support vector machines svm searches separating hyperplane introducing regularization term primal problem svm expressed qpp follow min denotes norm parameter quantitative meanings vector ones appropriate dimension slack vector diag note minimization regularization term equivalent maximize margin two parallel supporting hyperplanes structural risk minimization principle implemented problem pegasos pegasos considers strongly convex problem modifying follow min recasts problem min replaces negative components vector zeros tth iteration pegasos constructs temporary function defined random sample starting initial pegasos iteratively updates step size cyt sign terminate conditions satisfied last outputted new sample predicted sign proved average solution bounded optimal solution thus pegasos probability least find good approximation authors also pointed often used instead practice sample selected randomly replaced small subset belonging whole dataset subset including sample often used practice order extend generalization ability pegasos bias term svm appended pegasos replacing however modification would lead function strongly convex thus yield slow convergence rate twsvm twsvm seeks pair nonparallel hyperplanes expressed hyperplane close samples one class certain distance class find pair nonparallel hyperplanes required get solutions primal problems min min positive parameters slack vectors geometric meaning clear example objective function makes samples class proximal hyperplane together regularization term constraints make sample class distance away hyperplane solutions problems respectively obtained new point assigned class depends distance two hyperplanes arg min kwi absolute value sgtsvm section elaborate sgtsvm give convergence analysis together boundedness linear formation following notations section recast qpps twsvm unconstrained problems min min respectively order solve two problems construct series strictly convex functions selected randomly respectively functions obtained sign sign sign sign respectively sgtsvm starts initial updates given step size typically set terminated condition satisfied assigned assigned new sample predicted procedures summarized algorithm nonlinear formation extend sgtsvm nonlinear case kernel trick suppose predefined kernel function nonparallel hyperplanes expressed counterparts formulated min algorithm sgtsvm framework require given training dataset positive class negative class select parameters small tolerance tol typically tol ensure set zeros choose pair samples random respectively compute tth gradients update tol stop updating set tol stop updating set min construct series functions similar updates obtained details omitted large scale problem time consuming calculate kernel however reduced kernel strategy successfully applied svm twsvm also applied sgtsvm reduced kernel strategy replaces random sampled subset practice needs samples get well performance reducing learning time without loss generalization analysis subsection discuss two issues convergence solution sgtsvm relation solution sgtsvm optimal one twsvm convenience consider first qpp linear twsvm together sgd formation linear sgtsvm conclusions another qpp nonlinear formations obtained easily first one let notations subscripts sgtsvm also comply definition first qpp reformulated min next reformulate tth function sgtsvm samples selected randomly tth iteration respectively denoted sign given step size updated sign lemma upper bounds proof formation rewritten identity matrix sign note sufficient positive integer positive definite largest eigenvalue smaller equal based therefore max thus max max let largest norm samples dataset max max leads upper bound upper bound theorem iterative formation sgtsvm convergent proof one hand proof lemma lim indicates lim hand indicates following limit exists lim note infinite series vectors convergent norm series convergent therefore following limit exists lim combine conclude series convergent based theorem reasonable take terminate condition tol moreover reform order keep convergent fast suggested set following analyse relation solution sgtsvm optimal solution twsvm lemma let sequence convex functions sequence vectors belongs set suppose upper bounds respectively sufficiently large given proof since convex note combine multiplying leads conclusion hand suppose lim lim lim lim note lim given sufficiently large lnt ready bound average instantaneous objective theorem defined sgtsvm constructed optimal solution two constants actually upper bounds respectively sufficiently large given proof obviously convex let respectively upper bounds conclusions come lemmas following let discuss relation solutions sgtsvm twsvm uniform sampling corollary assume conditions stated theorem sample number respectively suppose integer sample selected times random sufficiently large given proof first prove formation since upper bound largest norm samples dataset first part second part third part right hand respectively therefore constant satisfying easy obtain thus since note objective twsvm based using theorem conclusion immediately modify sampling rule obtain result one corollary corollary assume conditions stated corollary suppose integer least common multiple sample selected times random one times random sufficiently large given note proof corollary corollary corollaries provide approximations sampling rule stated corollaries upper bounds longer holds however kakade tewari shown way obtain similar bounds high probability twsvm sgtsvm figure results twsvm sgtsvm cross planes black solid lines respectively experiments experiments compared sgtsvm svm pegasos twsvm several artificial large scale problems methods implemented intel core duo processor ghz ram artificial datasets artificial datasets pegasos twsvm sgtsvm implemented matlab corresponding sgtsvm matlab codes uploaded upon http first consider similarity twsvm sgtsvm two methods implemented cross planes dataset twsvm superior dataset figure shows proximal lines dataset obvious two proximal lines sgtsvm similar ones twsvm twsvm sgtsvm precisely capture data distribution thus obtain well classifier measure similarity quantitatively optimums twsvm calculated compared ones iteration sgtsvm cross planes uci datasets dataset australia includes samples features dataset creadit includes samples features dataset hypothyroid includes samples features linear twsvm sgtsvm tbsvm sgtsvm tbsvm sgtsvm objective objective tbsvm sgtsvm tbsvm sgtsvm iteration iteration cross planes australia sgtsvm tbsvm objective objective tbsvm sgtsvm tbsvm sgtsvm tbsvm sgtsvm iteration iteration creadit hypothyroid figure results linear twsvm sgtsvm four datasets vertical axis denotes objectives nonlinear versions implemented gaussian kernel exp used nonlinear versions parameters fixed figure shows results two linear classifiers figure corresponds nonlinear case figures horizontal axis denotes iteration sgtsvm vertical axis denotes objectives twsvm sgtsvm due objectives twsvm constant denoted two horizontal lines objectives sgtsvm iteration denoted two broken lines figures different datasets seen sgtsvm converges twsvm different iterations instance linear sgtsvm converges twsvm iterations figure whereas thing appears figure iterations generally sgtsvm converges twsvm iterations datasets either linear nonlinear case furthermore cross validation used datasets ran twsvm sgtsvm times tbsvm sgtsvm tbsvm sgtsvm objective objective tbsvm sgtsvm tbsvm sgtsvm iteration cross planes australia tbsvm tbsvm sgtsvm tbsvm objective sgtsvm tbsvm sgtsvm objective iteration sgtsvm iteration iteration creadit hypothyroid figure results nonlinear twsvm sgtsvm four datasets vertical one fig reported mean accuracy standard deviation table differences mean accuracies implies classifiers obtained twsvm sgtsvm significant difference secondly test stability sgtsvm compared pegasos datasets generated randomly dataset contain samples negative samples normal distribution positive ones best classification point zero implemented pegasos sgtsvm without restrictions datasets obtained classifiers shown figure upper right digit mean lines together standard deviation parameters pegasos sgtsvm fixed clear sgtsvm obtains much compact classification lines pegasos mean line sgtsvm closer zero standard deviation smaller table mean accuracy standard deviation twsvm sgtsvm cross validation data cross planes australia creadit hypothyroid linear case nonlinear case pegasos sgtsvm figure results pegasos sgtsvm artificial datasets upright black solid lines final classifiers pegasos sgtsvm figure results pegasos sgtsvm artificial datasets upright black solid lines final classifiers samples dash line invisible sampling pegasos order investigate effect sampling pegasos sgtsvm implemented datasets restricted sampling possible support vectors negative samples svm samples close support vectors invisible sampling figure shows results pegasos sgtsvm dash line denotes samples scope invisible sampling figure seen classification lines pegasos fall two regions sgtsvm obtains compact region thus means possible support vectors significantly influence pegasos sgtsvm relatively relies data distribution figures pegasos always acquires mean classification line zero larger standard deviation sgtsvm therefore sgtsvm stable pegasos datasets without restricted sampling show classifiers stability recorded classification accuracies pegasos sgtsvm one datasets pegasos sgtsvm implemented times dataset parameters set two methods iterated times every accuracies methods reported figure figure accuracies sgtsvm belong pegasos indicates sgtsvm stable pegasos aspect classification result although pegasos obtains highest accuracy test sgtsvm obtains higher accuracies pegasos cases finally test convergence pegasos sgtsvm dataset pegasos accuracy sgtsvm times figure accuracies pegasos sgtsvm normal distribution dataset method implemented times mean pegasos mean sgtsvm figure results pegasos sgtsvm normal distribution dataset method implemented times horizontal axis iteration vertical one classification location iteration pegasos mean sgtsvm mean pegasos std sgtsvm std time sec pegasos mean sgtsvm mean pegasos std sgtsvm std tol tol figure iteration time pegasos sgtsvm normal distribution dataset method implemented times contains samples generated randomly negative samples normal distribution positive ones pegasos sgtsvm implemented times method iterated times current classification locations different iterations reported figure horizontal axis iteration vertical one classification location figure seen initial selected samples affect pegasos sgtsvm iterating times iterating times classification locations two methods centralized zero error less iii important pegasos gets higher error iterating times sgtsvm indicates pegasos converges slower sgtsvm precisely discuss convergence pegasos sgtsvm implemented times method terminated solution error parameter tol details tol found algorithm tol selected corresponding iteration spent time reported figure clear figure sgtsvm converges faster pegasos tol moreover one needs smaller solution error tol tol iterations pegasos would times sgtsvm would times tol thus learning time pegasos sgtsvm hundredfold therefore sgtsvm converges much faster pegasos large scale datasets test feasibility methods large scale datasets ran svm pegasos sgtsvm six large scale datasets table shows table details large scale datasets data name skin gashome susy kddcup gas hepmass samples dimension ratio details large scale datasets ratio table sample number positive class negative one dataset split two subsets one including samples used training including samples used testing svm implemented liblinear pegasos sgtsvm implemented softwares written language corresponding softwares downloaded http nonlinear sgtsvm reduced kernel used kernel size fixed first let test influence parameter tol pegasos sgtsvm methods implemented large scale datasets tol respectively set parameters fixed testing accuracy learning time reported figure comparing figure bee seen sgtsvm including linear nonlinear cases stable pegasos tol order select high accuracy acceptable learning time figure tol set pegasos set sgtsvm compare svm pegasos sgtsvm fixed tol datasets methods accuracies recorded table validation accuracy obtained cross validation training subset testing accuracy obtained testing subset parameters svm pegasos sgtsvm selected gaussian kernel parameter nonlinear sgtsvm selected simplicity also set sgtsvm optimal parameters recorded table table obvious sgtsvm owns highest time sec accuracy skin gashome susy kddcup gas hepmass skin gashome susy kddcup gas hepmass tol tol pegasos pegasos skin gashome susy kddcup gas hepmass time sec accuracy skin gashome susy kddcup gas hepmass tol time sec accuracy skin gashome susy kddcup gas hepmass skin tol gashome susy kddcup gas hepmass tol tol figure accuracy learning time pegasos linear sgtsvm nonlinear sgtsvm six large scale datasets dashed box corresponds chosen parameter tol table results large scale datasets data skin gashome susy kddcup gas hepmass validation testing validation testing validation testing validation testing validation testing validation testing svm pegasos linear case nonlinear case memory table optimal parameters svm pegasos sgtsvm data skin gashome susy kddcup gas hepmass validation testing validation testing validation testing validation testing validation testing validation testing svm pegasos linear case nonlinear case linear sgtsvm nonlinear sgtsvm pegasos liblinear time sec figure learning time sgtsvm pegasos liblinear large scale datasets optimal parameters accuracies groups comparisons performs well svm pegasos groups however svm performs much worse sgtsvm dataset gashome work three much larger datasets though pegasos work datasets performs much worse sgtsvm susy gas comparing learning time methods report time figure optimal parameters obvious sgtsvm including linear nonlinear cases much faster others thus sgtsvm comparable svm pegasos large scale datasets addition softwares sgtsvm pegasos need much less ram liblinear software svm detail liblinear needs store entire training set ram pegasos sgtsvm store subset related iteration due required memory liblinear increases size dataset tends memory increasing data size thing appear pegasos sgtsvm conclusion stochastic gradient twin support vector machines sgtsvm based stochastic gradient decent algorithm proposed hiring nonparallel hyperplanes sgtsvm stable stochastic sampling pegasos theory prove sgtsvm convergent approximation twsvm uniform sampling experimental results confirmed merits sgtsvm shown sgtsvm better accuracy compared liblinear pegasos fastest learning speed practical convenience corresponding sgtsvm codes including matlab language downloaded http future work possible design special sampling sgtsvm obtain powerful performance together applying sgtsvm bigdata problems acknowledgment work supported national natural science foundation china nos natural science foundation inner mongolia autonomous region china zhejiang provincial natural science foundation china references bazarra sherali shetty nonlinear programmingtheory algorithms second wiley bennar monnez almost sure convergence stochastic approximation process convex set international journal applied mathematics vapnik learning rigorous support vector machines 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nonparallel support vector machine solver neural networks wang shao bai deng twin support vector machine clustering ieee transactions neural networks learning systems wang shao model selection smooth twin support vector machine pattern recognition wang shao proximal support vector classifier applications neural computing applications towards optimal one pass large scale learning averaged stochastic gradient descent arxiv preprint zhang tian deng new interpretation support vector machines statistical learning theory science china zhang solving large scale linear prediction problems using stochastic gradient descent algorithms proceedings international conference machine learning page acm
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mar symmetric surfaces sheng bai vanessa robins chao wang shicheng wang abstract suppose orientation preserving action finite group closed surface genus extends embedding upper bound achieved surfaces realizing maximum symmetries either unknotted knotted similar problems category also discussed connection minimal surfaces addressed maximum symmetric surfaces realized minimal surfaces identified contents introduction upper lower bounds extendable finite groups unknotted examples symmetric surfaces knotted examples symmetric surfaces minimal surfaces space groups proof main result references introduction let closed orientable surface genus closed surface genus three dimensional torus short consider following question smooth category suppose action finite group closed surface extend maximum order group maximum action look like similar problem addressed surfaces embedded simplest compact sense one point compactification three space universal spherical covering spherical see surfaces orientable category note orientable surfaces mathematics subject classification key words phrases maximum surface symmetry minimal surface second author supported grant nsfc last author supported grant nsfc sheng bai vanessa robins chao wang shicheng wang embedded another natural significant question universal compact euclidean sense covers compact euclidean moreover covered preimage closed surface covering triply periodic surface interesting object natural sciences engineering definition let finite group closed surface extendable respect embedding act embedding orientable surface define whether unknotted way usual definition knotting definition embedding unknotted splits two handlebodies otherwise knotted assume orientable manifolds note already oriented according whether action preserves orientation surface define four classes maximum orders definition let either define maximum order extendable maximum order extendable preserve orientation maximum order extendable preserve orientation maximum order extendable serve orientation prove following theorem theorem theorem gives upper bounds invariants defined definition theorem provides infinitely many realize upper bound theorem theorem suppose genus extendable realizes one upper bounds corresponding surface thought symmetric surface theorem suppose positive integer theorem achieved upper bound unknotted embedding knotted embedding divide symmetric surfaces upper bound theorem achieved unknotted embedding upper bound theorem achieved odd proof theorem using formula equivariant loop theorem given proof theorem occupies remaining three sections paper outlined construct various examples realize upper bounds theorem unknotted case knotted case therefore theorem follows verification knottiness embeddings examples explicit following sense obtained standard opposite face identification cube define surfaces cube identification indeed theorem construct embeddings image moment expect realizing see conjecture constructions embeddings realizing often rely understanding crystallographic space groups knottedness examples involved theorem well theorem could verified arguments topological reasoning present convenient interesting way first link verification knottedness examples minimal surfaces equivalently triply periodic minimal surfaces important topic see examples historically many minimal surfaces constructed using symmetry known minimal surfaces must unknotted identify examples known triply periodic minimal surfaces isotopy therefore unknotted hand simple criterion covering space theory constructions examples knotted therefore realized minimal surfaces also identify group actions known space groups well supergroups theorem proved corollary realizing upper bound realized minimal surfaces easy construct extendable action order see example end introduction following conjecture suppose genus listed realizing upper bound theorem moreover examples realizing listed runs finitely many quadratic cubic functions sheng bai vanessa robins chao wang shicheng wang upper lower bounds extendable finite groups need following two important results see prove theorem formula regular branched covering transformation group let branched points indices equivariant loop theorem let three manifold smooth action discrete group let equivariant subsurface respect inclusion admits compression disk nonempty subset disk compression disk bound disk proof suppose extendable preserves orientation cutting along get three manifold since orientable must therefore contains two boundaries clearly acts since preserves orientation surface since abelian abelian induced homomorphism injective least one injective respect inclusion suppose admits compression disk equivariant loop theorem exists reverses orientation choose regular neighbourhood homeomorphism also compression disk clearly preserves orientation hence assume element preserves also preserves orientation particular fixed point free since preserves orientations induced preserves quotient topology homeomorphic regular branched covering since action fixed point free previous discussion simple closed curve let branch points indices note formula symmetric surfaces must bound disk containing one branch point bound disk contradiction hence either notice elementary calculation moreover equality holds remark zimmermann first proved order orientation preserving finite group action handlebody genus bounded soon work proof actually following theorem suppose embedded three manifold finite group acts preserves two sides orientation follow proof suppose extendable let normal subgroup containing elements preserve orientations case index two case index four hence proof suppose extendable action preserves orientation choose equivariant regular neighbourhood also acts since orientable homeomorphic twisted orientable double cover bundle projection since follows homeomorphic clearly preserves two sides since action preserves orientation also preserves orientation result similar proof example let unit cube cross properly embedded shown right side figure put copies get cuboid antennae properly embedded shown figure identify opposite faces cuboid sheng bai vanessa robins chao wang shicheng wang get graph genus first identify right left faces get circular antennae bars easy see acts pair dihedral group acts standard way around clear action preserves pair faces identified therefore induces action let regular neiborhood order acts figure unknotted examples symmetric surfaces section give three classes examples realizing upper bound case first construct triply periodic graph euclidean space space group preserving choose rankthree translation normal subgroup space finite group acts preserving graph finally choose equivariant regular neighbourhood surface discussion see example complement also handlebody hence surface unknotted definition let group integer translations let element acts following define three classes subgroups integers presented uniquely form follows hntx nty ntz hnty ntz ntz ntx ntx nty nty ntz ntx nty ntz ntx nty ntz subscript equal volume symmetric surfaces clearly subgroups one easily verify since linear combinations nty ntz ntx ntx nty hence similarly one verify hence definition define five isometries follows rxy rxyz isometries rxy rotations rotation angle line respectively isometry rxyz positive rotation cube body diagonal rule rotation direction notations space groups used come example let besthe one skeleton unit cube figure let tessellation unit cube triply periodic graph called simple primitive cubic lattice pcu rcsr figure one skeleton let hry rxy rxyz isometric group order symbol let space group preserves graph one vertex three edges hence euler characteristic genus clearly preserving acts order hence choose equivariant regular neighbourhood get extendable action order sheng bai vanessa robins chao wang shicheng wang similarly graph since normal subgroup acts preserving hence genus get extendable action order notice also normal subgroup similar construction get extendable action order graph example superscript equal volume minimal also equal number vertices graph following examples use similar notations example let graph unit cube figure consists four edges let one check connected known scientists bonding structure diamond dia rcsr figure shows fundamental region figure let hry rxyz isometric group regular tetrahedron formed convex hull symbol let rxy space group preserves graph two vertices four edges hence euler characteristic genus normal subgroup order acts preserving hence choose equivariant regular neighbourhood get extendable action order similarly graph one check normal subgroup symmetric surfaces quotient acts preserving hence genus get extendable action order example let graph unit cube figure three edges let let one check connected chiral net known many names hop including srs rcsr figure shows fundamental region figure let rxyz rxy space group preserves graph four vertices six edges hence euler characteristic genus normal subgroup order acts preserving hence choose equivariant regular neighbourhood get extendable action order similarly graph one check normal subgroup quotient acts preserving sheng bai vanessa robins chao wang shicheng wang hence genus get extendable action order knotted examples symmetric surfaces section give six classes examples realizing upper case embedding knotted bound surface bound handlebody one side complement handlebody handlebody clear discussion constructions extendable actions surfaces similar graph disconnected space group preserving rank three translation normal subgroup graph connected finite group acts preserving choose equivariant regular neighbourhood surface example let graph thought dual two connected components named rcsr unit cube figure figure part let space group let preserves normal subgroup since changes two components graph connected euler istic order genus acts preserving hence choose ant regular neighbourhood get extendable action order similarly let graph connected since symmetric surfaces normal subgroup acts preserving hence genus get extendable action order example let graph thought dual two connected components called rcsr fundamental region figure figure part let space group preserves contains normal subgroup since changes two components graph connected euler order characteristic genus acts preserving hence choose equivariant regular neighbourhood get extendable action order similarly graph connected one check normal subgroup acts preserving hence genus get extendable action order example let four connected components unit cube figure sheng bai vanessa robins chao wang shicheng wang figure part one check defined example space group preserves graph connected euler characteristic genus order acts preserving hence choose equivariant regular neighbourhood get extendable action order similarly graph preserving connected acts hence genus get extendable action order example let eight connected components unit cube figure one check defined example space group serves graph connected euler characteristic genus order group acts preserving hence choose ant regular neighbourhood get extendable action order symmetric surfaces figure part similarly graph connected acts preserving get extendable action hence genus order example let graph unit cube figure three edges three edges separately let min one check connected components similar mirror image figure shows part fundamental region construction space group preserves one check connected four vertices six edges hence euler characteristic genus group acts preserving hence choose equivariant regular neighbourhood get extendable action order similarly graph connected quotient acts sheng bai vanessa robins chao wang shicheng wang figure part min preserving hence genus get extendable action order definition let define ntx ntx define rotation following hexagonal lattice rotation respect direction example let hexagon center vertex see figure let contains boundary let contains infinitely many connected components horizontal plane contains exactly one local picture figure let isometric group symbol let space group preserves connected graph two vertices three edges hence euler characteristic genus clearly acts preserving symmetric surfaces figure regular hexagon figure local picture order hence choose equivariant regular neighbourhood get extendable action order normal subgroup notice connected similarly graph acts preserving sheng bai vanessa robins chao wang shicheng wang hence genus get extendable action order minimal surfaces space groups proof main result section finish proof theorem need results triply periodic minimal surfaces space groups lemma given three classical triply periodic minimal surfaces admit high symmetries schwarz surface schwarz schw schoen gyroid surface illustrated figure denote fundamental region cube volume redraw fundamental region used figure translational unit cells minimal surfaces match translational unit cells associated graphs displayed viewing direction picture shows cube twice region used depict depicted viewing direction following results known checked boundaries equivariant regular neighbourhoods example realized respectively symmetric surfaces unknotted actually genus minimal orientable closed surface must unknotted notations space groups come example space groups respectively preserved space groups respectively following index two subgroup sequences lemma let lattice covering map embedded surface connected knotted proof note first definition unknotted embedding induced map fundamental groups must surjective let connected component let stable subgroup lattice translations preserve one component hence surjective hence knotted proof theorem results surfaces described unknotted since disconnected lemma surfaces described knotted combining examples finish proof theorem going prove theorem based examples results recall example told action equivalently order preserves order extendable realizing upper bound group contains two elements satisfying preserves orientation reverses orientation reverses orientation preserves orientation choosing index two subgroups get extendable action realizing upper bound respectively proved recall example told action equivalently order similarly also order extendable realizing upper bound sheng bai vanessa robins chao wang shicheng wang furthermore odd contains translation reversing orientation preserving orientation see example details better see figure modulo translation get order extendable action realizing upper bound choosing index two subgroup get extendable action realizing upper bound proved remark discussion regular neighbourhood homeomorphic twisted complement essentially regular hood example similarly odd get order extendable realizing upper bound choosing index two subgroup get extendable action realizing upper bound regular neighbourhood homeomorphic twisted complement essentially regular neighbourhood example similar construction regular neighbourhood gyroid minimal surface translation reverses orientation handlebodies side mirror images one another references hop rcsr brakke minimal surface page http frohman hass unstable minimal surfaces heegaard splittings invent math hahn international tables crystallography reidel publishing company hempel princeton university press hurwitz algebraische gebilde mit eindeutigen transformationen sich mathematische annalen hyde blum landh lidin ninham andersson larsson language shape role curvature condensed matter new york usa elsevier hyde keeffe proserpio short history elusive yet ubiquitous structure chemistry materials mathematics angewandte chemie int meeks conformal structure geometry triply periodic minimal surfaces bull amer math soc meeks yau equivariant dehn lemma loop theorem commentarii mathematici helvetici keeffe peskov ramsden yaghi reticular chemistry structure resource rcsr database symbols crystal nets accts chem res symmetric surfaces schw schwarz gesammelte mathematische abhandlungen berlin springer scho schoen nasa tech note wang wang zhang zimmermann extending finite group actions surfaces topology appl wang wang zhang zimmerman maximal orders extendable finite group actions surfaces groups geom dyn zimmermann uber homomorphismen henkelkrper und endliche erweiterungen von comment math helv school mathematical sciences peking university beijing china address bais department applied mathematics research school physics engineering australian national university canberra australia address school mathematical sciences university science technology china hefei china address chao wang school mathematical sciences peking university beijing china address wangsc
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hypergroups hyperfields universal algebra apr louis rowen abstract hypergroups lifted power semigroups negation yielding method transferring results semigroup theory applies analogous structures hypergroups hyperfields hypermodules permits transfer general theory espoused hypertheory introduction note companion grew conversation matt baker realized tropical hyperfield isomorphic extended tropical arithmetic izhakian dissertation university also hand many parallels two theories motivated see whether hyperfields general also studied semiring theory fits well theory universal algebra might amenable study viewing hyperfield group additive structure part power set want extend structure power set thereby making definitions tighter improving additive structure make standard tools available developing algebraic theory thus theme embed category hyperfields modules category semirings negation modules studied defined power sets tricky part passing power set distributivity must weakened times notion call weak distributivity thereby weaken semiring distributivity holds respect special subset treat hyperrings hyperfields context turns hyperrings injected naturally power sets negation map context whereby hyperring identified subset singletons one develop linear algebra hyperfields also making appropriate adjustments view matroids semifields negation direction henry defined hypergroup structure symmetrized monoids semigroups need identity element monoids usually written multiplicatively start semirings standard reference since distributivity weakened remove well element definition enrich monoid hyper operation resembling addition definition monoid acts set multiplication satisfying multiplicative monoid also possessing structure additive abelian semigroup acts group premodule monoid abelian group acts also satisfying condition date april mathematics subject classification primary secondary key words phrases hyperfield negation map tropical algebra tropical geometry power set semiring supertropical algebra rowen premodule premodule monoid also satisfying condition left distributivity without would satisfying usual distributive laws various versions distributivity relevant later definition define following notions premodule right distributivity distributivity left right distributivity iii double distributivity generalized distributivity remark properties order increasing formal strength although induction double distributivity implies generalized distributivity motivation power sets semigroups let denote power set set set subsets sets called singletons theorem given monoid extend operation elementwise putting also monoid identity element acts given semigroup define addition elementwise defining also semigroup thus resp semiring via action generally relevant concepts universal algebra outlined including signatures defined via operators identical relations lift operators algebra follows given operator define via let call identical relation multilinear appears exactly definition via operators multilinear identical relation holding also holds proof first verify associativity generalized distributivity hypergroups hyperfields aik aik aik aik generalize proof easy induction applied definition formula satisfies thus multilinear identical relation holding elementwise also holds particular distributivity generalized distributivity lift semiring relation multilinear encounter difficulties due repetition example group indeed defining identical relation quadratic defining inverse unary operation left side fact invertible elements singletons hypermonoids semirings next step formulate extra structure terms addition possibly operations premodule easy semiring let pause review hypermonoids see much semiring structure would need intuitive definition hypermonoid triple analog associativity holds fundamental difficulty definition set element technically defined difficulty exacerbated considering generalized associativity example mean rectify passing definition hypermonoid triple commutative binary operation also associative sense define neutral element always think sort addition write note thus natural embedding given transfer addition defining definition hypermonoid closed repeated addition makes difficult check basic identical relations associativity many hypermonoids satisfy extra property property whenever singleton hyperinverse element hypermonoid element denoted hyperzero hypermonoid element form hypergroup hypermonoid satisfying extra property every element unique hyperinverse hypergroup satisfies condition rowen reversibility definition viro calls multigroup definition hypermodule monoid hypermonoid together action distributivity holds hyperring hypermonoid also hypermodule words hyperring additive hypermonoid also monoid respect associative multiplication distributes addition two operations distributivity holding elements hyperfield hyperring group definition defines hypermonoid morphism map hypergroups satisfying yields category hypergroups morphisms matches definition morphism easy instances associativity fails example consider natural algebra define sup element unique hyperinverse associative distinct single elements taking max whereas define sup element unique hyperinverse lemma monoid hypermonoid acts via action proof power set hyperfield proceed need associativity level sets need following definition make precise hopefully manageable since calculations power set definition every operator extended operator sense hypermonoid monoid respect commutative associative binary operation compatible operation singletons sense subset containing singletons sums neutral element lemma multiplicative monoid invertible element must singleton hypergroups hyperfields proof two elements invertible contains implying theorem generalizes easily theorem given hyperfield define addition elementwise means monoid whose identity element case viewed set singletons set invertible elements proof need verify associativity repeating proof theorem replacing set invertible elements must contained set singletons finite set write makes sense since already associativity remark viro showed general transition universal relations straightforward may seem analogous argument theorem unravels hyperrings since distributivity pass elements sets theorem hyperring argument shows finite sets theorem recall viro triangle hyperfield defined formula words iff exists euclidean triangle sides lengths triangle hyperfield satisfy double distributivity distributive words analog theorem fails distributivity overcome setback need modify underlying algebraic structure hyper level power set level crux matter weaken generalized distributivity sets actually many important examples hyperfields doubly distributive could pass distributive power semirings without ado even absence doubly distributivity formulate weaker theory terms universal algebra order techniques disposal face problematic since hypersum set could arbitrarily large however get around focusing monoid singletons using operators instead elements motivated fact hyperring multiplicative monoid bring following definition keeping power set mind definition set additive abelian semigroup multiplicative monoid necessarily satisfying usual distributive laws given distinguished multiplicative submonoid premodule also satisfies distributivity conditions line call tangible conditions hold call rowen described framework universal algebra elaboration details given define left multiplication maps unary operator rewrite rules identical relations example hyperring taking categorical approach improve results slightly working category weaker definition morphism definition multiplication weakly distributes addition subset finite case call weak power semiring semiring restriction definition hyperrings definition suppose monoid hypermonoid multiplication weakly distributes case call weak hyperring proposition suppose hyperring multiplication weakly distributes proof need verify writing since reverse inclusion fails since simultaneously encounter varying varies thus interested weak power semirings strictly speaking quite semirings far power set hyperring weak power semiring repeat proof theorem show bottom level lost anything theorem suppose weak hyperring also group hyperring identified set singletons proof theorem obtain distributivity need reverse inequality singleton given multiplicative inverses thus theory hyperfields embeds theory weak power semirings definition weak morphism weak power semirings multiplicative homomorphism satisfying induction finite set thus arbitrary described universal algebra definition note multiplicative version yields equality since since singletons hypergroups hyperfields weak power modules often case one gets deeper understanding turning modules done definition take slightly weaker definition line earlier categorical considerations definition suppose weak power semiring weak module set element monoid scalar multiplication satisfying following iii leads slight modification hypermodules definition weak hypermodule weak hyperring hypergroup together binary operation satisfying following properties iii finite submodule definition weak module morphism map weak modules satisfying weak module morphism example morphism given theorem suppose weak power semiring also group weak module proof repeat proof theorem first show indeed obtain distributivity need reverse inclusion given multiplicative inverses taking singleton given taking left multiplication element precisely weak distributivity words weakly distributive iff left multiplication weak module morphism every element group distributive negation maps also want treat negation maps perspective review definition definition negation map additive semigroup semigroup homomorphism order written thus operators including multiplication different perspective definition negation map operator addition satisfies rowen lemma negation map induces negation map via proof special case theorem viewing properties negation map identical relations see note likewise implies hyperfield negation map way embedded theory hyperfields theory negation map pushed even major examples let see applies major examples since examples important pay special attention set corresponding hyperring although theory presented formally passes weak power semiring one actually gets distributivity underlying hyperring satisfies generalized distributivity happens many examples even better often identify semiring already recognize many good examples put framework remark example let commutative semiring multiplicative monoid together surjection multiplicative monoids induced hyperring structure given hyperaddition law extends naturally via generalized distributivity thus inherited generalized distributivity explicitly yielding opposite direction given reverse get thus semiring theory embedded semiring theory complications arise example applicable vii viii next example example tropical hyperfield define define product max thus multiplicative identity element additive identity hyperfield satisfying property called tropical hyperfield proposition easily seen isomorphic hyperfields izhakian extended tropical arithmetic expounded supertropical algebra identify natural hyperfield isomorphism tropical hyperfield hypergroups hyperfields isomorphism semirings thus example applicable example krasner hyperfield let usual operations boolean algebra except krasner hyperfield satisfies property generates subsemiring three elements supertropical algebra monoid identify example applicable example valuative hyperfields example also isomorphic extended semirings sense way example hyperfield signs let usual multiplication law hyperaddition defined hyperfield satisfying property called hyperfield signs case though natural interpretation means positive denoted means negative denoted iii means means denoted means denoted means could anything note denoted familiar identifications iii six elements constitute example triangle hyperfield remark doubly distributive satisfy property since since interval since example phase hyperfield let denote complex unit circle say points antipodes multiplication defined usual corresponds addition angles call arc less degrees short hypersum given points short arc antipodes hyperfield satisfying property called phase hyperfield power set level given one least two points define union short arcs point point together makes connected arc together contains antipode note arc obtained taking single point middle two endpoints set short arcs possibly adjoined subin words semiringl need set arcs concatenation filling rest arcs half way around adjoining arcs contain antipode double distributivity fails take almost antipodes arc connecting passes antipode arc little semicircle whereas already viro also somewhat different version distributive rowen example another example suggested lopez also consider addition given interval extends addition intervals given min max clearly associative inverse unique since contains hand satisfy restriction every set form form modify condition stipulate must form unique essentially general condition set forth references akian gaubert guterman linear independence tropical semirings beyond tropical idempotent mathematics litvinov sergeev eds contemp akian gaubert guterman tropical cramer determinants revisited tropical idempotent mathematics applications contemp math amer math providence baker matroids hyperfields blachar rowen symmetrized rank matrices connes consani monoids hyperstructures search absolute arithmetic casimir force casimir operators riemann hypothesis walter gruyter berlin gaubert des dans les phd dissertation school mines paris july golan semirings applications business dordrecht previously published kluwer acad gondran minoux graphs dioids semirings volume operations science interfaces series springer new york henry izhakian tropical arithmetic matrix algebra comm algebra izhakian rowen supertropical algebra adv preprint jaiung algebraic geometry hyperrings preprint available arxiv pages mazariegos vargas cifuentes group structures families subsets group niv matrices characteristic polynomials supertropical algebra linear algebra appl niv factorization tropical matrices algebra plus linear systems max proceedings conference decision control honolulu reutenauer straubing inversion matrices commutative semiring algebra rowen graduate algebra noncommutative view semigroups combinatorial applications american mathematical society rowen algebras negation map originally entitled symmetries tropical algebra rutherford inverses boolean matrices proceedings glasgow mathematical association straubing combinatorial proof theorem discrete viro hyperfields tropcial geometry hyperfields dequantization arxiv department mathematics university israel address rowen
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general video game framework evaluating agents games content generation algorithms feb diego jialin liu ahmed khalifa raluca gaina julian togelius simon lucas video game playing gvgp aims designing agent capable playing multiple video games human intervention general video game gvgai competition framework created released purpose providing researchers common easy use platform testing methods potentially infinity games created using video game description language vgdl framework expanded several tracks last years meet demand different research directions agents required either play multiples unknown games without access game simulations design new game levels rules survey paper presents vgdl gvgai framework existing tracks reviews wide use gvgai framework research education competitions five years birth future plan framework improvements also described ntroduction benchmarks competitions used testing artificial intelligence capabilities since inception research field first four five decades testing almost exclusively based board games however since early number competitions benchmarks based video games sprung include competition mario competition simulated car racing competition arcade learning environment starcraft competitions far competitions game benchmarks challenge agents play single game exception arcade learning environment contains several dozens games far almost published studies deal learning agents one game time leads overspecialization overfitting agents individual games reflected outcome individual competitions example five years simulated car racing competition ran submitted car controllers got better completing races fast incorporated engineering arguably less general machine learning algorithms similarly bots starcraft competitions generally highly domain even display little way learning adaptation come surprise whenever possible researchers incorporate domain knowledge game design agent yet trend threatens negate usefulness competitions spurring testing development stronger general general video game competition founded belief best way stop researchers relying engineering agents make impossible idea researchers develop agents without knowing games playing submitting agents competition agents evaluated using unseen set games every competition event requires design new set games reusing previous games would make possible adapt agents existing games order make possible video game description language vgdl developed easily create new games played agents vgdl designed would easy create games humans algorithms eventually allowing automated generation testbed games game engine also created allow games specified language visualized played humans agents forming basis competition gvgai competition initially focused benchmarking algorithms playing game competition associated software multiple uses addition competition tracks dedicated agents competition tracks focused generating game levels rules also potential use vgdl gvgai game prototyping system rapidly growing body research using framework everything building design tools demonstrating new concepts game design paper gives overview various tracks gvgai competition gvgai software various uses software put start describing basics vgdl gvgai framework describe various competition tracks launched far planning level generation rule generation learning next sections devoted discussing various kinds methods used submissions track especial consideration given planning track existed longest received submissions date followed section cataloging noncompetition research uses gvgai software final sections look forward future use development framework competition used teaching open research problems specifically related planning tracks future evolution competition framework vgdl gvgai ramework idea video game description language vgdl initially proposed ebner dagstuhl seminar artificial computational intelligence games first vgdl game space invaders created tom schaul continued work completing improving language python framework modelbased learning released first game engine vgdl text description language allows definition arcade gridbased physics generally stochastic games levels ontology defined framework vgdl permits description objects sprites properties consequences colliding additionally termination conditions reward signals form game scores defined games total four different sets defined sprite set defines kind sprites take part game interaction set rules govern effects two sprites colliding termination set defines game end mapping set specifies characters level file map sprites sprite set ebner levine described dagstuhl need interest framework could accommodate competition researchers tackle challenge general video game playing gvgp perez implemented version schaul initial framework java organized first general video game gvgai competition following years framework extended accommodate games level rule generation physics games framework provided api creating bots would able play game defined vgdl without giving access rules games behaviour entities defined game however agents access forward model order roll current state forward given action thinking time set competition settings controllers also access every game tick game state variables game status winner time step score state player also referred paper avatar position orientation resources health points history collisions positions different sprites game identified unique type additionally sprites grouped categories attending general behaviour characters npc static movable portals spawn sprites game behave entry exit point levels resources collected player information also provided agents learning setting framework competition exception forward model last date modification framework gvgai included compatibility creating learning agents java python would learn play game given episodic manner moment writing framework counts games contest leg winner olets type tree search method section yolobot ybcriber hybrid yolobot tree search method hybrid hybrid yolobot hybrid table inners editions gvgai lanning competition indicates layer track ybrid denotes techniques combined single algorithm yper heuristic high level decision maker decides sub agent must play see ection table extended iii gvgai ompetition tracks section summarizes different tracks featured gvgai competition planning first competition track planning one agent also referred bot controller plays games aid forward model controller second initialization game tick decision time gvgai tracks follow similar structure several public sets games included framework allowing participants train agents edition one validation one test set sets private stored competition games set different levels participants submit entries time submission deadline training validation sets preliminary rankings displayed competition website names validation set games anonymized controllers run test set submission deadline determine final rankings competition executing agent times level rankings computed first sorting entries per game according victory rates scores time steps order rankings award points first entries first tenth position winner competition submission sums points across games test set detailed description competition rules reader referred table shows winners editions date along section survey method included paper describes approach depth planning planning track added aim testing general agents environments intel core machine memory complex present direct player interaction games setting played simultaneous move fashion rules games included competition still secret players similar track additional piece information hidden whether game competitive cooperative types games ensures agents tuned compete opponents instead correctly judge various situations identify goal intelligent presence game however question appropriate opponent model remains open competition entries far employed random simple techniques two legs track organized first time ieee world congress computational intelligence wcci ieee conference computational intelligence games cig although first leg second winner adrienctx overall championship showed general agent may best particular subset problems perform high level many different ones win overall competition organized ieee congress evolutionary computation cec final results competition seen table details reader referred learning track gvgai learning track started among gvgai tracks one accepts submission agent written java also python order accommodate popular machine learning libraries written language learning track briefly introduced technical details framework competition procedure explained track technical manual learning track based gvgai framework key difference forward model provided controllers hence learning needs achieved episodic repetition games played note even forward model accessible learning validation phases controllers receive observation current game state via stateobservation object provided gson object game screen png format agent free select either forms game state observation game tick similar planning tracks controllers either java python inherit abstract class constructor three methods implemented init called beginning every game must finish cpu time act called every game tick determines next action controller cpu time result called end every game time limit allow processing outcome perform learning planning track games sets count games levels execution given set split two phases learning phase allows training first levels game validation phase uses agent kkunan dontunderestimateuchiha ercumentilhan yolobot training set score ranking table test set score ranking table shows score ranking submitted learning agents obtained training test set denotes sample controller planning learning best score best score agent kkunan dontunderestimateuchiha kkunan kkunan table iii table compares best scores single player planning learning agents test set ote one game games test set removed final ranking due bugs game denotes sample controller available levels execution controller set therefore phases learning phase games controller limited amount time minutes learning first levels play levels free choose next level play time remaining controller able send action called abort finish game moment method result called end every game regardless game finished normally aborted controller thus controller play many games desired potentially choosing levels play long respects minutes time limit validation phase immediately learning phase controller plays times levels sequentially total time limit agent respects time limits init act result continue learning game playing first gvgai learning track held ieee cig besides two sample random agents written java python one sample agent using sarsa written java three submissions written java one python table illustrates score ranking learning agents training test sets table iii compares score achieved best planning agent games test set score achieved best learning agent level generation level generation track started aim track allow competitors develop general level generation algorithm participate track competitors must implement level generator competitors provide least one function responsible generating level framework provides generator information needed game game sprites interaction set termination conditions level mapping additionally framework also provides access forward model order allow testing generated levels via agent simulation levels generated form matrix characters character representing game sprites specific location determined matrix competitors choice either use provided level mapping provide one first level generation competition held international joint conference artificial intelligence ijcai received four participants participant provided month submit new level generator three different level generators provided order help users get started system see section vii description three four participants level generators remaining based cellular automata competition day people attending conference encouraged try pairs generated levels select level liked one neither finally winner selected based generator votes winner contest easablade generator cellular automata described section competition run ieee cig configuration previous year one month implementation followed site judging unfortunately one submission received hence competition canceled submission used model generate new constrained levels using recorded player keystrokes rule generation rule generation track introduced held cig aim track generate interaction set termination conditions certain level fixed set game sprites fixed amount time participate track competitors provide rule generator framework provides competitors game sprites certain level forward model simulate generated games level generation case generated games represented two arrays strings first array contains interaction set second array contains termination conditions first rule generation competition intended run cig three different sample rule generators provided discussed section viii contest ran month period unfortunately submissions received track ethods ingle layer lanning section describes different methods implemented single player planning gvgai controllers face challenge common possibility using forward model sample future states current game state plus fact limited time attempts abide decision time imposed competition efforts literature compel agents obey maximum number uses forward model section briefly introduces basic methods found within framework section describes different tree search methods implemented settings community followed evolutionary methods section often one method combined algorithm gives place hybrid methods section algorithms section discussion methods common included section basic methods gvgai framework contains several agents aimed demonstrating controller created singleplayer planning track competition therefore methods particularly strong simplest methods without much doubt donothing agent returns action nil every game tick without exception next agent complexity samplerandom returns random action game tick finally onesteplookahead another sample controller rolls model forward one available actions order select one highest action value determined function tries maximize score minimizing distances npcs portals tree search methods one strongest influential sample controllers samplemcts implements monte carlo tree search algorithm games initially implemented closed loop version states visited stored tree node without requiring use forward model tree policy phase mcts achieved third position participants first edition competition winner edition adrien implemented open loop expectimax tree search olets open loop states visited never stored associated tree node version mcts include rollouts uses open loop expectimax ole tree policy ole substitutes empirical average reward weighted sum empirical average rewards maximum children values torsten schuster msc thesis analyzes several enhancements variations mcts different sets gvgai framework modifications included different tree selection expansion policies results show combinations sampling technique mast selection technique nst progressive history provided overall higher rate victories counterparts without enhancements although result consistent across games simpler algorithms achieving similar results different study dennis soemers explored multiple enhancements mcts progressive history nst tree selection steps tree reuse starting game tick subtree grown previous frame corresponds action taken rather new root node tree initialization direct successors root note explored mcts starts safety prune nodes high number game loses found loss avoidance mcts ignores game lose states found first time choosing better alternative pruning states features rarely seen less likely pruned knowledge based evaluation deterministic game detection authors experimented enhancements games framework showing improved performance mcts significantly combination increased average win rate sample agent percentage points best configuration winner one editions competitions see table frydenberg studied yet another set enhancements mcts authors showed using mixmax backups weighing average maximum rewards node improved performance games combination reversal penalty penalize visiting location twice offers better results vanilla mcts enhancements repeating action several times sequence partial expansion child node considered expanded children also expanded improve results obtained perez implemented version mcts two main enhancements first distance different sprites considered features linear combination weights evolved bias mcts rollouts secondly knowledge base kept interesting player different sprites interesting measure curiosity rollouts biased towards unknown sprites experience bias getting entities results applying algorithm first set games framework showed combination two components gave boost performance games first training set work extended researchers field also put special effort biasing monte carlo simulations authors modified random action selection mcts rollouts using potential fields bias rollouts making agent move direction akin field authors showed kbmcts provides better performance potential field used instead euclidean distance sprites implemented additionally similar study authors substituted euclidean distance measure calculated pathfinding algorithm addition achieved improvements original although authors noted study using pathfinding provide competitive advantage games another work park kim tackles challenge determining goodness sprites game computing influence map based using bias simulations occasion adding third term upper confidence bound ucb equation tree policy mcts although compared resultant algorithm improves performance sample controllers several games framework albeit performing worse games used study biasing rollouts also attempted dos santos introduced redundant action avoidance raa policy ndp raa analyzes changes state avoid selecting sequences actions produce alteration position orientation properties new sprites avatar ndp makes recommendation policy ignore children root node found least one game loss simulation state children marked defeat normal higher number visits recommendation followed modifications able improve performance mcts games waard introduced concept options gvgai option associated goal policy termination condition selection expansion steps mcts modified search tree branches option finished allowing deeper search amount time results show option mcts outperforms mcts games small levels number sprites loses comparison mcts games bigger due options becoming large similar line perez employed gvgai games used continuous rather gridbased physics games larger state space turn delays effects player actions modifies way agents navigate level defined sequence repetition action steps arguably simplest kind devised mcts performed better without macroactions average across games particular games mcts needs avoid losing every attempt authors also concluded length impacts different games distinctly although shorter ones seem provide better results larger ones probably due fine control movement agents studies brought optimization challenge instance perez implemented version mcts concretely maximizing score level exploration simultaneously games tested rate victories grew normal mcts version showing great promise approach different study khalifa apply concepts evolving parameters tree selection confidence bounds equation previous work bravi also discussed later section provided multiple ucb equations different games work evolved using selection evolutionary optimization algorithm linear weights ucb equation results combining single one components respond different conflicting objectives results show possible find good solutions games tested significant exception mcts regards tree search methods gvgai geffner winner one editions competition ybcriber indicated table implemented iterated width concretely search crucial alteration new state found search pruned make true new tuple atom atoms boolean variables refer position orientations case avatars changes certain sprites specific locations authors found performed better mcts many games exception puzzles pruning according pairs atoms showed better performance agent declared winner ceec edition planning track badabi implemented several versions enforced hill climbing ehc search method looks successor current state better heuristic value ehc obtained similar results first set games framework disparities specific games set finally mark nelson ran study mcts order investigate giving higher time budget algorithm increasing number iterations mcts able master games words nature gvgai framework competition reason different approaches fail achieve high victory rate study provided times budget agent performance mcts increased marginally even level fact improvement achieved means losing less often rather winning games paper concludes aspect factor challenge also diversity games words increasing computational budget answer problem gvgai poses least mcts evolutionary methods second big group algorithms used planning evolutionary algorithms concretely use eas problem mostly implemented form rolling horizon eas rhea family algorithms evolves sequences actions use forward model sequence individual fitness value state found end sequence time budget first action sequence highest fitness chosen applied time step gvgai competition includes samplerhea sample controller samplerhea population size individual length implements uniform crossover mutation one action sequence changed another one position new action chosen uniformly random gaina analyzed effects rhea parameters performance algorithm games chosen among existent ones order representative set games framework parameters analyzed population size individual length results showed higher values parameters provided higher victory rates study motivated inclusion random search samplers sample framework equivalent rhea infinite population size one generation evaluated budget consumed achieves better results rhea games also compared rhea mcts showing better performance individual length high population sizes different evolutionary computation agent proposed jia consists genetic programming approach authors extract features screen capture game avatar location positions distances nearest object type features inputs system using arithmetic operands nodes determines action execute result three trees horizontal vertical action use authors report different variations inputs provided algorithm give similar results mcts three games tested study hybrids previous studies feature techniques one technique predominant agent created albeit may include enhancements place boundary hybrids section describes approaches opinion authors would right considered techniques mix one approach single algorithm example one approaches presented gaina analyzed effects seeding initial population rhea using different methods part decision time budget dedicated initialize population sequences promising determined onesteplookahead mcts results show seeding options provide boost victory rate population size individual length small benefits vanish parameters large enhancements rhea proposed incorporating mutation statistical tree shift buffer rollouts end sequences banditbased mutation breaks uniformity random mutations order choose new values according suggestions given armed bandit however authors reported improvement performance noticed statistical tree previously introduced keeps game tree visit count accumulated rewards root node subsequently used recommending action take time step enhancement produced better results shorter individuals smaller population sizes shift buffer enhancement provided best improvement performance consist shifting sequences individuals population one action left removing action previous time step variation similar keeping tree frames mcts combined addition rollouts end sequences provided improvement victory rate percentile points vanilla rhea scores similar previous study conducted horn particular study features rhea rollouts rhea mcts alternative actions mcts determine action exception one recommended rhea rhea rollouts sequence planning approach shift buffer rhea rollouts occlusion detection removes needed actions sequence reaches reward rhea rollouts npc attitude check rewards sequences terms proximity sprites provide positive negative reward results show rhea rollouts improved performance many games although variants additions performed worse sample agents interesting see case shift buffer provide improvement victory rate although may due use different games torsten schuster proposes two methods combine mcts evolution one proposed evolves vector weights set game features order bias rollouts towards interesting parts search space rollout becomes evaluation individual weight vector using value final state fitness second algorithm based stronglytyped stgp uses game features evolve state evaluation functions embedded within mcts two approaches join mast nst see section larger comparison study concludes different algorithms outperform others distinct games without overall winner terms superior victory rate although superior vanilla mcts cases idea evolving weight vectors game features mcts rollouts introduced explored jim van eeden msc thesis particular author added pathfinding algorithm replace euclidean distance used accurate measure changing evolutionary approach used weight pair action chosen step softmax equation work combines move actions single weight picks action using gibbs sampling author concludes improvements achieved modifications marginal likely due inclusion pathfinding additional improvements proposed chu authors replace euclidean distance feature sprites grid view agent surroundings also approach bias mcts rollouts making algorithm update weights step rollout proposed modifications improved victory rate several sets games framework also achieved highest average victory rate among algorithms compared approach could also considered hybrid given influence tree approaches also partially described section implemented combination mcts true online sarsa authors use mcts rollouts episodes past experience executing true online sarsa iteration selection policy weights learnt features taken smallest euclidean distance sprites type results showed proposed approached improved performance vanilla mcts majority games used study evolution mcts also combined different ways one bravi used system evolve different tree policies mcts concretely authors evolve different policy one five games employed study aiming exploit characteristics game particular results showed tree policy plays important role performance mcts agent although cases performance poor none evolved heuristics performed better default ucb mcts finally sironi designed three mcts tuned parameters mcts playout depth exploration factor using naive montecarlo evolutionary algorithm bandit evolutionary algorithm results show tuning algorithms improve performance mcts vanilla mcts performs poorly keeping similar rate victories mcts performs well algorithm selection several authors also proposed agents use several algorithms rather combining single one higher level decision process determines one used time blaine ross msc thesis proposes agent combination two methods approach uses enforced hill climbing navigate game high level switches mcts close proximity goal work highlights problems computing paths short time budget allowed indicate goal targeting combined local maneuvering using mcts provide good performance games tested joppen implemented yolobot arguably successful agent gvgai date several editions competition approach consists combination two methods heuristic best first search bfs deterministic environments mcts stochastic games initially algorithm employs bfs game deemed stochastic optimal solution found certain game tick threshold reached extending several consecutive frames needed search unless optimal sequence actions found agent execute enhanced mcts consistent informed priors rollout policies backtracking early cutoffs pruning resultant agent shown consistently good level play multiple game sets framework another approach also winner one editions competition see table determines first game deterministic stochastic case former used direct agent sprites interest otherwise random walks employed navigate level fact type portfolio agents shown promising results triggered research hyperheuristics game classification work bontrager used cluster games algorithms attending game features derived type sprites declared vgdl description files resulting classification seemed follow difficulty pattern clusters grouped games agents different rates mendes built selected automatically agent portfolio agents playing individual game tested gvgai framework approached employed features train different classifiers support vector machines svm perceptrons decision trees among others order select agent used playing game results show svm obtained higher victory rate single agents separately horn described section also includes analysis game features difficulty estimation authors suggest multiple enhancements constantly attempted many algorithms could potentially switched depending game played objective dynamically adapting present circumstances ashlock suggest possibility creating classification games based performance multiple agents variations different enhancements heuristics objectives furthermore classification needs stable order accommodate collection games within gvgai framework also flexible enough allow algorithm choose version better adapts unseen games ethods layer lanning section approaches agents developed researchers within planning setting entries submitted planning track competition two methods stood base entries received far monte carlo tree search mcts evolutionary algorithms one hand mcts performed better cooperative games well showing ability adapt better asymmetric games involved role switch matches environment eas hand excelled games long lookaheads puzzle games rely specific sequence moves identified counterparts basic methods described section available framework track well difference one step lookahead agent requires action supplied opponent simulating game states opponent model used sample agent assumes perform random move exception actions would cause loss game tree search methods competition entries season based mcts see section entries employed open loop version would store statistics nodes trees game states therefore needing simulate actions iteration potentially accurate evaluation possible game states due unnecessarily costly deterministic games entries yolobot switched search games initial analysis game type method shown ability finding optimal solution game lasts long enough enhancements brought mcts include generating value maps either regarding physical positions level concepts higher values assigned states agent closer objects interacted interesting targets determined controllerspecific heuristics winner wcci leg also employed dynamic monte carlo length adjustments increased number iterations encourage lookahead budget allows weighted weights per action generated randomly beginning agents use online learning one way another simplest form base monte carlo tree search backups used gather statistics action multiple simulations overall championship winner adrienctx uses offline learning training set supplied tune parameters stochastic gradient descent function employed learning rate mini batch size evolutionary methods two competition entries used technique base alternative mcts catlinux winner cig leg controller placing overall championship uses genetic algorithm one population containing individuals represent action sequences main improvement features top base method generation value used encourage agent exploration towards interesting parts level initialized based inverse frequency object type therefore lower value higher object number including range influence nearby tiles event history used evaluate game objects simulations update value map catlinux top controller either individual legs run placed overall championship agent uses rolling horizon evolutionary algorithm rhea shift buffer enhancement used boost performance specifically keeping population evolved one game tick next instead discarding action sequence shifted one action left therefore removing previous game step new random action added end complete individual fixed length offline learning used agents although could scope improvement parameter tuning offline online opponent model agents submitted competition use completely random opponent models entries adopted method integrated within sample one step lookahead controller choosing random action competition webpigeon assumed opponent would always cooperate therefore play move beneficial agent used advanced model time remembered opponent actions simulations added statistics stored mcts tree nodes policy used select opponent actions based recorded provided boost performance games wcci leg improve controller position rankings following cig leg opponent models found area explore gonzalez perez looked different models integrated within sample mcts agent provided framework alphabeta builds tree incrementally returning best possible action time tick minimum returns worst possible action average uses similar tree structure computes average reward actions returns action closest average fallible returns best possible action probability action minimum reward otherwise probabilistic involved offline learning games gvgai framework order determine probability mcts agent select action using determine opponent action playing online action returns action agent plays mirror returns opposite finally limitedbuffer records last actions performed player builds probabilities selecting next action based data unlimitedbuffer records entire history actions game opponent models tested round robin tournament probabilistic models achieve highest win rates two models probabilistic unlimitedbuffer outperforming random opponent model ethods ingle layer earning gvgai framework also used agent learning perspective setting agents use forward model plan ahead actions execute real game instead algorithms learn games repeatedly playing multiple times episodes reinforcement learning ideally improving performance progressively section first describes approaches tackled challenge set learning track competition described section move ones adhered competition format section competition entries entries available random agent sample random agent selects action uniform random every game tick included framework java python purposes testing agent also meant taken baseline learner expected perform better agent acts randomly undertake learning bandit algorithms dontunderestimateuchiha kunanusont based two popular multiarmed bandit mab algorithms greedy algorithm ucb game tick current best action probability picked otherwise action uniformly randomly selected best action time defined equation log arg max denotes number times action denotes empirical mean selected increment score applying action far interesting combination selection equation greedy algorithm aim balancing exploiting action exploring others additionally set decreases slowly along time formalized according competition setting games last longer game ticks result random decisions made approximately time samplelearner ercumentilhan based sarsa algorithm agents choose use subset whole game state information build new state reduce amount information saved take account similar situations main difference former uses square region fixed size centered avatar position latter uses view fixed distance kkunan kunanusont simple qlearning agent using avatar current information features exceptions avatar health screen size elements vary greatly game game reward game tick defined difference game score game tick one learning rate discounted factor manually set learning phase random performed probability otherwise best action selected validation phase best action always selected despite simplicity edition track date entry ranked sample random agent tree search methods yolobot adaption yolobot planning agent described previously section forward model accessible https learning track mcts substituted greedy algorithm pick action minimizes distance chosen object according authors poor performance yolobot learning track contrary success planning tracks due collision model created work well learning agents one first works used framework learning environment carried samothrakis employed games benchmark concretely authors experimented separable natural evolution strategies using two different policies versus softmax linear function approximator versus neural network state evaluation function features like score game status avatar sprites information used evolve learners episodes results show linear function approximator better combination learn maximize scores game braylan miikkulainen performed study objective learn forward model games objective learn next state current one plus action state defined collection attribute values sprites spawns directions movements etc means logistic regression additionally authors transfer learnt object models game game assumption many mechanics behaviours transferable experiments showed effective value object model transfer accuracy learning forward models resulting agents stronger exploration also learning setting kunanusont developed agents able play several games via screen capture particular authors employed deep games framework increasing complexity included several enhancements gvgai deal different screen sizes game mode results showed approach allowed agent learn play deterministic stochastic games achieving higher winning rate game score number episodes increased vii ethods level generation different researchers used different approaches generate levels gvgai framework following subsection describes known generators either included framework developed competition constructive methods constructive generators designed generate level based general knowledge example enemies away avatar walls divide world islands etc based game generator adjusts couple parameters rules fit game example number characters npcs generated level constructive generators need simulations generating level following known constructive generators sample random generator naive method generate level generator first identifies solid sprites solid sprites block avatar npcs moving react anything generator adds selected solid sprite border generated level prevent sprites wandering outside game screen followed adding one character level mapping section random location level step ensures game playable finally adds random amount random sprites level mapping random locations level sample constructive generator generator uses general game knowledge generate level first generator calculates level dimensions number sprites level labels game sprites based interactions sprite types constructs level layout using solid sprites later add avatar random empty location knowing avatar position generator adds harmful sprites kill avatar far location avatar adds sprites random free locations finally generator makes sure number goal sprites sufficient prevent winning losing automatically game starts easablade constructive generator winner generator first level generator competition generator similar sample constructive generator uses cellular automata generate level instead layering objects randomly cellular automata run multiple layers first layer design map obstacles followed exit avatar goal sprites harmful sprites others constructive generator generator uses model generate level generator records player actions previous action sequence used generate levels using predefined rules constraints example player uses use action quite often generator include enemies level used specify rules instead reacting separate action model reacts actions generation process algorithm keeps track number position every generated object ensure generated sprites overpopulate level single avatar sprite placed lower half level methods generators use simulations make sure generated level playable better placing random objects following known generators sample genetic generator level generator based feasible infeasible population genetic algorithm genetic algorithm uses populations one feasible chromosomes infeasible chromosomes feasible population tries increase difference olets agent see section look ahead infeasible population tries decrease number chromosomes violate problem constraints least one avatar game avatar must die first steps population evolves children transfer two populations generator initializes population using sample constructive generator genetic generator generator built top sample genetic generator main idea generate level fits certain suspense curve suspense calculated point time calculating number actions leads death tie using olets agent algorithm modifies levels make sure suspense curve constant life time game good generators aimed producing suspense peeks values half actions average lead losing game one advantages using technique makes sure generated level winnable games hard win higher peak suspense curve valued highly generator jnicho genetic generator generator uses standard genetic algorithm similar crossover mutation operators sample genetic algorithm fitness function used combination score difference constraints specified sample genetic generator score difference calculated monte carlo tree search agent one step look ahead agent score difference normalized make sure overshadow constraint values genetic generator modified version sample genetic generator modifications includes using adaptive crossover mechanism adaptive mutation rate better agent olets allowing crossover feasible infeasible population allowed sample genetic generator methods asp generator generator uses answer set programming asp generate levels main idea generate asp rules generate suitable levels current game generated rules consists three different types first type basic rules based specific decisions keep levels simple instance levels one sprite per tile second type game specific rules extracted game description file example identification singleton sprites one sprite level last type additional rules minimize search space rules limit minimum maximum number sprite type rules evolved using evolutionary strategy algorithm performance difference samplemcts random agent fitness function viii ethods rule eneration section describes different algorithms included framework found literature toward generating rules framework constructive methods constructive methods algorithms generate rules one pass without need play game constructive methods often incorporate knowledge game design generate interesting games sample random generator simplest generator provided framework main idea generate game compiles errors example game contain interactions killing end screen eos sprite algorithm starts generating random number interactions selecting two random sprites including eos random interaction rule one one algorithm checks every interaction valid adding generated game generating random interactions algorithm generates two termination conditions one winning one losing losing condition fixed avatar killed winning either winning game random amount frames winning game number certain sprite reaches zero sample constructive generator complex generator utilizes knowledge vgdl language level design generate interesting games algorithm starts classifying game sprites different categories wall sprites sprites surround level sprites immovable sprites cover around level spawner sprites sprites spawn another sprite type etc type sprite algorithm rules generate interactions based example harmful sprites kill avatar collision wall sprites either prevent movable object passing kill movable object upon collision etc details rules reader referred game interactions generated two termination conditions generated one winning one losing losing condition fixed avatar death winning condition depends current sprites example collectible sprites exist current definition winning condition set collect methods methods use search based algorithm find game based certain criteria ensure generated game better rules randomly choosing sample genetic generator similar level generation track search based algorithm uses feasible infeasible population genetic algorithm evolve new games discussed keeps two populations one feasible games infeasible games infeasible games tries become feasible satisfying multiple constraints minimizing number bad frames frames contains sprites outside level boundaries certain threshold avatar die first etc hand feasible chromosomes try maximize fitness fitness consists two parts first part maximize difference performance olets mcts agents difference mcts random agent second part maximize number interaction rules fires simulation generated game thorbjrn generator generator inspired sample genetic generator similar sample genetic generator tries maximize difference performance different algorithms generator uses evolutionary strategies mutation crossover operators generate entire game instead interaction set termination conditions esearch builds gvgai game design machado implemented recommender system based vgdl recommend game elements sprites mechanics recommender system expanded cicero game design tool built top gvgai cicero statistics tool interactions help figuring unused game rules visualization system illustrate information game objects events mechanics recommender query system data retrospective analysis application seekwhence gameplay sessions human players agents recorded every single frame every game tick easily extracted study analysis recently liu applied simple random mutation hill climber rmhc bandit rmhc together resampling methods tune game parameters automatically games instances significant evolved using gvgai agents futhermore kunanusont evolved simultaneously gvgai agents part game opponent models guerrero explored five gvgai agents using four different heuristics separately playing twenty gvgai games allowing different behaviors according diverse scenarios presented games particular work explored heuristics focused winning game explore level interact different sprites games agents used evaluate generated games thus help evolve preferences particular behaviors khalifa modified mcts agents editing uct formula used agent human playing data used modeling make modified agents playing way primary results showed one studied agents achieved similar distribution repeated actions one human players work extended bravi data used evolve effective uct alternatives specific game mcts agents using new formulas none limited domain information compared standard implementation mcts samplemcts agent gvgai game missile command applying uct alternatives evolved using data standard mcts significantly improved performance besides designing games agents used automatic generation video game tutorials aimed helping players understanding play game also interesting study green pointed gvgai framework provides easy testbed tutorial generation game rules gvgai defined vgdl therefore tutorial generation easily achieved reading translating vgdl files recent work anderson focused designing deceptive games deceive agents lead agents away globally optimal policy designing games helps understand capabilities weaknesses existing agents serve preparation step designing gvgp combines advantages different agents authors categorized deceptions imported various types deception existing gvgai games editing corresponding vgdl files agents submitted gvgai planing competition tested new games interestingly final ranking agents games differed significantly rankings gvgai competition new designed deceptive games successfully explored weaknesses agents performed well test set official competition game generation rapp nielsen proposed relative algorithm performance profile rapp measure relative performance agents tested approach different general gameplaying agents using gvgai framework authors showed games clear thus able distinct good bad players words strong agent human player perform significantly better weak agent human player multiple playings games instance skillful agent expected perform better random agent one move nielsen integrated differences average game scores win rate agent random agent evaluation new games either randomly generated generated editing existing gvgai games though resulted games interesting play exceptions core challenge game removed instance enemy heart player makes enemy still provides useful starting points human designers kunanusont extended idea rapp five gvgai agents deterministic agent designed tested space battle game used candidate opponent considered part game evolved two gvgai agents one step look ahead weak mcts strong deterministic agent mediocre used play multiple times evolved game evaluation evaluation function defined minimum difference game scores strong mediocre agents difference game scores mediocre weak agents aiming generating games clearly distinguish stronger agents weak agents recently proposed bandit evolutionary algorithm compare simple rmhc two human players tested generated games preferred new game evolved using bandit evolutionary algorithm robustness testing perez ran study winners editions single player planning competition order analyze robust changes environment regards actions rewards aim work analyze different type generality controllers framework developed play multiple games certain conditions authors investigated could effect breaking compromises inaccurate noisy forward model agent execute move decided algorithm score penalties incurred performing certain actions interesting conclusion study conditions altered sample agents climb top rankings good controllers behave worse agents rely best first search yolobot already described paper handled noise badly mcts also showed quite robust regard rolling horizon agents could cope well changes work also reinforced idea gvgai framework competition also robust despite changes performance agents controllers better others practically conditions opposite rankings depending noise factors instance would mean framework quite fragile ducational use gvgai gvgai framework used provide engaging assignments taught modules basis many msc dissertation projects descriptions give idea educational uses gvgai intended exhaustive list interesting games assignment done good effect university copenhagen students produced number challenging puzzle games later used planning track training validation sets similar approach taken module game design university essex planning track games also produced msc dissertation projects gvgai offers extensive range interesting research challenges addressed msc dissertation projects majority ones aware focused planning track surprising first track developed planning track also benefit providing good sample agents starting points work either sense extending samples achieve higher performance using samples useful source comparison good example work mcts options used showed version options significantly outperformed sample mcts agent games studied many cases began msc thesis later published conference paper experience usually provides excellent educational experience student planning track theses include enhancements variants approaches beyond planning track examples already described survey include applying programming asp genetic algorithms level generation track learning screen capture essentially learning track approach learning track running finally another approach extend framework way developing learning track iscussion open research problems single two player planning taught modules gvgai used least two distinct ways within taught modules typical way use design specific aspects course around framework teaching students main concepts gvgai examples write agents selected tracks followed assignment significant weight given well student group entry performs league several institutions run private leagues including otto von guericke magdeburg university essex university muenster universidad carlos iii madrid universidad malaga new york university running private league means course supervisor full control setup league including students enter thoroughly entries evaluated set games evaluate track also allows control opponents chosen another taught modules teach vgdl part framework set development novel single planning versions gvgai ones received attention research despite popularity efforts best approaches rarely surpass approximately victory rate competition game sets low victory percentage great number games similarly different mcts rhea variants including many enhancements studied literature struggle achieve higher victory rate hundred games framework therefore increasing performance great proportion games probably challenging problem moment literature shows multiple enhancements algorithms methods aiming improve performance vast majority cases improvements affect subset games certain configurations algorithms understandable due nature general video game playing also shows current approaches work order reach truly general approaches work across board work described survey shown however interesting insights point right direction instance several studies show using sophisticated methods potential fields distances sprites features works better euclidean distances downside computing measurements take important part decision time budget used case needed games states best action take general one could say one main points address use decision time wisely approaches tried make every use forward model count like agents attempt learn facts game rollouts mcts attempts direction provided marginal improvements problem may trying general feature extractor words try learn influenced know existing games sprites good bad spawn entities things resources avatar collect games may require features thought especially challenge presents games seen participants another improvement tried several studies use cases repetition action several consecutive steps make action space coarser make better use time budget modifications improved performance certain games including algorithm previously either impact others made performance worse likely different games benefit different lengths work could done trying automatically dynamically adapt number times action repeated also complex structures allow planning fact games require planning still open problem solved setting games classification use also interesting area research best approaches date yolobot make differentiation stochastic deterministic games later use one another algorithm open challenge make classification accurate detailed approach could count portfolio algorithms adapt every game attempts made classify game features results suggest classifications algorithms used strong enough devising general features clustering maybe focused agent gameplay experience rather game features line future research unsolved issues apply single two player settings although latter case adds difficulty opponent compete collaborate two open problems arise first study made tries identify game behaviour opponent collaborative competitive analysis player intentions seen case add general game playing component secondly advancements done using opponent models beyond random investigation complicated opponent models better capture learn behaviour player could potentially yield better results beside development agents game playing aiassisted game design automatic game testing debugging using gvgai agents attracted researchers attention work around evolving game using relative performance gvgai agents done recently work focused relative algorithm performance profiles rapp performance measured terms well agents play given games however sensible explore aspects agent game play influence game design factors like amount level explored different agents generator favours levels games allow wider exploration maybe progressive one decisiveness selecting best action take entropy moves also used end xii uture irections gvgai framework competition constant development opportunities benchmark provides different lines research education varied section outlines future directions planned ahead following years new tracks new challenges proposed possibility organizing tracks competition arise listed possible new tracks attract interesting areas research automatic game design game design involves limited game generation level generation rule generation playtesting playing experience game feeling fun etc study game market user interface design audio design automatic game design becomes active research topic since late though best knowledge barney pell firstly automatically generated rules games review state art automatic game design found game generation track would aim providing controllers automatically generate totally new games game instances varying game parameters parameter tuning achieve former open question way would providing particular theme database game objects searching spaces game rules participants generate new games ideal case would controllers automatically create totally new games nothing though yawning gulf aspiration reality interdisciplinary field combining automatic game design automatic programming expected latter automatic game tuning relatively easier searchbased methods applied game parameter optimisation aiming maximising depth game variants finding playable games gvgai games drawn people attention instance real time strategy games starcraft board games mahjong study gvgai fruitful research topic atari games also extended games particular seen game related competitions held since recent ghost team competition included partial observability held cig nevertheless general track favorable interface planning track initially developed two players potential expanded planning track agent allowed control one player players controlled separate agent future track expanded learning framework providing player learning track turing test gvgai determining agent playing game human bot challenge subject study many years idea applying general video game setting new offers interesting opportunity extend framework turing test track participants create agents play like humans game given albeit understandable difficulty problem interest research area significant features make agent play like human game general directions several improvements additions framework done would potentially affect existent future competition tracks one continuous modifications constant enlargement games library new games added new edition competition work done automatic game design using gvgai framework potential create infinite number games integrated perfectly framework adding games also complemented compatibility systems general frameworks like openai gym ale microsoft count great number single model based tasks interfacing systems would increment number available games gvgai agents could play via common api would also open framework games important section environments current benchmark cover regards agents another possibility could provide wider range available actions instance player could able apply one action simultaneously actions could form continuous action space pressing throttle range general would enhance number legal combinations agent choose decision step beside framework website gvgai could also improved provide better faster feedback competition participants data analysis features added visualization score changes game playing action entropy exploration game world visited positions related work provide better flexible support game play metric logging better support data mining results together visualization better data saving help enabling upload replays action logs agents human another envisaged feature able play game web browser without download installation agent human visualize analyzed features game playing real time bonus easy parametrisation options thus player agent easily set parameters rules define desired game inserting values directly generate level play using automatic game tuning techniques given particular goals features xiii onclusions gvgai framework offers comprehensive system date evaluating performance general video game playing agents testing general purpose algorithms creating new games creating new content novel games framework used multiple international competitions used evaluate performance hundreds general video game agents agent tracks cater planning agents able exploit fast forward model learning agents must learn react sensibly without benefits forward model planning track already comes single versions learning track currently version development although longterm learning may also used within planning track agents far know yet done success alphagozero indicates achieved combining learning planning applying similar system within gvgai interesting prospect main alternative gvgai atari learning environment ale time writing ale offers games gvgai commercial games decades ago gvgai terms ale offers two tracks learning singleplayer planning learning track widely used future work machine learning video games predict tracks become important offer challenges based arms race intelligence new players developed also outside current scope ale addition offering wider range tasks gvgai also benefits much easily extensible ale though ale far greater uptake within sectors machine learning community immediate priority test rich set ale agents equivalent gain sense relative difficulty environment learn relative challenges offered one content creation tracks offer extremely hard challenge creating rules levels unseen games promising directions include development exploitation range general game evaluation measures greater use best gvgai agents perform novel rules levels video game description language vgdl important part gvgai date since makes possible rapidly concisely specify new games however also source limitation limited expressivity makes hard make games fun humans play vgdl also limits ease complex game mechanics embedded games turn limits depth challenge posed gvgai agents hence important future direction authoring gvgaicompatible games suitable language conform necessary gvgai api order ensure compatibility desired gvgai track finally discussion provides compelling case future gvgai tool academic study also believe reaches higher level maturity provide important tool game designers vision provide army intelligent agents range abilities diverse set metrics analyse range important functional aspects game acknowledgements authors would like thank participants tracks competition work submitted controllers generators work partially funded epsrc cdt intelligent games game intelligence iggi eferences philipp rohlfshagen jialin liu lucas conquers academia two decades research using classic arcade game ieee transactions computational intelligence games karakovskiy togelius mario benchmark competitions ieee transactions computational intelligence games vol loiacono lanzi togelius onieva pelta butz lonneker cardamone perez simulated car racing championship ieee transactions computational intelligence games vol bellemare 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video game learning screen capture ieee conference evolutionary computation cec ieee mnih kavukcuoglu silver rusu veness bellemare graves riedmiller fidjeland ostrovski control deep reinforcement learning nature vol kimbrough koehler wood infeasible genetic algorithm constrained optimization distance tracing free lunch european journal operational research vol nichols use genetic algorithms automatic level generation master thesis university essex neufeld mostaghim procedural level generation answer set programming general video game playing computer science electronic engineering conference ceec ieee nielsen barros togelius nelson towards generating arcade game rules vgdl computational intelligence games cig ieee conference ieee machado bravi wang nealen togelius shopping game mechanics machado nealen togelius cicero game design tool proceedings international conference foundations digital games ser fdg new york usa acm seekwhence retrospective analysis tool general game design 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may consideration publication theory practice logic programming abstract diagnosis tccp using linear temporal logic marco comini laura titolo dimi degli studi udine italy alicia villanueva dsic universitat spain villanue submitted january revised january accepted january abstract automatic techniques program verification usually suffer state explosion problem classical approaches based browsing structure form model represents behavior program check given specification valid implies part model built sometimes needed fragment quite huge work provide alternative automatic decision method check whether given property specified linear temporal logic valid tccp program proposal based abstract interpretation techniques require build model results guarantee correctness usual using abstract semantics completeness lost keywords concurrent constraint paradigm linear temporal logic abstract diagnosis decision procedures program verification introduction concurrent constraint paradigm ccp saraswat simple logic model different concurrent programming paradigms mainly due notion replaces classical model based underlying constraint system handles constraints variables deals partial information within family boer introduced timed concurrent constraint language tccp short adding original ccp model notion time ability capture absence information features one specify behaviors typical reactive systems timeouts preemption actions modeling verifying concurrent systems hand extremely hard task thus development automatic formal methods essential one known techniques formal verification model checking originally introduced clarke emerson queille sifakis automatically check system satisfies given property consisted exhaustive analysis system thus problem main drawback reason many proposals literature try mitigate proposals model checking common part model target program built sometimes needed fragment quite huge work propose completely different approach formal verification temporal ltl properties concurrent reactive systems specified tccp formalize method validate specification expected behavior tccp program expressed linear temporal formula require build model linear temporal logic use express specifications csltl adaptation propositional ltl logic concurrent constraint framework logic also used basis abstract domain new abstract semantics language brief method extension abstract diagnosis tccp comini abstract domain formed csltl formulas use original abstract diagnosis framework comini since complete lattice contributions work following new abstract semantics tccp programs based csltl formulas novel effective method validate csltl properties based ideas abstract diagnosis proposal intuitively consists viewing formula transformer means abstract immediate consequence operator works csltl formulas decide validity check implies automatic decision procedure csltl properties makes method effective technique check instance railway crossing system time train approaching gate whenever train crossed gate property non valid method identifies buggy process declaration technical results sections found comini operational behavior tccp language tccp language boer particularly suitable specify reactive time critical systems languages ccp paradigm saraswat parametric cylindric constraint system handles data information program terms constraints computation progresses concurrent asynchronous activity several agents accumulate information store query information briefly cylindric constraint algebraic structure false true var composed set constraints complete algebraic lattice lub operator false true respectively greatest least element var denumerable set variables existentially quantifies variables constraints entailment inverse given cylindric constraint system set process symbols syntax agents given grammar skip tell ask else see boer saraswat details cylindric constraint systems false ask false else false else else else false fig transition system finite constraints denotes generic tuple variables tccp program object form agent called initial agent set process declarations form agent notion time introduced defining discrete global clock operational semantics tccp defined boer formally described transition system conf configurations conf pairs representing agent executed current global store transition relation conf least relation satisfying rules figure transition step takes exactly one example guiding example paper use guiding example part full specification railway crossing system introduced alpuente let call following tccp declaration master near tell near tell master else tell tell master else master due monotonicity store streams written way used model variables boer master process uses input channel implemented stream receives signals environment trains output channel sends orders gate process checks input channel near signal guard first agent case sends tells order links future values stream restarts check following time instant recursive call master near signal detected else branch looks signal present behaves dually first branch finally signal detected current time instant last else branch process keeps checking following time instant auxiliary agent makes explicit local store auxiliary agent linked principal hiding construct setting initial local store true thus work prove correctness technique denotational concrete semantics comini correct complete operational behavior tccp also csltl interpreted denotational model thus introduce relevant aspects semantics denotational semantics tccp program consists set conditional timed traces represent compact way possible behaviors program manifest fed input initial store conditional traces seen hypothetical computations time instant condition representing information global store satisfy order proceed next time instant briefly conditional trace possibly infinite sequence conditional states three forms conditional store pair condition store stuttering construct stutt true end process construct intuitively conditional store means provided condition satisfied current store computation proceeds following time instant store condition pair called positive negative condition respectively condition represents information given store entail thus consistent sense stuttering construct models suspension computation none guards agent satisfied set guards agent conditional traces monotone consistent store trace entail negative conditions following conditional state denote domain conditional trace sets plete lattice prefix define sequence resulting removing information variable distinguish two special classes conditional traces said first condition true store satisfies successive condition moreover thus definition demands tional traces additional information agents needed order complete semantics definition based semantics evaluation function ajaki comini given agent interpretation builds conditional traces associated interpretation function associates process symbol set conditional traces modulo variance semantics set process declarations fixpoint jdk lfp djdk continuous immediate consequences operator djdki ajaki proof full abstraction operational behavior tccp given comini example consider process declaration example given interpretation semantics master graphically represented figure set conditional traces considered generalization using conditional states place stores traditional strongest postcondition semantics cnear cdown master true cnear cout true cout cnear cup master master fig tree representation master example used shortcuts characteristic constraints namely cnear near near cdown cout cup branch left represents computation near signal arrives first conditional state requires cnear holds thus constraints cdown concurrently added store computational step recursive call also concurrently invoked process calls modify store invoked affect store following time instant graphically represented triangle labeled interpretation process branch middle taken cout entailed cnear entailed initial store occurs negative condition first conditional state branch finally branch right represents case cnear cout entailed initial store abstract semantics tccp csltl formulas section present novel abstract semantics formulas approximates semantics described section therefore operational behavior tccp program end first define abstract domain logic formulas variation classical linear temporal logic manna pnueli following palamidessi valencia boer boer valencia idea replace atomic propositions constraints underlying constraint system definition csltl formulas given cylindric constraint system var formulas constraint system linear temporal logic false true csltlc set temporal formulas omit clear context false classical logical meaning atomic formula true states entailed current store existential quantification set variables var usual use shorthand true constraint formula atomic formula negation formulas called next formulas constraint next formulas said elementary called eventualities formulas finally formulas form define abstract domain domain formed csltl formulas true false modulo logical equivalence ordered algebraic lattice always exist finite sets formulas complete since semantics temporal formula typically defined terms infinite sequence states validates use conditional traces instead usually done context temporal logics define satisfaction relation infinite conditional traces implicitly transform finite traces end replicating last store infinite times definition semantics given function defined csltl satisfaction relation defined true false iff stutt iff iff iff iff iff iff exists say sound approximation said satisfiable exists valid cases fairly standard except conditional trace prescribes entailed current store thus models constraint formulas note monotonicity store tccp computations positive conditions conditional traces contains information previously added constraint store furthermore definition condition since contradiction holds neither contradiction thus conditional trace stutt models constraint formulas contradiction set holds continuation monotonicity lemma function monotonic injective csltl abstract semantics technical core semantics definition csltl agent semantics evaluation function given agent interpretation process symbols builds csltl formula sound approximation concrete behavior sequel denote set agents set sets process declarations built signature constraint system definition let distinct variables function modulo two functions variants exists renaming semantic domain set ordered extension denotes starting state family elements indexed modulo variance definition csltl semantics given define csltl semantics evaluation structural induction follows true ask else given define immediate consequence operator sound approximation sound approximation theorem correctness let example consider process declaration example let use abbreviate repetition given definition compute master near near master near master near master three disjuncts match three possible behaviors master signal near emitted train emitted signal arrives abstract diagnosis tccp csltl formulas since complete lattice impossible find function adjoint function forms galois connection therefore use abstract diagnosis framework tccp defined comini thus propose section new weaker version abstract diagnosis works given set declarations specification abstract intended behavior say abstractly partially correct jdk abstractly complete jdk differences jdk usually called symptoms many actually proposal defined using sake simplicity could easily defined parametrically suitable family concretization functions symptoms consequence originating ones direct consequence errors abstract diagnosis determines exactly originating symptoms case incorrectness faulty process declarations captured definitions abstractly incorrect process declaration abstract uncovered element definition let process declaration process abstractly incorrect testimony false uncovered element false informally abstractly incorrect derives wrong abstract element intended semantics dually uncovered declarations derive intended semantics theorem let abstractly incorrect process declarations partially correct partially correct abstract uncovered elements complete absence abstractly incorrect declarations sufficient condition partial correctness necessary approximation happen concretely correct declaration abstractly incorrect hence abstract incorrect declarations general warning possible source errors however abstract correct declaration contain error thus manual inspection needed declarations abstractly incorrect moreover shown following theorem concrete visible indeed detected lead abstract incorrectness abstract uncovered intuitively concrete error visible express formula whose concretization reveals error logic expressive enough theorem let process declaration concrete specification sound approximation exists false abstractly incorrect testimony exists abstract uncovered element exists point says concrete error abstract symptom hidden approximation moreover exists formula express following examples borrow alpuente notation last entailed value stream holds last instantiated value stream example verify example time near signal arrives train order sent gate model property define specification property master worth noticing although notions defined section similar defined standard approach formal definitions proofs different due weaker framework interesting property namely addition gate eventually verified comini simplified property due space limitations check whether program implies specification check defined example three disjuncts implies thus theorem partially correct check process declaration specification fails method reports partially correct occurs formula testimony possible incorrectness gives useful information fix process declaration check whether corresponds false positive example show technique detects error buggy set declarations remove instruction tell process declaration example avoid misunderstandings call modified process let new process declaration aim verify order sent whenever signal received master need compute one step semantics buggy version process master master near near master near near master detect incorrectness process testimony since false testimony suggests channel signal see corresponding signal channel technique behaves negatively sets declarations one fixpoint happens programs loops produce contributes sense non meaningful programs situations actual behavior model specification fixpoint since fixpoint detect abstractly incorrect declaration shown following example example pathological cases let else specification compute thus see partially correct however explicitly added process note assumed hold process defined jdk satisfies automatic decision procedure csltl order make abstract diagnosis approach effective defined automatic decision procedure check validity formulas involved definition table rules form adapt csltl tableau construction propositional ltl gaintzarain gaintzarain comini contains preliminary version method intuitively tableau consists tree whose nodes labeled sets formulas root labeled set formulas checked satisfiability branches built according rules defined syntax formulas see table defining formulas basic idea formula node selected depending form rule table applied formulas generate bifurcation tree specific rules next existential quantification formulas branches tree closed definition formula models otherwise obtain model open branches definition csltl tableau csltl tableau finite set formulas tuple nodes nodes finite set nodes nodes initial node nodes csltl labeling function associates node formulas true node initial node labeled set branches exactly one following points holds every branch every exists another branch form existential quantified formula fresh variable case set formed elementary formulas next next rules standard replacing one two formulas according matching pattern rules table except rule uses context defined following next operator used rule different corresponding one pltl since also preserves constraint formulas needed guaranteeing correctness particular setting tccp store monotonic finally rule specific case removed renaming fresh definition node tableau inconsistent contains couple formulas formula false constraint formula merge csltl existential quantification correspond one logic serves model local variables seen information local positive constraint formulas node branch closed contains inconsistent node last condition inconsistence node particular ccp context describe algorithm automatically builds csltl tableau given set formulas see comini pseudocode construction consists selecting step branch extended using rules elimination none applied next operator used pass next stage dealing eventualities determine context rule necessary distinguish eventuality unfolded path given node rule applied distinguished otherwise true use contexts mechanism tuality set detect loops allows one mark branches containing eventuality formulas open generate inconsistent nodes mark branches closed node marked closed inconsistent marked open last node branch contains constraint formulas branch cyclic eventualities cycle already distinguished order ensure termination algorithm necessary use fair strategy distinguish eventualities sense every eventuality open branch must distinguished point assumption fact given finite set initial formulas exists finite set possible labels systematic tableau imply termination tableau construction moreover constructed tableau sound complete therefore check validity formula form build tableau negation check closed abstractly correct otherwise extract explicit testimony abstract incorrectness construction linear size systematic tableau construction said gaintzarain worst case however believe bound asymptotic behavior quite meaningless context since realistic think formulas specification grow much big formulas difficult comprehend real situations people would hardly try even imagine moreover note tableau explosion due nesting eventualities practice really eventualities used specifications therefore real situations expect extremely big tableaux built related work constraint linear temporal logic defined valencia verification different timed concurrent language called ntcc shares tccp concurrent constraint nature behavior restricted negation fragment logic negation allowed state formulas shown decidable however efficient decision procedure given apart proof moreover verification results given fragment ntcc avoids original language contrast work address problem checking temporal properties full tccp language techniques defined tccp past falaschi villanueva alpuente alpuente falaschi worth noting notions correctness completeness works defined terms jdk terms concrete semantics therefore check requires potentially infinite fixpoint computation contrast notions abstractly incorrect declarations abstract uncovered elements defined terms one application moreover since defined compositionally checks defined process declaration isolation hence proposal used partial sets declarations property falsified model checking provides counterexample terms erroneous execution trace leaving user problem locating source bug contrary identify faulty process declaration falaschi first approach declarative debugging ccp language presented however cover particular extra difficulty nonmonotonicity behavior common timed concurrent constraint languages makes approach significantly different moreover although propose use ltl specification properties formulation based concretization function complicates task efficient implementation finally proposal clearly relates abstract diagnosis framework tccp defined galois insertions comini work compete precision model checking main drawback fact abstract domain allow specify temporal properties compact way fact specifications consisted sets abstract conditional traces thus specifications big unnatural written use temporal logic proposal certainly overcomes problem conclusion future work defined abstract semantics tccp based domain linear temporal logic constraints semantics correct behavior language using abstract semantics defined method validate csltl formulas tccp sets declarations since abstract semantics defined means galois connection use abstract diagnosis framework tccp defined comini thus devised scratch weak version abstract diagnosis framework based concretization function works applying abstract specification checking validity resulting implications whether computation implies abstract specification computational cost depends essentially cost check implication also presented automatic decision procedure csltl logic thus effectively check validity implication currently finishing implement proof concept tool available online url http realizes proposed instance would able compare tools assess real life goodness proposal future also plan explore instances method based logics decision procedures semi automatic tools exists proposal also immediately adapted concurrent languages like tcc ntcc suitable fully abstract semantics developed references alpuente falaschi villanueva symbolic model checker tccp programs first international workshop rapid integration software engineering techniques rise revised selected papers lecture notes computer science vol alpuente gallardo pimentel villanueva semantic framework abstract model checking tccp programs theoretical computer science alpuente gallardo pimentel villanueva verifying realtime properties tccp programs journal universal computer science clarke emerson design synthesis synchronization skeletons using temporal logic logic programs kozen lecture notes computer science vol springer comini titolo villanueva abstract diagnosis timed concurrent constraint programs theory practice logic programming comini titolo villanueva condensed bottomup fixpoint modeling behavior tccp tech dsic universitat available http comini titolo villanueva towards effective decision procedure ltl formulas constraints workshop methods programming environments wlpe corr comini titolo villanueva abstract diagnosis tccp using linear temporal logic tech universitat available http boer gabbrielli meo timed concurrent constraint language information computation boer gabbrielli meo temporal logic reasoning timed concurrent constraint programs time proceedings eighth international symposium temporal representation reasoning time ieee computer society washington usa boer gabbrielli meo proving correctness timed concurrent constraint programs corr falaschi olarte palamidessi valencia declarative diagnosis temporal concurrent constraint programs logic programming international conference iclp proceedings dahl eds lecture notes computer science vol falaschi policriti villanueva modeling concurrent systems specified temporal concurrent constraint electronic notes theoretical computer science falaschi villanueva automatic verification timed concurrent constraint programs theory practice logic programming gaintzarain hermo lucio navarro systematic semantic tableaux pltl electronic notes theoretical computer science gaintzarain hermo lucio navarro orejas dual systems tableaux sequents pltl journal logic algebraic programming manna pnueli temporal logic reactive concurrent systems specification springer palamidessi valencia temporal concurrent constraint programming calculus international conference principles practice constraint programming lecture notes computer science vol springer queille sifakis specification verification concurrent systems cesar symposium programming montanari eds lecture notes computer science vol springer saraswat concurrent constraint programming languages thesis university saraswat concurrent constraint programming mit press cambridge mass valencia decidability timed ccp processes ltl theoretical computer science
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superrosy fields valuations aug krzysztof abstract prove every valuation infinite superrosy field positive characteristic divisible value group algebraically closed residue field fact prove following general result let field every finite extension every natural number index finite char given index also finite either definable valuation every valuation divisible value group char algebraically closed residue field zero characteristic case get partial results kind also notice minimal fields property every valuation divisible value group algebraically closed residue field introduction motivation work comes open structural questions concerning various contexts fundamental theorem says superstable algebraically closed important generalization superstable theories class supersimple theories yet general class superrosy theories superrosy theories nip non independence property also form generalization superstable theories orthogonal supersimple theories sense supersimple theory nip superstable known perfect pac pseudo algebraically closed small absolute galois group absolute galois group possessing many closed subgroups every index supersimple conjecture predicts converse conjecture infinite supersimple field perfect pac small absolute galois group research supported ncn grant mathematics subject classification key words phrases superrosy field valuation complementary conjecture superrosy nip formulated conjecture infinite superrosy field nip either algebraically real closed recall algebraically closed real closed superrosy nip dropping nip assumption one extend list possibilities conclusion conjecture namely since perfect pac small absolute galois group well orderable prc pseudo real closed small absolute galois group known superrosy appendix following conjecture strongest possible see section prc chosen pac prc conjecture infinite superrosy field perfect prc small absolute galois group known pac simple absolute galois group small supersimple perfect small absolute galois group similarly prc superrosy perfect absolute galois group small see fact thus conjectures know pac resp prc rest conclusion automatically also easy see conjecture implies conjecture one show orderable prc strict order property simple see remark fact conjecture also implies conjecture also interesting questions conjectures concerning nip without assuming superrosiness nip closed artinschreier extensions hasson shelah formulated dichotomies nice algebraic properties question existence valuations particular one expect following true conjecture suppose infinite field nip property every finite extension every natural number index finite either definable valuation either algebraically real closed note pure algebraically real closed valuation superrosy fact notice also facts conjecture implies conjecture another interesting problem classify strongly dependent section independently questions hasson shelah nip context approach attack conjectures assume question satisfy conclusion try produce valuation existence contradicts rosiness fact approach led following conjecture whose assumptions generalize situations conjectures whose conclusion weaker conclusions conjectures see section explanations one could say common approximation conjectures conjecture let field every finite extension every natural number index finite char given index also finite either definable valuation every valuation divisible value group either algebraically real closed residue field main result proof conjecture positive characteristic case fact prove following theorem theorem let field every finite extension every natural number index finite char given index also finite either definable valuation every valuation divisible value group case char algebraically closed residue field facts one gets following corollary corollary every valuation superrosy field positive characteristic divisible value group algebraically closed residue field since nip closed extensions also get following corollary corollary suppose field positive characteristic satisfying nip property every finite extension every natural number index finite either definable valuation every valuation divisible value group algebraically closed residue field proof theorem relies appropriate results existence valuations presence certain multiplicative additive subgroups established contrast directly minimality obtain following variant corollary minimal theorem every valuation minimal field divisible value group algebraically closed residue field recall famous podewski conjecture predicts minimal algebraically closed results tells various situations valuations question divisible value groups algebraically closed residue ultimate goal show original residue respect trivial valuation algebraically closed pac questions arises information question deduced information valuations value groups divisible residue algebraically closed discussion questions included last section paper constructed follows first give prelimnaries concerning rosy theories valuations listing facts used course proof theorem section devoted proof theorem partial results concerning conjecture zero characteristic case short easy section yields proof theorem last section discuss facts questions concerning potential applications results prove original conjectures katharina dupont currently working around conjecture project supervision salma kuhlmann collaboration assaf hasson approach one presented paper although also based details mentioned last section author would like thank thomas scanlon discussions suggestions concerning superrosy visit berkeley preliminaries valuations subsection list facts useful paper let recall valuation basic notions good reference valuations example definition valuation field surjective map ordered group satisfying following axioms min let valuation valuation ring subring either say valuation trivial equivalently group units equals finally unique maximal ideal called residue valuation associated valuation ring conversely starting valuation ring one valuation one way operations inverses appropriate value groups namely valuation ring order valuation definition say valuation field possibly additional structure definable ordered group interpretable interpretation graph definable discussion equivalent fact definable extension valuation say extends isomorphic embedding value group value group equivalently situation write let value group value group treated subgroup called index extension similarly since get called residue degree extension fact whenever finite one following inequality two valuations say coarsening situation prime ideal way one gets correspondence overrings prime ideals going one gets correspondence set prime ideals set convex subgroups value group details see chapter let add two valuations said comparable perspective important question exists valuation given deep insight question provided particular additive resp multiplicative subgroup author associates certain valuation ring denoted gives complete characterization order recall results use later definition let valuation additive resp multiplicative subgroup compatible resp weakly compatible resp coarsely compatible weakly compatible proper coarsening fact lemma let valuation either multiplicative subgroup char additive subgroup char fully compatible weakly compatible fact proposition additive resp multiplicative subgroup field two coarsly compatible valuations comparable unique finest coarsly compatible valuation ring denote moreover whenever proper resp admits weakly compatible valuation author concludes additive resp multiplicative subgroup exactly one following possibilities holds groups case valuation case coarsely compatible valuation ring property weakly compatible valuations fully compatible weak case weakly fully compatible valuation case valuation ring property weakly compatible valuations coarsenings valuation residue case weakly compatible valuations fully compatible valuation case fully compatible valuation ring going recall theorem main tool paper complete characterization given additive resp multiplicative subgroup ring language fact applications use positive part characterization namely cases one formulate full characterization emphasis mean structure existence formula every model fact theorem let field additive resp multiplicative subgroup denote maximal ideal subgroup induced residue field definable following cases group case iff discrete always weak case iff discrete iff discrete residue case always iff ordering rosy theories paper need two properties rosy groups although folklore reader convenience recall fundamental concerning rosiness give proofs two properties fact let rosy field definable valuation fact let commutative superrosy group every natural number finite index also finite denotes subgroup consiting powers particular superrosy field every index finite char image function defined subgroup finite index since finite extension elementary extension also superrosy holds finite extensions elementary extension details rosy theories reader referred rosy groups information rosy groups found subsection work ceq monster model theory language motivation consider rosy theories fact sense largest class theories allows application techniques stability theory especially basic forking calculus class contains stable generally simple theories well theories rosiness said following rosy ternary relation small subsets ceq satisfying basic properties forking independence simple theories except independence theorem relation called independence relation concrete particularly useful independence relation rosy theories called going formula strongly divides set formulas say tuple strongly divides formula implies disjunction formulas say type formula implied similarly say denoted rosy theories weakest independence relation sense implies independence relation rosy group mean group possibly additional structure whose theory rosy rosy theories also means local definition given formula finite set formulas object variables parameter variables finite set formulas variables natural number define inductively follows consistent limit ordinal parameter infinitely many given partial type define minimum recall theory rosy local thorn rank one could prove fact using characterization however one give immediate proof using another characterization rosiness terms equivalence ranks considered section definition let partial type let finite set formulas variables define follows consistent limit ordinal equivalence relation defined ceq representatives different equivalence classes section know rosy every proof fact suppose contradiction valuation rosy assume monster model sort sort consider formula formula equivalence relation denote put enough show rosy argue induction case trivial suppose consider saturation many classes say representatives taking get using way stable theories means namely unique function collection complete types ordinals together property ordinal tuple type rosy theories nice properties stable theories lascar inequalities assume rosy set sup course supremum maximum turns group supremum also attained remark set also lascar inequalities groups groups say superrosy every type group superrosy theory superrosy proof fact let claim see take course acl since get acl thus lascar inequalities get using lascar inequalities groups immediately conclude definition say theory nip formula sequence hai every iff main result says nip extensions char function surjective proof general uses algebraic geometry assuming image function index immediate consequence existence smallest subgroup bounded index wlog assume monster model indeed one easily checks ideal since image subgroup index get contains equal superrosy fields section devoted proof theorem observations concerning conjecture zero characteristic case fact use superrosiness assumption theorem consequence superrosiness fact given assumption following every extension natural number index char given index also start preparatory observations remark algebraic field extension valuation also proof suppose contradiction trivial consider let minimal monic polynomial contradiction remark let finite extension field suppose valuation property residue field algebraically closed residue field valuation either algebraically real closed proof consider valuation chevalley extension theorem extension valuation assumption algebraically closed since fact extension conclusion follows lemma let valuation field let finite extension residue field finite extension extension field isomorphic residue field proof enough prove extensions let choose monic polynomial minimal monic polynomial denotes polynomial obtained reducing modulo consider root put take extension valuation since leading equal one easily gets since irreducible get deg deg everywhere equalities fact therefore isomorphic next lemma recalls standard method showing given algebraically closed lemma let field every finite extension every prime number case char function given onto algebraically closed proof assumption perfect algebraically closed proper galois extension minimal degree intermediate gal prime number char galois theory tells splitting polynomial form assumption polynomial least one zero zeros contradiction assume char since primitive root unity galois extension degree less choice conclude galois theory conclude splitting polynomial form assumption polynomial zero zeros contradiction remark let infinite field characteristic either index multiplicative groups infinite index additive groups also infinite particular index finite perfect proof suppose let linearly independent cosets modulo index similarly elements additive cosets modulo index also proof theorem suppose valuation part value group valuation divisible proof suppose contradiction divisible put subgroup generated union family claim proper definable subgroup proof claim since index subgroup get union many cosets contains fact follows directly see proper suppose contradiction write contradiction claim know proper fully compatible fact conclude also get group case fact tells contradiction completes proof part part assuming char residue field valuation algebraically closed proof lemmas done prove following two claims claim every finite extension every extension valuation every prime number one claim every finite extension one function given onto proof claim first notice remark assume indeed since living cartesian power remark get valuation also clear consider case subclaim proof subclaim suppose true proper subgroup fully compatible fact since see valuation let residue corresponding valuation ring since char fact gives either group case residue case fact group case impossible residue case either impossible positive cone ordering however last thing also impossible char subclaim indeed since get consider case suppose contradiction proper subm already proved whenever char see thus remark implies moreover otherwise either either contradiction implies contradicts proof claim proof claim remark assume let given subclaim proof subclaim suppose true proper subgroup fully compatible fact valuation since fact ensures either group case residue case residue case fact contradiction group case since know fact yields adopt argument proof theorem reader convenience give details let largest fractional contained aim certain fractional properly contain contained contradict choice denote valuation value group corresponding real number mean rationals rationals makes sense since divisible part proof mean negation course fractional since properly contains easily implies check inclusion proper take natural number otherwise one easily gets yot generates valuation ring properly containing contained contradicts fact coarsely compatible choose proof subclaim completed show enough prove every element belongs since immediately get therefore thus claim hence obtain therefore hence subclaim indeed proof part whole theorem completed pointed introduction corollary follows theorem facts corollary follows theorem fact nip closed extensions proposition let field characteristic zero satisfying containing either definable valuation every nontrivial valuation char every char prime number primes different proof suppose valuation argument based facts proof claim proof theorem works noting assumption eliminates possibility positive cone ordering suppose prime proper subgroup char get contradiction proof subclaim proof claim theorem char consider prime let hand since fact implies belong group case thus conclude hence char however impossible char case one gets contradiction proof subclaim proof claim theorem char goal show case proper subgroup facts yield char belong group case since conclude char get char contradicts assumption relatively prime one able strengthen conclusion proposition showing primes using lemma one would get assumption proposition either valuation valuation char residue algebraically closed arguing proof remark one would also get assumption implies either valuation valuation char residue either algebraically real closed fact strengthening slightly assumption would imply full conclusion conjecture proposition suppose conclusion proposition strengthn ened primes let field every finite extension elementary extension every natural number index finite char given index also finite either definable valuation every valuation divisible value group either algebraically real closed residue field proof suppose valuation using remarks assume let valuation divisibility proved theorem theorem remains consider case char discussion proposition done case char remains consider case char take monster model let maximal valuation ring let maximal ideal containing ove characteristic claim valuation ring see recall corresponds proper convex subgroup put convex subgroup properly saturation valuation containing ring since hence residue maximal ideal characteristic algebraically closed conclusion proposition holds residue zero characteristic case place ove implies residue also algebraically closed algebraically closed minimal fields prove theorem contrast theorem proof relies results proof theorem trivial consequence minimality recall minimal whose every one variable subset proof theorem consider valuation minimality every natural number see notice multiplicative subgroup subgroup everything clear divisible consider monic polynomial positive degree let since takes many values exists otherwise implies contradiction fact therefore root final comments say given property every valuation divisible value group algebraically closed residue say property every valuation divisible value group either algebraically real closed residue remark property property results conjectures discussed paper natural questions arises said structure property particular one deduce results conjectures least positive characteristic podewski conjecture recall bounded absolute galois group small absolute galois group many closed subgroups index equivalently every natural number many extensions degree isomorphism orderable equipped order making ordered equivalently formally real sum squares recall pac pseudo algebraically closed every absolutely irreducible variety point prc pseudo real closed every absolutely irreducible variety point every real closed containing point pac prc know orderable prc perfect pac lemma fact prc field superrosy perfect bounded proof proved appendix orderable prc pac see converse note argument theorem yields boundedness finally suppose contradiction perfect remark contradicts fact remark orderable prc field strict order property simple particular conjecture implies conjecture proof let formally real prc claim element sum many squares sum two squares consider element sum many squares polynomial zero real closed containing since absolutely irreducible eisenstein criterion prc polynomial zero sum two squares relation sum squares many elements claim partial order chain fact antisymmetric follows fact formally real transitivity clear existence chain seen follows take since paragraph proof sum two squares see fact non separably non real closed prc field nip particular conjecture implies conjecture proof main result says non separably closed pac nip consider non separably non real closed prc suppose contradiction nip still nip interpretable pac separably closed contradiction perfect pac property lemma prc property one could ask whether property implies question prc non formally real case prc replaced pac virtue results fact positive answer would imply conjectures positive characteristic unfortunately answer negative explain hasse principle brauer groups considered shown theorem perfect pac extension relative transcendence degree hasse principle brauer groups hand shown theorem whenever perfect hasse principle brauer groups holds extensions relative transcendence degree property also asked question whether non formally real perfect hasse principle brauer groups holds extensions relative transcendence degree necessarily pac constructed authors introduced class weakly pac proved weakly pac non formally real assuming perfectness satisfy hasse principle brauer groups extensions relative transcendence degree constructed perfect weakly pac pac particular perfect weakly pac property discussion leads following questions context question exist weakly pac pac superrosy resp supersimple field positive answer would refute conjecture resp conjecture negative answer would support prove conjectures question let infinite perfect field nip satisfying property true either algebraically real closed applying trick adding see versions question equivalent corollaries positive answer would imply conjectures positive characteristic question exist non algebraically closed perfect weakly pac field superrosy nip positive answer would yield negative answer question positive answer extended version would refute conjecture notice since know non separably closed pac nip answer question positive witness would weakly pac pac case order conjectures one could try construct suitable non prc property possible candidates could among perfect weakly pac pac try prove conjectures showing generalization corollary characteristic zero case open problem course thereom yields divisibility value groups problem residue conjecture every valuation superrosy field divisible value group either algebraically real closed residue field another idea attacking conjecture means corollary rather conjecture suggested hrushovski try show superrosy nip residue respect valuation another superrosy nip use corollary conjecture zero characteristic case conclude original either algebraically real closed mentioned introduction dupont approach conjecture also conjecture adding goal show either every extension every prime number algebraically closed lemma valuation assume possibility fails put follows assumptions remark perfect char goal show valuation ring dupont deduced family sets subbasis second goal project show using nip assumption last condition holds references adler geometric introduction forking preprint chatzidakis simplicity independence closed fields models computability leeds london math soc lecture note vol cambridge univ press cambridge chatzidakis pillay generic structures simple theories ann pure appl logic cherlin shelah superstable fields groups annals mathematical logic duret les corps faiblement clos ont model theory algebra arithmetic proc karpacz lecture notes mathematics springer vol ealy pillay superrosy dependent groups finitely satisfiable generics annals pure applied logic ealy onshuus characterizing rosy theories journal symbolic logic efrat hasse principle function fields pac fields israel journal mathematics engler prestel valued fields springer netherlands geyer jarden non pac fields whose henselian closures separably closed mathematical research letters hrushovski fields related structures preprint kaplan scanlon wagner extensions nip simple fields israel journal mathematics koenigsmann definable valuations delon dickmann gondard editors seminaire structures paris vii fields interpretable rosy theories israel journal mathematics fields interpretable superrosy groups nip case journal symbolic logic macintyre theories fields fundamenta mathematicae onshuus rosy theories thesis berkeley onshuus properties consequences journal symbolic logic podewski minimale ringe math phys semesterber shelah strongly dependent theories preprint wagner simple theories kluwer academic publishers dordrecht netherlands address instytut matematyczny uniwersytet grunwaldzki poland address kkrup
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real time impact control charging energy storage shipboard power systems jan yusheng luoa sanjeev srivastavab manish mohanpurkara svetomir stevica rob hovsapiana idaho national laboratory idaho falls idaho siemens corporate technology college princeton abstract medium voltage based integrated power system projected one solutions powering ship faces significant challenges accurately energizing advanced loads especially pulsed power load rail gun high power radar state art equipment energy storage based supercapacitors proposed technique buffering direct impact pulsed power load power systems however high magnitude charging current energy storage pose disturbance distribution generation systems paper presents fast switching device based real time control system achieve desired balance maintaining required power quality fast charging energy storage required time test results shown verify performance proposed control algorithm keywords medium voltage based integrated power system pulsed power load power quality disturbance metric real time control introduction research related navy shipboard power system raise critical concern regarding system stability due diverse loads similar microgrids navy shipboard power systems slack bus viewed microgrid always operating islanding mode compared typical terrestrial microgrid ratio overall load generation much higher although new avenues zonal load architecture high preprint submitted international journal electrical power energy systemsjanuary density energy storage systems high efficiency generator developed supporting stability control new challenges also arise due diverse demand load side order reduce ship size visibility future battlefields integrated power system ips based ship developed electric ship characterized using electrical motor driven propulsion system replace traditional mechanical transmission propulsion system therefore propulsion system integrated shipboard power system termed ips application electrical propulsion waives need slow speed gear providing power mechanical propulsion remaining fast gear drive generation electrical propulsion therefore space occupied transmission system largely reduced nevertheless electrical propulsion high efficiency means less fuel needed ship size reduced shape ship optimized weapons system also improved significantly radars capable providing detection higher resolution wider area pulsed power load features fast powerful accurate attacks electromagnetic ejector inject high launch dynamics shipboard aircrafts state art devices share common characteristic power demand appears large pulse due integration power electronics interface electrical propulsion pulsed power load ppl place huge burden generation distribution capacity shipboard power systems address challenge reliably powering ship several efforts made different distribution architectures medium voltage alternative current mvac high frequency alternative current hfac medium voltage direct current mvdc analyzed research efforts reached conclusion mvdc system projected optimal distribution architecture ips account mvdc concerns regarding generator synchronization harmonics distortion frequency stability electrical devices related control strategy supporting mvdc distribution architecture also developed besides developing ips contributions also made test performance ips powering advanced loads propulsion loads occupy large portion total shipboard load consumption rest loads ppls usually critical due indispensable role battle one challenging issues controlling ips operation providing expected power energy ppl without affecting operation critical loads impact ppl ips various distribution architectures analyzed analysis shows impact energizing ppl around power seriously influence system stability power quality noteworthy work presented innovative application power electronics interfaced supercapacitor energy storage system sess energy storage found wide application hybrid energy systems augmenting limited generation modern loads beginning development ips energy storage used auxiliary power supply supercapacitor well known high power density enable fast switching power source shipboard generation sess direct impact ppls ips eliminated however charging sess induce high current still destabilize ips control methods driving sess charging circuit within desired limits rates proposed supercapacitor control proposed although method effective requires precise understanding system current power limits order prevent abrupt disturbance system control sess charging introduced control effective trapezoidal load profile fits system requirement vice versa control pbc features minimal impact system stability within fixed time frame however still requires prerequisite information system current limit furthermore challenging set fixed charging time essential systems papers focus coordinating sess charging control generation control mitigate possible disturbance caused sess charging however mission time especially battle time quality communication supporting coordination may guaranteed due weather battle damage nevertheless coordination strategies mostly applied mvac system highlight coordination generation control load control mvdc system proposed waiver concerns generation control coordination another word open decentralized control strategy henceforth decentralized control strategy considered suitable method sess charging mvdc system proposed work paper based contribution proposed disturbance metric control dmc dmc decentralized technique developed control sess charging according disturbances monitored real time sess charging mvdc system however needed prove existing hardware capable realizing proposed real time control theory achieve expected performance dmc study integrates cutting edge igbt technology dmc proposed real time dmc rtdmc paper comprises following sections section describes experimental system section analyzes disturbance due charging section presents dmc strategy guided two major concerned disturbance metrics mvdc system section describes hardware implementation proposed rtdmc strategy section test results based proposed rtdmc discussed conclusion summarized section system description test environment unpredicted unaccounted sess damage hardware leading failures making uneconomical test applications hence digital real time simulator drts used assess performances generate result environment simulating electrical circuit power electronics simulation reduced real time computation speed power systems test whether proposed control strategy fast enough adjust sess charging expected control performane achieved mvdc system maturity technology circuit breakers cables mvdc system deemed advantageous power distribution following introduction mvdc simulated studied shipboard ips mvdc distribution architecture shown figure topology main power source two main gas turbine generators mtgs mtg generation capacity case need mtg equipped auxiliary gas turbine generator atg generation output mtgs atgs fed controllable rectifier mvdc distribution bus two distribution buses constitute mvdc distribution system since ship equally divided two parts starboard port bus feeds starboard called starboard bus whereas one called port bus bus directly connected mtg atg propulsion motor two buses zonal loads stern circuit breaker bow circuit breaker zonal loads use circuit breaker choose drawing power one bus buses stern bow circuit breakers used connect disconnect starboard port buses starboard port bus connected generation four generators merged together mode connection called ring mode starboard port bus separated power supplied bus come one mtg one atg call operation split plant mode spm ppl module directly connected port bus means power supply charging sess much spm circuit breaker rectifier circuit breaker port propulsion motor disconnect rectifier disconnect mvdc starboard bus drive inverter zone load center stern circuit breaker zone load center zone load center zone load center disconnect ppl buck converter zone load center disconnect starboard propulsion motor disconnect bow circuit breaker mvdc port bus drive inverter rectifier circuit breaker circuit breaker rectifier figure system topology mvdc based ips pulsed power load module inside ppl module ppl sess power electronics interface serving charging circuit sess shown figure charging circuit buck converter controlled switch commutation period turn time ton ton power goes enters sess opened current goes sess continued passing buck converter mvdc bus vin sess ppl mvdc bus figure charging circuit sess charging reaches steady state voltage sess maintained fixed value shown ton vin energizing ppl power sent sess finally reaches ppl prevent ppl directly draw power mvdc bus allowed closed simultaneously therefore always opened charging remained opened ppl energized study totally energy expected injected sess amount energy allows ppl least fired times without recharging according analysis disturbance due sess charging figure simplified diagram mvdc system using single line representation used analyze possible disturbance caused sess charging figure emf generated single phase generator reactance resistance generator ignored transient analysis switches forming rectifier convert power rline line resistance distribution system rload resistance loads sess set analysis time begins positive turned generator terminal current charging expressed rline rload terminal voltage also mvdc bus voltage calculated vbus iload rline vbus rload sess figure simplification diagram mvdc based ips right sess charging initiated since voltage supercapacitor suddenly change supercapacitor zero initial voltage equal insert short circuit fault system rload henceforth removed calculation changed rline transient introduced system generator stator reactance replaced transient reactance smaller value compared got rline compared real imaginary parts denominator reduced magnitude denominator smaller moment value reviewed constant current magnitude therefore increased since usually times rload rline got following equation rline rload rline means vbus reduced conclude reactive power generator also increases according rline rload rline reduced voltage influences operation loads increased reactive power reduce fuel efficiency generator two disturbances mitigated within tolerated range sess charging allowed disturbance mitigation control strategy critical impact mainly correlated charging current higher charging current severe impact expected charging control systems must strike balance rapid charging tolerable impact according analysis presented previous section power quality impacts sess charging reflected mvdc bus generator reactive power output two metrics developed evaluate impact sess charging lim vbus lim input limit mvdc bus voltage vbus real time measured mvdc bus voltage real time measured generator reactive power designing power systems standards employed guiding principles one applicable standards related expected power quality range limitations device operating within system control designs ensure long specified power quality maintained devices properly operate well thus sess charging controller ensure power quality lowered limit required standard loads function normally performance acceptable forming basis dmc proposed paper assuming dmc restrain disturbance set limits adopters proposed work need modify limits defined applicable standards customized upper limits two metrics input dmc dmc maintains metric values limit therefore expected balance reached shown figure dmc measures metric value determines control signal sent charging circuit charging current defined current injected mvdc bus positive node sess module assume charging process mvdc bus voltage settle steady state rated value expected maintain charging current minor variation charging power close constant value facilitate ips reaching new steady state charging due variations characteristics mvdc bus voltage mtg reactive power two different procedures controlling charging current according changes control procedure mitigating impact mvdc bus voltage charging current directly affect mvdc power therefore maintain charging current constant value bus voltage also stay close constant value hence key part mitigating impact mvdc bus voltage find maximum charging current proposed control procedure set alert value less close preset upper limit bus voltage deviation start controlled charging process alert metric value metric value reaches preset alert value stop charging record corresponding charging current set recorded maximum charging current reference charging circuit let controller closely tracking stop charging sess voltage reaches desired value firing dmc pulse charging circuit metric value expected region metric value metric curve charging time figure dmc control strategy control procedure mitigating impact mtg reactive power output existing test result shows impact charging current mtg reactive power output indirect delay charging current change consequent reactive power change therefore control procedure mitigating bus voltage applied mitigating reactive power increment adaptive control strategy developed reactive power output limitation set alert value less close preset upper limit start charging stop charging metric value first time hit alert value record current value moment suspension set current reference value based recorded value resume charging regulate charging acquired reference value stop charging sess stored voltage reaches desired value figure flow chart mitigating mvdc bus voltage sag bus voltage deviation start controlled charging process metric value reaches preset alert value stop charging record corresponding charging current set charging current value maximum charging value implementing algorithm metric value continues increase exceeds upper limit use suitable attenuation factor alpha revise value maximum charging value using formula attenuationf actorxrecordedchargingvalue resume charging charging current increases revised maximum charging value stop charging monitor metric change metric value still reach upper limit repeat step repeat iteratively metric exceed upper limit set last value product maximum charging current let upper limit value times positive coefficient less lower limit charging current coefficient proposed work set reference current value switched lower upper limit based actual monitored value charging current although current oscillating oscillation impact tolerated due difference upper limit lower limit stop charging sess stored voltage reaches desired value performance proposed control strategy depends quickly critical current value captured precisely actual charging current track reference current immediately charging current value switched one state another state desired controller hardware implementation meeting aforementioned prerequisites expected implemented fast switchable igbt based hardware implementation real time disturbance metric controller expected hardware capable immediately turning sess charging also capable withstanding high voltages turn period furthermore switch also allow monetary overshoots current based requirements fast response controller conducted literature review related fast switchable power electronics devices inferred chopper igbt module infineon necessay functionalities realizing proposed control strategy according turn delay device reduced sustain exerted voltage turn connecting start charging stop charging metric value first time hit alert value record current value moment suspension charging current reaches upper limit stop charging monitor trend mtg reactive power increase resume charging exceeds preset limit yes let last recorded current value multiply attenuation factor generate upper limit fix upper limit lower limit charging current resume charging regulate charging acquired limit value stop charging sess stored voltage reaches desired value figure flow chart limit mtg reactive power output multiple switches parallel buck chopper capable fast turn high current implemented concerned disturbance maintained within desired regions igbt module may similar products necessary functionalities expected energy stored sess reduce size capacitor stored voltage high possible hand highest voltage stored sess high affect turning performance figure according best performance verified vce switch device allow increase test voltage final charged voltage set according capacitance sess study case test result order validate whether proposed test properly mitigate impact ips ensure fast charging test case setup drts simulation time step reaction test system well approach actual real time power system response overall initial generation output study cases set mvar generation margin left sess charging mva test cases run limit limit limit mvar limit mvar sess charging begins fifth second test initialized test case mvdc bus voltage sag mitigation figure shows test result limit set alert value set bus voltage goes reaches value charging suspended current value marked charging controller enables actual current value stay charging process lasts second stops sess stored voltage reaches minimum bus voltage average bus voltage value charging maximum value charging expected bus voltage value maintained figure case study limit test case mtg reactive power output restrain figure used demonstrate effect dmc limit set mvar alert value set mvar attenuation factor set charging current first reaches single mtg reactive power output reaches mvar hence charging suspended mtg output continues increase peaks mvar therefore set upper limit charging current set lower limit seen setting reactive exceeds limit twice caused adjustment generator excitation short transient reactive power stays mvar average value mvar takes seconds inject energy sess attenuation factor set table shows summarization test cases test results show proposed dmc strike optimal balance point fast charging required power quality acquisition figure case study limit mvar table test cases result summarization test setting limit limit limit mvar limit mvar maximum metric value minimum metric value average metric value mvar mvar mvar mvar mvar mvar charging current value charging time conclusions paper dmc strategy proposed based analysis sess charing mvdc based ips proposed control strategy focus reaching reasonable balance fast charging sess maintaining required power quality based first stage analysis impact sess charging power quality assessed studied underlying reason impact sess charging power quality large charging current challenging quantify relation charging current power quality one feasible way limiting impact charging current online real time monitoring charging current power quality based real time monitoring result optimal charging current value generated next challenging part drive charging current rapidly precisely track reference value fast switching charging circuit based latest igbt technology development proposed test results shows using fast igbt implement sess charging system accurately follow control command sent dmc realization fast sess charging maintain required power quality future greater disturbances caused sess charging side harmonic influence mvdc bus expected analyzed proposed control strategy highly depends rapid monitoring action paper using drts effect real time control simulated simulation result shows device monitoring controlling fast enough rapid sess charging realized without sacrificing required power quality future test real time performed order validate proposed control strategy actual hardware finally decentralized coordination sess charging control generator control also studied future work acknowledgements proposed work supported water power technology office department energy inl laboratory directed research development ldrd program doe 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homology littlewood complexes feb steven sam andrew snowden jerzy weyman abstract let symplectic vector space dimension given partition parts associated irreducible representation representation admits resolution natural complex call littlewood complex whose terms restrictions representations parts representation defined littlewood complex still makes sense purpose paper compute homology find either acyclic unique homology group forms irreducible representation homology group exists computed rule reminiscent occurring theorem result interpreted computation derived specialization irreducible representations categorifies earlier results universal character rings prove analogous results orthogonal general linear groups along way see two topics commutative algebra minimal free resolutions determinantal ideals koszul homology contents introduction preliminaries symplectic groups orthogonal groups general linear groups references introduction statement main theorem let symplectic vector space complex numbers dimension associated partition parts irreducible representation irreducible representations uniquely form space defined quotient usual schur functor sum images obvious maps obtained removing vertical strip size two words presentation presentation admits natural continuation resolution call littlewood complex characterized minimal resolution representations extend date february mathematics subject classification sam supported ndseg fellowship miller research fellowship snowden partially supported nsf fellowship weyman partially supported nsf grant steven sam andrew snowden jerzy weyman number parts exceeds still makes sense speak complex even though longer associated irreducible see simple example however typically longer exact higher degrees natural problem compute homology exactly main theorem paper accomplishes theorem homology either identically zero else exists unique irreducible representation fact procedure called modification rule see allows one compute exactly homology group irreducible representation rule phrased terms certain weyl group action way theorem reminiscent classical theorem also combinatorial description rule terms border strips see theorem precise statement prove analogous theorems orthogonal general linear groups clarity exposition concentrate symplectic case introduction analogous result symmetric group found proposition elaborated upon example let give simplest example theorem namely irreducible representation quotient line spanned symplectic form identify via form complex thus differential multiplication form complex clearly makes since even differential surjective trivial representation differential isomorphism homology vanishes differential injective involved examples found representation theory proper context theorem lies representation theory explain connection noting however somewhat exotic objects discussed used proof theorem occur remainder paper let rep denote category algebraic category first identified dps denoted also studied slightly different point view shown specialization functor rep rep functor right exact exact category rep littlewood complex makes sense defines complex rep exact positive degrees simple object simple objects uniquely form schur functor object rep two important properties projective specializes thus see projective resolution specializes therefore following observation explains significance littlewood complex point view proposition computes derived specialization simple object thus rephrase theorem follows theorem let irreducible algebraic representation either acyclic else unique irreducible representation use pro version category opposite usual ind version category see details homology littlewood complexes symplectic schur functors exhibit stability large dimensional vector spaces explained see also htw general contrast usual schur functors general strategy dealing problems involving symplectic schur functors pass stable range work take advantage simpler behavior apply specialization functor return unstable range strategy viable one must understand behavior specialization functor one source motivation project accomplished theorem relation results theorem categorifies results explain universal character ring defined ring homomorphism specialization representation ring given basis given shown image either plus minus character irreducible representation fact grothendieck group rep map induced specialization functor class simple object grothendieck group thus shadow theorem precisely result specialization however note proof depends koszul homology classical invariant theory theorem reinterpreted calculation homology groups koszul complex generators ideal arises classical invariant theory explained see also orthogonal general linear groups remark koszul homology seems remarkably difficult calculate even classes ideals determinantal ideals cases worked explicitly point case codimension perfect ideals case codimension gorenstein ideals classes ideals determinantal singled koszul homology modules property fails determinantal ideals resolutions determinantal ideals see interpretation theorem terms koszul homology also interpreted terms minimal free resolutions certain modules supported determinantal varieties defined pfaffians generic matrix coordinate ring determinantal variety module computation resolution becomes special case theorem therefore realizes classical resolution first piece much larger structure orthogonal group general linear group correspond determinantal varieties generic symmetric matrices generic matrices respectively refer reader wey calculation minimal free resolutions coordinate rings determinantal varieties overview proof three main steps proof first establish combinatorial result relating bott algorithm calculating cohomology irreducible homogeneous bundles modification rule appearing thevmain theorem introduce certain module polynomial ring sym auxiliary vector space compute minimal free resolution main tools step theorem geometric method third author lastly identify piece ring sym results step specialization homomorphism see koi wen used get enough control identification made results step give minimal free resolution theorem follows littlewood complex identified piece minimal free resolution steven sam andrew snowden jerzy weyman notation conventions always work complex numbers possible work field characteristic seem advantage write number parts partition rank partition denoted rank number boxes main diagonal write transpose partition occasionally use frobenius coordinates describe partitions recall let rank let resp denote number boxes right resp ith diagonal box including box frobenius coordinates denote coefficients coefficient schur function product relevant background partitions schur functions schur functors refer mac chapter wey chapters preliminaries geometric technique let smooth projective variety let exact sequence vector bundles trivial let another vector bundle put sym sym ring fact symmetric algebra following proposition encapsulates need geometric technique third author proof stronger result see wey proposition assume natural isomorphism tora theorem let set integer sequences eventually identify partitions sequences sequences necessarily let transposition switches let group automorphisms generated group coxeter group fact infinite symmetric group admits length function definition length minimum number exists expression alternatively number inversions interpreted permutation define second action denoted follows put terms generators action let element precisely one following two possibilities occurs exists unique element partition case call regular exists element case call singular bott algorithm wey procedure determining regular goes follows find index index exists partition regular singular otherwise apply repeat keeping track produces minimal word element definition particular important note choice index pick first step different choices lead different minimal words resulting partition permutation independent choices homology littlewood complexes let vector space let grassmannian rank quotients assume dim obviously tautological sequence rank partition parts partition let element given theorem wey theorem let partition parts let partition let vector bundle suppose regular write partition otherwise suppose singular remark rank similarly dim problems disappear dim sufficiently large compared fact situation becomes completely uniform dim resolution second veronese ring treatment odd orthogonal groups need know resolution second veronese ring relative setting state relevant result interrupt discussion later let variety let vector bundle let associated projective space bundle one dimensional quotients let universal rank one quotient define kernel map result need following proposition let sum partitions rank mod rank proof relative version calculation minimal free resolution sym second veronese ring sym odd veronese module vector space case contained wey theorem calculate case note functorial due stability properties schur functors calculate resolution rank result also holds rank enough handle case rank odd arbitrarily large consider total space structure map define also consider map spec sym using setup wey theorem see ext dual wey proposition rank odd partitions free resolution fit square size rank level partitions index schur functors appearing free resolution duality amounts taking complements within square reversing direction arrows hence result follows remark let exact sequence steven sam andrew snowden jerzy weyman ring sym identified sym projective coordinate ring second veronese avrelative version geometric method minimal locally free resolution computed sheaves appearing proposition criterion degeneration certain spectral sequences let map proper varieties let coherent sheaf say degenerate leray spectral sequence degenerates second page following simple criterion degeneracy applies one case interest lemma suppose group acts oni suppose furthermore degenerate proof differentials spectral sequence thus forced vanish lemma commutative algebra give simple lemma allows interpret koszul homology groups tor useful since ultimately interested certain koszul homology groups geometric technique computes tor let graded let homogeneous subspace form koszul complex basis familiar koszul complex let sym natural homomorphism following result lemma natural identification tora proof resolve using koszul resolution gives also one computes tora tensoring symplectic groups representations let symplectic space dimension symplectic form gives isomorphism stated introduction irreducible representations indexed partitions see call partitions admissible admissible partition write corresponding irreducible representation littlewood complex let vector space put sym sym consider inclusion given first inclusion multiplication inclusion defines algebra homomorphism put quotient ideal generated maps spec spec spec natural identification spec space hom linear maps spec space forms map spec spec takes linear map form space spec call littlewood variety fiber map consists maps words spec consists maps image isotropic subspace homology littlewood complexes let koszul complex littlewood variety decompose complex action littlewood complex independent long dim complex theorem zeroth homology admissible otherwise lemma tora decomposition tora applied obtain admissible remark possible compute terms explicitly let set partitions whose frobenius coordinates satisfy see discussion set admissible higher homology vanishes see proposition taking euler characteristics get equality representation ring significance identity expresses class irreducible terms representations restricted due littlewood lit see also prop name complexes special case main theorem main theorem computes homology complex formulate prove theorem particularly simple case mention worthwhile know argument needed prove main theorem proposition suppose admissible otherwise proof choose dimension lemma higher homology follows either computation given lemma suppose dim spanned regular sequence proof suffices show dim spec dim spec dim put dim observe locus spec injective open let igr variety isotropic subspaces comes rank tautological bundle oigr natural birational map total space hom spec thus spec dimension dim spec dim result follows steven sam andrew snowden jerzy weyman modification rule associate partition two quantities used describe homology recall dim give two equivalent definitions quantities one via weyl group action one via border strips begin weyl group definition following wen recall defined automorphisms set integer sequences define additional automorphism negates let group generated coxeter group type equipped length function defined let define new action action agrees one defined despite difference action given given partition exactly one following two possibilities hold exists unique element partition admissible put exists element put leave undefined note admissible partition first case give border strip definition following sun based kin put suppose recall border strip connected skew young diagram containing square let connected border strip length starts first box final row exists exists nonempty partition put denotes number columns occupies otherwise put leave undefined remark alternative way think removing terms hooks given box young diagram recall book set boxes either directly directly right including border strips begin last box first column property young diagram naturally bijection boxes first column take box row ends important point size border strip size hook removing removing hook shifting boxes hook one box northwest direction illustrated following diagram shaded boxes indicate border strip left diagram hook right diagram agreement two definitions may known experts unaware reference provide proof proposition two definitions agree proof suppose removing border strip size begins first box final row let number columns sequence homology littlewood complexes column lengths conversely use weyl group modification rule expression must begin apply increase number inversions sequence write wsi hum theorem resulting expression minimal choose maximal replaced first column moved right much possible adding time pass column subtracting column passed resulting shape young diagram removing border strip length columns finally third modification rule defined need know statement rule cite results need know rule equivalent previous two equivalence rule border strip rule comes fact rules derived determinantal formulas see theorem kin footnote main theorem main theorem following theorem partition integer otherwise particular exact remark consider coordinate ring rank matrices identifying dual quotient ideal generated pfaffians description resolution found wey jpw hand part theorem combined shows appears resolution thus modification rule gives alternative description resolution pleasant combinatorial exercise show directly two descriptions agree description terms modification rule complicated advantage readily generalizes situation proof theorem take remainder section follow plan outlined throughout space fixed dim step let set partitions whose frobenius coordinates satisfy set admits inductive definition useful describe empty partition belongs partition belongs number rows one number columns partition obtained deleting first row column belongs significance set plethysm see mac let partition write place section define regular partitions steven sam andrew snowden jerzy weyman lemma let partition let partition obtained removing first row column also belongs furthermore let resp unique element resp partition border strip defined see following identities proof suppose applying bott algorithm number moves places left thus first steps algorithm reach sequence notice subsequence starting subsequence starting except added entry former follows bott algorithm runs exactly manner particular regular would either shows belongs suppose bott algorithm terminates steps discussion bott algorithm terminates steps following formula since border strip boxes using remark see exists since box row first column hook size furthermore since result follows lemma let belong suppose partition let partition defined exists partition element partition obtained removing first row column proof reverse steps lemma proposition unique bijection maps exists case proof let element let partition show induction belongs clear suppose follows tacitly employ lemma let partition obtained removing first row column belongs choose partition induction since furthermore completes induction thus shown defines map sets show map injective suppose two elements map sequences identical multisets numbers particular rearrange sequence numbers right bar get sequence numbers right bar sequences right bar strictly decreasing see finally show surjective partition empty partition preimage suppose belongs let induction size homology littlewood complexes find mapping applying lemma find partition mapping completes proof step let vector space dimension least let grassmannian rank quotients let tautological bundles put define exact sequence finally partition put sym note sym lemma let partition correspond otherwise proof follows immediately proposition theorem lemma let partition let integer sum partitions particular left side vanishes proof result follows previous lemma proposition tora sum partitions proof follows immediately previous lemma proposition step partition parts put homsp note compatible action goal show isomorphic lemma spaces isomorphic representations finite multiplicities proof let enough show isomorphic representations finite multiplicities fact enough show multiplicity spaces isomorphic representations finite multiplicities steven sam andrew snowden jerzy weyman multiplicity space decomposition representation ring computed applying specialization homomorphism thm result note fixed finitely many values make richardson coefficient establishes finiteness multiplicities equality tora representation ring applying proposition find tora see mac obtain therefore find given result follows proposition isomorphism proof according proposition tora tora since must agree everywhere except possibly first column therefore presentation note occurs multiplicity one thus occur therefore occur either since isomorphic representations lemma tora coordinate ring littlewood variety multiplicity space therefore surjection since occur copy lies kernel therefore induces surjection finally since two isomorphic representations finite multiplicity spaces surjection isomorphism combining proposition proposition obtain following corollary corollary tora combining yields main theorem choose arbitrarily large homology littlewood complexes remark arguments step made use construction module simply satisfied proposition precisely say type admissible partition tora sum partitions arguments step establish following statement type module exists isomorphic thus type actually argument bit weaker since works step thought simply providing construction module type examples give examples illustrate theorem example suppose complex differential multiplication symplectic form treated element differential injective irreducible representation differential isomorphism homology vanishes differential surjective complex identically example suppose complex differential multiplication symplectic form treated element differential injective irreducible representation complex exact homology vanishes finally reader check easily instances description homology agrees rule given weyl group action example suppose dim modification rule using border strips proceeds follows start left supposed remove border strip size border strip shaded result second displayed partition border strip remove length result removing strip third partition border strip length result removing final partition satisfies algorithm stops thus see follows steven sam andrew snowden jerzy weyman illustrate modification rule using weyl group action write idea getting weyl group element apply first column length long sort result repeat necessary start subtracting result get orthogonal groups representations let orthogonal space dimension orthogonal form gives isomorphism write even odd recall representation theory see details irreducible representations indexed partitions first two columns boxes total call partitions admissible admissible partition write corresponding irreducible representation given admissible partition let partition obtained changing number boxes first column minus present value call conjugate conjugation defines involution set admissible partitions irreducible representations conjugating partition corresponds twisting sign character sgn follows isomorphic restricted fact restrictions remain irreducible unless equivalent case decomposes sum two irreducible representations admissible partition exactly one element set boxes first column denote element thus otherwise littlewood complex let vector space put sym sym consider inclusion given first inclusion multiplication inclusion defines algebra homomorphism put quotient ideal generated maps spec spec spec natural identification spec space hom linear map spec space symmetric forms map spec spec takes linear map form space spec call littlewood variety fiber map consists maps words spec consists maps image isotropic subspace let koszul complex littlewood variety decompose complex action homology littlewood complexes complex littlewood complex independent long dim proposition zeroth homology admissible otherwise lemma tora decomposition tora applied obtain admissible special case main theorem main theorem computes homology complex formulate prove theorem particularly simple case mention worthwhile know argument needed prove main theorem proposition suppose admissible otherwise proof choose dimension lemma higher homology follows either computation given lemma suppose dim spanned regular sequence proof proof lemma difference worth pointing affect proof dim dim grassmannian isotropic subspaces two connected components variety cut two irreducible components modification rule symplectic case associate partition two quantities give two equivalent definitions begin weyl group definition following wen let automorphism set negates swaps first second entries let group generated coxeter group type let length function defined note group subgroup aut equal one length function different since using different set simple reflections let define new action agrees one defined action given definitions exactly first half give border strip definition motivated sun based kin sun focuses special orthogonal group modify definition get correct answer full orthogonal group one given except three differences border strip length definition use instead total number border strips removed odd replace end result steven sam andrew snowden jerzy weyman one stop applying modification rule either becomes admissible resulting values instance admissible one stop immediately instead one could remove border strip border strip occupies first column removed yields thus since removed odd number strips proposition two definitions agree proof suppose removing border strip size begins first box final row let number columns first two column lengths two cases depending two quantities bigger first suppose remove another border strip size begins first box final row let number columns sequence gives column lengths suppose removed border strip odd number replace according case sequence gives column lengths conversely use weyl group modification rule expression must begin apply increase number inversions sequence write wsi hum theorem resulting expression minimal choose maximal replaced first two columns moved column length right much possible adding time pass column subtracting column passed resulting shape minus first column young diagram two possibilities whole shape young diagram result first removing border strip length columns replacing resulting otherwise first column length resulting shape less second column length choose maximal moved first column right much possible adding time pass column subtracting column passed resulting shape young diagram case removed two border strips size columns respectively finally third modification rule defined symplectic case need know statement rule cite results equivalence rule border strip rule comes fact rules derived determinantal formulas see theorem kin footnote remark rules stated special orthogonal group enough purposes matter notation write place main theorem main theorem exactly symplectic case homology littlewood complexes theorem partition integer otherwise particular exact remark consider coordinate ring rank symmetric matrices identifying dual quotient ideal generated minors resolution known see wey jpw hand part theorem combined shows appears resolution thus modification rule gives alternative description resolution pleasant combinatorial exercise show directly two descriptions agree symplectic case description terms modification rule complicated advantage generalizing situation separate proof theorem two cases according whether even odd case follow plan even case throughout section dimension space step let set partitions whose frobenius coordinates satisfy set admits inductive definition follows empty partition belongs partition belongs number columns one number rows partition obtained deleting first row column belongs significance set plethysm see mac let partition write place define regular partitions lemma let partition let partition obtained removing first row column also belongs furthermore let resp element resp partition defined following identities proof suppose applying bott algorithm number moves places left thus first steps algorithm reach sequence bott algorithm sequence runs like algorithm regular belongs suppose algorithm terminates steps algorithm terminates steps following formula steven sam andrew snowden jerzy weyman since border strip boxes using remark see exists since box row first column hook size furthermore since done proposition unique bijection maps exists case rank even rank odd proof except computation proof exactly like proposition proof lemma see number border strips removed rank determination follows step let vector space dimension least let grassmannian rank quotients let tautological bundles put define exact sequence finally partition put sym note sym proposition tora sum partitions proof proof exactly like proposition step admissible partition put homo note compatible action let partition rows put otherwise put note decomposition holds lemma let partition spaces isomorphic representations finite multiplicities proof put enough show isomorphic representations finite multiplicities fact enough show multiplicity spaces isomorphic representations finite multiplicities multiplicity space decomposition representation ring computed applying specialization homomorphism thm result fixed finitely many values make coefficient establishes finiteness multiplicities equality tora homology littlewood complexes representation ring applying proposition find tora see mac obtain therefore find multiplicity space given result follows proposition let partition isomorphism proof suppose first let partition given proposition provides following presentation note occurs multiplicity one rule thus occur therefore occur either since isomorphic representations since tora see follows surjection since occur see induces surjection since two isomorphic representations finite multiplicities surjection isomorphism consider case define define using recipe applied thus proposition provides following presentation occur multiplicity one middle module richardson rule thus neither occurs therefore neither occurs either see tora therefore surjection since neither occurs see induces surjection since spaces isomorphic representations finite multiplicities surjection isomorphism remark second case proof occur occur follows presentation direct sum decomposition argument proof shows would interesting modules could constructed directly combining proposition proposition obtain following corollary steven sam andrew snowden jerzy weyman corollary tora representations combining shows otherwise representations thus see isomorphic restricted therefore either isomorphic fact isomorphic wen theorem finishes proof main result odd case throughout section dimension space step let set partitions whose frobenius coordinates satisfy equivalently set partitions set admits inductive definition follows empty partition belongs partition belongs number rows columns equal partition obtained deleting first row column belongs let partition write place define regular partitions lemma let partition let partition obtained removing first row column also belongs furthermore let resp element resp partition defined following identities proof suppose applying bott algorithm number moves places left thus first steps algorithm reach sequence bott algorithm sequence runs like algorithm regular belongs suppose algorithm terminates steps algorithm terminates steps following formula since border strip boxes using remark see exists since box row first column hook size furthermore since done proposition unique bijection maps exists case rank rank even rank odd homology littlewood complexes proof proof exactly proposition step let vector space dimension least let partial flag variety quotients ranks thus vector bundles ranks surjections let kernel let kernel put let grassmannian rank quotients let natural map let usual bundles space naturally identified universal rank one quotient let kernel let contains subbundle define exact sequence let admissible partition let otherwise put sym note finally put wish compute lemma sum partitions rank mod proof since schur functors commute pullback projection formula gives result follows proposition lemma let correspond rank otherwise proof follows proposition theorem lemma sum partitions proof follows immediately lemmas proposition lemma pair degenerate sense proof representation criterion lemma applies steven sam andrew snowden jerzy weyman lemma sum partitions particular proof lemma isomorphism representations result follows lemma proposition tora sum partitions proof follows immediately previous lemma proposition step admissible partition put homo note compatible action lemma spaces isomorphic representations finite multiplicities proof let sum admissible partitions enough show isomorphic representations finite multiplicities fact enough show multiplicity spaces isomorphic representations finite multiplicities proof goes exactly lemma proposition let admissible partition isomorphism proof arguing exactly proof proposition obtain surjection course also surjection previous lemma isomorphism isomorphisms combining proposition proposition obtain following corollary corollary tora representations examples give examples illustrate theorem example suppose complex symi differential multiplication symmetric form treated element differential injective irreducible representation differential isomorphism homology vanishes differential surjective trivial representation trivial group homology littlewood complexes example suppose complex differentials multiplication symmetric form treated element differential injective irreducible representation complex exact homology vanishes finally reader check easily instances description homology agrees rule given weyl group action example let consider situation example modification rule using border strips proceeds follows starting remove border strip size obtain partition supposed remove border strip size border strip shaded however upon removing strip young diagram follows homology vanishes weyl group version amounts nontrivial stabilizer transposition swaps first fifth entries stabilizer contains element example suppose border strip algorithm runs follows removed three border strips thus according rule final partition conjugate thus see illustrate modification rule using weyl group action write idea getting weyl group element apply sum first two column lengths big sort result repeat necessary start subtracting result get steven sam andrew snowden jerzy weyman general linear groups representations let vector space dimension irreducible representations indexed pairs partitions see koi details call pairs admissible given admissible pair denote corresponding irreducible representation identifying weights elements representation irreducible highest weight representation usual schur functor representation dual littlewood complex let vector spaces put sym sym let inclusion given first inclusion multiplication identity element inclusion defines algebra homomorphism let quotient ideal generated maps spec spec spec natural identification spec space hom hom pairs maps space spec naturally identified space bilinear forms map spec spec takes pair maps form trace map space spec call littlewood variety fiber map consists pairs maps remark one modify definitions rings replacing dual everywhere space spec identified hom spec identified set pairs maps map spec spec takes space spec consists pairs thus spec space complexes form let koszul complex littlewood variety decompose complex action littlewood complex independent long complex dim dim bry theorem zeroth homology admissible otherwise lemma tora decomposition tora applied obtain admissible homology littlewood complexes special case main theorem main theorem computes homology complex formulate prove theorem particularly simple case mention worthwhile know argument needed prove main theorem proposition suppose admissible otherwise proof choose dim dim lemma either computation higher homology follows given lemma suppose dim dim spanned regular sequence proof suffices show dim spec dim spec dim put dim dim observe locus spec injective surjective open let variety partial flags subscript indicates dimension subspace variety comes tautological partial flag ofl natural birational map total space hom hom spec thus spec dimension dim spec dim result follows modification rule associate pair partitions two quantities usual give two equivalent definitions begin koike definition weyl group definition discussion preceding koi prop first consider sequence obtained taking reversing negating entries add entry call result example note weakly decreasing sequence nonnegative integers makes sense reverse procedure define set let group permutations index set coordinates differ identity permutation finitely many places given permutation define usual one two possibilities occurs exists unique element weakly decreasing case set exists element case put leave undefined give modified equivalent description weyl group action group acts set via action define new involution let subgroup aut generated isomorphic infinite symmetric group comes equipped length function respect set generators given pair partitions exactly one following two possibilities hold exists unique element pair partitions admissible put notation simply means exists element put leave undefined steven sam andrew snowden jerzy weyman always already admissible first case give border strip definition could find literature admissible define assume admissible let resp border strip length starting first box final row resp exists exist partitions put otherwise put leave undefined proposition two definitions agree proof consider diagram complement rectangle allowing possibility case need consider negative column lengths purposes explanation assume according koike rule supposed apply bott algorithm columns union describe two procedures start shape single column length could negative union partition apply bott algorithm end single column length union shape obtained removing border strip size possible otherwise dually start shape union column length apply bott algorithm get shape union column length obtained adding border strip length starting bottom box last column exists otherwise back shape apply column last column last column becomes removed border strip size process used simple reflections apply minus last column result shape obtained adding border strip starting right left length union column length also describe shape follows added border strip starting right size second step used simple reflections finally note adding border strip starting right equivalent removing border strip usual sense main theorem main theorem following theorem pair partitions integer otherwise homology littlewood complexes exact particular remark consider coordinate ring rank matrices identifying dual quotient ideal generated minors resolution known see wey las hand part theorem combined shows appears resolution thus modification rule gives alternative description resolution pleasant combinatorial exercise show directly two descriptions equivalent previous situations modification rule complicated advantage readily generalizes situation proof take rest section follow plan given step fix integers let pair partitions write place place define partitions regular pairs partitions lemma let partition let partition obtained removing first row column also belongs furthermore let resp elements resp partitions defined identities proof reasoning proof lemma see belongs suppose applying bott algorithm resp number resp moves resp places left let resp number steps bott algorithm applied resp proof lemma trace bott algorithm find examine border strips used modification rule boxes using remark see resp exists since box resp row first column hook size furthermore completes proof proposition unique bijection maps exists case steven sam andrew snowden jerzy weyman proof proof essentially proposition step let vector spaces dimensions least respectively let resp grassmannian rank resp quotients resp let resp tautological bundles resp regard four bundles put define exact sequence finally pair partitions put sym note sym lemma let partitions let correspond let vector bundle given otherwise proof formula shows definition regular find partitions theorem get otherwise otherwise proposition completes proof lemma let partitions let integer sum pairs particular left side vanishes proof course need sum result follows lemma homology littlewood complexes proposition let partitions tora sum pairs proof follows immediately lemma proposition lemma module independent choice provided proof applying proposition shows presentation left map unique scalar multiple since occurs multiplicity one middle group lemma follows thus admissible pair step admissible pair put homgl note compatible action goal show isomorphic lemma let admissible pair spaces isomorphic representations finite multiplicities proof done put admissible enough show isomorphic representations finite multiplicities fact enough show multiplicity spaces isomorphic representations finite multiplicities multiplicity space decomposition representation ring computed using koi thm result notation koi computing zeros lead massive simplification general formula given note fixed finitely many values make product coefficients establishes finiteness multiplicities equality tora representation ring applying proposition find tora steven sam andrew snowden jerzy weyman obtain therefore find component given result follows proposition let admissible pair isomorphism equivariant proof proof lemma presentation note occurs multiplicity one middle module thus occur therefore occur either since isomorphic representations since tora see follows surjection since occur see induces surjection since two isomorphic representations finite multiplicities surjection isomorphism combining proposition proposition obtain following corollary combined proves main theorem corollary tora examples give examples illustrate theorem complex example suppose positive differential multiplication identity treated element let dim differential injective irreducible representation highest weight vanishes differential isomorphism homology vanishes identically finally complex reader easily check description homology given agrees rule given weyl group action example suppose supposed remove border strips size partition homology littlewood complexes let border strips picture follows partition left shaded right shaded let unshaded boxes diagrams pair admissible algorithm continues border strips size picture thus removing border strips obtain partitions pair admissible algorithm terminates follows irreducible highest weight illustrate modification rule using koike original weyl group action using notation sort get permutation sorting length made consecutive swaps subtracting get first partition second partition references luchezar avramov herzog koszul algebra codimension embedding math bry ranee kathryn brylinski matrix concomitants mixed tensor model adv math dps elizabeth ivan penkov vera serganova koszul category representations finitary lie algebras thomas enright jeb willenbring hilbert series howe duality branching classical groups ann math william fulton joe harris representation theory first course graduate texts mathematics new york roger howe perspectives invariant theory schur duality actions beyond israel mathematical conference proceedings htw roger howe tan jeb willenbring stable branching rules classical symmetric pairs trans amer math soc arxiv hum james humphreys reflection groups coxeter groups cambridge studies advanced mathematics cambridge university press cambridge jpw pragacz weyman resolutions determinantal varieties tensor complexes associated symmetric antisymmetric matrices young tableaux schur functors algebra geometry soc math france paris kin king modification rules products irreducible representations unitary orthogonal symplectic groups mathematical phys koi kazuhiko koike decomposition tensor products representations classical groups means universal characters adv math kazuhiko koike itaru terada methods representation theory classical groups type algebra las alain lascoux syzygies des adv math steven sam andrew snowden jerzy weyman lit mac sun wen wey dudley littlewood theory group characters matrix representations groups reprint second edition ams chelsea publishing providence macdonald symmetric functions hall polynomials second edition oxford mathematical monographs oxford steven sam andrew snowden modules polynomial rings infinitely many variables steven sam andrew snowden stability patterns representation theory preparation steven sam jerzy weyman koszul homology codimension gorenstein ideals proc amer math appear sheila sundaram tableaux representation theory classical lie groups invariant theory tableaux minneapolis ima vol math springer new york hans wenzl quotients representation rings represent theory jerzy weyman cohomology vector bundles syzygies cambridge university press cambridge department mathematics university california berkeley address svs department mathematics mit cambridge address asnowden department mathematics northeastern university boston address
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insulin regimen control patients oct mark shifrin hava siegelmann october abstract model individual patient blood glucose level bgl stochastic process discrete number states mainly solely governed medication regimen insulin injections bgl states change otherwise according various physiological triggers render stochastic statistically unknown yet assumed nature process order express incentive desired healthy bgl heuristically define reward function returns positive values desirable levels negative values undesirable levels state space consists sufficient number states order allow memoryless assumption turn allows formulate markov decision process mdp objective maximize total reward summarized long run probability law found reinforcement learning optimal insulin treatment policy retrieved mdp solution introduction diabetes mellitus type makes diabetes cases increasing rates since early goal therapy bring average blood glucose close normal range possible done various oral medications insulin injections exact dosage frequency medication intakes coordinated blood glucose measurements normally performed designated glucose meters see recent recommendations patients advised measure blood glucose level bgl least times day work address individually adopted insulin regimen control purpose harness tools area machine learning stochastic control theory particular build markov decision process blood glucose level solution defined mdp expressed policy assigns action according measured bgl note pose contradiction known methods employed medical care theory agnostic physiological processes biochemical insulin impacts renal hepatic effects insulin resistance effect possibly malfunctioning processes inside impairments insulin pathways merely taking different approach modeling insulin treatment controllable stochastic process contrary known including works exploit mathematical biological models express chemical dependencies one difficulties related insulin regimen control associated short ultradian insulin secretion peaks recent time well understood bgl known increase meal intake decrease fasting periods physical activity however peaks render bgl vary according pattern hard mathematically characterize particular patients insulin resistance especially progressed abnormal insulin oscillations first phase phase insulin secretion even important oscillations may exactly follow glucose oscillations like known healthy people order characterize bgl state patient assume continuous bounded axes perform discretization dividing entire region finite states example see state refers continuous region region assigned corresponding state state augmented additional individually measured parameters particular may also account glucose absorption rate gat glycosylated haemoglobin values patient bgl hence viewed stochastic process varies according physiological dynamics according patient activities process controllable subject control applied means medications insulin injections finesse discretization set way assure next state solely depend previous state action taken state bgl process clearly continuously fluctuating time measurements performed discrete time slots hence view state patient certain time marks interested transition law transition probabilities one state time mark next state patient sees feels right next measurement occurs time mark law expressed transition probability state associated corresponding bgl given set actions taken patient state next state aforementioned state transition probabilities learned reinforcement learning input data taken existing samples measurements individually performed effect medication applied physician administration assume samples exist individual patient alternatively individual assigned category patients initial insulin policy already defined policy improved henceforth individually adjusted applying reinforcement learning individual measurements document show results database open purposes academic usage methods already addressed treatment type diabetes mellitis paper proposes linearizion mdp solution model based high dependence insulin glucose levels however assumption necessarily true treatment facilitated frequent every several minutes measurements actions measurements normally performed several times day makes control appropriate candidate stochastic modeling discrete mdp discrete actions control prediction bgl addressed references therein giving stronger emphasis neural network based model presented aimed predict bgl patients training performed patients purpose convergence work crucial learning part performed using individual data objective individually adjust patient authors model suggest convergence sample data patients may cause bias dependency specific data note use data samples various patients research phases algorithm presented paper adopting specific patient fulfilled phase control wiener modeling prediction done previous works apply context diabetes employ great deal system parameters especially helpful prediction approach designed completely hide complex impact multiple parameters behind stochastic process bgl consider main indication insulin control purpose allow assumption bgls stochastic process embodies impact relevant physical parameters including extremely simplified cases carbohydrates intakes physical activities hence rely learning statistical properties process using states actions transition laws formulate markov decision process mdp aims maximize average reward time purpose heuristically define mapping transfers bgl state reward positive reward obtained healthiest states negative rewards obtained undesired levels also introduce components aim fine dangerous states overly high bgl hypoglycemia formally define mdp describe algorithm next section probabilistic model start defining state space next define reward function reward functional action space definition spaces measurement space consider blood glucose interval glmin glmax lower bound means extremely low bgl possibly corresponding hypoglycemia upper bound means extremely high level glucose blood bgl finesse coefficient bfc defines size interval mapped single state example glmin lowest interval corresponds lowest state set possible bgl forms space glmax glmin denote allow augment state space additional measured parameters glucose absorption rate gat similarly bgl perform discretization gat defining bounds gat axes gamin gamax gat finesse coefficient gfc used order define gamax gamin measurement space defines set possible measurements patient case bgl gat measured times simplest scenario component measurement space bgl merely daytime space differentiate bgl process instantiations day different measurement points points expressed daytime space denoted number measurements per day simplicity assume number constant example consider approximately constant measurement timings corresponding morning noon afternoon evening night morning noon afternoon evening night detailed time denoted couple sense represents date represents time day corresponding one measurements absolute measurement points counted denoting consider carbohydrates intake history space carbohydrates finesse coefficient cfc used order define quantity carbohydrates consumed time mark patient chmax chmin order wisely incorporate carbohydrates intake history cih assume last intakes carbohydrates relevant insulin treatment measurement point care measurement range measurements older points presumed longer impact individual cih time denoted chi see example time noon noon noon morning night evening afternoon ternoon noon morning night structure allows impact past carbohydrate intakes one hand preserves memoryless property hand depends physical activity history space assume physical activities prolonged effect similarly cih measurements energy spent given activity done metabolic equivalents mets see table translation activity mets physical activity finesse coefficient pfc used order define number mets corresponding energy spent individual measurement period two consecutive measurements phmax phmin pfc define physical activity history pah denote time set phi size relevant history patient activity space define patient activity space consider reason differ space measurement space summarized follows directly known patient foreseen conducted activities contrary assumed purely stochastic measured insulin medication applied order control bgl may effect dosage aim apply optimal control given optionally given seen secondary means control however maintaining certain level activities part insulin policy see definition overall state space definition ready define state space state denote denote state absolute measurement point state consists bgl measurements according definition patient activities history according definition daytime according definition note simplest scenario component measurement space bgl patient activities accounted merely also note even though already carries information still sometimes explicitly specify clarity reward function denote optimal healthy bgl glh define reward function maps bgl given time value quantifies quality healthiness patient bgl linear reward component normalized given glh gll define subset dangerous states lhyp close hypoglycemia lhyp normally small defines number states critically low bgl define reward component fines close hypoglycemia igl gllc gllc gllc standard indicator function joint instantaneous reward given action space action space consists set possible insulin related medical actions allow finite number treatments particular known insulin preparations insulin preparations denote sets possible dosages corresponding insulin medications note first values spaces equal meaning mediation type given long medications administered concurrently assume action space given state action chosen time mark triplet transition probabilities transitions states occurs every absolute measurement point according transition probabilities given probability getting next state given previous state action taken clearly summation possible states clearly probabilities unknown priori hence objective statistically learn using patient measurements previously applied insulin regime regime could applied according doctor purely medical considerations probabilities learned new optimal insulin policy derived total reward functional define total discounted reward positive discount factor discount factor two important interpretations first one follows known analytical property allows sum converge second one advocates reasoning feeling better important feeling better distant future therefore bgl rewards distant time marks discounted geometrically decreasing discount multiplicative alternative approach values equally feeling throughout day employs following definition mdp solution bellman equation write equation given initial state follows denote policy selects action state state set time mark according transition probability state given action chosen according define value function given initial state max indicates optimal total reward obtained bgl throughout time starting bgl use next dynamic programming principle expand follows applying value function gives bellman equation see max denote vector bellman equation solved value function result mapping state space space total discounted reward complete metric space denote equipped suitable metric kva observe written operator result see operator strict contraction kva therefore fixed point see theorem fixed point constitutes unique solution bellman equation reasoning means found method value iteration see iteratively applied starting initial values till arbitrarily close convergence fixed point algorithm formulation observe solved using transition probabilities defined previous section probabilities constitute dynamics stochastic process bgl response actions insulin preparations straightforward approach calculate mere statistical learning existing vector samples measurements bgl paired corresponding actions formulate complete insulin optimization policy algorithm next see figure complete formulation phase initialization vector statistics counts occurrences state acting applying passing next state phase comprises learning phase acting done initial policy normally taken samples stems doctor heuristic consideration treating individual patient output phase transition probabilities probabilities exploited phase perform value iteration retrieve optimal policy number iterations depends size vector heuristically set order achieve desired level convergence finally phase exploitation part exploits latest found optimal policy phase keeps tracking transition occurrences updates policy every transitions final phase reflects possible changes bgl process properties time optimal policy result section use data order demonstrate activation algorithm described previous section examples presented section solely based short vectors several hundreds measurements bgl anything else carbohydrates intakes physical activities presumed embodied bgl process clearly obtained policy presumable outcome learning done vectors bgl simply denoted glucose graphs scale presented starting minimal measurement ending maximal bgl measurement insulin optimization policy algorithm initialization initialize statistical data triplets initialize value function initial states learning training sequence length set visit see action taken state next state update increment calculate value function policy calculation use perform value iteration till desired level convergence apply max operator state find best action set argmaxa optimal policy policy tracking calibration initialize set visit see action taken state next state update increment calculate new use new perform value iteration update new points bgl scale policy policy indicated reason stems limited data size particular action outcome learned case occurrence action specific bgl ever recorded reason preliminary policy carefully selected best anticipated practice would assessment particular individual assigning specific category policies successful assignment done correction expected effective even limited period individualized measurements offered elaborate future work section future work future work aimed two directions producing set categories individuals individuals would assigned finite number policy sets set contain patients resembling clearly similar insulin regime figure policy example measurement points figure policy example measurement points patients may also resembling parameters body weight age etc categorizing process done heuristic assignment correlation study second direction relates tot pure approach directed towards coping size purpose approach approximate mdp amdp introduced see methods described chapter references bertsekas dynamic programming optimal control volume athena scientific belmont daskalaki diem mougiakakou machine learning biomedicine feasibility study type diabetes plos one georga protopappas fotiadis glucose prediction type type diabetic patients using data driven techniques applications data mining intech harvard school public health measuring physical activity https holt cockram flyvbjerg goldstein textbook diabetes john wiley sons blood glucose joslin diabetes center chart http lichman university california irvine school information computer sciences uci machine learning repository https mayo clinic blood sugar testing http moscou getting word advocacy social marketing policy development enforcement public health nursing page powell approximate dynamic programming solving curses dimensionality volume john wiley sons reed simon methods modern mathematical physics functional analysis volume gulf professional publishing ripsin kang urban management blood glucose type diabetes mellitus fam physician rollins bhandari kleinedler kotz strohbehn boland murphy andre vyas welk inferential modeling blood glucose level using noninvasive inputs journal process control national library medicine blood glucose https world health organization diabetes facts list https http zitar towards neural network model informatica
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deformations fundamental group representations earthquakes surface groups sep son lam bstract article construct type deformations representations arbitrary lie group large class manifolds including cat manifolds deformations defined based codimension hypersurfaces certain conditions also disjoint union hypersurfaces show commutativity deforming along disjoint hypersurfaces application consider anosov surface groups show construction extended continuously measured laminations thus obtaining earthquake deformations surface groups ntroduction twist one fundamental tools studying deformation space hyperbolic surface equivalently teichmueller space geometrically described cutting surface along simple closed geodesic twist length glue back however deformations also described algebraically perhaps naturally point view paper define type algebraic deformations representations arbitrary lie group manifold contractible universal cover turns many geometric deformations special cases construction consider holonomy representation example bending quasifuchsian surface group defined thurston bending deformation convex real projective surface goldman etc interestingly starting representation deform need discrete construction results let oriented surface genus directed simple closed curve case interest even though construction applies generally aspherical manifolds higher dimensions aspherical hypersurfaces let faithful suppose centralizer assign lift universal cover transformation cyclic subgroup preserve deck transformation assignment also satisfy condition equivariance means choice determines rest fix base construction informally described follows given point consider directed path image deck transformations path cross ordered collection lifts define new function mathematics subject classification key words phrases cat manifold deformation surface group quasifuchsian hyperbolic convex cocompact earthquakes bending son lam intersection sign path fact result depend particular path long path intersects transversely show resulting map indeed homomorphism thus example holonomy representation hyperbolic surface choose inside group hyperbolic transformations obtain family representations corresponding twists section generalize construction surfaces multihypersurfaces manifolds also prove commutativity result theorem state let union disjoint hypersurfaces satisfying generic conditions suppose component lifts chosen equivariantly deformations defined theorem theorem implies following important corollaries deformations along hypersurface flow deformations along disjoint hypersurfaces commute example given surface group bending along curve twisting along another disjoint curve commutative construction theorem presented section large part paper section switch attention case surface groups isom study limit deformations along weighted simple closed curves approach measured lamination let surface genus thurston introduced space measured laminations thought completion space weighted simple closed fact dense classical earthquake deformation defined limit twists sequence approach measured lamination aim generalize method anosov surface groups order define earthquakes showed surface groups anosov equivalent convex cocompact see also work bowditch state reduced version theorem theorem let finitely generated word hyperbolic group real semisimple lie group real rank representation following equivalent anosov exists continuous injective map iii ker finite convex cocompact indeed since closed surface coincide notion group condition implies exists continuous whose image limit set injective map moreover purely loxodromic element attracting repelling fixed point centralizer element must contain group hyperbolic transformations pair fixed points use groups define twist deformations along weighted simple closed curves weight determines translation length pass simple closed curves measured laminations use geodesic currents view measured laminations thus measured lamination measure convergence weighted simple closed curves deformations representations earthquakes surface groups together homeoto measured lamination becomes convergence measures show following convergence result morphism theorem corollary anosov surface group representation neighborhood two weighted simple closed curves corresponding representations close means simply define eli whenever continuous map space representations near direct generalization classical earthquake hyperbolic surfaces unlike dimension simple dimension count implies earthquakes take nearby points moduli space implications directions mcmullen defined notion complex earthquake starting fuchsian group combines classical earthquake thurston deformation deform original group become quasifuchsian theorem corollary implies starting quasifuchsian along measured lamination earthquake thurston bending defined commutative thus combine deformations complex parameters like one ask whether possible connect quasifuchsian surface groups operations thurston showed starting fuchsian group theorem thurston projective grafting map homeomorphism teichmuller space hyperbolic structures whose holonomy fuchsian space structures whose holonomy include quasifuchsian groups anosov surface groups form open subset moduli space hom representations conjugations general whole component space interesting question whether earthquake paths outside closure another interesting direction question coordinates surface groups particular tan kourouniotis constructed complex fenchelnielsen coordinates case situation complicated since cases bending parameter rotation abelian moreover dimension count may dimensions internal parameter decomposition parameters specify arrangement rotation axes work done study groups dimension whole picture remains mysterious acknowledgement would like thank schlenker virginie charette mentors writing article thank advisor bill goldman whose works inspiration eformations along hypersurfaces construction manifold said aspherical universal cover tractible equivalently rest section let connected aspherical manifold possibly boundary dimension least finitely generated fundamental group let connected properly embedded aspherical hypersurface injective son lam let representation arbitrary lie group define deformations along hypersurface important example hyperbolic surface simple closed geodesic generally totally geodesic hypersurface hyperbolic manifold cases construction corresponds bending denote preimage universal cover disjoint union let countable collection connected components copy contractible codimension two components common boundary definition define dual tree tree whose vertices connected edges adjacency vertices edges corresponds components adjacency components vertex tree possibly infinite number edges attached deck transformation induce action transitive free set vertices action set edges corresponding base point choose base point right action deck transformations notation act define algebraic twist deformation representation let base point choose path path concatenation induces injective homomorphism whose image subgroup component path lifts uniquely path label component action preserves subgroup preserves component following crucial point construction define map following properties centralizer rest article reserved denote map note property implies property since want check centralizer satisfying exists claim choosing centralizer uniquely determines suppose use property define note unique since chose centralizer depending different choices suppose let thus deformations representations earthquakes surface groups property therefore satisfies ready define twist deformation representation respect first choose orientation induces orientation definition let defined given minimal path direction cross sequence hypersurfaces name nak order define nak equals intersection sign directed path say deformation remark note necessary path minimal since overlapping would cancelled expression tree structure intersection sign path depends chosen orientation dual definition obtained changing orientation changing orientation equivalent switching left right twist case surface groups definition map minimal path induces minimal path dual tree unique sequence lines nak uniquely determined following proposition shows homomorphism proposition let defined therefore homomorphism proof use notations section let nap sequence hypersurfaces nbq sequence hypersurfaces let ncr sequence hypersurfaces suppose definition gives nai nbi nci signs determined corresponding intersection signs indeed elements associated hypersurfaces nbq elements associated hypersurfaces nap hand elements associated sequence hypersurfaces consider tree structure dual tree see figure indeed difference minimal path son lam igure concatenation minimal paths segment back tracking along segment crosses let say hypersurfaces must cancellation therefore similar argument shows case simple closed curves surfaces following proposition let closed surface genus let disjoint simple closed curves distinct homotopy classes let representation centralizers deformations constructed definition along distinct proof cut surface along obtain surface boundary still contain choose base point component containing separating curve note representation changes conjugation base point changes choose represented curve intersecting touching boundary thus thus deformation along section define deformations along set disjoint hypersurfaces generalization deforming surface group along multicurves theorem commutativity deformations shown deformations representations earthquakes surface groups let collection mutually disjoint connected hypersurfaces satisfies properties previous section moreover require pair homotopic let collection components subgroup preserving let suppose order indeed determined following definition proposition straight forward generalization previous section definition let defined given direction cross sequence hypersurfaces minimal path name lak order define lak equals intersection sign directed path proposition deformation along multiple disjoint hypersurfaces gives homomorphism need following discussion prove commutativity deformations let different base points choose path determine lift deform two different ways first using definition base point resulting second way deforming using base point resulting representation obtained using base point follows let use possibly path obtained going remark see taking path change deformation result show difference conjugating transformation lemma let minimal path cross intersection sign proof hypersurfaces order let lar hypersurfaces path crosses following hypersurfaces lar using property get lai sai son lam sai lai get result lemma let hypersurface subgroup preserving suppose minimal path crosses order including proof choose near minimal path cross adjacent closest still component path independence definia means tion applying lemma get following theorem implies commutativity deforming along disjoint hypersurfaces also shows choosing values subgroup centralizer induces flow path representations theorem let maps satisfying proof note left hand side different one right starting representations different distinction required write right hand side left hand side similarly equation written unambiguously note yet define since lemma transformation naturally define similarly relate way let let hypersurfaces subgroup preserving since lemma moreoever definition get analogous formula shown together commutativity assumption theorem follows corollary deformations along disjoint hypersurfaces commute deformations representations earthquakes surface groups infinitesimal deformation let lie algebra let definition suppose analytic path exp suppose let called derivative may also write notation previous section suppose subgroup etx contained centralizer thus family choices write etx applying construction get differentiable family deformations representation hom composition adjoint representation gives aut infinitesimal deformation given see let simplicity notation suppose family deformations previous section given etxi group course collection depends let exp words definition smooth deformation map defined called infinitesimal deformation corresponding indeed corresponding family representations exp etxk formula coefficients higher order right hand side include various combinations nested lie bracket therefore considering linear coefficients get remark lgebraic earthquake along measured laminations switch attention case closed hyperbolic surface lie group isom lie algebra rest section let anosov representation section define twist deformations along weighted simple closed curves earthquakes along measured laminations theorem let finitely generated word hyperbolic group real semisimple lie group real rank representation following equivalent anosov exists continuous injective map iii ker finite convex cocompact considering rank word hyperbolic use condition definition anosov representations case thus homeomorphism boundary infinity limit set rest article reserved denote limit set map son lam following measured laminations let view measure acts swapping two coordinates indeed space unoriented geodesic also considering geodesic arc define measure space geodesics intersecting transversely support contained inside interior compact set disjoint closed intervals indeed separated neighborhoods end points extended geodesic containing sequence converges converges weak topology space measures given anosov representation recall homeomorphism gives particular thus measure pushes forward measure particular induces measure deforming anosov representations along weighted simple closed curves suppose weighted simple closed geodesic anosov representation define right twist along following construction section let base point lying lift choose orientation orientation way construct turn orientation matter notation let hyperbolic transformation loxodromic without rotation fixing translation length direction repelling attracting fixed point preserve geodesic whose translate along geodesic distance suppose subgroup preserving let lift universal cover deck transform action infinite cyclic group moreover properties anosov representations loxodromic thus written uniquely composition hyperbolic transformation direction elliptic transformation among fixed points important note choose generator respectively starting ending point directed infinite geodesic commutes conjugation acts scalar multiplication acts rotation choose weight indeed commutes required condition section thus choice gives lift note also respectively starting ending points following definition constructed family algebraic twist deformations definition let elt family representations obtained deforming along weighted simple closed curve choosing simply write considering boundary infinity mapped homeomorphically limit set picture deformations similar classical twist deformations representations earthquakes surface groups igure respectively deformation hyperbolic surface let let geodesic arc definition elt let indeed end points lift changed orientation signs would flipped would also invert choice hence thus would keep deformation show repelling fixed points one side attracting fixed points otherside let endpoints geodesic extended union two disjoint open intervals name intervals geodesic intersects positive intersection point refer definition starting point makes starting point remark proof convergence rest section let goal prove sequence weighted simple closed curves converges eli converges limit defined approach proof convergence loosely follow kerckhoff notation point metric space denote open ball radius centered notation subset topological space denote closure notation let set denote son lam set unordered distinct pairs points notation let note remark let stereographic projection coordinate let varies analytically vary map analytic represented either also analytic point work chosen positive definite inner product corresponding left invariant metric standard metric induces product metric lemma let fixed constant let depending continuously ktv also depends continuously proof let ensures since analytic locally lipschitz thus lipschitz compact subsets particular let infimum lipschitz constant compact set indeed sup depends continuously let let scale sides get first inequality inverse exponential map neighborhood analytic bijective lipschitz compact subsets particular set let infimum lipschitz constants inverse exponential map depends continuously fixed ktv let notation let geodesic path let path purpose constant lemma following discussion section choice base point determines compact set anosov representation determines limit set homeomorphism deformations representations earthquakes surface groups lemma compact set exists constant weights cdg proof map analytic respect reasonable coordinates result follows locally lipschitz argument compactness domain note depends lemma constant depending real positive weights proof first note compactness locally lipschitz argument exists constant depending let let constant max compact subset distance using lemma get therefore triangle inequality replacing terms product one one right depends theorem given anosov exists neighborhood two weighted simple closed curves proof recall notation eli representation obtained algebraic deformation along weighted simple closed curve notation abbreviate induced measure lemma continuous function compact set reaches minimum value min lemma depends continuously centers balls considered let kmax constant works compact set following proof lemma let compact subset contains transformation form long let constant lemma works son lam choose also kmax reason choosing apparent end compact set partitioned disjoint intervals interval diameter less particular interval lie inside open ball radius center point may use topologically intervals ensure disjoint partitioned squares name notation square let center since finite number geodesic leaf positive measure assume continuity set respect boundaries choose neighborhood containing measured laminations measure measure close every square choose constant lemma note depends turn depends let weighted simple closed curves weights respectively close since simple curve lifts disjoint suppose contains endpoints lift contain endpoints lift either words contain endpoints lifts either squares containing endpoints lifts complete ordering set squares statements made geodesic lamination let ordered set squares contain endpoints lifts similarly define set set also complete ordering compatible note similarly lemma following order products determined ordering therefore deformations representations earthquakes surface groups choice show close similarly order pairs end points corresponding lifts intersect geodesic path let weight hand recall thep compact set applying lemma estimate distance replacing terms one one right time adding distance kmax analogous inequality holds well therefore following choose set generators say two representations close induced topology space representations independent choice generating set following corollary applying theorem generator taking finite intersection open neighborhoods corollary anosov surface group representation neighborhood two weighted simple closed curves corresponding representations close implies sequence weighted simple closed curves converging corresponding sequence representations eli cauchy thus converges moreover limit sequence depend choice sequence converging define eli also representation thus continuous map connected components representations near son lam eferences aramayona leininger hyperbolic structures surfaces geodesic currents lecture notes bonahon geometry space via geodesic currents invent math bonahon shearing hyperbolic surfaces bending pleated surfaces thurstons symplectic form ann fac sci toulouse math bowditch geometrical finiteness variable negative curvature duke math goldman symplectic nature fundamental groups surfaces adv math goldman convex projective structures compact surfaces differential geom guichard wienhard anosov representations domains discontinuity applications invent math kamishima tan deformation spaces geometric structures aspects manifolds kinokuniya tokyo kapovich kleinian groups higher dimensions progress math vol kapovich leeb porti lectures anosov representations dynamical geometric characterizations preprint kerckhoff nielsen realization problem annals math kourouniotis complex length coordinates quasifuchsian groups mathematika johnson millson deformation spaces associated compact hyperbolic manifolds discrete groups geometry analysis vol progress mcmullen complex earthquakes teichmuller theory amer math tan complex coordinates structures internat math tan wong zhang generalized formulas oriented augmented rightangled hexagons hyperbolic adv math thurston geometry topology princeton university lecture notes online http epartment athematics niversit herbrooke herbrooke anada
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nonparametric intensity estimation indirect point process observations unknown error distribution martin mar january synopsis consider nonparametric estimation intensity function poisson point process circular model indirect observations observations emerge hidden point process realizations target intensity contamination additive error assumption error distribution unknown available means additional sample derive minimax rates convergence respect sample sizes abstract smoothness conditions propose orthonormal series estimator attains optimal rate convergence performance estimator depends correct specification dimension parameter whose optimal choice relies smoothness characteristics intensity error density since priori knowledge characteristics strong assumption propose choice dimension parameter based model selection show adaptive estimator either attains minimax optimal rate suboptimal logarithmic factor key words phrases poisson point process intensity function nonparametric estimation minimax theory adaptive estimation statistical inverse problem ams subject classification introduction point process models used wide variety applications including amongst others stochastic geometry extreme value theory queueing theory realization point process random set points alternatively represented random measure denotes dirac measure concentrated poisson point processes ppps particular importance since serve elementary building blocks complex point process models let locally compact second countable hausdorff space corresponding borel locally finite measure measurable space relatively compact sets random set points resp random measure called poisson point process intensity measure number points located follows poisson distribution parameter relatively compact mannheim institut mathematik mannheim germany financial support deutsche forschungsgemeinschaft dfg research training group rtg gratefully acknowledged indebted jan johannes martin schlather helpful comments paper disjoint sets random variables nan independent distribution ppp completely determined intensity measure hence statistical point view estimation intensity measure derivative intensity function respect dominating measure observations point process fundamental importance inference testing problems poisson general point processes tackled wide range scenarios monographs offer comprehensive overview discuss parametric nonparametric methods methodological point view approach paper related article estimation intensity function direct observations considered present paper analysis adaptive estimator based appropriate concentration inequalities poisson point processes whereas concentration inequalities developed applied represent analogues results random variables due one main tool approach analogues concentration inequalities due derived separate manuscript approaches nonparametric intensity estimation direct observations without making claim exhaustive found performance histogram estimator hellinger loss analysed using testing approach model selection using minimum complexity estimator aalen model suggesting wavelet estimator multiplicative intensity model theoretical work intensity estimation recently motivated applications genomic data model considered article motivated data arising throughout processing dna data article takes motivation analysis genomic data well paper author studies nonparametric inference reproduction function one realization aggregated point process additional observations related model observations paper type ones consider although estimation problem also ascribed area intensity estimation indirect observations addition let mention two articles development nonparametric statistical methods analysis point processes inspired applications biology first motivated dna sequencing techniques article introduces model nonhomogeneous poisson processes occurring molecular biology second article considered nonparametric inference cox process data means kernel type estimator usually one aims estimate intensity function direct observations realizations ppp target intensity paper however assume interested nonparametric estimation intensity function without access observations instead setup poisson inverse problem observe given identity yij xij indirect observations related hidden definition yij fractional part additively contaminated xij yields circular model means usual topological identification interval circle perimeter assumptions concerning intensity errors discussed detail note circular definition general model convenient model case periodic intensity functions particular importance applications periodic intensity functions suitable model occurence events subject natural temporal period day week instance financial transactions gun crimes refer interested reader references concerning wide range applications model closely related circular deconvolution models circular deconvolution problem cdp one aims estimating density random variable values repeated observations error density cdp analogue extension real line observations treated wide range research articles comprehensive introduction subject note contrast approach existing papers poisson inverse problems assume error distribution known conservative assumption also present research literature dealing circular deconvolution problems error density unknown even identifiability statistical model guaranteed thus several remedies introduced overcome problem instance possible impose additional assumptions statistical model instance article deals blind convolution additive centred gaussian noise unknown variance alternatively one consider framework panel data finally one assume availability additional sample error density instance guarantee identifiability enable inference paper stick last option let assume errors contributing general model stationary sense unknown error density assumption two models particular interest errors case referred poisson model throughout paper errors coincide case referred cox model names poisson resp cox model justified note classical circular deconvolution problem differentiation distinct models point process case possible study nonparametric estimation intensity function poisson cox model observations given natural aim detect optimal rates convergence terms sample sizes construct adaptive estimators attaining rates remarked cox model already studied intensively authors paper exclusively consider case error density known making sample needless ordinary smooth fourier coefficients polynomially decreasing note investigation paper allows consider error densities faster rates decay stated assumptions article provides remarkable proof minimax lower bound besov spaces suggests wavelet series estimator automatically adapts unknown smoothness attains rate optimal logarithmic factor moreover authors show theorems impossible construct consistent estimator intensity basis preceding alignment step methodological point view approach inspired one conducted contrast paper proof minimax lower bound setup deal different nature observations key argument hellinger distance poisson point processes bounded hellinger distance corresponding intensity measures consider orthonormal series estimators form appropriate estimator fourier coefficient exp corresponding basis function see section details course estimator motivated representation crucially depends choice turn performance estimator dimension parameter optimal value depends smoothness characteristics intensity usually available practice order choose completely datadriven manner follow approach based model selection instance select dimension parameter minimizer penalized contrast criterion pen argmin knm depending samples sizes random observations determines set admissible models contrast function penk penalty term specified detail choice penalty term random already partially adaptive case one assumes smoothness intensity unknown smoothness error density known theoretical analysis adaptive estimator need talagrand type concentration inequalities tailored framework ppp observations directly transferred results applied usual density estimation deconvolution frameworks remark already mentioned inequalities already derived separate manuscript specific results needed application paper stated appendix article organized follows section introduce methodological approach section consider nonparametric estimation problems minimax point view section considers adaptive estimation intensity poisson model whereas section deals adaptive estimation cox model methodology notation throughout work assume intensity density belong space functions interval let complex trigonometric basis given exp fourier coefficients function denoted follows strictly positive symmetric sequence introduce weighted norm defined via corresponding scalar product denoted throughout paper use notation numerical constant independent minimax point view means mean inwe evaluate performance arbitrary estimator take minimax point view tegrated weighted squared loss consider maximum risk defined sup sup classes potential intensity functions densities respectively note case known error density one would consider supremum fixed error density one able show upper risk bound holds uniformly belonging class one obtains upper bound maximum risk order show minimax lower bounds terms sample size turn sufficient reduce class even subset two elements see proof theorem minimax risk defined via inf sup sup estimator called rate optimal infimum taken estimators sup sup inf sup sup classes intensity functions densities considered article specified section derive lower bounds minimax risk specific choices prove lower bound attained numerical constant suitably defined orthonormal series estimator sequence space representation poisson model orthogonal series estimator introduce next subsection crucially motivated sequence space representation considered models representation forms fourier coefficients point origin definition estimators address derivation sequence space model poisson model observed point processes generated intensity function independent independent poisson point processes random contaminations individual points emphasize unobserved contaminations assumed follow probability law given unknown density understood additively modulo thus observations poisson model given realization poisson point process intensity function errors note poisson model realization poisson point process whose intensity function given circular convolution modulo error density precisely given formula convolution theorem campbell theorem chapter theorem deduced measurable functions dni provided integral side exists exploiting equation setting dni precisely thus obtain centred random variables dni sequence space representation cox model cox model permits sequence space representation similar one poisson model obtained cox model independent point processes generated random contamination individual points poisson model contrast unobservable poisson point processes poisson model however additive error points one realization observations given realization poisson point process intensity function alternatively generation point processes described following procedure first step random shifts generated second step conditionally drawn realization poisson point process whose intensity function thus second model observations follow distribution cox process directed random measure random intensity note integrable functions dni implies dni denotes circular convolution function density thus mean measure generic realization obeying cox model radonnikodym derivative respect lebesgue measure note mean measures observed point processes poisson cox model coincide observations defined cox model stem cox instead poisson process relation holds model thus get following representation sequence space dni centred random variables connection sequence space model hand stated sequence space model formulation statistical linear inverse problems discussed detail section orthonormal series estimator recall represented fourier series representation hence view natural estimator given defined note directly available thus estimated sample observed contrast setup cox model note substituted assume density known hence expectation suggested since additional sample hand natural idea seems substitute leads estimator poisson model additional threshold occurring indicator function set compensates small definition absolute values imposed order avoid unstable behaviour estimator definition event accordance chosen threshold corresponds parametric rate estimated sample optimal choice dimension parameter minimax framework determined section completely depends classes thus estimator unacceptable applications degree smoothness functions usually available advance choice dimension parameter discussed sections minimax theory model assumptions let strictly positive symmetric sequences fix section derive minimax rates convergence concerning maximum risk defined respect classes intensity functions error densities respectively regularity conditions imposed sequences summarized following assumption assumption strictly positive symmetric sequences sequences finally note special case weighted norm coincides usual additional assumption contained requirement sequence minimax lower bounds poisson model first derive lower bounds terms sample sizes poisson model first theorem provides lower bound terms sample size mild assumptions theorem set argmin max max let assumption hold assume inf min inf sup sup min infimum taken estimators based observations poisson model proceeding lower bound let state remarks concerning theorem firstly follows immediately proof lower bound holds already case known error density secondly assuming convergence series condition necessary order establish hypotheses considered proof finally noteworthy proof theorem adapted case direct observations poisson point processes intensity function theorem corresponding lower bound obtained replacing remark key ingredient proof theorem fact hellinger distance two ppps bounded hellinger distance corresponding intensity measures since relation holds special case poisson processes directly extended cox model given proof fails thus derivation minimax lower bounds based fourier expansions cox model framework remains open needs addressed future research note give lower bound proof case known ordinary smooth error density following theorem provides lower bound terms sample size theorem set max min let assumption hold addition assume exists density inf sup sup based infimum taken estimators min observations poisson model let state remarks concerning theorem first proof theorem sufficient construct two hypotheses statistically indistinguishable already establish lower bound contrast proof theorem construct hypotheses second condition imposed order guarantee hypotheses considered proof belong easy check condition satisfied satisfies max remark proof theorem transferred cox model neither proof equality would imply equality mean measures two cox process hypotheses equality distributions holds true ppps next corollary immediate consequence theorems corollary assumptions theorems inf sup sup upper bound let establish upper bound maximum risk terms estimator suitable choice dimension parameter precisely following defined theorem establishes upper bound rate convergence poisson cox model analysis proofs indicates slightly different numerical constant thus due lower bound proofs preceding subsection shown attains minimax rates convergence terms samples sizes poisson model note optimal choice dimension parameter depend sample size theorem let assumption hold assume samples drawn accordance poisson cox model sup sup restrictions pol pol exp pol log pol exp log log exp exp log log table exemplary rates convergence nonparametric intensity estimation rates given framework theorems impose given restrictions examples whereas choices pol exp sequences explained section examples convergence rates order flesh abstract results section consider special choices sequences state resulting rates convergence respect sample sizes sequence assume throughout resulting weighted norm corresponds usual sth weak derivative choices sequence narios concerning sequence distinguish following two pol corresponds case unknown intensity function belongs sobolev space exp exp case belongs space analytic functions choices sequence concerning sequence consider following scenarios pol corresponds case error density ordinary smooth exp exp table summarises rates corresponding different choices rates respect coincide classical rates nonparametric inverse problems see instance table error variance corresponds setup case considered adaptive estimation poisson model estimator considered theorem obtained specializing orthonormal series estimator defined thus procedure suffers apparent drawback depends smoothness characteristics namely sequences since characteristics typically unavailable advance need adaptive selection dimension parameter require priori knowledge order reach adaptive definition poisson model cox model dealt section proceed two steps first step treated subsection assume class unknown assume class potential error densities known assumption allows define partially adaptive choice second step treated subsection dispense knowledge smoothness propose fully adaptive choice dimension parameter partially adaptive estimation unknown known subsection aim choosing equal contrast section longer depend sequence sequence definition terminology introduced let max log log setting define log inf inf log consider contrast set knm via define random sequence penalties pen pen additional tuning parameter parameter finds way upper risk bound numerical constant effect rate convergence dependence adaptive estimator specific choice suppressed sake convenience sequel building definition contrast penalty define partially adaptive selection dimension parameter pen argmin following theorem provides upper bound partially adaptive estimator theorem let assumption hold sup sup min max observations stem poisson model fully adaptive estimation unknown also dispense knowledge smoothness error density propose fully selection dimension parameter resulting estimator adapts unknown smoothness attains optimal rate convergence wide range scenarios case partially adaptive estimation introduce notation first let max set log log log inf log inf cnm consider contrast function partially adaptive case penalities define random sequence pen pen note definition depend knowledge sequence using definition completely penalty define fully adaptive selection dimension parameter means pen argmin bnm introduce order state prove upper risk bound estimator notation keep definition subsection slightly redefine log log also define max log log regarded analogues subsection case known error density finally define inf log inf log inf log inf log contrast proof theorem set knm impose additional assumption proof upper risk bound assumption exp additional assumption following theorem establishes uniform upper risk bound completely estimator theorem let assumptions hold sup sup min max observations stem poisson model note additional prerequisite theorem contrast validity assumption examples convergence rates continued subsection consider configurations sequences subsection particular assume different configurations investigated following compare also minimax rates convergence assumption satisfied considered cases given table note additional max realizes best compromise let define squared bias penalty scenario pol pol scenario holds log case rate first assume minimax optimal rate case holds rate log minimax optimal logarithmic factor assume estimator obtains optimal rate respect otherwise log yields contribution log rate scenario exp pol log scenario pol pol since log holds optimal rate respect holds case otherwise tradeoff generates contribution exp rate scenario pol exp holds sample size obstacle optimal rate holds well attaining optimal rate convergence get rate log coincides optimal rate respect sample size scenario exp exp log solution exp solution exp thus computation resp shows loss logarithmic factor occur contribution rate arising far exp deteriorates squared bias penalty determined optimal rate respect logarithmic factor remark paper considered case fourier coefficients error density obey decay exp assumption satisfied arbitrary indeed definition quantity case shortage removed considering elaborate choice quantities considered include adaptive estimation cox model unfortunately approach section transferred order obtain upper risk bound adaptive estimator case cox observations thus section follow another approach price pay obtain rates contain additional logarithmic factor split investigation partially adaptive fully adaptive case partially adaptive case define might interpreted dimension model corresponding linear subspace spanned considered well contrast inverse problem addition define quantities function exactly section however replace definition penalty given case poisson observations pen log ddk log based updated definition penalty define adaptive choice dimension parameter case cox observations means pen argmin theorem let assumption hold sup sup log min max observations stem cox model note proof theorem intricate one theorem due fact need introduce additional terms proof order deal terms apply talagrand type concentration inequalities poisson processes random variables fully adaptive case fully adaptive case replace dimension parameter previous subsection estimate moreover poisson case adapt definition penalty pen log log define contrast function exactly subsection set log inf log inf analogy approach cnm define fully choice poisson model via pen argmin bnm statement proof following theorem define quantities inf log inf log inf log inf log note proof following theorem knm requires validity assumption theorem let assumptions hold sup sup min max log observations stem cox model remark using approach presented subsection able dispense additional logarithmic factor rates case cox model note case error density known informally equivalent regain adaptive rate established case unknown intensity ordinary smooth fourier coefficients obey polynomial decay however results general since exclusively consider case polynomially decreasing fourier coefficients remark needless say numerical constants definition penalty ridiculously large makes rate optimal estimator hardly applicable small sample sizes hence still research necessary establish estimator performs well theoretical point view also yields good results simulations relatively small sample sizes another approach would calibrate numerical constants penalty means simulation study way done example remark course approach presented subsection also applied case poisson observations since logarithmic factor rates unavoidable would obtain worse rates using approach section examples convergence rates continued subsections note scenarios considered table denotes optimal squared bias term log computations similar ones leading rates table show rates respect sample size minimax framework table replaced log determines rate exactly long contribution seems noteworthy mention scenario pol exp attain upper bound minimax theory theorem case pol pol however one observe loss logarithmic factor also observed case adaptive estimation proofs section proof theorem let define statement theorem function function definition since moreover holds due estimate estimate together imply let fixed let denote joint distribution samples true parameters respectively let denote corresponding marginal distributions expectation respect arbitrary estimator key argument proof following let reduction scheme sup sup sup element defined consider hellinger affinity arbitrary estimator conclude means elementary inequality introduce hellinger distance two probability measures two finite analogously hellinger distance measures necessarily total mass equal one usual integral formed respect measure dominating let denote intensity measure poisson point process whose derivative respect lebesgue measure given note estimate due realized analogy shown obtain since distribution sample depend choice obtain first estimate follows lemma second one due theorem applied since poisson point process poisson model thus relation implies finally putting obtained estimates reduction scheme leads sup sup arbitrary finishes proof theorem since proof theorem following reduction scheme follows along general strategy establishment lower bounds nonparametric estimation detailed account chapter note markov inequality arbitrary estimator arbitrary specified reduction two hypotheses implies sup sup sup sup sup denotes distribution true parameters specific hypotheses specified chosen application triangle inequality yields denotes minimum distance test defined arg hence obtain inf sup inf sup sup inf sup infimum taken functions based observations thus remains find hypotheses allow bound universal constant independent arg max purpose set min min defined statement theorem take note inequalities combination imply inequalities used without reference define ekm note definition furthermore together imply identity shows condition satisfied let existence guaranteed condition define since holds estimate since thus trivially moreover hence obtain lower bound defined consider joint distribution samples note due construction thus due fact distribution poisson point process determined intensity hellinger distance depend distribution sample precisely proceed bounding recall used obtain estimate hence application statement theorem implies finishes proof theorem proof theorem give proof poisson model proof cox model follows complete analogy exploiting statement instead part lemma set proof consists finding appropriate upper bounds quantities estimate obtain upper bound using identity definition independence using estimate get var var var applying statements lemma together yields using holds due assumption implies definition consider using estimate yields var notice theorem implies existence constant using inequality combination assertion lemma implies var combination implies addition min exploiting fact definition obtain putting together estimates yields upper bound decomposed implies lemma yields estimate together imply putting obtained estimates finishes proof theorem auxiliary results proof theorem lemma notations introduced main part article following assertions hold poisson model var cox model var var min proof proof statement given identity var var implies prove identity var estimate var easily derived var analogy proof part order bound notice assertion follows combining obtained bounds var assertion follows proof note var estimate var proof consider two cases statement evident otherwise implies applying chebyshev inequality exploiting definition yields var statement follows proofs section proof theorem define events identity provides decomposition establish uniform upper bounds ellipsoids three terms side separately uniform upper bound denote linear subspace spanned functions holds since identity using identity obtain definition yields knm pen pen pen denotes projection subspace elementary computations imply pen pen addition defined introduce abbreviations knm well deduce using abbrevations identity pen pen define every knm estimate implies sup sup combining last estimate get pen pen sup sup sup since note due assumption specializing obtain sup sup sup definition combining facts obtain knm estimate hence obtain thus sup sup event obtain exploiting definition penalty pen max knm sup sup applying lemma yields sup dkf exp exp definition obtain using statement lemma fact knm knm sup log exp exp log last estimate due fact note exp log log numerical constant implies knm sup last term bounded means lemma immediately yields sup combining preceeding estimates hold uniformly conclude equation sup sup min max uniform upper bound define note consequently since knm obtain estimate knm knm due assumption lemma easily seen using definition obtain knm knm last estimate follows applying theorem two times lemma implies knm exploit definition bound otherwise knm first term second term bounded noting thanks assumption obtain log thanks logarithmic increase harmonic series lemma last estimate implies knm thus independent actual value knm using obtained estimates conclude uniform upper bound order find uniform upper bound first recall definition consider estimate using estimate obtain means lemma controls second term side bound first term means side knm inequality theorem easily seen need following estimate easily verified otherwise knm start bounding first term side using definition obtain since inequality combination rem implies second term side obtain exploit definition together knm log logarithmic increase harmonic series lemma conclude finally third last term side independent actual value knm bounded way exploiting definition obtain putting together derived estimates obtain finally statement theorem follows combining obtained uniform upper bounds proof theorem consider event cnm addition event introduced proof theorem slightly redefined event defined defining identity motivates decomposition establish uniform upper risk bounds four terms side separately thus uniform upper bound estimate last estimate implies log log log log log log log log log log log log conclude estimate putting penk penk pen penk observe note implies holds pen penk penb pen pen pen penk proceed mimicking derivation proof theorem used proof pen using definition precisely replacing penalty term pen penk obtain sup pen sup pen sup penk sup knm proof theorem second third term bounded applying lemmata respectively hence means obvious adaption statement lemma replaced estimates log log log log obtain analogy way proceeding proof theorem sup sup min max upper bound uniform upper bound derived analogy bound proof theorem using assumption instead statement lemma proof lemma hence obtain sup sup upper bound term also bounded analogously bound established proof theorem exploit additional assumption get sup sup upper bound find uniform upper bound term one use exactly decompositions proof uniform upper bound theorem replacing probability one obtain means lemma sup sup result theorem follows combining auxiliary results lemma let assumption hold following assertions hold true exp nothing show otherwise proof case definition log definition implies log log log log consider two cases first case log directly implies estimate second case log log therefrom log log log thus cases division yields assertion lemma sufficiently large note due assumption sufficient show desired inequality values definition log implies exp log exp assertion follows take note observation min min log log lemma let sequences max sup exp exp positive numerical constants proof proof combination proofs lemma deals case lemma general sequences considered framework random variables instead point processes precisely one apply proposition statement replaced makes proposition applicable also functions setting lemma let sup sup proof note implies sup sup thus recalling definition suffices show min realised means identity bound min already derived proof theorem corresponding upper bound obtained statement lemma lemma let assumption hold consider event defined theorem exp numerical constant proof note two terms side bounded chernoff bounds poisson distributed random variables see theorem precisely exp exp log log lemma let assumption hold consider event defined proof theorem proof complement owing statement lemma direct calculation using reverse triangle inequality shows case case one obtains thus together hoeffding inequality implies exp exp statement lemma follows statement lemma estimate holds definition lemma let assumptions hold event defined satisfies proof let consider random sets cnm cnm establish bounds separately owing upper bound use equality definition log yields log similar way obtain definition thus since applying hoeffding inequality proof lemma exploiting assumption yields exp particular upper bound first note knm knm obtain log using log analogously knm yields thus knm knm application hoeffding inequality exploiting assumption yields exp knm exp statement lemma follows combining equations proofs section proof theorem define quantities appearing proof exactly proof thereom unless otherwise stated use decomposition established proof theorem use exactly arguments proof bound terms thus remains find appropriate bound order get bound first proceed proof theorem order obtain estimate sup sup let introduce quantity vector containing unobserved shifts using setting decomposition obtain sup sup sup taking expectations obtain analogy proof theorem min max log knm knm log log sup sup sup sup log log sup apply lemma order obtain sup log exp log hence sup log exp log hence definition definition knm log obtain last estimate thanks logarithmic increase harmonic series due obtain knm knm sup log applying lemma obtain sup log exp log exp log using relation obtain knm sup log finally bounding last term means lemma obtain using obtained estimates min max log shows desired lower bound combining bounds yields result proof theorem use decomposition proof theorem particular quantities arising sequel defined proof theorem unless otherwise stated terms bounded exactly proof theorem remains find appropriate bound set penk log log one immediately obtains definition penk pen penk pen one follows pen penk penb pen pen pen combining arguments proofs theorem one show sup sup min max log statement theorem follows combining bounds established auxiliary results following result conditional version proposition since proof exactly one unconditional case poisson processes instead cox processes omit proof proposition let cox processes onra polish space driven finite random measures respectively set dnk contained countable class measurable functions denote exist constants sup exp exp sup sup sup var lemma let sequence sup log dnk exp log exp log strictly positive numerical constants proof putting easy check given dni thus framework proposition remains find suitable constants satisfying preconditions condition concerning sup krt sup sup one choose condition concerning sup log one choose condition concerning holds var log one choose statement lemma follows applying proposition proposition let random variables values polish space let belonging countable set measurable complexvalued functions constants sup exp exp sup sup sup var proof proof consists easy adaption proof given case statement follows bookkeeping occuring numerical constants lemma let sequence sup log exp log proof define coincides definition proof lemma setting framework proposition remains find suitable constants satisfying preconditions proposition condition concerning note definition definition proof lemma thus obtain sup krt sup sup sup take condition concerning sup log var log set condition concerning holds define statement lemma follows proposition references antoniadis bigot poisson inverse problems ann statist baraud estimating intensity random measure histogram type estimators probab theory related fields barron massart risk bounds model selection via penalization probab theory related fields bigot intensity estimation poisson processes shifted trajectories electron stat model selection poisson processes ims lecture notes monograph series institute mathematical statistics point processes queues martingale dynamics springer cavalier nonparametric statistical inverse problems inverse problems chagny estimation adaptative avec des application des survie phd thesis url https cavalier koo poisson intensity estimation tomographic data using wavelet shrinkage approach ieee inform theory comte lacour pointwise deconvolution unknown error distribution math acad sci paris comte lacour density estimation presence additive noise unknown distribution density estimation roy statist soc ser comte estimation spartacus paris comte rozenholc taupin penalized contrast estimator adaptive density deconvolution stat diggle hall fourier approach nonparametric deconvolution density estimate roy stat soc ser convergence rates minimum complexity estimator counting process intensities nonparametric statistics helmers mangku zitikis consistent estimation intensity function cyclic poisson process multivariate anal johannes deconvolution unknown error distribution ann statist johannes schwarz adaptive circular deconvolution model selection unknown error distribution bernoulli karr point processes statistical inference marcel dekker new york klein rio concentration around mean maxima empirical processes ann probab kroll concentration inequalities poisson point processes application intensity estimation indirect observations arxiv arxiv kutoyants statistical inference spatial poisson processes springer new york massart constants talagrand concentration inequalities empirical processes ann probab meister deconvolution problems nonparametric statistics springer berlin mitzenmacher upfal probability computing randomized algorithms probabilistic analysis cambridge university press cambridge neumann deconvolution panel data unknown error distribution multivariate anal neumann effect estimating error density nonparametric deconvolution nonparametr stat petrov limit theorems probability theory clarendon press oxford patil wood counting process intensity estimation orthogonal wavelet methods bernoulli adaptive estimation intensity inhomogeneous poisson processes via concentration inequalities probab theory related fields reiss approximate distributions order statistics springer new york reiss course point processes springer new york resnick extreme values regular variation point processes springer sansonnet wavelet thresholding estimation poissonian interactions model application genomic data scand statist serfozo basics applied stochastic processes springer berlin stoyan stochastic geometry applications third edition john wiley sons schwarz van bellegem consistent density deconvolution partially known error distribution statist probab lett shen zhang model nonhomogeneous poisson processes application copy number profiling dna sequencing ann appl statist tsybakov introduction nonparametric estimation springer new york zhang kou nonparametric inference doubly stochastic poisson process data via kernel method ann appl statist
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community detection spiking neural networks neuromorphic nov kathleen hamilton oak ridge national laboratory computing computational sciences dir one bethel valley road oak ridge tennessee hamiltonke neena imam oak ridge national laboratory computing computational sciences dir one bethel valley road oak ridge tennessee imamn travis humble oak ridge national laboratory computing computational sciences dir one bethel valley road oak ridge tennessee humblets abstract present results related performance algorithm community detection incorporates computation define mapping takes graph system spiking neurons using fully connected spiking neuron system inhibitory excitatory synaptic connections firing patterns neurons within community distinguished firing patterns neurons different communities random graph vertices known community structure show using binary decoding hammingdistance based metric individual communities identified spike train similarities using bipolar decoding finite rate thresholding verify inhibitory connections prevent spread spiking patterns graph partitioning community detection ubiquitous tasks encountered diverse set sciences many methods developed sort vertices graph nodes network classes based similarity measures methods utilize graphical characteristics structures transitivity modularity betweenness analysis often require large scale matrix analysis methods parallelized many algorithms exist analysis large networks however emergence unconventional processors neuromorphic processors requires approaches utilize computation present work paper related development algorithm incorporates computation identification related vertices networks graphs inspired recent work shaub different approaches community detection organized according problem designed solve partitioning problems clustering problems dynamical problems describe approach hybrid dynamical clustering method incorporates discrete time signals spiking neuron system paper present results serve related mapping recurrent neural networks based interacting spin dynamics hopfield networks spiking neural systems goal construct system spiking neurons used generate set spike responses identify vertex communities graph choose characterize community graph subset vertices density edges internal subset higher density edges connecting remainder graph similar definitions used cluster based models well known aphorism hebbian learning attributed siegrid neurons fire together wired use statement inversion neurons wired together fire together construct approach community detection vertices edges given graph mapped network symmetrically connected spiking neurons selectively driven external currents must degree similarity resulting spike trains used distinguish individual communities binary decoding bipolar decoding spike trains similarity trains measured using hamming distance metric ability identify individual communities dependent linear ccs concepts computing graph algorithms random graphs emerging tools methodologies neural systems keywords neuromorphic community detection spiking neural networks acm reference format kathleen hamilton neena imam travis humble community detection spiking neural networks neuromorphic hardware proceedings ornl neuromorphic workshop oak ridge tennessee usa july neuromorphic computing pages doi work supported united states department defense used resources computational research development programs oak ridge national laboratory manuscript authored llc contract department energy united states government retains publisher accepting article publication acknowledges united states government retains irrevocable license publish reproduce published form manuscript allow others united states government purposes department energy provide public access results federally sponsored research accordance doe public access plan permission make digital hard copies part work personal classroom use granted without fee provided copies made distributed profit commercial advantage copies bear notice full citation first page copyrights components work must honored uses contact neuromorphic computing oak ridge tennessee usa copyright held doi introduction neuromorphic computing july oak ridge tennessee usa separability hamming distance metric values controlled size bin width used binary decoding individual spike trains approach incorporates hopfield networks spin glass models shown recurrent networks steady states used find solutions problem graph partitioning use positive negatively weighted edges needed drive system solution meets optimization conditions however approaches often limited finding partitions equal sizes graph require prior knowledge number communities find hopfield networks implemented using spiking neurons task content addressable memories pattern retrieval work focus application recurrent neural network model task related graph partitioning show paper conjunction carefully chosen parameters neuron model based models used find sized groups establish spiking data used identify communities recent works community detection focused networks may ambiguous community structure limitations much information inferred metadata based approaches work focus discussion spiking data used identify communities work graphs generated known communities fixed labels graphs analyzed paper instances specific class random graphs benchmark graphs graphs vertices organized equalorder communities vertices average degree hdi using graphs demonstrate approach community detection focusing binary decoding bipolar decoding spike trains graph mapped system spiking neurons driven manner generated spike trains used reconstruct known community labels original graph two neurons must firing patterns exhibit degree similarity distinguishes spike trains generated neurons section derive theoretical structure underlies main components approach begin graph mapped spiking neural network snn discuss physical parameters associated spiking neuron system must set selective driving snn must done order generate set spike trains generated distinguish single community remaining sections describe spike responses decoded discuss metrics use analyze decoded spike trains paper presents results establish mapping driving pattern exists used generate spike trains characteristic graph known community structure section introduce spiking neural networks incorporated existing community detection algorithms hamilton spiking neuron model construction focus community detection undirected unweighted graphs graph defined vertex set edge set multiple edges allowed initial work study graphs clearly delineated communities snns dynamical systems compute without use steady states information transmitted electrical pulses neurons compose spiking network nonlinear units exhibit rich set dynamics firing patterns based physical parameters use parameters build snns fire neurons defined threshold voltage vth refractory period time constant full spiking network defined set homogeneous neurons set symmetrically connected synapsesw weighted edges connect neurons graph assume set known vertex communities vertex exists one community remainder paper develop spiking systems identify individual communities mapping undirected unweighted graph snn analogous construction hopfield recurrent neural network using symmetric connections either positive negatively weighted mapping first defines snn defining spiking neuron vertex graph edge graph defines symmetric pair excitatory synapses snn transformed fully connected snn addition symmetric pairs inhibitory synapses edges exist original graph magnitudes excitatory inhibitory synapses equal leaky integrate fire neurons simplified models neuronal behavior work models close proximity behavior ibm truenorth processor resulting networks spiking neurons require extensive detail biological behavior attempt describe cortical network dynamics focus model membrane potential potential single neuron function changes depending discrete continuous inputs discrete signals measured upon arrival positively negatively weighted spikes external driving force term vex assumed continuous additionally use existence leak continuously relax equilibrium value vex vex iex neuron potential exceeds firing threshold vth fires spike along synaptic connections enters refractory period potential changed choice community detection spiking neurons neuromorphic computing july oak ridge tennessee usa neuron system parameters determined whether used generate similar spike trains enhance dissimilarity spike trains refractory period needed impose sense directionality spread spiking patterns aided positively weighted synapses spike responses spread neurons yet fired correspond connected vertices graph lead similarities spike trains neurons community parameters lead dissimilar spike trains help impede spread spiking patterns across several neurons system primarily negatively weighted synaptic connections many parameters play dual role necessary creating similar spike trains inhibit spike pattern spread time constant balanced ensure arriving spikes accumulate lead secondary firing also ensuring effects spike impulses firing threshold ensures neuron fires response incoming spike impulses multiple impulses arrive short time window set homogeneous spiking neurons many system parameters tunable order generate spiking patterns informative original graph community structure previously studied simplest hopfield network could mapped system spiking networks considered graphs fully connected communities connected single bridge bond barbell graphs networks could driven using pair sinusoidal currents phase degrees negative driving current well careful tuning neuron system parameters refractory period time constant firing threshold synaptic weight sufficient generate spike trains characteristic two communities work generalize approach system connected spin glass role negative driving current replaced negatively weighted synapses inhibit growth limit occurrences spike cascades neuron dynamics square pulse driving relatively simple one neuron actively driven square pulse time effects refractory period allow equations motion separated set equation describe firing dynamics active driving one set equations describe firing dynamics reaction neighbor actively driven equation square pulse found integrating equation motion vex amax tanh tanh shape square pulse determined pulse height amax pulse width parameter determines sharpness pulse rise gap subsequent pulses applied different neurons sufficiently large induced vex neuron decayed away neuron actively driven parameter controls sharpness square pulse sharp step needed ensure spikes fired external current constant driving force imax however function vex must remain continuous active square pulse driving effect constant driving force leads constant firing rate vex found terms time constant square pulse amplitude amax reset voltage spike threshold voltage vth log vth firing rate used determine synaptic weight assumption neuron fire spike response one nearest neighbors actively driven spikes arrive short time span setting first spike arrives increases potential arrival second spike happens resulting potential must exceed spike threshold vth integrating dynamical equation time straightforward since active driving inequality becomes vth log binary decoding spike train similarity neuron associated set spike times called spike train spike train data analyzed using comparison matrix similarity trains quantified similar metric one used ref binary decoding converts spike trains binary vectors using discrete time step value least one spike occurs time window label assigned given neuron included completeness set binary vectors length compared pairwise construct entries comparison matrix entries defined normalized hamming distance two binary decoded spike trains introduce modified version comparison matrix inclusion terms weights entries matrix spike train firing neuron compared neuron later see sec begin develop scalable methods analysis terms omitted hamming metric defined either corresponds neuron initial tests algorithm use benchmark graphs fortunato see communities minimal overlap neuromorphic computing july oak ridge tennessee usa hamilton figure benchmark graph instance generated software available used generate spiking data analyzed different communities distinguished vertex color community red community blue community black community grey see fig using software available generate instances random graphs known community memberships map snn homogeneous neurons simulate spiking dynamics using python library snn constructed parameters vth spiking data analyzed paper generated graph instance shown fig three four communities driven neurons driven individual square pulses maximum height amax width gap subsequent pulses complete set neurons driven ordered community driven first driving primary firing neuron fires spikes separated nearly uniform time intervals secondary firing neuron fires spikes nearly uniform time intervals slight variations time interval spikes possible due spikes fired vex imax approximations introduced numerical integration equations motion generated spike trains comparison matrix constructed using binary decoding sec fig three driven communities identified magnitude block matrix structure along diagonal value significantly higher driven neuron neuron target community compared block driven neuron target neuron different communities neurons exhibit strong similarity driven neurons bipolar decoding firing rate thresholding concept community section used decode spiking output driving sequence used section conjecture neurons undergoing active figure comparison matrix benchmark graph instance community three driven difficult discern noisy background coherent spiking pattern able spread one community next hamming metric used generate matrix included terms low spiking neurons driving contained community exhibit higher firing rates outside community two neuronal states corresponding hopfield network states allowed states mapped spike train patterns use time window binning variation binary code discussed section consider total number spikes fired neuron hopfield network state mapped neuron firing rate fixed time window exceeds threshold value considered active state hopfield network state mapped neuron firing threshold fixed time window exceed threshold value inactive state similar approaches mapping hopfield networks spiking neural networks investigated task pattern retrieval using fitzhughnagumo neuron models upper bound found looking spiking response community fully connected clique neuron actively driven maximum number spikes neuron fire fmax active firing rate response firing rate minimum firing threshold value still investigation use simple heuristic approximate girvannewman benchmark graphs studied paper communities known order hdi lower bound minimum number spikes neuron fire still counted member community make assumption least half vertex neighbors must community neuromorphic computing july oak ridge tennessee usa spike count spike count neurons label spike count neurons label spike count community detection spiking neurons time interval figure full spike raster benchmark graph instance communities driven neuron communities actively driven vertical lines mark sec start driving sec end driving start driving sec end driving start driving sec end driving replace factor equation fmin study robust lower bound would require stricter definition community task reconstructing known set labels community structure benchmarks heuristic suffice inhibitory synapses impede spread spike cascades throughout entire neuron system shown fig instance benchmark graph shown communities driven similarity spike trains driven community significantly higher similarity spike trains different communities additionally community never exhibits significant degree spike response showing spread spiking synchronicity throughout entire network impeded identifying unknown communities future development community detection dependent well spiking neuron systems incorporated existing algorithms exists near linear algorithm community identification called label propagation introduced raghavan albert kumara similarity method potts spin model noted consider label propagation algorithm incorporate data however several obstacles overcome spiking neuron systems incorporated community identification tasks first need know neuron spike response affected neurons driven random order use girvannewman benchmark graph shown fig instead time interval time interval spike count neurons label spike count spike count spike count neurons label time interval figure spike counts neurons labelled community three time windows width seconds showing neurons community react active driving applied neurons respectively red dashed horizontal upper bound total number spikes would generated square pulse driving community black dashed line lower bound generated using mean degree hdi assumption least half neuron neighbors community label driving used sections randomly permute drive entire neuron set randomly permuting neuron set reduces usefulness bipolar decoding spiking data analysis section uses binary decoding additionally randomly driving neurons must carefully chosen response subsequent driven neurons covered single time window happens distinction signals belonging individual neurons may different community labels may lost upper bound time window second scalability hamming metric quite poor analysis larger graphs require driving large number neurons length binary decoded spike trains rapidly increase length binary decoded spike train increases similarity two neurons fire frequently become quite large remedy return original implementation hamming metric defined returning hamming metric originally defined affect comparison matrix structure see fig using hamming metric show spiking data used label propagation algorithms label propagation done choosing vertex act source fixing label updating neuromorphic computing july oak ridge tennessee usa figure comparison metric graph instance vertices complete neuron set first randomly permuted every neuron individually driven square pulse time window chosen hamming metric include terms scale limited range along matrix diagonal seen labels neighboring vertices spiking data incorporated method need ensure choice source significantly impact linear separability hamming metric values role tuning quality shown fig sec mean hamming metric value defined value excluded source target different labelled communities mean metric nearly overlapping source target community mean nearly linear separability mean metric value dependent size seen fig conclusions approach community detection using computing depended vertices within community exhibiting similar spiking patterns studied snns constructed driven external current decoded order generate spiking patterns undirected graph mapped fully connected set homogeneous spiking neurons excitatory inhibitory synapses selectively driven square pulse currents neurons contained within densely connected regions exhibit synchronicity firing patterns spike trains decoded time window binning using varying window widths resulting binned vectors hamilton converted binary codes entries binary vectors based metric construct comparison matrix dimension graph order degree similarity spike trains quantified used infer membership communities original graph resulting vectors tabulate number spikes fired use threshold determine appropriate label assign neurons using binary decoding driven communities could distinguished hamming metric similarity simulations spiking neuron dynamics demonstrated ability identify nonoverlapping communities equal order see figs graphs known community structure fig seen use inhibitory synaptic connections prevented large scale spike cascade hamming metric distinguish single community neuron system driven sequentially according community neuron set randomly permuted bipolar decoding assigning community labels according spike counts distinguish driven communities dominant spiking behavior however bipolar decoding used neuron system driven sequentially according community results presented generated one instance benchmark graph shown fig multiple instances tested similar results additional simulations also tested graphs communities explore method performs data randomly permuting driving entire neuron set fig linear separability hamming metric value shows small enough value hamming metric distinguish one community remaining communities results show spiking data used identify communities undirected graphs separability hamming similarity measure see fig could incorporated label propagation workflow community detection unique use neuromorphic hardware utilizing neural systems hidden units require training large data sets future work related community detection investigate two areas applicability scalability method applicability focuses quantifying limits spikebased approach effectiveness graph classes identification overlapping communities establishing resolution limit smallest community identified large network scalability addresses method applied large network analysis particular presented work binary analysis potential poor scaling system vertices mapped densely connected system neurons produces matrix dimension references marcelo blatt shai wiseman eytan domany superparamagnetic clustering data phys rev lett apr issue https stefano boccaletti vito latora yamir moreno martin chavez hwang complex networks structure dynamics physics reports andrew cassidy paul merolla john arthur steve esser bryan jackson rodrigo pallab datta jun sawada theodore wong vitaly feldman cognitive computing building block versatile efficient community detection spiking neurons neuromorphic computing july oak ridge tennessee usa figure instance graph vertices complete neuron set randomly permuted every neuron individually driven square pulse mean hamming metric source target community black distinguished mean hamming metric mean hamming metric source target different communities red blue green bin width small standard deviation mean shown shaded regions digital neuron model neurosynaptic cores neural networks ijcnn international joint conference ieee aaron clauset newman cristopher moore finding community structure large networks phys rev dec issue https santo fortunato community detection graphs physics reports santo fortunato santo fortunato website software https accessed brian gallagher hanghang tong tina christos faloutsos using ghost edges classification sparsely labeled networks proceedings acm sigkdd international conference knowledge discovery data mining kdd acm new york usa https wulfram gerstner werner kistler spiking neuron models single neurons populations plasticity cambridge university press michelle girvan mark newman community structure social biological networks proceedings national academy sciences dan goodman romain brette brian simulator spiking neural networks python frontiers neuroinformatics kathleen hamilton neena imam travis humble community identification spiking neural networks poster presented siam network science workshop july pittsburgh john hertz anders krogh richard palmer introduction theory neural computation santa institute studies sciences complexity vol john hopfield neural networks physical systems emergent collective computational abilities proceedings national academy sciences john hopfield neurons graded response collective computational properties like neurons proceedings national academy sciences john hopfield david tank neural computation decisions optimization problems biological cybernetics mark humphries communities finding groups similar spike trains journal neuroscience takashi kanamaru yoichi okabe associative memory retrieval induced fluctuations pulsed neural network physical review takashi kanamaru yoichi okabe memory retrieval pulsed neural network storing sparse patterns hierarchical correlations physical review andrea lancichinetti santo fortunato benchmarks testing community detection algorithms directed weighted graphs overlapping communities physical review andrea lancichinetti santo fortunato filippo radicchi benchmark graphs testing community detection algorithms physical review lowel singer selection intrinsic horizontal connections visual cortex correlated neuronal activity science https arxiv http wolfgang maass christopher bishop pulsed neural networks mit press wolfgang maass thomas networks spiking neurons emulate arbitrary hopfield nets temporal coding network computation neural systems fragkiskos malliaros michalis vazirgiannis clustering community detection directed networks survey physics reports paul merolla john arthur rodrigo andrew cassidy jun sawada filipp akopyan bryan jackson nabil imam chen guo yutaka nakamura million integrated circuit scalable communication network interface science mark newman michelle girvan finding evaluating community structure networks physical review newman networks introduction oxford university press peel learning relational networks arxiv leto peel daniel larremore aaron clauset ground truth metadata community detection networks science advances usha nandini raghavan albert soundar kumara near linear time algorithm detect community structures networks physical review reichardt stefan bornholdt detecting fuzzy community structures complex networks potts model physical review letters reichardt stefan bornholdt statistical mechanics community detection phys rev jul issue https michael schaub delvenne martin rosvall renaud lambiotte many facets community detection complex networks applied network science https hideki tanaka takashi morie kazuyuki aihara associative memory operation spiking neural network modulation resting membrane potential int symp nonlinear theory applications bruges belgium gergely equivalence label propagation method community detection potts model approach physica statistical mechanics applications david van den bout thomas miller graph partitioning using annealed neural networks ieee transactions neural networks
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generalized sure optimal shrinkage singular values matrix denoising oct bigot charles deledalle delphine institut bordeaux cnrs umr bordeaux october abstract consider problem estimating signal matrix noisy measurements assumption distribution data matrix belongs exponential family setting derive generalized stein unbiased risk estimation sure formulas hold spectral estimators shrink threshold singular values data matrix leads new spectral estimators whose optimality discussed using tools random matrix theory numerical experiments spiked population model asymptotic setting dimensions data matrix let going infinity theoretical properties approach compared recent results asymptotically optimal shrinking rules gaussian noise also leads new procedures singular values shrinkage matrix denoising measurements keywords matrix denoising singular value decomposition model gaussian spiked population model spectral estimator stein unbiased risk estimate random matrix theory exponential family optimal shrinkage rule degrees freedom ams classifications introduction low rank matrix denoising exponential family various applications interest estimate signal matrix noisy data typical examples include case data produced matrix form others concerned observations multiple samples organized matrix form setting typical inference problem involves estimation unknown signal matrix noisy data matrix satisfying model noise matrix real entries assumed independent random variables var paper focus situation signal matrix assumed low rank structure consider general setting distribution belongs continuous exponential family parametrized entries matrix discrete observations count data also consider specific case poisson noise low rank assumption often met practice exists significant correlation columns case columns represent images different wavelength hyperspectral data since images nearby wavelengths strongly correlated applications modeling relevant found genomics nmr spectroscopy collaborative filtering medical imaging among many others matrix estimation classically done setting additive noise gaussian homoscedastic variance general case observations sampled exponential family less developed exists increasing research interest study low rank matrix recovery beyond gaussian case examples matrix recovering poisson distributed observations found applications count data network traffic analysis call center data theory matrix recovery completion case poisson observations also recently proposed matrix completion low rank assumption additive errors subexponential distribution belonging exponential family also considered recent work proposes novel framework approximate low rank matrix tabular data set made numerical boolean categorical ordinal observations class spectral estimators standard approach estimate low rank matrix relies singular value decomposition svd data matrix min denote singular values denote associated singular vectors paper propose consider class spectral estimators possibly mapping acts singular values data matrix leaving singular vectors unchanged precisely estimators take form min min real positive values may depend hence write whole matrix investigated spectral estimators typical examples spectral estimators include classical principal component analysis pca applied matrix denoising defined min implicitely understood another typical spectral estimator matrix denoising gaussian measurements softthresholding corresponds choice min soft min possibly threshold parameter max finaly consider general class shrinkage estimators encompassing pca perform min possibly shrinking weight main contributions assumption distribution belongs exponential family goal paper derive choices weights construct estimators via procedure first active set singular values defined second step weights associated singular values optimized shown reach desired asymptotical properties gaussian spiked population model main contributions paper following ones aic inspired criterion rank singular values locations estimation priori available rank signal matrix optimizing weights min lead estimators large variance overfitting noise propose automatic rule prelocalize subset singular values active set min singular values defined minimizer penalized criterion inspired akaike information criterion aic arg min log cardinal likelihood data given exponential family estimated parameter case gaussian measurements homoscedastic variance one denotes frobenius norm matrix show active set singular values boils gamma poisson measurements resort greedy optimization procedure described section active set determined subsequent shrinkage estimator obtained optimizing weights within subset setting ones zero novel shrinkage rules minimizing formulas use principle stein unbiased risk estimation sure derive unbiased estimation formulas mean squared error mse risk mean mkl risks spectral estimators minimizing formulas appropriate class spectral estimators shown lead novel shrinkage rules singular values matrix particular approach leads novel spectral estimators situations variances entries noise matrix necessarily equal may depend signal matrix illustrative example let consider spectral estimators form act first singular value data setting ones zero paper examples choices weight following ones gaussian measurements known homoscedastic variance gamma measurements see section precise definition min min lmn lmn poisson measurements see section precise definition min beyond case rank one solutions weights obtained except case gaussian measurements homoscedastic variance latter case rule generalizes eigenvalues min gamma poisson distributed measurements propose fast algorithms get numerical approximations weights see section details asymptotic properties gaussian spiked population model another contribution paper discuss optimality shrinking weights gaussian noise asymptotic setting dimensions matrix let going infinity theoretical results obtained spiked population model introduced literature random matrix theory covariance matrix estimation see theoretical asymptotic results paper derivation proposed estimators assume model definition gaussian spiked population model corresponds following setting iid gaussian random variables zero mean variance entries unknown matrix low rank structure meaning admits svd left right singular vectors associated singular value rank matrix assumed fixed dimensions data matrix let going infinity asymptotic framework sequence gaussian spiked population model asymptotic locations empirical singular values well understood random matrix theory details given section note setting rank held fixed allowed grow min different see references therein gaussian spiked population model contributions follows prove convergence sure formula dimensions tend infinity shown minimizing asymptotic value sure leads estimator limiting value estimator obtained minimizing sure model allows show novel spectral estimators derived paper asymptotically connected existing optimal shrinkage rules matrix denoising setting also able connect choice penalty function stein notion degrees freedom see spectral estimators numerical experiments publicly available source code theoretical properties estimators studied asymptotic setting report results various numerical experiments analyze performances proposed estimators matrices experiments allow comparison existing shrinkage rules measurements also used shed lights finite sample properties method measurements also exhibit settings signal matrix either easy difficult recover experiments main findings following ones use appropriate active set singular values essential step quality shrinkage estimators whose weights estimators taking min leads poor results choice minimizing aic criterion appears yield best performances gaussian noise performances approach similar obtained asymptotically optimal spectral estimator proposed true rank signal matrix sufficiently small large moderate values ratio approach may perform better existing methods literature gamma poisson distributed measurements spectral estimators proposed paper give better results estimators based pca restricted active set singular values beyond case gaussian noise implementation estimators straightforward thus provide publicly available source code https reproduce figures numerical experiments paper related results literature early work singular value thresholding began work best approximation fixed rank data matrix spectral estimators different amounts shrinkage singular value data matrix proposed case gaussian measurements homoscedastic variance problem estimating assumption recently received lot attention literature statistics see recent works also consider general setting distribution additive noise matrix orthogonally invariant entries iid random variables zero mean finite fourth moment papers authors focused spectral estimators shrink threshold singular values singular vectors left unchanged setting main issue derive optimal shrinkage rules depends class spectral estimators considered loss function used measure risk estimator appropriate assumptions distribution additive noise matrix organization paper section devoted analysis data matrix whose entries distributed according continuous exponential family formula first given mean squared error risk risk example discrete exponential family also derive risk estimators poisson distributed measurements computation datadriven shrinkage rules discussed gaussian gamma poisson noises section restrict attention gaussian spiked population model order derive asymptotic properties approach study asymptotic behavior sure formula proposed spectral estimators using tools rmt result allows make connection spectral estimators minimizing sure gaussian noise asymptotically optimal shrinkage rules proposed section study penalized criterion used select active set singular values connection degrees freedom spectral estimators rank estimation matrix denoising discussed various numerical experiments finally proposed section illustrate usefulness approach developed paper denoising compare performances existing methods proofs main results paper gathered technical appendix numerical implementation details described appendix formulas exponential families introduction exponential families refer idea unbiased risk estimation exponential families dates back recently generalized sure formulas proposed estimation mse risk denoising various continuous discrete distributions inverse problems whithin continuous exponential families formula derived estimation risk applies continuous discrete exponential families follows borrow ideas results works first treat case continuous exponential families focus poisson data discrete case data sampled continuous exponential family recall matrix independent real entries assume random variable sampled continuous exponential family sense admits probability density function pdf respect lebesgue measure real line pdf thus written general form exp link function smooth function function twice differentiable mapping known function unknown parameter interest belonging open subset throughout paper suppose following assumption holds assumption link function function denotes first derivative since exponential families general form assumption implies thus data matrix satisfies relation centered noise matrix agreement model let also convenient consider expression pdf canonical form exp usually called canonical parameter exponential family finally recall relation var denotes second derivative denote matrix whose entries examples data satisfying model following ones gaussian noise known variance exp var exp measurements known shape parameter exp var log matrix estimated via matrix whose entries given spectral estimator defined rest section follow arguments derive formulas exponential family estimators using either meansquared error mse risk risk unbiased estimation mse risk consider following mse risk provides measure discrepancy space natural parameters indirectly space interest definition squared error risk meani squared error mse risk defined mse using mse risk compare implies discrepancy estimator matrix interest measured quantity mse different mse gaussian noise xkf gamma distributed measurements known shape parameter follows following proposition gives sure formula mse risk introduced definition proposition suppose data sampled continuous exponential family assume function definition exponential family twice continuously differentiable following condition holds quantity div gsure div unbiased estimator mse note gsure unbiased estimator mse mse shown section results proposition coincide approach derivation sure formula case gaussian noise smooth spectral estimators case gamma noise assuming implies conditions function proposition satisfied hence assuming conditions holds well using fij follows gsure fij unbiased estimation risks following terminology let introduce two different notions kullbackleibler risk arise discrepancy measure definition let smooth spectral function consider estimator defined matrix whose entries independent random variables sampled exponential family canonical form synthesis kls risk defined fij kls log mean kls risk defined mkls kls analysis kla risk defined kla log mean kla risk defined mkla kla key advantage risk measures discrepancy unknown distribution estimate thus invariant respect reparametrization unlike mse risk may also write mkls mkls mkla mkla suggested mkla risk represents well distribution explain random variable sampled pdf mkla risk natural loss function many statistical problems since takes reference measure true distribution data see mkls risk represents well one may generate independent copy sampling random variable pdf mkls risk also considered various inference problems statistics simple calculation follows mkls mkla hence case gaussian measurements known easily retrieve variance mkls mkla mse case gamma distributed measurements known shape parameter follows log mkls mkla log use results whose main contributions derivation new unbiased estimators mkls mkla risks continuous exponential family risk estimate derived unbiased mkls risk asymptotically unbiased mkla risk respect ratio data sampled continuous exponential family makes simpler use mkls risk derive shinkage low rank matrix denoising therefore chosen concentrate study risk setting following proposition establishes sure formula estimate mkls risk continuous case proposition suppose data sampled continuous exponential family assume function definition exponential family continuously differentiable suppose function definition exponential family twice continuously differentiable following condition holds quantity sukls div div unbiased estimator mkls key difference formula unbiased estimates mse risks pmin computation divergence term smooth spectral estimator sense function assumed almost everywhere differentiable min setting divergence term expression gsure depends upon matrix therefore nonlinear mapping generally possible obtain simpler expression div contrary sukls divergence term div following expression smooth spectral estimators min div min min min thanks results theorem note sukls sure gaussian measurements hence gsure sukls strategies match case case gamma measurements assuming implies conditions function proposition satisfied assuming condition holds well follows fij sukls log fij lmn div expression div given note implicitly understood definition div mapping fij differentiable differentiability spectral function thus components fij consequence assumption functions fmin acting singular values supposed differentiable details differentiability fij refer section arguments follows formula divergence also valid assumption function differentiable except set lebesgue measure zero case poisson data poisson data key result obtain unbiased estimate given risk following lemma dates back work lemma let measurable mapping let denote fij measurable function let matrix whose entries independently sampled poisson distribution fij fij etj fij denotes entry matrix resp denotes vector resp entry resp entry equals one others equal zero hudson lemma provides way estimate unbiased way expectation frobenius inner product matrix matrix see usefulness result one may consider following error mse therefore lemma one immediately obtains pure fij etj unbiased estimate quantity mse poisson data one may also define following risks mkls log mkla log agreement definition risks data sampled poisson distribution arguments currently exist approach derive sure formula mkls risk poisson case since unbiased formula log nevertheless shown hudson lemma provides unbiased estif mator log possible unbiasedly estimate mkla risk follows proposition data sampled poisson distribution quantity pukla log fij etj unbiased estimator mkla log shrinkage matrix denoising matrix entries consider shrinkage estimators form min underlying matrix constrained positive entries gamma poisson cases consider instead estimators form max priori lower bound smallest value matrix max max construction subset postponed section focus selecting weights way fixed given following denote complementary set let found considering estimators form appropriate trying find shrinking weights entries matrix positive given subset gaussian noise known homoscedastic variance applying gsure formula gaussian distributed measurements thanks expression divergence smooth spectral estimators obtain defined sure expression given min sure unbiasedly estimate mse hence differentiating expression respect follows weight empirical singular value given min fullfils requirement note sukls sure gaussian measurements exact weight would obtained minimizing estimate mkls case estimators rank one consider case estimators rank let follows given min gamma poisson distributed measurements gamma poisson cases possible follow strategy gaussian case derive optimal weights using established formulas shall investigate shinkage approximated section numerical experiments using fast algorithms nevertheless estimator restricted rank optimizing risk estimators lead expressions assumption entries data matrix strictly positive case estimators rank one gamma noise consider case mators rank let let denote pca approximation rank entries matrix strictly positive theorem entries first singular vectors strictly positive therefore entries belong set consider defined instead assuming sukls formula hold follows simple calculations sukls mnl log log lmn min hence differentiating expression respect monotonic sides unique minimum optimal value minimizing sukls given min min lmn lmn yields shrinking rule stated introduction paper note possible obtain optimal value weight minimizes criterion gsure case estimators rank one poisson noise using assumption entries positive consider theorem defined instead pure formula proposition apply estimator log yield pure pukla log log largest singular value matrix etj resp denotes left resp right singular vectors therefore differentiating expression respect monotonic sides unique minimum optimal value minimizes pure given min however optimal shrinking rule used practice since evaluating values feasible computational point view large values nevertheless fast algorithm find numerical approximation optimal value proposed section contrary using positive theorem value minimizing pukla min straightforward compute corresponds shrinkage rule given introduction gaussian spiked population model section restrict analysis gaussian spiked population model asymptotic setting introduced definition asymptotic location empirical singular values summarize asymptotic behavior singular values data matrix pmin gaussian spiked population model case well known empirical distribution singular values converges quarter circle distribution generalized version distribution supported compact interval bulk right edge low rank structure asymptotic behavior singular values also well understood generalizations noise matrix whose distribution orthogonally invariant also recently considered recall results needed paper end let introduce function defined following result holds see theorem proposition proposition assume ispa random matrix sampled gaussian spiked population model fixed one almost surely lim otherwise moreover lim follows shall also use relation holds consequence results section existing asymptotically optimal shrinkage rules briefly summarize results construction asymptotically optimal spectral estimators let min given smooth spectral estimator consider standard squared error measure risk set spectral functions minimizing given min however used practice since obviously unknown first alternative suggested rather study asymptotic risk lim almost sure sense gaussian spiked population model proposed find asymptotically optimal choice minimizing among given class smooth spectral functions results show among spectral estimators form pmin continuous shrinker whenever asymptotically optimal shrinkage rule given choice otherwise proposed consider spectral estimators form positive weights theorem follows weights minimize given follows shrinkage rules shown equivalent serve reference asymptotic optimality stressed estimators equivalent indeed method requires estimate rank approach applies shrinker empirical singular values nevertheless shrinkage function applied significant singular values either bulk edge given rank asymptotic behavior estimators based sure following principle sure second alternative choose smooth spectral estimator form study problem selecting set functions minimize unbiased estimate mse recall fij denotes entry matrix condition fij follows results equivalently proposition gaussian noise sure div unbiased estimate mse divergence div admits expression sure formula used find value case singular values shrinkage corresponds choice min study asymptotic behavior sure formula end shall use proposition also need following result whose proof found appendix study terms expression divergence proposition assume ispa random matrix sampled gaussian spiked population model fixed one almost surely lim follows restrict analysis following class spectral estimators terminology definition borrowed pmin definition let smooth spectral estimator given min estimator said spectral shrinker order collapses bulk whenever reason restricting study spectral estimators whenever linked choice active set singular values gaussian case detailed section spectral shrinker order collapses bulk study asymptotic behavior terms expression depend namely sure reason studying sure finding optimal shrinkage rule minimizes sure equivalent minimizing expression spectral shrinkers order collapses bulk since sure sure using proposition proposition assumption continuously differentiable functions immediately obtain following result lemma assume random matrix sampled gaussian spiked population model let spectral shrinker order collapses bulk function continuously differentiable moreover assume one almost surely lim sure asymptotically optimal shrinkage singular values thanks lemma one may determine asymptotic optimal spectral shrinker one minimizing sure purpose let define class estimators given integer positive weights practice estimator computed replacing bulk edge approximation moderate large values quantities close replacement change numerical performances provided follows lemma lim sure differentiating expression respect weight leads following choice asymptotically optimal weights therefore singular values matrix estimated sufficiently large namely using proposition one asymptotically optimal spectral shrinker order given choice functions otherwise using relation one may also express asymptotically optimal shrinking rule either function otherwise function using equivalent otherwise therefore spectral shrinker order remark shrinkage rule coincides rule obtained similarly quantity expressed function retrieve shrinking rule derived therefore minimizing either asymptotic behavior sure sure limit risk leads choice asymptotically optimal spectral estimator shrinkage empirical singular values results section principle sure minimisation leads following choice spectral shrinker order collapses bulk given proposition proposition follows almost surely lim therefore spectral estimator asymptotically leads optimal shrinking rule singular values given obtained minimizing asymptotic behavior sure note suffices replace condition definition yields shrinking rule stated introduction paper estimating active sets singular values exponential families section propose formulate new akaike information criterion aic select appropriate set singular values apshrinkage procedure might applied end shall consider estimator defined subset min address problem selecting optimal subset data case gaussian measurements shrinkage estimators use numerical experiments form possibly shrinkage functions set based knowledge approximation bulk edge thanks proposition bulk edge interpreted threshold allows distinguish locations significant singular values data due presence additive noise interestingly following result shows active set may interpreted prism model selection using minimisation penalized criterion proposition assume entries iid gaussian variables zero mean standard deviation arg min mky min cardinal proof remark results mky otherwise using follows set definition therefore criterion mky minimum concludes proof model entries iid gaussian variables zero mean variance well known degrees freedom dof given estimator defined dof cov dof widely used statistics define various criteria model selection among collection estimators see low rank matrix denoising following proposition shows possible derive asymptotic behavior dof spectral estimators proposition assume random matrix sampled gaussian spiked population model let spectral shrinker order collapses bulk function continuously differentiable moreover assume one almost surely lim dof proof thanks derivation sure formula divergence spectral estimators one dof div proposition proposition assumptions one almost surely lim div completes proof hence gaussian spiked population model proposition using follows lim dof hence quantity asymptotically upper bound dof normalized given set let consider general case entries sampled exponential family best knowledge extending notion bulk edge nongaussian data sampled exponential family considered far literature random matrices low rank perturbation model therefore except gaussian case far trivial find appropriate threshold value define active set form nevertheless select appropriate active set singular values introduce following criterion inspired previous results dof estimator gaussian case statistical literature well known aic model selection definition aic associated aic log cardinal likelihood data general form estimated parameters definition aic quantity approximation degree freedom numbers free parameters justified proposition case gaussian measurements aic allows define optimal subset active variables arg min aic gaussian measurements proposition gives value optimal set following arguments section gamma poisson measurements given subset consider estimator max priori value satisfy positivity constraint entries estimator setting however contrary case gaussian noise search optimal subset arg min aic becomes combinatorial problem context numerical experiments thus choose construct approximation greedy search strategy reads follows aic aic gaussian measurements since optimisation problem becomes separable numerical experiments found selects relevant set active singular values separates well structural content removing noise component details given section gaussian noise computation active set singular values may also interpreted way estimate unknown rank signal matrix setting one suggests choice max estimator exists abundant literature problem estimating rank empirical covariance matrix purpose selecting appropriate number significant components kept pca factor analysis much beyond scope paper give overview topic point review summary existing methods determine number components pca grouped three categories subjective methods test tools computational procedures recent contributions matrix denoising model gaussian noise refer works references therein example gaussian data know variance optimal hard thresholding singular values suggest take max simple method estimate rank remarked problem estimating true rank model somewhat gaussian spiked population model proposition implies one may expect estimate effective rank reff max see section numerical experiments shall compare different choices active set singular values form either given oracle choices reff methods based hypothesis testing could used rank estimation gaussian model beyond purpose paper give detailed comparison poisson gamma noise difficult interpret computation way estimate rank since numerical experiments found cardinality generally equal max moreover best knowledge much work estimation true rank noisy matrix beyond gaussian case therefore included numerical comparison methods choice active set singular values two cases numerical experiments section assess performance srhinkage rules various numerical experiments involving gaussian gamma poisson measurements case signal matrix rank one consider simple setting rank matrix known equal one meaning vectors unit norm fixed numerical experiment positive real let varying also choose fix purpose sampling data gamma poisson take distribution took singular vectors positive entries entry resp entry resp chosen proportional resp pmin let matrix whose entries sampled model satisfying gaussian measurements first consider case gaussian measurements var context compare following spectral shrinkage estimators pca shrinkage soft asymptotically optimal shrinkage proposed weighted estimator derived section formula follows results section using numerical experiments value obtained numerical solver order minimize sure benchmark also consider oracle estimator performs shrinkage using knowledge true defined corresponds asymptotically optimal shrinking rule function setting note form formula necessary range considered estimators figure compare estimated estimated weights functions four aforementionned estimators estimators subject noise variance display estimators median values confidence intervals obtained noise realizations seen median curves eigenvelues weights coincide variations slightly larger former agreement asymptotic analysis shrinkage rules carried section spectral estimator obtained also leads optimal shrinkage rule figure four spectral estimators display noise realizations functions following normalized mse nmse normalized mse estimators soft values larger differ values close threshold corresponding values bulk edge remarkably offer similar nmse values oracle shrinkage estimator terms median also terms variability assessed confidence intervals performances estimator standard pca clearly poorer numerical experiments also illustrate finitedimensional low rank matrix denoising spectral estimators obtained minimizing sure criterion achieve performances similar asymptotically optimal shrinkage rules gamma poisson distributed measurements let consider case entries data matrix independently sampled gamma poisson distribution mean satisfy constraint estimators must matrices positive entries consider estimators form context compare following spectral shrinkage estimators set pca shrinkage max soft max estimated singular value estimated singular value singular value estimated weight estimated weight singular value singular value singular value estimated weight estimated singular value singular value singular value figure case gaussian measurements estimated first singular value function true underlying one proposed estimator soft asymptotical one compared first singular value one oracle asymptotical estimator corresponding weight curves computed noise realizations median confidence interval represented respectively stroke shadded area color weighted estimator max approximated active subset defined section value obtained numerical solver order minimize either gsure sukls criterion gamma case either pure pukla criterion poisson case weight obtained numerical solver order minimize gsure pure described section according section weight nmse nmse nmse singular value singular value singular value figure fig normalized mse corresponding estimators minimizing sukls criterion following formula min lmn lmn pukla criterion min evaluate performances estimators perform study involving noise realizations gamma case shape parameter results reported figure ranges poisson case results reported figure generate data poisson distribution mean value took ranging context entries average ranging entries equals correspond extreme level noise entries concentrate around standard deviation correspond simpler noisy setting experiments seen spectral estimators achieve comparable results really small errors terms mse mkl risks performances similar meaning optimizing either criteria leads spectral estimator closed correspond matrix denoising ordinary pca however unlike gamma case might observed poisson case reaching stronger noise level small value nmse estimator increases denoising problem becomes challenging nevertheless weight driven pukla present drop wich allows reaching slightly smaller mkla gamma case noise level proportional signal level remain constant singular value singular value mkls singular value nmse estimated weight gsure singular value gsure singular value estimated singular value mkls nmse estimated weight estimated singular value sukls sukls singular value singular value singular value figure case gamma measurements estimated first eigenvalue function true underlying one proposed estimator soft guided gsure compared first singular value corresponding weights nmse risk mkls risk exact esperiments proposed estimator guided sukls curves computed noise realizations median confidence interval represented respectively stroke shadded area color finally mentionned use estimator gaussian model homoscedastic variance one may take estimator hence provided variance entries data matrix known always possible use scaled version shrinkage rule however setting gamma poisson noise variance additive noise varies one entry another depends entries unknown signal matrix recover reason possible use scaled version shrinkage rule would require use scaling factors depending unknown values entries therefore comparison approach asymptotically optimal shrinkage proposed gaussian noise possible case gamma poisson measurements note gamma measurements one var numerical experiments assumed constant known typical example assumption reasonable one statistical models singular value singular value singular value pukla pukla singular value mkla nmse pure singular value estimated weight estimated singular value pure mkla nmse estimated weight estimated singular value singular value singular value singular value figure case poisson measurements estimated first eigenvalue function true underlying one proposed estimator soft guided pure compared first singular value corresponding weights nmse risk mkla risk exact esperiments proposed estimator guided pukla curves computed noise realizations median confidence interval represented respectively stroke shadded area color speckle used coherent imagery synthetic aperture radar sar sound navigation ranging sonar imagery imaging systems observed irradiance pixel indices obtained square modulus complex signal modeled circular complex gaussian distributed consequence central limit theorem follows exponential mean corresponding underlying irradiance estimated order improve contrast images namely signal noise ratio average independent identically distributed images often performed resulting pixel value becomes gamma distributed parameter number images averaged chosen practitioner parameter absolutely known without uncertainties reason require estimated nevertheless variance var remains unknown exponential distribution particular instance gamma distribution parameter estimators behave similarly rank setting see next section significantly differ rank let larger case signal matrix rank larger two consider complex realistic setting rank matrix unknown potentially larger two vectors unit norm fixed numerical experiment positive real values also fixed experiment also choose fix true rank shown red curve figure pmin let matrix whose entries sampled model satisfying gaussian distributed measurements first consider case gaussian measurements var following numerical experiments study behavior spectral estimator pca shrinkage min soft asymptotically optimal shrinkage proposed weighted estimator derived section xsoft xrw nmse singular values nmse rank restricted bulk edge indices nmse nmse rank restricted rank nmse rank restriction rank restricted effective rank rank restricted figure zoom matrix single realization corrupted version gaussian noise oracle soft rmax soft pca full rank rmax min rmax oracle full rank approximation full rank estimation rmax rmax corresponding singular values nmse various approximations function rank without knowledge bulk edge namely active set singular values form either given rank reff effective rank figures solid curves correspond oracle estimators dashed curves correspond estimators obtained noise realizatrions grey areas represent confidence interval min value obtained numerical solver order minimize sure otherwise specified consider max estimator rank using knowledge bulk edge hence discussed section compare iments influence rank estimation analyzing performances estimators either rmax min without knowledge bulk edge namely rank reff effective rank singular values order assess quality sure estimator mse also compare aforementioned approach oracle counterparts given min soft minimizes squared error risk sets softthresholding approximations respectively note soft ideal approximations used practice serve benchmarks evaluate performances estimators soft order shed light variance estimators indirectly variance sure perform experiments independent realizations results reported figure estimator rank given either max knowledge bulk edege rank reff effective rank observed achieve comparable performances min even though two first rely unknown matrix similarly soft soft achieve also comparable performances showing sure accurately estimates mse terms error bands nmse outperform soft soft provided large enough moreover performance plateaus optimum rank becomes large allows choose min priori true effective rank interestingly fig shows estimators used without thepknowledge bulk edge taking computation instead corresponds choice rmax min performance actually decreases rank becomes large indeed clear fig error band nmse becomes much larger rank increases illustrates sure suffers estimation variance case parametrization becomes large thus used estimate jointly large number weights therefore knowledge appropriate estimator rank using bulk edge seems provide relevant upper bound number weights jointly robustly estimated sure xsoft gsure kls xsoft sukls xrw gsure kls xrw sukls nmse rank restricted active set indices rank restriction mkls mkls singular values nmse rank restricted active set rank restriction figure single realization corrupted version gamma noise zoom matrix oracle soft datadriven soft respectively gsure kls sukls rmax pca full rank approximation rmax min oracle full rmax rank approximation rwmax full rank estimation respectively gsure kls sukls corresponding singular values averaged noise realizations nmse averaged noise realizations function rank without using active set respect mkls xsoft pure kla xsoft pukla xrw pure kla xrw nmse pukla rank restricted active set indices rank restriction mkla mkla singular values nmse rank restricted active set rank restriction figure single realization corrupted version poisson noise zoom matrix oracle soft datadriven soft respectively pure kla pukla pca rmax full rank approximation rmax min oracle full rank rmax approximation rwmax full rank estimation respectively pure kla pukla corresponding singular values averaged noise realizations nmse averaged noise realizations function rank without using active set respect mkla matrix entries displayed better visual assessment gamma poisson measurements let consider case entries data matrix independently sampled gamma poisson distribution mean consider estimators form context compare following spectral shrinkage estimators set pca shrinkage max driven min soft max driven weighted estimator max min approximated active subset defined section value obtained numerical solver order minimize either gsure sukls criterion gamma case either pure pukla criterion poisson case shown section case gamma resp poisson measurements value minimizes gsure resp pure sukls resp pukla obtained closed form alternative adopt greedy optimization strategy starting matrix next updating weights sequentially starting min constraint weight set zero end resort optimization techniques interval using matlab command fminbnd strategy used gsure sukls pure pukla evaluating described section gaussian setting compare spectral estimators oracle counterparts given min soft max max wkoracle wkoracle minimizes one objective kls kla nonexpected risks set matrices sharing first left right singular vectors approximations respectively note soft ideal approximations used practice serve benchmarks evaluate performances estimators soft results gamma noise reported figure gaussian setting observed achieve comparable performances well soft soft showing gsure resp sukls accurately estimates resp kls visual inspection restored matrices tends show estimators driven gsure produce less relevant results compared kls sukls confirmed curves nmse mkls performance terms nmse also illustrates minimizers coincides gaussian setting outperform soft soft standard pca provided large enough moreover performance obtained objectives plateaus optimum rank becomes large allows choose min priori true rank results poisson noise reported figure conclusions similar gaussian gamma cases obviously nmse smaller approximations minimizes pure minimizing kla pukla however visual inspection obtained matrices tends demonstrate minimizing objectives might less relevant minimizing objectives setting performance par one soft based pukla fact choices matrices based pukla might improve terms mkls much soft might improve much based pure nevertheless whatever observed driven pukla always reaches least good performance terms mkls best driven soft fig fig fig fig show estimators used without active set choosing min performance actually decreases rank becomes large gaussian setting explained fact gsure sukls pure pukla suffer estimation variance case parametrization hence used estimate jointly large number weights active set manner bulk edge seems provide relevant selection weights jointly robustly estimated data driven way signal matrix equal singular values increasing rank propose highlight limitations approach situation prpotential rank matrix let growing positive singular values equal namely vectors unit norm fixed centered random matrix whose entries iid gaussian variables variance choose fix true rank let growing min following numerical experiments constant chosen larger hence corresponds gaussian spiked population model setting positive singular values equal larger threshold choice motivated results proposition given value true rank performed experiments involving realizations model compare nmse estimators oracle rmax soft soft pca full rank rmax min oracle rmax rmax rmax full rank approximation full rank estimation estimators introduced section figure report results numerical experiments displaying errors bars nmse estimators functions true rank low values true rank rmax rmax estimators approach shrinkage rule achieve best performances similar term median value nmse however approach limitations respect performances estimator setting signal matrix equal positive singular values rank increasing moreover error bands nmse approach becomes significantly larger estimators true rank increases illustrates sure minimization may lead estimators high variance case parametrization exists large number significant close singular values signal matrix influence dimension ratio section used simulated data consisting signal matrix equal positive singular values increasing rank setting likely empirical weights pmin used approach high variance due term expresk sion however numerical experiments carried section correspond specific configuration signal matrix many equal singular values high rank likely encountered real data conclude numerical experiments finally analyze influence dimension data ratio performances approach estimator realistic setting gaussian noise two estimators ones giving best results thus interest compare experiments use real square signal matrices relatively fast decay singular values see figure figure choose let varying define root ratio rsnr rsnr value rsnr ranging performed experiments involving realizations model compare nmse estimators full rmax rmax rank estimation approach shrinkage rule rmax figure figure report results numerical experiments displaying true rank xrw true rank xrw true rank true rank xsoft nmse nmse xrw xsoft true rank true rank xsoft xrw nmse xsoft true rank nmse nmse nmse true rank xsoft nmse xrw xrw xsoft nmse nmse true rank figure comparison nmse function true rank model different values estimator roracle soft max soft pca full rank rmax min oracle full rank rmax rmax rmax approximation full rank estimation active set set singular values form max estimator rank using knowledge bulk edge median value nmse various estimators gaussian noise realizations model function true rank grey areas represent error bands nmse oracle estimators signal matrix singular values nmse nmse nmse rsnr size size rsnr size rsnr figure comparison nmse function dimension model rmax rmax square matrix different values rsnr estimator rmax active set set singular values form max estimator rank using knowledge bulk edge signal matrix size decay singular values rmax rmax scale median value nmse gaussian noise realizations model function dimension orange grey areas represent error bands nmse two estimators errors bars nmse estimators functions dimension seen approach dominates numerically estimator values rsnr settings likely encountered practice simulated data used section proof main results proof proposition let first introduce notation definitions used proof let eigenvalues namely fixed signal matrix size nmse nmse singular values nmse size rsnr rsnr size rsnr figure fig another signal matrix let introduce function defined supp supp support random measure denotes dirac measure clear main difficulty proof show almost surely lim purpose follows matrix denote singular values hence one recall fixed matrix rank random matrix iid entries sampled gaussian distribution zero mean variance first step proof show random measure behaves asymptotically almost sure limit empirical spectral measure wishart matrix definition eigenvalues thus defined well know see theorem almost surely empirical spectral measure converges weakly distribution deterministic following recall convergence density also characterized cauchy stieltjes transform defined probability measure outside support one obtains almost surely lim cauchy transform moreover proposition convergence uniform compact subset follows weyl interlacing inequalities see theorem convention thanks results recalled asymptotic properties one may use inequalities prove almost surely random measure converges weakly distribution assumptions proposition using proposition shown exists almost surely sufficiently large fornany recall support supp random measure supp hence sufficiently large one supp supp therefore thanks weak convergence using ascoli theorem one may prove lim sup almost surely thanks assumptions one almost surely proposition hence almost surely sufficiently large one sup therefore using uniform convergence continuity one obtains almost surely lim since using equation relation follows immek almost surely diately lim lim completes proof technical result prove formulas recall key lemma needed prove formulas exponential family continuous case similar results already formulated different papers literature see review proposed lemma let random matrix whose entries independently sampled continuous exponential family canonical form distribution absolutely continuous respect lebesgue measure suppose function continuously differentiable let denote fij continuously differentiable function following relation holds fij fij proof using expression pdf random varibles one dyk fij yij exp yij dyij fij thanks condition follows dyk fij yij exp yij dyij exp therefore given exp yij integration part imply yij exp yij dyij dyk fij yij yij fij yij finally obtain fij fij since completes proof proof proposition remark mse using lemma fij condition follows definition exponential family remark yij exp yij dyij hence using integration parts twice arrive yij exp yij dyij complete proof suffices insert equalities proof proposition thanks one mkls using lemma fij condition follows thus inserting equality implies sukls recall fij unbiased estimator mkls assumption therefore thus sukls completes proof proof proposition thanks expression mkla risk data sampled poisson distribution follows log log mkla case poisson data one exp yij therefore applying hudson lemma fij log follows log log fij etj completes proof implementation details discuss algorithmic approach find spectral estimators first discuss compute spectral estimators expression risk estimators sukls continuous exponential families sure gaussian case provide respectively solution evaluated linear time contrary computations gsure beyond gaussian case pure pukla given respectively evaluated reasonable time rely respectively divergence div computation fij log fij without assumptions quantities requires operations general standard approach computation divergence suggested unbiasedly estimate simulations sampling following relation div random directions satisfying following similar first order approximation used two quantities fij etj fij log fij etj log fij entries chosen bernoulli distributed parameter advantage three approximations computed linear time making use results provide expression directional derivative given matrices whose columns matrices defined otherwise otherwise otherwise extended min references alter brown botstein singular value decomposition genomewide expression data processing modeling proceedings national academy sciences pnas august anderson guionnet zeitouni introduction random matrices volume cambridge studies advanced mathematics cambridge university press cambridge akaike new look statistical model identification automatic control ieee transactions bydder noise reduction data sets using singular value decomposition magn reson imaging bazerque mateos giannakis inference poisson count processes using tensor data pages icassp ieee international conference acoustics speech signal processing proceedings nadakuditi singular values vectors low rank perturbations large rectangular random matrices multivariate analysis brown fundamentals statistical exponential families applications statistical decision theory institute mathematical statistics baik silverstein eigenvalues large sample covariance matrices spiked population models journal multivariate analysis bai silverstein spectral analysis large dimensional random matrices springer series statistics springer new york second edition recht exact matrix completion via convex optimization found comput trzasko unbiased risk estimates singular value thresholding spectral estimators ieee trans signal choi taylor tibshirani selecting number principal components estimation true rank noisy matrix preprint cao xie poisson matrix recovery completion ieee trans signal processing deledalle estimation losses noisy recovery problems within exponential family electronic journal statistics donoho gavish minimax risk matrix denoising singular value thresholding ann dozier silverstein empirical distribution eigenvalues large dimensional matrices multivariate deledalle vaiter fadili dossal risk estimation matrix recovery spectral regularization presented icml workshop sparsity dictionaries projections machine learning signal processing edinburgh united kingdom edelman matrix jacobians wedge products mit handout efron estimation prediction error covariance penalties crossvalidation journal american statistical association pages eldar generalized sure exponential families applications regularization ieee transactions signal processing efron morris empirical bayes vector observations extension stein method biometrika efron morris multivariate empirical bayes estimation covariance matrices ann eckart young approximation one matrix another lower rank psychometrika gavish donoho gavish donoho optimal hard threshold singular values ieee trans information theory girard fast procedure large least squares problems noisy data numerische mathematik goodman fundamental properties speckle josa hall loss density estimation ann horn johnson topics matrix analysis cambridge university press cambridge new york melbourne suite matrix analysis hannig lee poisson signal estimation discrepancy squared risk statist plann inference optimal shrinkage singular values preprint hudson natural identity exponential families applications multiparameter estimation ann jolliffe principal component analysis springer series statistics springerverlag new york second edition josse sardy adaptive shrinkage singular values statistics computing pages lafond low rank matrix completion exponential family noise proceedings conference learning theory colt paris france july pages lam babacan haldar schuff liang denoising diffusionweighted magnitude image sequences using low rank edge constraints isbi pages ieee lewis sendov twice differentiable spectral functions siam journal matrix analysis matrix analysis applications ledoit wolf nonlinear shrinkage estimation covariance matrices ann nadakuditi optshrink algorithm improved signal matrix denoising optimal singular value shrinkage ieee trans inform theory nguyen peng liang spatiotemporal denoising spectroscopic imaging data approximations isbi pages ieee pastur lejay matrices statistique asymptotique des valeurs propres xxxvi volume lecture notes pages springer berlin ramani blu unser sure optimization regularization parameters general denoising algorithms ieee trans image processing raphan simoncelli learning bayesian without supervision advances neural inf process syst nips volume pages mit press shen huang analysis call centre arrival data using singular value decomposition research articles appl stoch model bus may shabalin nobel reconstruction matrix presence gaussian noise multivariate sun sun nonsmooth matrix valued functions defined singular values technical report department decision sciences national university singapore stein estimation mean multivariate normal distribution ann ulaby dobson handbook radar scattering statistics terrain norwood artech house udell horn zadeh boyd generalized low rank models foundations trends machine learning wall dyck brettin value decomposition analysis microarray data bioinformatics yanagimoto risk stein estimator conditional mle ann inst statist
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complexity computing graph luis pedro ignasi sep montpellier montpellier france lpmontejano algco project team cnrs lirmm montpellier france abstract graph denoted defined minimum size edge set whose removal leaves exactly two connected components containing least vertices graph invariant seen generalization minimum extensively studied combinatorial point view however little known complexity computing recently parameterized complexity community notion good edge separation graph defined happens essentially motivated relevance invariant combinatorial algorithmic points view article initiate systematic study computational complexity special emphasis parameterized complexity several choices parameters provide number results well keywords graph cut good edge separation parameterized complexity polynomial kernel introduction motivation graph invariant widely studied literature combinatorial point view since classical may suffice measure accurately connected graph deleting edges esfahanian hakimi proposed notion restricted given graph set least two connected components called restricted isolated vertices restricted minimum cardinality restricted inspired definition fiol proposed notion positive integer generalizing notion called every component extended abstract article appeared proceedings international workshop concepts computer science pages volume lncs garching germany june montejano sau least vertices assuming denoted defined minimum cardinality min note graph size minimum connected graph called exists let denote set edges two disjoint vertex sets let denote complement vertex set clear cut size graph exactly two connected components recently chitnis defined notion good edge separation algorithmic purposes two positive integers partition vertex set connected graph called edge separation connected holds admits edge separation thus notions defined independently existed connection far essentially good edge separations turned useful designing parameterized algorithms cut problems using technique known recursive understanding basically consists breaking input graph highly connected pieces considered problem efficiently solved mentioned kawarabayashi thorup defined similar notion introduced idea recursive understanding technique also subsequently used little known complexity computing edgeconnectivity graph spite extensive study combinatorics article initiate systematic analysis topic special emphasis parameterized complexity problem nutshell main idea identify relevant parameters input problem study running time algorithm solving problem depends chosen parameters see introductory textbooks area results consider following two problems concerning krestricted graph existential restricted erec instance graph positive integer question restricted rec instance connected graph positive integer output correct report computing graph latter problem seen generalization computing minimum cut graph solvable section prove even restricted graphs section study parameterized complexity rec problem precisely given connected graph two integers consider problem determining whether existing results concerning good edge separations imply problem fpt parameterized prove parameterized fpt parameterized unlikely admit polynomial kernels moreover prove erec problem fpt parameterized finally section also consider maximum degree input graph parameter prove erec problem remains graphs rec problem fpt parameterized note implies particular rec problem parameterized fpt graphs bounded degree table summarizes results article problem classical complexity parameterized complexity parameter npc even fpt fpt thm thm thm nph even fpt fpt thm fpt thm thm poly kernels thm thm thm table summary results denotes maximum degree input graph npc resp nph stands resp symbol denotes problem defined parameter research open questions determining existence polynomial kernels rec problem parameters fpt algorithm theorem quite inefficient improving bound maximum degree theorem studying parameterized complexity rec problem planar graphs sparse graph classes notation use standard notation see instance graph let denote maximum degree vertex degree denoted define define unless stated otherwise throughout article denotes number vertices input graph problem consideration always assume input graphs connected montejano sau preliminary results clearly connected graph computed polynomial time minimum cut algorithm however exist infinitely many connected graphs graphs containing cut vertex every component vertices graphs called flowers literature correspond exactly stars moreover erec problem hard indeed given graph even theorem determine whether contains two connected subgraphs order summarize discussion follows remark erec problem section strengthen hardness result case maximum degree input graph note remark implies rec problem problem furthermore even input graph guaranteed computing remains hard shown following theorem theorem rec problem restricted graphs proof prove even reduction minimum bisection restricted connected graphs known given connected graph even number vertices instance minimum bisection construct instance rec adding two universal vertices note since bipartition containing different parts induces two connected subgraphs claim necessarily belong different connected subgraphs optimal solution indeed let bipartition assume contradiction since connected vertex least one neighbor let note connected since least one neighbor adjacent vertices checked contradicting definition therefore solving rec problem corresponds exactly solving minimum bisection problem concluding proof given graph even number vertices minimum bisection problem consists partitioning two parts minimizing number edges one endpoint part computing graph parameterized analysis results previous section naturally lead considering parameterized versions problem section consider following three distinct parameterizations parameterized restricted instance connected graph two integers parameter integers parameter integer parameter integer question mentioned introduction determining whether corresponds exactly determining whether admits edge separation latter problem recently shown solvable time min log log chitnis lemma theorem chitnis problem fpt parameterized would like note improvement running time algorithm behind theorem would answer open question raised would direct consequences improve algorithms described pointed problem solved time roughly idea guess two sets vertices inducing connected subgraph contract two vertices call minimum cut algorithm words parameterized following theorem shows essentially best algorithm hope parameter indeed since parameter reduction linear fact solved time unless unlikely collapse occurs parameterized complexity theory implies problem solved time either theorem problem parameterized proof reduce known parameterized reduction one given downey theorem show cutting vertices graph problem analysis changes let graph wish determine whether construct graph follows start clique size representative vertices corresponding bijectively vertices montejano sau every representative vertex connected arbitrary vertices representative vertices fig illustration graph proof theorem see fig illustration consider take claim suppose first obviously connected vertices hand also connected least vertices finally straightforward check direction suppose edges exists connected two cases need distinguished case every vertex must representative vertex hence since every representative vertex degree hypothesis follows hence must case note every bipartition suppose consider bipartition contradiction therefore proof concludes applying case instead contrast theorem prove erec problem remark fpt parameterized proof uses technique splitters introduced naor also recently used designing parameterized algorithms main tool following lemma lemma chitnis exists algorithm given set size two integers outputs time min log set min log every two sets exists set computing graph theorem erec problem fpt parameterized precisely solved algorithm running time log log proof use easy property contains two trees order apply lemma take obtaining time log desired family subsets vertices trees exist necessarily exists set therefore order determine whether suffices check set whether contain connected component least vertices note set done linear time indeed set exists clearly otherwise property family contain two disjoint trees size therefore concerning parameterized complexity problem view theorems remains settle case parameter following theorem provides answer question need following result reformulation corollary lemma van bevern given graph vertices integer determining whether solved time explicit function depending theorem problem fpt parameterized proof solve problem using fpt algorithm given theorem parameter otherwise case proceed problem particular case fptwith parameter lemma follows vertex input graph integer let gpv graph obtained adding clique vertices edges vertex vertices claim holds exist vertex integer gpv proof assume first let partition achieving assume without loss generality let let vertex construction holds gpv conversely suppose exist vertex integer gpv let partition gpv achieving gpv assume without loss generality claim clique entirely contained indeed suppose contradiction let since gpv connected montejano sau necessarily distinguish two cases contradicting definition gpv otherwise use number edges least minimum cut clique size induced equal contradicting hypothesis partition gpv achieving gpv claim partition given defines edges concluding proof first note connected since gpv gpv connected hypothesis removal clearly preserves connectivity hand since integers latter inequality implies finally since holds note claim yields theorem implies problem deciding whether solved invoking times fpt algorithm given lemma complement theorem next theorem prove problem admit polynomial kernels parameterized unless conp theorem unless conp problem admit polynomial kernels parameterized proof proof strongly inspired one given van bevern theorem prove minimum bisection problem admit polynomial kernels turn resembles proof given garey prove minimum bisection main difference respect proof given need make appropriate modifications guarantee parts left connected issue minimum bisection problem first rule existence polynomial kernels generalization edges integer weights objective decide whether input graph partitioned two connected subgraphs least vertices removing set edges whose total weight exceed call problem remain get rid edge weights done end proof theorem shown bodlaender order prove admit polynomial kernels parameterized assuming conp sufficient define cross composition nphard problem case problem maximum cut see defined follows given graph integer one decide whether partitioned two sets computing graph least edges endpoint endpoint cross composition maximum cut parameterized consists algorithm given instances maximum cut constructs instance one instances maximum cut polynomially bounded function similarly may safely assume arguments given instances instances odd otherwise add consisting edgeless graph vertices given create follows see fig illustration let graph add vertices clique vertices whose edges weight add edge weight vertex vertex pair vertices add edge weight weight otherwise let two arbitrary distinct vertices call link vertices add two edges weight two edges weight completes construction edges among distinct called chain edges thicker edges fig finally set note polynomially bounded terms problem since parameter consider bounded construction clearly performed polynomial time claim exists maximum cut fig illustration graph proof theorem montejano sau assume first exists yesinstance maximum cut assume without loss generality let least edges proceed partition two sets connected total weight edges one endpoint one endpoint set contains set vertices containing exactly one possible since since odd let see connected set connected connected since contains exactly one link vertices graph clearly connected chain edges implies indeed connected proof connectivity similar using connected since turn connected since contains exactly one link vertices finally let show total weight edges note first two chain edges incident two chain edges incident belong cut defined chain edge belongs cut beside chain edges edges graph cut note clique vertices contains vertices since least edges weight belong cut therefore total weight cut conversely assume maximum cut want prove edgeweighted let partition connected weight edges minimized among partitions let since maximum cut bipartition cuts edges therefore total weight edges least since odd necessarily least one graphs cut assume first exactly one graph cut since value cut least thus claim always exactly one graph cut assume contradiction case two strictly positive values symmetry may assume total weight edges cut least computing graph construct another solution vertices cut one say note possible since order far exists pair proceed follows entirely contained otherwise choose contains exactly one end procedure exactly one graph cut finally arrange consecutively put vertices vertices indices counted cyclically connectivity guaranteed chain edges choice selector vertices let two indices procedure applied taking account four incident chain edges total weight edges cut new solution weight cut defined minus weight cut defined least used words defines cut strictly smaller weight contradicting definition conclude proof theorem remains deal edge weights show convert instance constructed equivalent instance resulting parameter remains polynomial given define instance unweighted graph obtained follows replace vertex clique size edge weight add pairwise disjoint edges cliques since cut size separate clique introduced vertex follows finally clear desired cut size still polynomial considering maximum degree parameter towards understanding parameterized complexity rec problem one may wonder whether considering maximum degree input graph montejano sau extra parameter turns problem easier classical approach parameterized complexity see instance first prove classical complexity point view bounding degree input graph turn problem easier stating hardness result need define matching problem short instance consists set disjoint sets set triples question whether exists matching covering element occurs exactly one triple instance represented bipartite graph see fig elements triples fig representation instance known even element appears triples theorem proved partitioning graph two connected subgraphs equal size using reduction worth noting graph constructed reduction contains two vertices degree greater five theorem appropriately modify reduction theorem constructed graph maximum degree theorem erec problem even maximum degree input graph proof given instance element appears triples define graph maximum degree follows see fig illustration set vertices computing graph elements triples fig construction graph proof theorem set triples every set edges eta etb eta tai etb tbi tai tbi every note maximum degree indeed vertices could get degree vertices degree since observe next show rec problem contains matching covering one direction easy suppose first contains matching covering let straightforward check connected conversely suppose partitioned connected subgraphs assume follows montejano sau since since every hence let since finally connected follows hence must matching covering order understand extent vertices high degree make complexity computing restricted graph hard also consider maximum degree input graph parameter problem theorem problem fpt parameterized maximum degree input graph proof algorithm based simple exhaustive search use property graph two integers contains two trees exists edge set trees belong different connected components hence determine whether trees exist every pair distinct vertices exhaustively consider trees vertices containing respectively note number trees every pair trees proceed follows contract tree resp single vertex resp keeping edge multiplicities run resulting graph minimum cut algorithm size returned desired trees otherwise continue searching clear overall running time algorithm acknowledgement would like thank anonymous referees helpful remarks improved presentation manuscript particularly grateful ideas prove theorem result left open question conference version article references balbuena carmona fiol extraconnectivity graphs large minimum degree girth discrete mathematics berman karpinski approximation hardness bounded degree mincsp electronic colloquium computational complexity bodlaender jansen kratsch kernelization lower bounds siam journal discrete mathematics bonsma ueffing volkmann leaving components order least three discrete mathematics computing graph chen huang kanj xia strong computational lower bounds via parameterized complexity journal computer system sciences chitnis cygan hajiaghayi pilipczuk pilipczuk designing fpt algorithms cut problems using randomized contractions siam journal computing cygan fomin jansen kowalik lokshtanov marx pilipczuk pilipczuk open problems school parameterized algorithms complexity http cygan fomin kowalik lokshtanov marx pilipczuk pilipczuk saurabh parameterized algorithms springer cygan kowalik pilipczuk open problems update meeting graph separation problems http cygan lokshtanov pilipczuk pilipczuk saurabh minimum bisection fixed parameter tractable proc acm symposium theory computing stoc pages diestel graph theory berlin edition downey fellows prieto rosamond cutting hard parameterized complexity related problems electronic notes theoretical computer science downey fellows parameterized complexity dyer frieze complexity partitioning graphs connected subgraphs discrete aplied mathematics dyer frieze planar journal algorithms esfahanian hakimi computing conditional graph information processing letters fiol extraconnectivity graphs large girth discrete mathematics flum grohe parameterized complexity theory garey johnson computers intractability guide theory freeman garey johnson stockmeyer simplified graph problems theoretical computer science holtkamp connectivity graphs digraphs maximizing edgeand emphasis local connectivity properties phd thesis rwth aachen university holtkamp meierling montejano graphs discrete applied mathematics kawarabayashi thorup minimum cut bounded size tractable proc annual symposium foundations computer science focs pages kim paul sau thilikos parameterized algorithms minmax multiway cut list digraph homomorphism proc international symposium parameterized exact computation ipec volume lipics pages marx parameterized graph separation problems theoretical computer science montejano sau marx pilipczuk everything always wanted know parameterized complexity subgraph isomorphism afraid ask proc international symposium theoretical aspects computer science stacs volume lipics pages naor schulman srinivasan splitters derandomization proc annual symposium foundations computer science focs pages niedermeier invitation algorithms oxford university press rai ramanujan saurabh parameterized algorithm mixedcut proc latin american symposium theoretical informatics latin volume lncs pages stoer wagner simple algorithm journal acm van bevern feldmann sorge parameterized complexity computing graph bisections proc international workshop concepts computer science volume lncs pages yuan liu sufficient conditions graphs discrete mathematics zhang yuan proof inequality concerning edgeconnectivity discrete mathematics
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efficient deep learning hashing neural network mobile visual search heng liu liang liu oct beijing key laboratory intelligent telecommunication software multimedia beijing university posts telecommunications beijing china qiheng liuwu liangliu abstract mobile visual search applications emerging enable users sense surroundings smart phones however particular challenges mobile visual search achieving high recognition bitrate becomes consistent target previous related works paper propose deep hashing approach constructing binary hash codes mobile visual search first exploit architecture mobilenet model significantly decreases latency deep feature extraction reducing number model parameters maintaining accuracy second add layer mobilenet train model labeled mobile visual data evaluations show proposed system exceed accuracy performance terms map importantly memory consumption much less deep learning models proposed method requires memory neural network achieves map mobile location recognition dataset used testing index mobile visual search supervised hashing binary code deep learning introduction proliferation mobile devices becoming possible use mobile perception functionalities cameras gps perceive surrounding environment among techniques mobile visual search plays key role mobile localization mobile media search mobile social networking however rather simply porting traditional visual search methods mobile platforms mobile visual search one must face challenges large auralvisual variance queries stringent memory computation constraints network bandwidth limitations desire instantaneous search experience work partially supported open research fund national natural science foundation china fundamental research funds central universities input image convolutional layers latent layer loss kernel depthwise convolutional filters kernel pointwise convolutional filters fig architecture proposed deep learning hashing neural network mobile visual search research mobile visual search predominantly focused achieving high recognition bitrates recently increasing number researchers attempting exploit feature signatures produced hashing mobile visual search good balance achieved among computation memory requirements training efficiency quantization complexity search performance however existing mobile visual search methods attempt compress existing classical handcrafted features binary code methods focus decrease loss suffered compression attempt automatically learn effective binary code features image dataset using deep neural network two main reasons lack effective deep learning hashing methods mobile visual search high computational complexity existing deep neural networks traditional visual search convolutional neural networks become ubiquitous studies shown deep features learned using networks capture rich image representations enable better performance handcrafted features visual classification object detection semantic segmentation image retrieval general trend research deep learning methods construct deeper complicated networks achieve higher accuracy field mobile visual search however deeper neural network model consumes memory time easily supplied mobile devices adapt deeper neural networks mobile devices new network architecture called mobilenet recently proposed google mobilenet model standard convolutional layer decomposed depthwise convolutional layer pointwise convolutional layer thereby greatly reducing amount calculation necessary model size model works well image classification problems inspired works presented develop effective efficient mobile visual search system paper proposes combine network architecture hash function incorporate hash function latent layer image representations classification outputs mobilenet allows maintain accuracy visual search also adapt model mobile environment overview system illustrated figure final binary hash codes learned minimizing objective function defined classification error experimental results mobile location recognition dataset show method achieves superior performance compared hashing approaches architecture section first describe mobilenet network structure convolution layer structure based depthwise separable filters pointwise separable filters describe hash layer convolutional neural network train convolutional neural network model exploits semantic labels automatically create binary codes mobilenet mobilenet model built depthwise separable convolutions convolutions factorize standard convolution depthwise convolution convolution called pointwise convolution mobilenet architecture defined table mobilenet depthwise convolution involves application single filter input channel pointwise convolution involves application convolution combine outputs depthwise convolution standard convolutional layer takes input feature map produces feature map spatial width height square input feature map number input table hashing network mobile visual search type stride conv conv conv conv conv conv conv conv conv conv conv conv conv conv conv conv conv conv conv avg pool sigmoid softmax filter shape pool place classifier input size nels input depth spatial width height square output feature map number output channel standard convolution following computational cost corresponding cost depthwise separable convolutions follows sum costs depthwise pointwise convolutions expressing standard convolution process filtering combining achieve following reduction computational cost mobilenet uses depthwise separable convolutions require times less computation standard convolutions small reduction accuracy table compares mobilenet popular models terms accuracy assessed imagenet database number million number parameters million mobilenet nearly accurate times smaller times less compute intensive accurate googlenet smaller requiring times less computation table comparisons mobilenet popular deep learning models model accuracy parameters alexnet squeezenet googlenet mobilenet mobilenet also accurate squeezenet alexnet nearly identical model size less computation hash function hash mapping required learning images based following principles hash codes respect semantic similarity image labels images share class labels mapped similar binary codes let denote images let associated label vectors total number class labels entry value image belongs corresponding class otherwise goal learn mapping maps images binary codes preserving semantic similarity relationships among image data network built mobilenet layer followed batchnorm layer nonlinear relu layer exception fully connected layer classification final average pooling layer reduces spatial resolution fully connected layer incorporate learned features binary codes add latent layer units top layer illustrated figure latent layer fully connected uses sigmoid units activations bounded let denote weights projection matrix latent layer given image feature vector layer activations units computed bias term logistic sigmoid function defined exp real value binary encoding function given sign sign sign sign otherwise sign performs operations matrix vector experiments evaluate proposed method conducted experiments mobile location recognition dataset presented table maps different hashing methods mobile location recognition method vhb ssfs dlbh ssdh method map dataset contains images captured locations used experimental parameters implemented approach using caffe package initialized network parameters adopting parameters mobilenet trained million images imagenet dataset parameters hash layer randomly initialized learning rate initialized decreased previous value every iterations entire training procedure terminated iterations learning network parameters conjunction backpropagation exploited stochastic gradient descent algorithm size images minimize classification error model lightweight modification mobilenet thus easy implement evaluation used mean average precision map evaluation criterion ranked images according hamming distances query image selected top images ranked list retrieval results computed map retrieved images set experiments used class labels ground truth adopted common settings computing map examining whether retrieved images query shared common class labels compared method visual hash bits vhb fingerprint hash codes ssfs traditional hashing methods mobile location recognition two methods considered comparison deep learning binary hash codes dlbh supervised deep hashing ssdh hashing methods table compares performances dataset different hashing methods different hash code lengths results see accuracies deep learning hashing methods greatly exceed traditional hashing methods demonstrates power deep learning technology binary code learning furthermore ssdh learns feature representations binary codes simultaneously imposes constraints binary code learning achieves higher performance dlbh method achieves best results dataset different hash code lengths especially hash code length relatively short method achieve extremely high accuracy attributed fact mobilenet networks enable joint learning representations hash functions images moreover learned representations effective stable models dlbh ssdh finally sizes dlbh ssdh models greater whereas mobilenet model size thus memory demand mobile device reduced factor approximately chuang gan ming lin yang gerard melo alexander hauptmann concepts alone exploring pairwise relationships video activity recognition aaai conclusion girshick donahue darrell malik rich feature hierarchies accurate object detection semantic segmentation cvpr present effective deep learning framework based mobilenet creating binary codes mobile visual search framework incorporate hash function latent layer feature layer output layer mobilenet network optimizing objective function defined classification error method jointly learns binary codes features classification results evaluate performance proposed network applied mobile location recognition dataset experimental results demonstrate achieve map improvement compared methods model requires memory references liu zhao chen deep learning hashing mobile visual search eurasip image video processing vol liu mei zhang instant mobile video search layered indexing progressive transmission ieee trans multimedia vol chandrasekhar takacs chen compressed histogram gradients descriptor ijcv vol feng liu mobile product search bag hash bits boundary reranking cvpr russakovsky deng krause imagenet large scale visual recognition challenge ijcv vol liu mei zhang che luo multitask deep embedding video thumbnail selection cvpr xia pan lai supervised hashing image retrieval via image representation learning aaai gan yao yang yang mei lead exceed video concept learning jointly exploiting web videos images ieee computer vision pattern recognition gan yang zhu zhao zhuang recognizing action using name approach international journal computer vision vol ciresan meier schmidhuber deep neural networks image classification cvpr sermanet eigen zhang overfeat integrated recognition localization detection using convolutional networks corr vol krizhevsky sutskever hinton imagenet classification deep convolutional neural networks nips chu jiang wang zhang huang robust spatial consistency graph model partial duplicate image retrieval ieee trans multimedia vol simonyan zisserman deep convolutional networks image recognition corr vol szegedy vanhoucke ioffe rethinking inception architecture computer vision corr vol gan wang yang yeung hauptmann devnet deep event network multimedia event detection evidence recounting cvpr howard zhu chen kalenichenko mobilenets efficient convolutional neural networks mobile vision applications corr vol szegedy liu jia going deeper convolutions cvpr iandola moskewicz ashraf squeezenet accuracy fewer parameters model size corr vol yang lin chen supervised learning hashing via deep neural networks image search corr vol glorot bordes bengio deep sparse rectifier neural networks ais wang zhao ssfs spacesaliency fingerprint selection framework crowdsourcing based mobile location recognition amip lin yang hsiao deep learning binary hash codes fast image retrieval cvpr workshops jia shelhamer donahue caffe convolutional architecture fast feature embedding acm
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boltzmann machines takayuki osogami sep ibm research tokyo osogami version abstract review boltzmann machines extended models often recurrent structure back propagration time bptt used learn parameters perstep computational complexity bptt online learning however grows linearly respect length preceding learning rule local time limits applicability bptt online learning review dynamic boltzmann machines dybms whose learning rule local time dybm learning rule relates dependent plasticity stdp postulated experimentally confirmed biological neural networks introduction boltzmann machine stochastic model representing probability distributions binary patterns paper review boltzmann machines studied stochastic generative models boltzmann machines define probability distributions binary patterns also modified deal patterns similar boltzmann machines modified patterns gaussian boltzmann machines see section follow probabilistic representations intuitive interpretations terms probabilities section start conditional restricted boltzmann machine crbm conditional boltzmann machine section gives conditional probability next pattern given fixed number preceding patterns limitation crbm take account dependency within fixed horizon increase length horizon complexity learning grows accordingly overcome limitation crbms researchers proposed boltzmann machines recurrent structures review section include spiking boltzmann machines temporal restricted boltzmann machines trbms recurrent temporal restricted boltzmann machines rtrbms extensions models standard approach learning models recurrent structures back propagation time bptt however bptt undesirable learn online manner update parameters model every time new pattern arrives online learning needed want quickly adapt changing environment sufficient memory store unfortunately computational complexity bptt online learning grows linearly respect length preceding computational complexity limits applicability bptt online learning section review dynamic boltzmann machine dybm extensions dybm computational complexity online learning independent length preceding discuss learning rule dybm relates dependent plasticity stdp postulated experimentally confirmed biological neural networks survey paper based personal note prepared third four parts tutorial given international joint conference artificial intelligence held melbourne australia august see tutorial information tutorial survey corresponding first part tutorial boltzmann machines models found follow definitions notations used learning models consider possibly denotes binary pattern vector time use denote patterns time goal learning maximize given collection multiple respect distribution defined model consideration use denote set parameters model log log denotes conditional probability pattern time given patterns time denotes probability pattern time interpreted empty history study models probability represented energy follows exp exp summation respect possible binary length summation respect possible hidden values gradient represented follows see represents random represents random hidden values denotes expectation respect model distribution etarget denotes expectation respect target distribution case empirical distribution single given target reduced https words maximized maximizing sum log exp exp conditional energy given gradient given analogously target single boltzmann machines model represent conditional probability used section start boltzmann machine used model order markov model arbitrarily determined order markov models conditional probability represented conditional restricted boltzmann machines figure shows particularly structured boltzmann machine called conditional restricted boltzmann machine crbm crbm represents conditional probability side figure set crbm consists layers visible units layer hidden units units within layer connections units different layers may connected visible layer corresponds pattern time crbm conditional boltzmann machine shown figure particular structure represent visible layers corresponding input visible layer corresponding output parameters crbm independent formally energy crbm given whv output input hidden represent conditional probability follows see hidden single hidden layer visible output input output input visible hidden multiple hidden layers figure conditional restricted boltzmann machines exp exp summation respect possible binary hidden patterns summation respect defined analogously one learn parameters whv model following method section extensions conditional restricted boltzmann machines crbm extended various ways taylor study crbm multiple layers hidden units see figure memisevic hinton study crbm extended interactions higher order boltzmann machine refer gated crbm specifically energy gated crbm involves denotes input values denotes output values denotes hidden values drawback gated crbm increased number parameters due interactions taylor hinton study factored crbm interaction represented reduced number parameters follows summation respect set factors consideration forward connections required purposes extrapolating hidden input visible output time structure spiking boltzmann machine used figure form temporal figure spiking boltzmann machine studied use figure version available http boltzmann machines recurrent structures spiking boltzmann machines spiking boltzmann machine studied shown essentially equivalent boltzmann machine illustrated figure boltzmann machine consists input units output units hidden units input units represent historical values visible units hidden units although hidden units random simply given input hinton brown make approximation using sampled values input hidden units specifically given visible values sampled hidden values time energy visible values hidden values time represented follows whv whh wvh whv wvv arbitrarily chosen function target learning namely boltzmann machine infinite number units characterized finite number parameters wvv wvh whv whh figure shows specific used notice boltzmann machine figure seen restricted boltzmann machine rbm whose bias weight depend represented although clear descriptions labels horizontal axis probably shifted one bias hidden units bias visible units weight visible units hidden units whv wvh wvv represent conditional probability follows exp exp discuss choice appears case figure case energy reduced connections visible units hidden units time hidden values affect distribution visible values role hidden units sampled hidden values used update bias problem mechanism allows learn appropriate values wvh whh observe succeeding visible values namely hidden values sampled dependency wvh whh whether sampled hidden values good known hidden values used input helps learn appropriate values whv wvh whh see discussion temporal restricted boltzmann machines sutskever hinton study model related crbm refer temporal restricted boltzmann machine trbm crbm defines conditional distribution visible hidden values time given visible values time trbm defines corresponding conditional probability given visible values hidden values time see figure similar crbm parameters trbm depend time unlike crbm trbm conditional rbm trbm single parameter used every distribution hidden values shared among trbm varying particular hidden values trbm depend future visible values dependency makes learning inference hard ignored namely values time conditionally independent values time given values time particular distribution hidden values time completely determined visible values time distribution hidden values time thus considered input use trbm define conditional distribution values time see figure sampled values hidden units trbm crbm visible output input hidden trbm visible output input hidden trbm approximations figure temporal restricted boltzmann machines gray circles indicate expected values used input hidden units furthermore expected values see section used hidden values input hidden units figure takes real values completely determined visible values time formally approximations trbm parameter defines probability distribution visible hidden values follows exp exp conditional distribution defined boltzmann machine shown figure expected hidden values specifically used compute subsequently used expectation respect notice computed deterministic manner dependency however dependency ignored learning trbms recurrent temporal restricted boltzmann machines overcome intractability trbm without approximations sutskever study refined model trbm refer recurrent temporal restricted boltzmann machine rtrbm hidden input visible output figure recurrent temporal restricted boltzmann machine rtrbm simplifies trbm removing connections visible layers connections hidden layers separated one lag means visible hidden values time conditionally independent visible values time hidden values time given hidden values time similar approximation made trbm figure rtrbm uses expected values hidden values time defines conditional distribution visible hidden values time binary values see figure formally let denote expected values hidden units time random vector representing hidden values time represents conditional expectation given visible values time probability distribution values time given exp exp conditional energy given parameters bias visible units bias hidden units weight matrix visible units hidden units weight matrix previous expected value hidden units hidden units inference marginal conditional distribution visible values time represented follows exp exp conditional given log exp visible values time given conditional probability distribution hidden values time represented follows exp exp term canceled numerator denominator conditional energy given corollary hidden values time conditionally independent given exp binit exp element follow allow hidden units time bias binit differ expected values thus given exp operations defined elementwise figure recurrent temporal restricted boltzmann machine unfolded time learning parameters rtrbm trained back propagation time analogous recurrent neural networks contrastive divergence understand train rtrbms figure shows rtrbm unfolded time recall expected values hidden units deterministically updated according hence understood hidden values recurrent neural network rnn rtrbm seen rnn gives rbm output instead real values would given output standard rnn derive learning rule rtrbm closely following using notations rtrbm unfolded time energy represented follows binit maximize given approach need gradient energy respect parameter caveat energy depends turn depends recursive manner also expectation respect heta needs computed approximation contrastive divergence see section first study last term let last term taking partial derivative respect obtain denotes row denotes element last equality follows notations write denotes elementwise multiplication function therefore partial derivative respect given recursively follows take derivative respect parameters starting dui last equality follows notations write given gradient respect parameters derived follows gradients used show following gradients energy given gradient given follows extensions rtrbm extended various ways mittleman study structured rtrbm units partitioned blocks connections particular blocks allowed lyu replaces rnn rtrbm one long memory lstm schrauwen buesing replaces rnn rtrbm echo state network slightly generalizes rtrbm relaxing constraint rtrbm must expected value namely rnn gives rbm output rnn rbm share parameters rtrbm shares parameters rnn rbm dynamic boltzmann machines bptt desirable online learning update every time new pattern observed computational complexity bptt online learning grows linearly length preceding online learning however needed example store observed patterns memory want adapt changing environment dynamic boltzmann machine dybm proposed model allows efficient online learning computational complexity learning rule dybm independent length preceding section start reviewing dybm introduced relation learning rule dependent plasticity stdp section study relaxation constraints dybm required way becomes suitable inference learning primary purpose constraints mimic particular form stdp relaxed dybm generalizes original dybm allows interpret form logistic regression data section review dybms dealing dybms analogous gaussian boltzmann machines deal patterns opposed boltzmann machines binary values gaussian dybm related vector autoregressive var model specifically show special case gaussian dybm var model additional variables capture long term dependency additional variables correspond dybm eligibility traces represent recently frequently spikes arrived neuron another also review extension gaussian dybm deal patterns continuous space wij figure dynamic boltzmann machine unfolded time figure models learning algorithms section implemented python java https dynamic boltzmann machines finite dynamic boltzmann dybm defined limit sequence boltzmann machines consisting layers tends infinity see figure formally defined crbm see section layers visible units layers input units one layer output units hidden units conditional energy defined weight assumes particular parametric form finite number parameters independent discuss following parametric form weight motivated observations biological neural networks leads particularly simple exact efficient learning rule biological neural networks stdp postulated supported experimentally particular weight neuron neuron strengthened neuron fires generates spike shortly neuron fires long term potentiation ltp weight weakened neuron fires shortly neuron fires long term depression ltd dependency timing spikes missing hebbian rule boltzmann machine see learning rule characteristics stdp ltp ltd assumes weight form illustrated figure define weight sum two weights section closely follows weight wij wij dji dij wji figure figure illustrates equation particular forms equation figure horizontal axis represents vertical axis represents value solid curves dashed curves dotted curves notice defined discontinuous hand defined discontinuous respectively recall assume paper otherwise simplicity assume single decay rate single decay rate opposed multiple ones simplicity assume conduction delay uniform connections opposed variable conduction delay see also ways tune values conduction delay figure value high conduction delay unit unit namely neuron likely fire immediately spike unit arrives delay likelihood controlled ltp weight value gradually decreases increases effect stimulus spike arrived unit diminishes time value low suggesting unit unlikely fire immediately spike unit arrives unlikelihood controlled ltd weight decreases magnitude gradually decreases get smaller represents weight spike neuron generated spike neuron neural eligibility trace synaptic eligibility trace fifo queue neuron neuron figure connection neuron neuron dybm dynamic boltzmann machine limit sequence finite dynamic boltzmann dybm defined limit sequence crbm also define limit sequence conditional probability defined limit considered conditional probability defined dybm likewise conditional energy dybm defined limit sequence conditional energy specifically conditional energy converges convergence due parametric form conditional energy turn defines conditional distribution via hidden units although conditional energy dybm involves infinite sum evaluated finite sum parametric form fact dybm understood artificial model spiking neural network computation inference learning performed locally synapse using information available around synapse specifically neuron connected neuron via fifo queue synapse see figure discrete time neuron either fires spike travels along fifo queue reaches synapse conduction delay words fifo queue length stores time spikes generated neuron time time synapse dybm stores quantity called synaptic eligibility trace value synaptic eligibility increases spike arrives synapse fifo queue otherwise decreased constant factor specifically time value synaptic eligibility trace stored synapse neuron updated follows decay rate satisfies figure shows example value synaptic eligibility trace changes depending spikes arrived synapse observe represents recently frequently spikes arrived neuron represented follows eligibility spike values time dybm step figure value synaptic neural eligibility trace function time synaptic eligibility trace synapse bars represent spikes arrived fifo queue synapse neural eligibility trace neuron bars represent spikes generated neuron neuron dybm stores quantity called neural eligibility value neural eligibility increases neuron fires otherwise decreased constant factor specifically time value neural eligibility trace neuron updated follows decay rate satisfies observe represents recently frequently neuron fired represented follows neuron dybm fires according probability distribution depends energy dybm neuron likely fire energy becomes lower fires otherwise let energy associated neuron time depend whether fires time well preceding spiking activities neurons dybm firing probability neuron given exp exp specifically represented follows ltp ltd bias parameter neuron represents likely spikes likely fire section closely follows assume single neural eligibility trace opposed multiple ones neuron large positive value define ltp ltd represents soon frequently spikes arrive synapse fifo queues although also represented recursive manner recursively computed prone numerical instability summation respect neurons connected weight parameter represents strength long term potentiation ltp weight parameter thus referred ltp weight neuron likely fire large neuron connected spikes recently arrived corresponding positive large ltp strong summation respect neurons connected summation respect neurons connected represents strength long term depression referred ltd weight neuron less likely fire large neuron connected spikes soon frequently reach corresponding positive large ltd strong second term represents neuron less likely fire neuron recently frequently fired large strength ltd given notice timing spike measured respect spike reaches synapse spike neuron delay spike neuron reaches immediately learning rule dybm derived way maximizes log likelihood given respect probability distribution given specifically time dybm updates parameters according hxi neurons learning rate training data given time denotes expected value firing probability neuron time according probability distribution given following stochastic gradient methods learning rate may adjusted time relation dependent dependent plasticity stdp amount change weight two neurons fired together depends precise timings two neurons fired stdp supplements hebbian rule experimentally confirmed biological neural networks section closely follows increased ltp gets stronger given becomes likely fire spikes recently frequently arrived large amount change depends exhibiting key property stdp particular increased large amount spikes recently frequently arrived according second term side increased ltd gets stronger given neuron becomes less likely fire spikes expected reach soon large amount change large spikes fifo queue close according last term increased given amount change proportional frequently recently fired learning rule thus exhibits key properties ltd stdp increased given becomes likely fire accordance training data amount change small already likely fire dependency considered form homeostatic plasticity related work significant amount prior work towards understanding stdp perspectives machine learning example nessler show stdp understood approximating expectation maximization algorithm nessler study particularly structured network learning rule maximizing log likelihood given static patterns hand dybm assume particular structures network learning rule properties stdp applies synapse network also learning rule dybm maximizes log likelihood given learning rule involve approximations beyond assumed stochastic gradient methods giving flexibility shown dybm section capability associative memory anomaly detection sequential patterns applications dybm limited simple tasks relatively low dimensional relax constraints dybm way gives flexibility useful learning inference specifically observe first term side rewritten definition follows let represents unlikely fires time fired time parametric form assumes ltd weight decays geometrically interval two spikes increases following relax constraint assumes ltd weights take independent values energy dybm neurons represented section closely follows conveniently matrix vector operations vector matrix boldface letters defined analogously vector lowercase matrix uppercase eligibility traces append subscript explicitly represent dependency decay rate functional form energy completely determines dynamics dybm relaxing constraints allows dybm represent wider class dynamical systems notice last term divided two terms vector synaptic eligibility traces decay rate comparing find without loss generality energy dybm represented following form define energy reduces original energy one also incorporate multiple synaptic neural eligibility traces varying decay rates equivalently represent energy using neural eligibility traces instead synaptic eligibility traces follows learning rule notations learning rule corresponding representation follows conditional expectation respect exp exp specifically exp exp exponentiation division vectors elementwise form implies dybm kind logit model feature vector depends prior values applying learning rules given given learn parameters dybm equivalently parameters logit model dynamic boltzmann machines gaussian dynamic boltzmann section show dybm deal form gaussian dybm gaussian dybm assumes follows gaussian distribution exp given variance parameter gaussian distribution contrast bernoulli distribution dybm given conditional energy gaussian dybm represented follows term depend canceled numerator denominator omit conditional energy letting matrix whose element matrix whose element conditional energy gaussian dybm represented follows conditional energy gaussian dybm may compared energy gaussian bernoulli restricted boltzmann machine see section closely follows derive learning rule gaussian dybm way maximizes given log log summation time steps conditional independence given fundamental property dybm shown approach stochastic gradient update parameters gaussian dybm step according gradient conditional probability density log log equality follow derive derivative respect parameter follows log log log log parameters thus updated learning rate follows division exponentiation vectors elementwise given maximum likelihood estimator gaussian dybm given gaussian dybm thus understood modification standard vector autoregressive var model specifically last term side involves eligibility traces understood features historical values added new variables var model value eligibility traces depend infinite past gaussian dybm take account history beyond lag natural section study learning rule based natural gradient gaussian dybm consider stochastic model gives probability density pattern natural gradients parameters stochastic model updated follows log step learning rate denotes fisher information matrix log log due conditional independence suffices derive natural gradient gaussian unit consider parametrization mean variance probability density function gaussian distribution represented parametrization follows exp log likelihood given log log log hence gradient inverse fisher information matrix given follows log parameters updated follows context gaussian dybm mean given linear respect also variance given hence natural gradient gives learning rules parameters follows exponentiation vector elementwise compare standard gradient gives section closely follows using nonlinear features gaussian dybms gaussian dybm linear model limited capability modeling complex way take account features gaussian dybm apply nonlinear mapping input feed resulting features additional input gaussian dybm example mapping echo state network esn esn maps input sequence recursively follows tanh wrec win wrec win randomly chosen fixed leak parameter satisfying tanh hyperbolic tangent function may replaced nonlinear functions sigmoid function eligibility trace may considered linear counterpart nonlinear features created esn features generated mappings fixed parameters given input gaussian dybm learning rules gaussian dybm stay unchanged nonlinear dybm uses esn slightly different manner functional dynamic boltzmann machines review functional dybm models functions patterns continuous space recall gaussian dybm defines conditional distribution next vector given preceding sequence vectors functional dybm defines conditional distribution next function given preceding sequence partial observations preceding functions time set points observed number points observed functional dybm assumes conditional distribution given gaussian process whose mean varies time depending preceding functions follows functional bias functional weight functional eligibility trace covariance gaussian process consists two components arbitrary kernel delta function hyperparameter tractability kajino proposes particular parametrization functional bias functional weight let set arbitrarily selected points spectral radius wrec set smaller row vector column vector reproducing kernel hilbert space kernel substituting gives following expression column vector element vector recursively updated follows use collectively denote parameters observed kajino uses maximum posteriori map estimator matrix element column vector defined analogously objective learning functional dybm maximize log likelihood observed values conditional probability density functional values locations time given exp objective thus maximize log term independent gradient given figure dynamic boltzmann machine hidden units modified figure gives gradient implies following learning rule stochastic gradient learning rate estimated map estimator dynamic boltzmann machines hidden section study dybm hidden units see figure layer dybm corresponds time two parts visible hidden visible part layer represents values time hidden part represents values hidden units time units within layer connections let define analogously boltzmann machine figure bias parameter weight parameter let parameters connected visible units units past conditional energy boltzmann machine given follows section closely follows define assume following parametric form decay rate satisfying conditional energy represented follows corresponds eligibility trace dybm define eligibility trace hidden part analogously energy gives conditional probability distribution given specifically exp exp exp exp binary vectors learning dynamic boltzmann machine hidden units dybm hidden units gives probability denotes summation possible values hidden units time arbitrarily define seek maximize log likelihood given maximizing lower bound given jensen inequality log log log log log summation respect possible values learning weight visible units gradient lower bound respect given log side summation expected gradients suggests method stochastic gradient namely step sample according update basis log learning rule equivalent one model units visible except values hidden units given sampled values therefore learning rule follows directly section denotes conditional expectation respect make explicit computed sampled hidden values take gradient respect learning weight hidden units log log log plugging side obtain log log similar expression suggests method stochastic gradient time sample according update basis following stochastic gradient log log learning rules derived follows log log log log denotes conditional expectation respect computation involves mainly two interrelated inefficiencies first although approximately computed using sampled hidden values way samples reused updating sampled distribution previous parameter second since summand depends also recomputed update thus computational complexity grows linearly respect length contrast whose complexity independent length observe consists products log log without dependency log parameter updated way likely generated learning rule would equivalent visible units update rule undesirable sampled necessarily want sample dependency log suggests updated large amount sampled happens make future values likely intuitively weighting log log inevitable whether particular values hidden units good purpose predicting future values known seeing future values approximations one could approximately compute recursively log discount factor recursive update rule puts exponentially small weight log computed old value recursively computed related momentum gradient descent value computed latest values let value immediately step recursive computation learning rules approximated following learning rules log log log log sample according quantity computed recursively present alternative approach learning dybm hidden units bidirectional manner consider backward dybm shares parameters forward dybm key observation parameters difficult learn forward dybm relatively easy learn backward dybm training forward dybm backward dybm effectively learn parameters forward dybm conclusion reviewed boltzmann machines modeling boltzmann machines used prediction anomaly detection based observed may also used generate human motion music movies use boltzmann machines one approach modeling learning popular models include limited recurrent neural networks long short term memory autoregressive models hidden markov models seen examples best model particular application might obtained appropriately combining existing models acknowledgments work supported jst crest grant number japan author thanks diyuan pointing several typographical errors original version references abbott nelson synaptic plasticity taming beast nature neuroscience ackley hinton sejnowski learning algorithm boltzmann machines cognitive science amari nagaoka methods information geometry oxford university press bengio mesnard fischer zhang stdp presynaptic activity times rate change postsynaptic activity poo synaptic modifications cultured hippocampal neurons dependence spike timing synaptic strength postsynaptic cell type journal neuroscience bottou online learning stochastic approximations saad editor learning neural networks chapter pages cambridge university press bengio vincent modeling temporal dependencies sequences application polyphonic music generation transcription proceedings international conference machine learning pages dasgupta osogami nonlinear dynamic boltzmann machines prediction aaai conference artificial intelligence january dasgupta yoshizumi osogami regularized dynamic boltzmann machine delay pruning unsupervised learning temporal sequences proceedings international conference pattern recognition duchi hazan singer adaptive subgradient methods online learning stochastic optimization journal machine learning research hebb organization behavior neuropsychological approach wiley hinton brown spiking boltzmann machines advances neural information processing systems pages november hinton salakhutdinov reducing dimensionality data neural networks science hinton sejnowski optimal perceptual inference proc ieee conference computer vision pattern recognition pages june hochreiter schmidhuber long memory neural computation jaeger haas harnessing nonlinearity predicting chaotic systems saving energy wireless communication science kajino functional dynamic boltzmann machine proceedings international joint conference artificial intelligence pages kingma adam method stochastic optimization proceedings international conference learning representations iclr krizhevsky learning multiple layers features tiny images master thesis computer science department university toronto toronto canada lazar pipa triesch sorn recurrent neural network frontiers computational article new introduction multiple time series analysis berlin heidelberg lyu zhu polyphonic music modelling proceedings acm international conference multimedia pages october marks movellan diffusion networks products experts factor analysis proceedings third international conference independent component analysis blind source separation memisevic hinton unsupervised learning image transformations proceedings ieee conference computer vision pattern recognition cvpr pages mittelman kuipers savarese lee structured recurrent temporal restricted boltzmann machines proc annual international conference machine learning icml pages june nessler pfeiffer buesing maass bayesian computation emerges generic cortical microcircuits plasticity plos computational biology osogami learning binary via dependent plasticity corr presented computing spikes nips workshop barcelona spain december osogami boltzmann machines models technical report ibm research tokyo osogami dasgupta learning values hyperparameters dynamic boltzmann machine ibm journal research development appear osogami kajino sekiyama bidirectional learning models hidden units proceedings international conference machine learning icml pages august osogami otsuka learning dynamic boltzmann machines dependent plasticity technical report ibm research osogami otsuka seven neurons memorizing sequences alphabetical images via dependent plasticity scientific reports qian momentum term gradient descent learning algorithms neural networks official journal international neural network society rumelhart hinton williams learning internal representations error propagation rumelhart mcclelland pdp research group editors parallel distributed processing explorations microstructure cognition vol foundations chapter mit press scellier bengio equilibrium propagation bridging gap models backpropagation schrauwen buesing hierarchy recurrent networks speech recognition nips workshop deep learning speech recognition related applications sutskever hinton learning multilevel distributed representations sequences proceedings eleventh international conference artificial intelligence statistics volume pages journal machine learning research proceedings track sutskever hinton taylor recurrent temporal restricted boltzmann machine advances neural information processing systems pages december taylor hinton factored conditional restricted boltzmann machines modeling motion style proc annual international conference machine learning icml pages june taylor hinton roweis modeling human motion using binary latent variables platt hoffman editors advances neural information processing systems pages mit press tieleman hinton lecture divide gradient running average recent magnitude coursera neural networks machine learning turrigiano nelson homeostatic 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cascade locally dissipative realizations linear quantum systems pure gaussian state covariance assignment apr matthew woolley ian petersen naoki yamamoto school engineering information technology university new south wales australian defence force academy canberra act australia department applied physics keio university yokohama japan abstract paper presents two realizations linear quantum systems covariance assignment corresponding pure gaussian states first one called cascade realization given covariance matrix corresponding pure gaussian state construct cascaded quantum system generating state second one called locally dissipative realization given covariance matrix corresponding pure gaussian state satisfies certain conditions construct linear quantum system local interactions environment achieves assigned covariance matrix realizations illustrated examples quantum optics key words linear quantum system cascade realization locally dissipative realization covariance assignment pure gaussian state introduction stochastic systems many performance objectives expressed terms variances covariances system states large space structure example vibration certain points structure must reduced acceptable level objective fact involves keeping variances variables deflections within prescribed bounds one way achieve assign appropriate matrix value covariance state vector method referred covariance assignment extensively studied series papers skelton colleagues linear stochastic systems white noises covariance matrix computed solving lyapunov equation system case covariance assignment problem reduces designing system matrices corresponding lyapunov equation prescribed solution turning attention quantum case find work supported australian research council jsps material paper partially presented ieee conference control applications cca antibes france corresponding author tel email addresses matthew woolley ian petersen yamamoto naoki yamamoto preprint submitted automatica covariance matrix plays essential role well field quantum information particular linear quantum system importance covariance matrix stands fully characterize entanglement property indeed crucial conducting quantum information processing therefore great use investigate covariance assignment problem linear quantum systems fact several proposals studies quantum feedback control problem covariance assignment analyze systems generate pure gaussian state note since gaussian state zero mean uniquely determined covariance matrix aforementioned covariance assignment problem also known gaussian state generation problem thus linear quantum system achieves covariance matrix corresponding target gaussian state call system generates gaussian state let especially focus refs provide basis paper mentioned papers pure gaussian states examined particularly important subclass gaussian states highest performance gaussian quantum information processing realized provided several methods construct stable linear quantum system generating given pure gaussian state moreover conditions generating arbitrary pure entangled gaussian state given surely important results state serves essential resource gaussian january tum information processing tasks course literature several methods generating various pure entangled gaussian states proposed instance gives systematic method generate arbitrary pure entangled gaussian state idea construct coherent process applying sequence prescribed unitary operations composed beam splitters squeezers optics case initial state thus method essentially approach contrast approach take opensystem one aim construct dissipative processes system stable uniquely driven desired target pure gaussian state strategy categorized reservoir engineering method general approach clear advantage system good robustness properties respect initial states evolution time gate photon counting extracted gaussian state realize entanglement distillation universal quantum computation hand generated internal gaussian state necessarily extracted outside purpose precision measurement scenario quantum metrology instance spin squeezed state atomic ensemble directly used ultraprecise magnetometry notation matrix whose entries complex numbers operators define superscript denotes either complex conjugate complex number adjoint operator diag denotes diagonal matrix main diagonal permutation matrix defined column vector describe problem considered paper methods developed lead infinitely many linear quantum systems uniquely generate target pure gaussian state systems easy implement others natural question find linear quantum system simple implement still uniquely generates desired pure gaussian state preliminaries consider linear quantum system modes mode characterized pair quadrature operators collecting operatorvalued vector write canonical commutation relations paper provide two convenient realizations linear quantum system generating target pure gaussian state first one cascade realization typical system structure found literature show given covariance matrix corresponding pure gaussian state construct cascaded quantum system uniquely generating state cascaded system series connection several subsystems output one fed input next clear advantage cascade realization subsystems placed remote sites note cascade structure also widely studied classical control literature emphasize transpose operation applied matrix say exchanges indices matrix leaves entries unchanged therefore let hamiltonian system let lindblad operators represent interactions system environment convenience collect lindblad operators vector second one locally dissipative realization motivated specific system structure found note references notion quasilocality studied paper focus stronger notion locality locally dissipative means interactions act one system component implementations locally dissipative systems considerably easier systems interactions paper show given covariance matrix corresponding pure gaussian state satisfies certain conditions construct locally dissipative quantum system generating state call coupling vector suppose quadratic linear quantum system described following quantum stochastic differential equations qsdes lastly remark state generated method internal one confined system state optics rather external optical field state means aim perform quantum information processing gaussian state must extracted outside instance method developed particular acting operations chapter input represents independent quantum stochastic processes satisfying following tum rules corresponding given pure gaussian state since pure gaussian state zero mean uniquely specified covariance matrix linear quantum system achieves covariance matrix corresponding pure gaussian state simply say linear quantum system uniquely generates pure gaussian state problem expressed mathematically kronecker output satisfies quantum rules similar quantum expectation vector denoted covariance matrix given see time evolutions mean vector covariance matrix derived using quantum rule given find subject covariance matrix corresponding desired target pure gaussian state matrix said hurwitz eigenvalues strictly negative real parts system described said asymptotically stable matrix hurwitz matrix recently necessary sufficient condition developed solving pure gaussian state covariance assignment problem result summarized follows classical case gaussian state completely characterized mean vector covariance matrix since mean vector contains information noise entanglement restrict attention gaussian states gaussian state pure covariance matrix satisfies det fact gaussian state pure covariance matrix always factored hurwitz lemma let covariance matrix corresponding given pure gaussian state assume expressed factored form pure gaussian state uniquely generated linear quantum system example vacuum state special pure gaussian state seen pure gaussian state uniquely specified complex symmetric matrix referred graph corresponding pure gaussian state note also matrix satisfies means symplectic matrix symplectic nature guarantees mapping preserves canonical commutation relations xrx free matrices satisfying following rank condition rank remark see resulting coupling vector engineered system therefore components nullifiers desired target pure gaussian state special example one engineer purely dissipative system generate pure gaussian state case one could take lemma resulting coupling vector nullifier vector desired target pure gaussian state note system initially gaussian state system always gaussian mean vector covariance matrix obeying respectively shall particularly interested covariance matrix assume system initially gaussian state problem pure gaussian state covariance assignment find hamiltonian coupling vector corresponding linear quantum system described asymptotically stable achieves covariance matrix remark lemma simple interpretation terms symplectic transformations mentioned vacuum states special class pure gaussian states quantum systems uniquely generate given pure gaussian state based fact provide two feasible realizations linear quantum systems covariance assignment corresponding pure gaussian states section devoted first one cascade realization covariance matrix corresponding vacuum state using physical realizability conditions proved vacuum state generated passive linear quantum system converse also true passive linear quantum system asymptotically stable must evolve toward vacuum state recall passive linear quantum system hamiltonian always form coupling vector always form apply symplectic transformation define terms hamiltonian rewritten coupling vector rewritten also observe relation covariance matrix covariance matrix given follows cascade realization convenience denote linear quantum system hamiltonian coupling vector suppose two linear quantum systems feed output system input system obtain cascaded quantum system shown fig based quantum theory cascaded linear quantum systems hamiltonian coupling vector cascaded system respectively given passive linear quantum system asymptotically stable based result result gives desired pure gaussian state combining results conclude given pure gaussian state complete parametrization linear quantum system uniquely generates pure gaussian state given fig cascade connection two linear quantum systems result extended cascade connection harmonic oscillators suppose harmonic oscillators nian coupling vector system obtained cascade connection harmonic oscillators shown fig repeatedly using hamiltonian coupling vector cascaded system given following lemma form asymptotically stable passive lin ear quantum system substituting using additional matrix transformations obtain formulas respectively idea behind lemma rank constraint indeed gives sufficient necessary stability condition original passive linear quantum system result also guarantees stability linear quantum system based linear transformation theory control field fig cascade connection harmonic oscillators lemma suppose system obtained via cascade connection aforementioned onedimensional harmonic oscillators hamiltonian coupling vector system respectively given cascade realization seen lemma matrices free matrices although must satisfy rank condition varying obtain different linear symmetric block matrix whenever whenever stability lyapunov equation guarantee cascaded system constructed asymptotically stable achieves covariance matrix words cascaded system uniquely generates desired target pure gaussian state seen lemma due cascade feature hamiltonian matrix coupling matrix cascaded system depend complicated way nevertheless given pure gaussian state always construct cascade connection several onedimensional harmonic oscillators cascaded quantum system asymptotically stable achieves covariance matrix corresponding desired target pure gaussian state result stated follows example example consider generation squeezed states squeezed states highly symmetric entangled states useful several quantum information protocols quantum teleportation covariance matrix corresponding squeezed state cosh sinh sinh cosh cosh sinh sinh cosh theorem pure gaussian state uniquely generated constructing cascade onedimensional harmonic oscillators proof prove result construction recall arbitrary pure gaussian state corresponding covariance matrix factorization shown using matrices obtained construct cascaded system hamiltonian coupling vector given squeezing parameter using factoriza cosh sinh tion sinh cosh therefore graph corresponding squeezed cosh sinh state given sinh cosh next provide two different cascade realizations first one realization constructed based heuristic derivation second one realization constructed based proof theorem using lemma calculate hamiltonian coupling vector cascaded system find follows qsde realization cascade realization subsystems respectively given sinh sinh sinh cosh cosh proved cascaded system asymptotically stable achieves covariance matrix proof similar theorem hence omitted using result corresponding quantum optical realization provided fig subsystem hamiltonian realized nonlinear crystal pumped classical field coupling operator realized implementing auxiliary cavity auxiliary cavity interacts clearly hurwitz furthermore verified subsystem via cascade pumped crystal beam splitter fast mode adiabatically eliminated auxiliary cavity auxiliary cavity auxiliary cavity auxiliary cavity fig another optical cascade realization linear quantum system uniquely generates squeezed state fig optical cascade realization linear quantum system uniquely generates squeezed state square arrow represents pumped crystal symbol square represents phase shift solid dark rectangles denote perfectly reflecting mirrors unfilled rectangles denote partially transmitting mirrors dark line represents optical beam splitter locally dissipative realization generate pure gaussian states shown later class stabilizable states fairly broad locally dissipative realization realization second realization constructed according method shown proof theorem direct calculation subsystems respectively given cosh cosh sinh sinh sinh sinh cosh cosh noted section coupling vector vector consists elements element called lindblad operator represents interaction system environment lindblad operator said local acts one system mode example consider system depicted fig lindblad operator acts first system mode local operator hand lindblad operator acts two system modes definition local operator lindblad operators local system called locally dissipative quantum system locally dissipative quantum system could relatively easy implement practice therefore would like characterize class pure gaussian states generated using locally dissipative quantum systems result given following theorem using result corresponding quantum optical realization cascaded quantum system provided fig cascaded system two crucial features first implementations hamiltonians involve pumped crystals second first component coupling vector cosh cosh cosh denotes annihilation operator first mode operator represents standard linear dissipation cavity mode continuum field modes outside cavity similar case also occurs coupling vector seen fig realization requires two pumped crystals contrast case realization four pumped crystals used viewpoint realization constructed based result clear advantage realization fig illustration local lindblad operators local lindblad operator local one theorem let covariance matrix corresponding given pure gaussian state assume expressed factored form pure gaussian state uniquely generated locally dissipative quantum system exists integer locally dissipative realization section describe second realization linear quantum systems covariance assignment corresponding pure gaussian states unlike cascade realization denotes element graph matrix pure gaussian state proof prove sufficiency part construction equation implies ithere exists row vector using method create three matrices unitary matrix first column unitary matrix first column purely imaginary matrix diag operator local suppose acts mode system let verified moreover substituting yields substituting gives rank complex numbers follows rank rank since since equation holds completes proof remark basic idea theorem choice always exist matrices rank condition satisfied first specify matrix coupling matrix local structure obtaining determine two matrices get system hamiltonian rank constraint generally given nonzero matrix infinite solutions satisfy rank condition different choices lead different system hamiltonians optimization problem hamiltonians beyond scope paper considered next subsection show specific recipe determining matrices used full rank property vandermonde matrix hence rank condition satisfied based lemma obtain locally dissipative quantum system asymptotically stable achieves given covariance matrix coupling vector system consists one lindblad operator given remark suppose pure gaussian state generated dissipative quantum system mode locally coupled environment equation straightforward see mode entangled rest system modes system achieves steady state examples see acts mode hence local hamiltonian system also obtained directly substituting matrices completes sufficiency part proof example consider generation canonical gaussian cluster states serve essential resource quantum computation continuous variables mention interesting class cluster states called bilayer cluster states proposed recently class cluster states practical advantages canonical gaussian cluster states quantum computation sake simplicity use canonical gaussian cluster states illustrate developed theory covariance next prove necessity part suppose pure gaussian state uniquely generated locally dissipative quantum system based lemma exists coupling vector local let nonzero column matrix corresponding lindblad hamiltonian representing coupling jth kth optical modes beam splitter also coupling vector given matrix corresponding canonical gaussian cluster state given squeezing parameter note limit canonical gaussian cluster state approximates corresponding ideal cluster state using obtain graph corresponding canonical gaussian cluster state given acts fourth mode hence local finally using result corresponding optical realization linear quantum system shown fig note three pumped crystals used conjecture minimum number required constructing desired locally dissipative system let consider simple case matrices satisfy thus theorem corresponding canonical gaussian cluster state generated locally dissipative system construct system let take lemma next step determine system parameters practical implementation one basic requirements system mentioned system pumped crystals possible motivated structure passive quantum systems described remark choose result hamiltonian matrix block skew matrix additionally take block diagonal matrix interaction hamiltonian modes passive simply realized beam splitters according guideline seek diagonal matrix direct calculation obtain fig optical dissipative system uniquely generates canonical gaussian cluster state coupling vector acts fourth mode hence local example next consider canonical gaussian cluster state specified following matrices strength theorem readily tells canonical gaussian cluster state generated locally dissipative system nonetheless let take matrix follow guideline discussed example set seek diagonal matrix direct calculation find substituting matrices rank condition obtain resulting linear quantum system asymptotically stable achieves covariance matrix corresponding hamiltonian linear quantum system determined auxiliary cavity let take corresponding system hamiltonian given remark method based essentially idea given pure gaussian state graph instead generating directly enlarge system adding auxiliary system specify target state diag diag theorem target state uniquely generated locally dissipative system original pure gaussian state obtained reduced state target state verified rank condition satisfied hence system constructed asymptotically stable achieves desired covariance matrix corresponding though case system needs following interaction environment conclusion paper provided two feasible realizations linear quantum systems covariance assignment corresponding pure gaussian states cascade realization locally dissipative realization first shown given covariance matrix corresponding pure gaussian state construct cascaded quantum system achieves assigned covariance matrix cascaded quantum system constructed cascade connection several harmonic oscillators without direct interaction hamiltonians oscillators second given complete characterization class pure gaussian states generated using locally dissipative quantum systems particular shown specific recipe constructing system relatively simple hamiltonian coupling system modes results developed paper potentially useful preparation pure gaussian states examples provided realizations quantum optics using result circuit figures shown examples necessarily simplest realizations quantum optics also system could realized instances linear quantum systems atomic ensembles optomechanical systems optical realization yet contains abstract component corresponding interaction depicted fig practical implementation fig optical linear quantum system uniquely generates canonical gaussian cluster state coupling vector acts third fourth modes hence local action depicted fig could experimentally difficult nonetheless issue resolved taking following method add auxiliary system single mode specify target canonical gaussian cluster state appendix briefly review synthesis theory linear quantum systems quantum optics developed realization quadratic hamiltonian suppose quadratic hamiltonian given hamiltonian realized placing crystal classical pump inside optical cavity shown fig working frame rotating half pump frequency hamiltonian written since theorem construct locally dissipative system uniquely generates canonical gaussian cluster state choosing taking similar procedure case example obtain desired locally dissipative quantum system obtain important observation canonical gaussian cluster state graph matrix squeezing matrix always possible generate state locally dissipative quantum system adding auxiliary system specifying target state diag detuning cavity mode frequency half pump frequency measure effective pump intensity see choosing values one make note constant term affect dynamics linear quantum system hence ignored therefore desired hamiltonian realized scheme fig interaction hamiltonian realized placing beam splitter two incoming modes fig quadratic hamiltonian realized placing crystal classical pump inside optical cavity frame rotating half pump frequency interaction hamiltonian written realization interaction hamiltonian determines effective pump intensity determines parameters beam splitter assume coupling coefficient partially transmitting mirror large mode heavily damped adiabatically eliminated elimination resulting coupling operator given suppose hamiltonian given hamiltonian realized implementing beam splitter two incoming modes shown fig beam splitter following transformations see choosing values large make desired coupling operator realized scheme see details denote outgoing modes denote complex reflectance transmittance beam splitter respectively note satisfy following relations let parametrize interaction hamiltonian beam splitter given see choosing values one make desired interaction hamiltonian realized scheme fig realization dissipative coupling operator realization dissipative coupling references realize coupling operator consider configuration shown fig configuration consists ring cavity mode auxiliary ring cavity mode cavity modes interact crystal pumped classical beam beam splitter frequency auxiliary cavity mode matched half pump frequency working hotz skelton covariance control theory international journal control vol collins skelton theory state covariance assignment discrete systems ieee transactions automatic control vol skelton ikeda covariance controllers linear systems international journal control vol huang james jiang robust control nonlinear cascade systems systems control letters vol braunstein pati quantum information continuous variables springer weedbrook pirandola cerf ralph shapiro lloyd gaussian quantum information reviews modern physics vol liu huang global robust stabilization cascadeconnected systems dynamic uncertainties without knowing control direction ieee transactions automatic control vol ohki hara yamamoto equivalence linear systems control problems application quantum entanglement assignment proceedings ieee annual conference decision control cdc december kraus diehl kantian micheli zoller preparation entangled states quantum 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network synchronization nonlinear dynamics switching aug tao ziyang guodong yiguang karl henrik johanssonk abstract paper considers synchronization problem networks coupled nonlinear dynamical systems switching communication topologies two types nonlinear agent dynamics considered first one dynamics stable dynamics convex lyapunov function second one dynamics satisfies global lipschitz condition case show various forms joint connectivity communication graphs sufficient networks achieve global asymptotic also show leads state synchronization provided certain additional conditions satisfied globally lipschitz case unlike case joint connectivity alone sufficient achieving synchronization sufficient condition reaching global exponential synchronization established terms relationship global lipschitz constant network parameters also extend results networks keywords systems nonlinear agents switching interactions synchronization work supported part knut alice wallenberg foundation swedish research council yang pacific northwest national laboratory battelle boulevard richland usa meng institute control technische munich germany zmeng shi college engineering computer science australian national university canberra act australia hong key laboratory systems control institute systems science chinese academy science beijing china yghong johansson access linnaeus centre school electrical engineering royal institute technology stockholm sweden kallej introduction consider synchronization problem network coupled nonlinear agents agent set interactions communications network described directed graph piecewise constant signal finite set possible graphs state agent time denoted evolves according aij piecewise continuous continuous representing uncoupled inherent agent dynamics set agent neighbors time aij piecewise continuous function marking weight edge time systems form attracted considerable attention works focus case communication graph fixed shown case satisfies lipschitz condition synchronization achieved connected graph provided coupling strength sufficiently large however case communication graph synchronization problem becomes much challenging existing literature mainly focuses special cases linear case case neutrally stable case studies assume particular structures communication graph particular authors focus case adjacency matrices associated communication graphs simultaneously triangularizable authors consider switching communication graphs weakly connected balanced times general case switching communication graph frequently directed spanning tree considered special structures switching communication graph rather restrictive compared joint connectivity communication lost time paper aims investigate whether joint connectivity switching communication graphs render synchronization nonlinear dynamics distinguish two classes depending whether nonlinear agent dynamics expansive nonexpansive case focus case nonlinear agent dynamics stable convex lyapunov function show various forms joint connectivity communication graphs sufficient networks achieve global asymptotic function agent state converges common value also show implies state synchronization provided additional conditions satisfied expansive case focus nonlinear agent dynamics globally lipschitz establish sufficient condition networks achieve global exponential synchronization terms relationship lipschitz constant network parameters remainder paper organized follows section presents problem definition main results section provides technical proofs section extend results networks finally section concludes paper problem definition main results problem throughout paper make standard dwell time assumption switching signal lower bound two consecutive switching time instants also assume constants aij denote xtn rnn assume initial time initial state xtn rnn digraph strongly connected contains directed path every node every node joint graph time interval denoted communication graph introduce following definition definition uniformly jointly strongly connected exists constant strongly connected assume undirected infinitely jointly connected connected paper interested following synchronization problems definition system achieves global asymptotic continuously differentiable function initial state exists constant definition system achieves global asymptotic synchronization rnn system achieves global exponential synchronization exist max kxi max kxi rnn remark type output synchronization output agent chosen related different since function individual agent state function agent states inherent dynamics section focus nonlinear inherent agent dynamics indicated following assumption assumption continuously differentiable positive definite convex function satisfying following lemma shows assumption enforces dynamics lemma let assumption hold along dynamics state main results case theorem let assumption hold system achieves global asymptotic uniformly jointly strongly connected theorem let assumption hold assume undirected multiagent system achieves global asymptotic infinitely jointly connected remark linear case exists matrix satisfies assumption linear case condition equivalent matrix neutrally stable state synchronization following result establishes conditions may imply state synchronization theorem let fixed strongly connected digraph system achieves global asymptotic positive definite function let assumption hold moreover assume bounded iii strongly convex system achieves global asymptotic synchronization lipschitz inherent dynamics consider also case nonlinear inherent agent dynamics possibly expansive focus dynamics satisfies following global lipschitz condition assumption exists constant main result case given theorem let assumption hold assume uniformly jointly strongly connected global exponential synchronization achieved system constant depending network parameters remark assumption variants considered literature fixed communication graphs compared existing literature study challenging case communication graphs unlike fixed case global lipschitz condition sufficient guarantee synchronization theorem established sufficient synchronization condition related lipschitz constant network parameters proofs main results section provide proofs main results proof lemma recall upper dini derivative function defined lim following lemma useful proof lemma let continuously differentiable set indices maximum reached denote first note convexity property implies follows lemma assumption max max max aij aij last inequality follows proves lemma proof theorem follows lemma initial state rnn exists constant shall show required constant definition first note lemma exist constants lim inf lim sup also note follows exists proof theorem based contradiction argument relies following lemma lemma let assumption hold assume uniformly jointly strongly connected exists agent exists given definition dwell time proof let first define exists infinite time sequence pick one greater equal denote prove lemma estimating upper bound scalar function agent agent proof based generalization method proposed proof lemma substantial differences agent dynamics lyapunov function moreover convexity plays important role step focus agent using assumption obtain follows step consider agent existence agent shown follows since uniformly jointly strongly connected hard see exists agent obtain next estimate considering two different cases case using assumption obtain xkj preceding relation obtain therefore applying analysis obtained agent obtain combining inequalities obtain case exists time instant applying similar analysis obtained agent obtain leads combining preceding relation using follows obtain preceding relation follows cases preceding relation follows step consider agent exists edge set agent existence agent follows similarly argument step similarly bound considering two different cases obtain combining using obtain step repeating process time intervals eventually obtain result lemma follows choosing ready prove theorem contradiction suppose exists agent follows lemma provided contradicts fact thus exist agent hence proof theorem proof relies following lemma lemma let assumption hold assume infinitely jointly connected exists agent exist proof proof lemma similar lemma based estimating upper bound scalar quantity agent agent however since infinitely jointly connected method get order agents based intervals induced uniform bound used however apply strategy analysis shown step step focus agent since infinitely jointly connected define inf set agent neighbor follows assumption thus applying similar analysis obtained step step focus estimate considering two different cases case case exists time instant using similar argument analysis agent proof theorem eventually obtain given step view set subsystem define first time edge subsystem remaining agents accordingly using similar analysis agent step estimate upper bound agent set step since infinitely jointly connected continue process eventually result lemma follows choosing remaining proof theorem follows contradiction argument lemma way proof theorem proof theorem system reaches asymptotic exists desired conclusion holds trivially due positive definiteness remainder proof assume shall prove result contradiction suppose state synchronization achieved exist two agents lim words exist infinite time sequence constant given condition theorem divide following analysis three steps step step prove following crucial claim claim two agents establish claim via recursive analysis either result follows trivially otherwise pick another agent satisfying edge always exists since strongly connected either must hold thus either established claim continue select another agent different repeat argument since finite number agents desired claim holds furthermore since finite number agent pairs without loss generality assume given agent pair vary different otherwise always select infinite subsequence following discussions step step establish lower bound small time interval particular satisfying given assumption see bounded follows condition theorem without loss generality assume independent plugging fact theorem obtain using condition step first note strong convexity implies exists using assumption obtain using condition theorem obtain yields straightforward see take min however contradicts definition since arbitrarily chosen completes proof desired conclusion holds proof theorem proof based convergence analysis scalar quantity max vij vij kxi unlike contradiction argument used proof theorem convergence rate unclear explicitly characterize convergence rate proof relies following lemmas lemma let assumption hold along dynamics nonincreasing proof lemma establishes critical property along dynamics globally lipschitz case proof follows techniques proving lemma investigating dini derivative let set containing node pairs reach maximum time vij hard obtain max kxi max kxj kxi max kxi kxi vij vij first equality follows lemma first inequality follows second inequality follows lemma let assumption hold assume uniformly jointly strongly connected exists vij given proof proof based convergence analysis vij agent pairs several steps similar proof lemma without loss generality assume also sometimes denote vij vij respectively notational simplification step begin considering agent since uniformly jointly strongly connected know root exists time agent first note follows lemma vij taking derivative vij along trajectories obtain first inequality follows second inequality follows thus follows similarly obtain follows step since uniformly jointly strongly connected know exists time instant arc estimate upper bound considering two different cases eventually obtain follows step continuing process obtain step since uniformly jointly strongly connected holds using analysis eventually obtain vij hence result follows choosing given ready prove theorem using lemma obtain ntt denotes largest integer greater follows max kxi max kxi hence global exponential synchronization achieved provided concludes proof desired theorem networks focus far achieving synchronization leaderless networks section consider synchronization problem networks suppose additional agent labeled agent plays reference leader agents set view also call agent follower denote overall agent set overall communication network described directed graph sake simplicity continue use denote piecewise constant graph signal also make standard dwell time assumption switching signal communication graph introduce following definition definition leader connected follower agent directed path leader follower agent time moreover jointly leader connected time interval union graph leader connected uniformly jointly leader connected exists constant union graph leader connected iii infinitely jointly leader connected union graph leader connected networks evolutions follower state leader state given aij nonlinear function aij follow definitions leaderless case piecewise continuous function marking strength edge assume constant also assume initial time denote initial state leader networks interested following synchronization problems definition system achieves global asymptotic synchronization rnn system achieves global exponential synchronization exist exist max kxi max kxi rnn inherent dynamics section extend results case agent dynamics networks make following assumption agent dynamics assumption continuously differentiable positive definite function following conditions hold assumption similar assumption however possibly different functions assumption holds relative coordinate convexity property guarantees property along dynamics shown following lemma whose proof similar lemma thus omitted lemma let assumption hold along dynamics state results case theorem let assumption hold system achieves global asymptotic synchronization uniformly jointly leader connected theorem let assumption hold assume undirected multiagent system achieves global asymptotic synchronization achieved infinitely jointly leader connected proofs theorems given appendices respectively based generalization methods proposed however nonlinear agent dynamics results different lyapunov function analysis based estimating scalar function agent agent thus yields estimate convergence rate different contradiction arguments used proofs theorems convergence rate unclear remark scalar function satisfies additional condition exist theorem leads global exponential synchronization remark case theorems show global asymptotic synchronization achieved leaderless case global asymptotic achieved shown theorems case uniformly jointly leader connected theorem requires leader center node leaderless case uniformly jointly strongly connected theorem requires every node center node thus connectivity condition leaderless case stronger case lipschitz inherent dynamics section extend result case agent dynamics globally lipschitz networks main result case given following theorem whose proof found appendix theorem let assumption hold suppose uniformly jointly leader connected system achieves global exponential synchronization constant depending network parameters conclusions paper synchronization problems networks nonlinear inherent agent dynamics switching topologies investigated two types nonlinear dynamics considered globally lipschitz case found convexity lyapunov function plays crucial rule analysis showed uniformly joint strong connectivity sufficient achieving global asymptotic communication graphs undirected infinitely joint connectivity sufficient synchronization condition moreover established conditions implies state synchronization globally lipschitz case found joint connectivity alone sufficient achieve synchronization established sufficient synchronization condition proposed condition reveals relationship lipschitz constant network parameters results also extended networks interesting future direction study synchronization problem coupled nonlinear inherent dynamics general switching topologies references chua application graph theory synchronization array coupled nonlinear oscillators ieee trans circuits syst fundam theory vol synchronization complex networks nonlinear dynamical systems gapore world scientific singapore belykh belykh hasler connection graph stability method synchronized coupled chaotic systems physica vol delellis bernardo russo quad lipschitz contracting vector fields consensus synchronization networks ieee trans circuits syst reg papers vol chen cao consensus directed networks agents nonlinear dynamics ieee trans autom control vol papachristodoulou consensus systems coupling delays switching topology ieee trans autom control vol liu cao coupling strength allocation synchronization complex networks using spectral graph theory ieee trans circuits syst reg papers vol jadbabaie lin morse coordination groups mobile autonomous agents using nearest neighbor rules ieee trans autom control vol moreau stability multiagent systems communication links ieee trans autom control vol lin francis maggiore state agreement coupled nonlinear systems siam contr vol ren beard consensus seeking multiagent systems dynamically changing interaction topologies ieee trans autom control vol scardovi sepulchre synchronization networks identical linear systems automatica vol huang stability class linear switching systems applications two consensus problem ieee trans autom control vol zhao hill liu synchronization complex dynamical networks switching topology switched system point view automatica vol qin gao zheng exponential synchronization complex networks linear systems nonlinear oscillators unified analysis ieee trans neural netw learn vol wen duan chen consensus tracking systems node dynamics switching topologies ieee trans circuits syst reg papers vol liberzon morse basic problem stability design switched systems ieee control syst vol distributed algorithms reaching consensus general functions automatica vol wang hong distributed algorithms systems variable coupling topology journal systems science complexity vol danskin theory applications siam appl vol boyd vandenberghe convex optimization new york cambridge university press shi johansson hong reaching optimal consensus dynamical systems compute intersections convex sets ieee trans autom control vol shi johansson robust consensus dynamics siam contr vol proof theorem proof theorems relies following lemma whose proof similar lemmas lemma let assumption hold assume uniformly jointly leader connected exists max given proof without loss generality assume initial time similar proof lemma estimate agent agent subintervals several steps step step focus follower agent existence follower agent follows fact uniformly jointly leader connected convenience introduce relative state leader agent obtain following dynamics aij similar obtain max obtain max edge may longer exist nevertheless max follows max step step analysis follower agent either edge edge existence due uniform joint leader connectivity going similar analysis step proof lemma eventually obtain max step applying similar analysis subintervals obtain max follows together leads result lemma follows choosing ready prove theorem without loss generality assume follows lemma thus max max follows max max together fact positive definite given assumption implies system achieves global asymptotic synchronization proof theorem proof theorems relies following lemma whose proof similar lemmas lemma let assumption hold assume infinitely jointly leader connected exist max proof since infinitely jointly leader connected exist sequence time instants leader connected moreover edge exists least dwell time shall estimate agent agent subintervals interval step step focus follower agent inf existence due fact leader connected follows similar analysis obtained max given step step similar step proof lemma view set subsystem define first time edge subsystem remaining follower agents accordingly going similar analysis step proof lemma eventually obtain max given step since infinitely jointly leader connected proceed analysis max defined similarly preceding relation obtain easy see together implies max given result follows choosing defined ready prove theorem without loss generality assume follows follows lemma lemma fact follows definition given max max max thus together fact positive definite given assumption implies system achieves global asymptotic synchronization proof theorem proof similar theorem however relative coordinate whose evolution given without loss generality assume initial time proof based convergence analysis nonnegative scalar max let define similar lemma obtain along dynamics combining preceding relation following similar analysis proof theorem show given given follows max max hence global exponential synchronization achieved provided
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neural networks joint sentence classification medical paper abstracts franck mit francky young mit jjylee dec abstract existing models based artificial neural networks anns sentence classification often incorporate context sentences appear classify sentences individually however traditional sentence classification approaches shown greatly benefit jointly classifying subsequent sentences conditional random fields work present ann architecture combines effectiveness typical ann models classify sentences isolation strength structured prediction model achieves results two different datasets sequential sentence classification medical abstracts introduction million scholarly articles published jinha number articles published every year keeps increasing druss marcus larsen von ins approximately half biomedical papers repository human knowledge abounds useful information may unlock new promising research directions provide conclusive evidence phenomena become increasingly difficult take advantage available information due sheer amount therefore technology assist user quickly locate information interest highly desired may reduce time required locate relevant information researchers search previous literature example often skim abstracts order quickly check whether papers match authors contributed equally work peter szolovits mit psz criteria interest process easier abstracts structured text abstract divided semantic headings objective method result conclusion however significant portion published paper abstracts unstructured makes difficult quickly access information interest therefore classifying sentence abstract appropriate heading significantly reduce time locate desired information call sequential sentence classification task order distinguish general text classification sentence classification context besides aiding humans task may also useful automatic text summarization information extraction information retrieval paper present system based anns sequential sentence classification task model makes use token character embeddings classifying sentences sequence optimization layer learned jointly components model evaluate model nictapiboso dataset well new dataset compiled based pubmed database related work existing systems sequential sentence classification mostly based naive bayes ruch huang support vector machines svms mcknight srinivasan yamamoto takagi hirohata hidden markov models hmms lin conditional random fields crfs kim hassanzadeh hirohata often require numerous features based lexical tionaries cue words semantic synonyms hyponyms structural tags headings sequential sentence position surrounding features information hand recent approaches natural language processing nlp based artificial neural networks anns require manual features trained automatically learn features based word well character embeddings moreover models achieved results various nlp tasks classification many ann models use word embeddings socher kim kalchbrenner recent works based character embeddings zhang conneau xiao cho dos santos gatti use word character embeddings however existing works using anns classification use context contrast sequential sentence classification sentence text classified taking account context context utilized classification could surrounding sentences possibly whole text one exception recent work dialog act classification lee dernoncourt utterance dialog classified dialog act preceding utterances used system designed applications mind feed forward concatanate concatenate concatanate token embeddings character embeddings figure ann model sequential sentence classification token token embeddings ith character character embeddings token embeddings hybrid token embeddings sentence vector sentence label vector number classes sentence label numbers parenthesis indicate dimensions vectors token embeddings initialized glove pennington embeddings pretrained wikipedia gigaword parker model following denote scalars italic lowercase vectors bold lowercase matrices italic uppercase symbols use colon notations denote sequences scalars vectors respectively ann model ann model figure consists three components hybrid token embedding layer sentence label prediction layer label sequence optimization layer hybrid token embedding layer hybrid token embedding layer takes token input outputs vector representation utilizing token embeddings well character embeddings token embeddings direct mapping token vector large unlabeled datasets using programs mikolov mikolov mikolov glove pennington character embeddings also defined analogous manner direct mapping character vector let sequence characters comprise token character first mapped embedding resulting sequence input bidirectional lstm outputs token embedding output hybrid token embedding layer token concatenation token embedding token embedding sentence label prediction layer let sequence tokens given sentence corresponding embedding output hybrid token embedding layer sentence label prediction layer takes input sequence vectors outputs element denoted reflects probability given sentence label achieve sequence first input bidirectional lstm outputs vector representation given sentence vector subsequently input feedforward neural network one hidden layer outputs corresponding probability vector label sequence optimization layer label sequence optimization layer takes sequence probability vectors label prediction layer input outputs sequence labels label assigned token order model dependencies subsequent labels incorporate matrix contains transition probabilities two subsequent labels define probability token label followed token label score label sequence defined sum probabilities individual labels transition probabilities scores turned probabilities label sequences taking softmax function possible label sequences training phase objective maximize log probability gold label sequence testing phase given input sequence tokens corresponding sequence predicted labels chosen one maximizes score experiments datasets evaluate model sentence classification task using following two medical abstract datasets sentence abstract annotated one label table presents statistics dataset dataset introduced kim basis alta shared task amini pubmed rct assembled corpus consisting randomized controlled trials rcts pubmed database biomedical literature provides standard set sentence labels objectives background methods results conclusions dataset train validation test pubmed nicta table dataset overview denotes number classes vocabulary size train validation test sets indicate number number abstracts followed number sentences parentheses training model trained using stochastic gradient descent updating parameters token embeddings character embeddings parameters bidirectional lstms transition probabilities gradient step regularization dropout rate applied characterenhanced token embeddings label prediction layer results discussion table compares model several baselines well best performing model lui alta shared task competing research teams participated build accurate classifier nictapiboso corpus first baseline classifier based logistic regression using features extracted current sentence use information surrounding sentences second baseline forward ann uses model presented lee dernoncourt computes sentence embeddings sentence classifies current sentence given preceding sentence embeddings well current sentence embedding third baseline crf crf uses features output variable crf corresponds label sentence sequence crf considers entire abstract crf baseline therefore uses preceding succeeding sentences classifying current sentence lastly model presented lui developed new nicta background conclusion methods objectives results total support table detailed results model pubmed rct dataset end start objectives background conclusion rou sio ckg clu results jec tiv sta methods tho approach called feature stacking metalearner combines multiple feature sets best performing system published literature system performs honorably pubmed rct quite poorly suggests using surrounding sentences may important pubmed rct forward ann system performs better system worse crf unsurprising forward ann system uses information preceding sentences use information succeeding sentences unlike crf model performs better crf system lui system hypothesize following four factors give edge model features unlike systems model rely features systems heavily rely model maps token token embedding feeds input rnn helps combat data scarcity example chronic tendonitis chronic tendinitis two different bigrams token embeddings similar since share meaning structured prediction labels sentences abstract predicted jointly improves coherence predicted labels given abstract joint learning model learned features token embeddings jointly sequence optimization pubmed rct recall precision lts table test set several baselines best published method lui literature model since pubmed introduced work previous best published method dataset presented results models test set run highest validation set label pubmed model forward ann crf best published model figure transition matrix learned pubmed rows represent label previous sentence columns represent label current sentence figure presents example transition matrix model trained pubmed rct see effectively reflects transitions different labels example learned first sentence abstract likely either discussing objective background token sentence pertaining methods typically followed sentence pertaining methods results table details result model label pubmed rct main difficulty classifier distinguishing background sentences objective sentences conclusions article presented ann architecture classify sentences appear sequence demonstrate jointly predicting classes sentences given text improves quality predictions yields better performance crf model achieves results two datasets sentence classification medical abstracts references amini iman amini david martinez diego molla overview alta shared task australasian language technology association workshop volume page yoon kim convolutional neural networks sentence classification proceedings conference empirical methods natural language processing emnlp pages association computational linguistics acl conneau alexis conneau holger schwenk barrault yann lecun deep convolutional networks natural language processing arxiv preprint larsen von peder olesen larsen markus von ins rate growth scientific publication decline coverage provided science citation index scientometrics dos santos nogueira dos santos maira gatti deep convolutional neural networks sentiment analysis short texts international conference computational linguistics coling pages lee young lee franck dernoncourt sequential classification recurrent convolutional neural networks human language technologies conference north american chapter association computational linguistics naacl hlt druss benjamin druss steven marcus growth decentralization medical literature implications medicine journal medical library association hassanzadeh hamed hassanzadeh tudor groza jane hunter identifying scientific artefacts biomedical literature evidence based medicine use case journal biomedical informatics hirohata kenji hirohata naoaki okazaki sophia ananiadou mitsuru ishizuka manchester interdisciplinary biocentre identifying sections scientific abstracts using conditional random fields international joint conference natural language processing ijcnlp pages huang huang chiang furen xiao liao charles liu wong pico element detection medical text without metadata first sentences enough journal biomedical informatics arif jinha article million estimate number scholarly articles existence learned publishing lin jimmy lin damianos karakos dina sanjeev khudanpur generative content models structural analysis medical abstracts linking natural language processing biology towards deeper biological literature analysis marco lui feature stacking sentence classification medicine australasian language technology workshop alta shared task page mcknight larry mcknight padmini srinivasan categorization sentence types medical abstracts american medical informatics association amia mikolov tomas mikolov kai chen greg corrado jeffrey dean efficient estimation word representations vector space arxiv preprint mikolov tomas mikolov ilya sutskever kai chen greg corrado jeff dean distributed representations words phrases compositionality advances neural information processing systems pages mikolov tomas mikolov yih geoffrey zweig linguistic regularities continuous space word representations hltnaacl pages kalchbrenner nal kalchbrenner edward grefenstette phil blunsom convolutional neural network modelling sentences proceedings annual meeting association computational linguistics proceedings annual meeting association computational linguistics parker robert parker david graff junbo kong chen kazuaki maeda english gigaword fifth edition technical report linguistic data consortium philadelphia kim nam kim david martinez lawrence cavedon lars yencken automatic classification sentences support evidence based medicine biomed central bmc bioinformatics pennington jeffrey pennington richard socher christopher manning glove global vectors word representation proceedings empiricial methods natural language processing emnlp ruch patrick ruch celia boyer christine chichester imad tbahriti antoine paul fabry julien gobeill violaine pillet dietrich christian lovis using argumentation extract key sentences biomedical abstracts international journal medical informatics socher richard socher alex perelygin jean jason chuang christopher manning andrew christopher potts recursive deep models semantic compositionality sentiment treebank proceedings conference empirical methods natural language processing emnlp volume page citeseer xiao yijun xiao kyunghyun cho efficient document classification combining convolution recurrent layers arxiv preprint yamamoto yasunori yamamoto toshihisa takagi sentence classification system multi biomedical literature summarization international conference data engineering workshops icdew pages ieee zhang xiang zhang junbo zhao yann lecun convolutional networks text classification advances neural information processing systems nips pages
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quantum predicative programming anya tafliovich hehner arxiv feb university toronto abstract subject work quantum predicative programming study developing programs intended execution quantum computer look programming context formal methods program development programming methodology work based probabilistic predicative programming recent generalisation predicative programming supports style program development programming step proven correct made inherit advantages theory generality simple treatment recursive programs time space complexity communication theory quantum programming provides tools write classical quantum specifications develop quantum programs implement specifications reason comparative time space complexity framework introduction modern physics dominated concepts quantum mechanics today seventy years recognition scientific community quantum mechanics provides accurate known description nature behaviour surprisingly idea using quantum mechanical nature world perform computational tasks new less thirty years old quantum computation quantum information study information processing communication accomplished quantum mechanical systems recent years field grown immensely scientists various fields computer science discovered thinking physically computation yields new exciting results computation communication extensive research areas quantum algorithms quantum communication information quantum cryptography quantum adiabatic computation quantum computation theoretical quantum optics new quantum game theory experimental quantum information communication also fruitful field experimental quantum optics ion traps solid state implementations nuclear magnetic resonance add experimental successes quantum computation subject work quantum programming study developing programs intended execution quantum computer assume model quantum computer proposed knill classical computer access quantum device capable storing quantum bits performing certain operations measurements bits reporting results measurements look programming context formal methods program development programming methodology field computer science concerned applications mathematics logic software engineering tasks particular formal methods provide tools formally express software specifications prove correctness implementations reason various properties specifications implementability implementations time space complexity today formal methods successfully employed stages software development requirements elicitation analysis software design software implementation work theory quantum programming based probabilistic predicative programming recent generalisation predicative programming deem simplest elegant programming theory known today supports style program development programming step proven correct made inherit advantages theory generality simple treatment recursive programs time space complexity theory quantum programming provides tools write classical quantum specifications develop quantum programs implement specifications reason comparative time space complexity framework rest work organised follows section introduction quantum computation assumes reader basic knowledge linear algebra knowledge quantum computing section contains introduction probabilistic predicative programming reader assumed background logic background programming theory necessary contribution work section defines quantum system introduces programming quantum system several problems classical quantum solutions formal comparative time complexity analyses section states conclusions outlines directions future research related work traditionally quantum computation presented terms quantum circuits recently attempt depart convention reason classical computation generally presented terms classical circuits develop complex quantum algorithms need ways express concepts control structures readable fashion introduced first quantum programming language qcl following work bettelli developed quantum programming language syntax based two works involve verification techniques sanders zuliani introduced quantum language qgcl extension pgcl turn generalises dijkstra language include probabilism zuliani later extends attempt formal program development verification discusses treatment nondeterminism quantum programs attempt made build aharonov work reason mixed states computations zuliani also provides tools approach task compiling quantum programs large amount work area performed past two years process algebraic approaches explored tools developed field category theory successfully employed others reason quantum computation functional language semantics form term rewrite system introduced notion linearity pertains quantum systems examined functional language qml design guided categorical semantics defined following work provides sound complete equational theory qml weakest preconditions appropriate quantum computation introduced work interesting part diverts standard approach reducing quantum computation probabilistic one also provides semantics language interesting work authors include reasoning knowledge quantum systems developing formal model distributed quantum computation similar work introduced language cqp modelling communication quantum systems defined latter approaches advantage process algebraic approaches mentioned earlier explicitly allow quantum state transmitted processes building work defines higher order quantum programming language based linear typed lambda calculus similar work contribution approach quantum programming amenable formal analysis different almost described work one similar work contribution paper twofold firstly building theory inherit advantages offers definitions specification program simpler specification boolean probabilistic expression program specification treatment recursion simple need additional semantics loops treatment termination simply follows introduction time variable final value time variable program one correctness time space complexity proved fashion moreover proving separately naturally obtain conjunction secondly way probabilistic predicative programming extended quantum predicative programming simple intuitive use notation makes easy write specifications develop algorithms treatment computation mixed states require additional mechanisms quantum predicative programming fully preserves predicative programming treatment parallel programs communication provides natural extension reason quantum communication protocols distributed quantum algorithms distributed shor algorithm well time space entanglement complexity preliminaries quantum computation section introduce basic concepts quantum mechanics pertain quantum systems consider quantum computation discussion underlying physical processes etc interest concerned model quantum computation reader familiar quantum computing consult comprehensive introduction field dirac notation invented paul dirac often used quantum mechanics notation vector column vector convention written inside ket dual vector written inside bra inner products vectors vector value inner product scalar outer product operator corresponds matrix dirac notation clearly distinguishes vectors operators scalars makes possible write operators directly combinations bras kets quantum mechanics vector spaces interest hilbert spaces dimension convenient orthonormal basis called computational basis label basis vectors using binary strings length follows string corresponds number column vector position everywhere else tensor product written simply arbitrary vector hilbert space written weighted sum computational basis vectors postulate state space associated isolated physical system hilbert space known state space system system completely described state vector unit vector system state space postulate evolution evolution closed quantum system described unitary transformation postulate measurement quantum measurements described collection measurement operators act state space system measured index refers possible measurement outcomes state system immediately prior measurement described vector probability obtaining result case state system immediately measurement described vector operators satisfy completeness equation measurement important special class measurements projective measurements equivalent general measurements provided also ability perform unitary transformations projective measurement described observable hermitian operator state space system measured observable spectral decomposition projector onto eigenspace eigenvalue corresponds outcome measurement probability measuring case immediately measurement system found state given orthonormal basis measurement respect basis corresponding projective measurement given observable projectors ihvm measurement respect computational basis simplest commonly used class measurements terms basis projectors state system immediately measuring case single qubit example measurement state results outcome probability outcome probability state system immediately measurement respectively suppose result measurement ignored continue computation case system said mixed state mixed state actual physical state system rather describes knowledge state system example mixed state expressed equation equation meant say probability probability application operation mixed state results another mixed state postulate composite systems state space composite physical system tensor product state spaces component systems systems numbered excluding system prepared state joint state composite system always describe composite system given descriptions component systems reverse true indeed given state vector describes composite system may possible factor obtain state vectors systems example state component state called entangled state probabilistic predicative programming section introduces programming theory choice work quantum programming based probabilistic predicative programming briefly introduce parts theory necessary understanding section work course predicative programming reader referred introduction probabilistic predicative programming found predicative programming predicative programing specification boolean expression variables specification represent quantities interest prestate inputs poststate outputs computation time space use primed variables describe outputs unprimed variables describe inputs example specification one integer variable states final value initial value plus computation satisfies specification given prestate produces poststate pair makes specification true specification implementable input state least one output state satisfies specification use standard logical notation writing specifications conjunction disjunction logical implication equality boolean equivalence nonequality else lower precedence use standard mathematical notation mod use lowercase letters variables interest uppercase letters specifications addition use following notations prestate poststate specifies values variables unchanged assignment state variable unprimed expression unprimed variables domain specifications variables obtained substituting occurrences primed variables variables obtained substituting occurrences unprimed variables variables sequential composition defined various laws proven sequential composition one important ones substitution law states expression prestate state variable specification substitute specification refined specification satisfied whenever satisfied specifications equal satisfied simultaneously given specification allowed implement equivalent specification stronger one informally bunch collection objects different set collection objects package bunches simpler sets nesting structure see introduction bunch theory bunch one element element use denote arbitrary bunches denote elements element bunch one element denotes union bunches denotes bunch inclusion bunch included bunch use notation mean including excluding fresh previously unused name bunch arbitrary expression function variable parameter domain body function denotes domain applied corresponding element range function variables function variable whose body function variables predicate function whose body boolean expression relation function whose body predicate function function whose parameter function quantifier unary prefix operator applies functions predicate boolean result obtained first applying elements domain taking conjunction results taking disjunction results produces similarly numeric function numeric result obtained first applying elements domain taking sum ofpthose results example applyingpthe quantifier function functionp yields sake simplicity abbreviate addition allow simplifications example omit domain variable clear context canp also group variables several quantifications example abbreviated program implemented specification simplicity take following implemented assignment else sequential composition booleans numbers bunches functions given specification proceed follows program work done build program refines refinement proceed steps one best features hehner theory simple treatment recursion possible appear additional rules required prove refinement example trivial prove else specification says initial value final value must solution value zero nothing otherwise decrement repeat long computation take account time add time variable use denote time computation starts denote time computation ends case characteristic distinguish terminating programs ones see discussion treatment termination choose use recursive time measure charge time unit time called replace call include time increment follows else easy see incremented number times decremented otherwise prove else probabilistic predicative programming probabilistic predicative programming introduced developed generalisation predicative programming allows reasoning probability distributions values variables interest although work apply reasoning boolean integer variables theory change want work real numbers replace summations integrals probability real number inclusive distribution expression whose value probability whose sum values variables example positive natural distribution pvariable since probability two positive natural variables also distribution distribution several variables written product distributions individual variables variables independent example previous example independent given distribution several variables sum variables obtain distribution rest variables example distribution generalise boolean specifications probabilistic specifications use boolean true alse implementable deterministic specification distribution initial state distribution final state readers familiar notation notice take liberty equate specifications variables obtained substituting occurrences primed variables variables obtained substituting occurrences unprimed variables variables sequential composition defined probability distributions else various laws proven sequential composition one important ones substitution law introduced earlier applies probabilistic specifications well implement probabilistic specification use number generator since even theory produce real random number generator means traditional computing assume number generator generates truly random numbers simply refer random number generator positive natural variable say rand produces random natural number uniformly distributed reason values supplied random number generator consistently replace every occurrence rand fresh variable whose value probability rand occurs context rand replace equation rand occurs context loop parametrise introduced variables execution time recall earlier example let change program slightly introducing probabilism else rand new program iteration either decremented unchanged equal probability intuition tells revised program still work except take longer let prove replace rand time probability ignoring time prove else rand execution time prove takes least time units complete else rand long expect wait execution complete words distribution consider following distribution final states prove else since positive distributed according negative binomial distribution parameters mean value therefore expect wait time units computation complete quantum predicative programming section contribution paper define quantum system introduce programming quantum system several problems classical quantum solutions formal comparative time complexity analyses proofs refinements omitted sake brevity reader referred detailed proofs algorithms quantum system let set complex numbers absolute value operator complex conjugate operator system function point kind function operations referred lifting two states system inner product defined basis system collection quantum states hbi adopt following notation computational basis denotes corresponding binary encoding following quantum state state system state system tensor product following state composite system div mod write mean tensored times operation defined quantum system function whose domain range maps complex numbers identity operation state system defined linear operation adjoint written unique operation two states unitary transformations describe evolution quantum system operations defined system setting tensor product operators defined usual way state system state system operations defined systems respectively tensor product defined qubit system tensor products states write mean operation tensored times suppose system qubits state measure suppose also variable domain use record result measurement variables affected measurement measurement corresponds probabilistic specification gives probability distribution depend type measurement states variables unchanged general quantum measurement described collection measurement operators satisfy completeness equation specification measurem measurem abbreviation means variables unchanged obtain distribution say sum rest variables follows projective measurements defined observable projector eigenspace eigenvalue measureo given arbitrary orthonormal basis measurement basis measureb finally simplest commonly used measurement computational basis measure case distribution distribution quantum state precisely mixed quantum state results measurement order develop quantum programs need add list implemented things section add variables type quantum state allow following three kinds operations variables state quantum system natural variable collection measurement operators satisfy completeness equation program unitary transformation system program measurem program special cases measurements described section therefore also allowed observable measureo program orthonormal basis measureb program finally measure program algorithm problem extension deutsch problem example broad class quantum algorithms based quantum fourier transform task given function either constant balanced determine case without restrictions number calls write specification let call follows constant balanced constant boolean variable informally stated properties defined formally follows constant balanced easy show constant balanced constant setting need implement specification defined follows hadamard transform widely used quantum algorithms defined system setting function operation system applies every qubit system action zero state system general state action bitwise inner product modulo bitwise xor another important definition quantum analog classical oracle quantum solution one quantum variable integer variable measure let add specification restriction number calls oracle introducing time variable suppose new specification constant balanced constant charge unit time call oracle operations free clearly quantum solution works classically specification unimplementable fact strongest classically implementable specification constant balanced constant grover search grover quantum search algorithm quadratic speedup offers solutions problems algorithm optimal multiplicative constant task given function find simplicity assume single solution denote proofs different general case one solutions use general quantum oracle defined addition define inversion mean operator follows grover algorithm initialises quantum system equally weighted superposition basis states repeatedly applies followed system finally state measured probability error determined number iterations performed algorithm algorithm easily understood help geometric analysis operators let sum solutions let solution oracle performs reflection vector plane defined words similarly inversion mean operator reflection vector plane defined therefore result followed rotation plane quantum solution quantum variable natural variables time variable sin arcsin sin arcsin measure else specification carries lot useful information example tells probability finding solution iterations sin arcsin might ask many iterations performed minimise probability error examining first second derivatives find probability minimised arcsin integer course number iterations performed must natural number interesting note probability error periodic number iterations since gain anything performing extra iterations pick finally assuming obtain elegant approximation optimal number iterations probability error approximately computing mixed states discussed section state quantum system measurement traditionally described mixed state equation understood follows state probability probability contrast pure state mixed state describe physical state system rather describes knowledge state system framework need additional mechanism compute mixed states indeed mixed state system state distribution system states programming notions apply distributions mixed state following distribution quantum state expression tells possible value domain probability value example state probability probability also scalars equal probability one way obtain distribution measure equally weighted superposition measure measure sequential composition one point law distribution quantum state desired similarly need extend application unitary operators consider following toy program measure else second application hadamard quantum state mixed evident syntax program analysis final quantum state notion mixed state meaningful operator applied pure system state though unsure state measure else else sequential composition one point law distribution quantum state computation lot properties measurements mixed states proven definitions measurement sequential composition example fact measurement computational basis performed immediately following measurement basis change state system yields result first measurement probability proven follows measure measure measure sequential composition one point law measure measure case general quantum measurement proof similar little computationally involved conclusion future work presented new approach developing analysing proving correctness quantum programs since adopt hehner theory basis work inherit advantageous features simplicity generality elegance work extends probabilistic predicative programming fashion quantum computation extends probabilistic computation provided tools write quantum well classical specifications develop quantum classical solutions analyse various properties quantum specifications quantum programs implementability time space complexity probabilistic error analysis uniformly framework current research research immediate future involve reasoning distributed quantum computation current work involves expressing quantum teleportation dense coding various games involving entanglement way makes complexity analysis quantum algorithms simple natural issues described forthcoming paper easily express teleportation refinement specification distinct qubits fashion however interested possibilities simple proofs analysis programs involving communication via quantum channels exhibiting locc local operations classical communication paradigm future work involves formalising quantum cryptographic protocols framework provide formal analysis protocols naturally lead formal analysis distributed quantum algorithms distributed shor algorithm references abramsky methods quantum computation information proceedings annual ieee symposium logic computer science abramsky coecke categorical semantics quantum protocols lics abramsky duncan categorical quantum logic qpl pages adao mateus process algebra reasoning quantum security qpl altenkirch grattage functional quantum programming language proceedings annual ieee symposium logic computer science altenkirch grattage vizzotto sabry algebra pure quantum programming qpl arrighi dowek operational semantics formal tensorial calculus qpl pages arrighi dowek qpl bennet brassard quantum cryptography public key distribution coin tossing ieee int conf computers systems signal processing pages boyer brassard tapp tight bounds quantum searching fortschritte der physik pages coecke logic entanglement danos hondt kashefi panangaden distributed measurementbased quantum computation qpl deutsch quantum theory principle universal quantum computer proceedings royal society london pages deutsch jozsa rapid solution problems quantum computation proceedings royal society london hondt panangaden quantum weakest precondition qpl pages hondt panangaden reasoning quantum knowledge gay nagarajan communicating quantum processes proceedings acm symposium principles programming languages grover fast quantum mechanical algorithm database search annual acm symposium theory computing pages hehner practical theory programming springer new york second edition available free hehner probabilistic predicative programming mathematics program construction hehner retrospective prospective unifying theories programming symposium unifying theories programming jorrand lalire toward quantum process algebra proceedings acm conference computing frontiers jozsa quantum algorithms fourier transform proceedings royal society london pages knill conventions quantum pseudocode technical report los alamos national laboratory lalire jorrand process algebraic approach concurrent distributed quantum computation operational semantics qpl pages morgan mciver pqcl formal reasoning random algorithms south african computer journal nielsen chuang quantum computation quantum information cambridge university press quantum programming qcl master thesis vienna sanders zuliani quantum programming mathematics program construction pages selinger towards quantum programming language mathematical structures computer science selinger towards semantics quantum computation qpl tafliovich quantum programming master thesis university toronto valiron quantum typing qpl pages van tonder lambda calculus quantum computation siam journal computing yimsiriwattana lomonaco distributed quantum computing distributed shor algorithm zuliani quantum programming qpl pages zuliani compiling quantum programs acta informatica zuliani quantum programming mixed states qpl
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done object recognition icub robot perspective sep giulia pasquale carlo ciliberto francesca odone lorenzo rosasco lorenzo natale abstract report extensive study current benefits limitations deep learning approaches robot vision introduce novel dataset used investigation avoid biases currently available datasets consider interaction setting design protocol visual object recognition icub humanoid robot considering performance models trained image retrieval datasets show necessity knowledge transfer indeed analyze different ways last step done identify major bottlenecks robotics scenarios studying object categorization identification tasks highlight key differences object recognition robotics image retrieval tasks considered deep learning approaches originally designed nutshell results confirm also considered setting remarkable improvements yield deep learning pointing specific open challenges need addressed seamless deployment robotics keywords humanoid robotics icub machine learning deep learning object recognition transfer learning dataset invariance pasquale ciliberto rosasco natale icub facility laboratory computational statistical learning istituto italiano tecnologia via morego genova pasquale odone rosasco dipartimento informatica bioingegneria robotica ingegneria dei sistemi degli studi genova via dodecaneso genova introduction artificial intelligence recently progressed dramatically largely thanks advance deep learning computational vision specifically object classification perhaps obvious example deep learning achieved stunning results raise question whether problem actually solved krizhevsky simonyan zisserman szegedy case robotics would main field benefits could far reaching effect indeed lack reliable visual skills largely considered main bottleneck successful deployment robotics systems everyday life kemp perspective mind recently started effort isolate quantify benefits limitations deep learning approaches object recognition robotics pasquale clearly visual perception one possible sensory modalities enabling object recognition robotics montesano chitta dahiya natale gorges sinapov hosoda iwase moldovan indeed comparison human intelligence suggests vision object recognition metta fitzpatrick pinto nonetheless current deep learning based artificial vision systems perform well seems natural ask far perceptual needed end work mainly focus object recognition tasks robotics using visual cues investigate effectiveness deep learning approaches robotic applications designed dataset tailored reflect prototypical visual experience humanoid robot indeed remarkable performance deep learning methods object recognition primarily reported computer vision benchmarks griffin everingham russakovsky essentially designed image retrieval tasks hardly representative robotics scenario fact motivation common recent works lai oberlin borji new datasets proposed using icub robot metta devised interaction framework acquire corresponding dataset named icwt icubworld transformations dataset rich easy expand include data complex perceptual scenarios includes several object categories many instances per category hence allowing test categorization identification capabilities notably dataset segmented different sets views purpose testing specifically robustness invariance properties recognition systems within realistic robotic scenario provided icwt dataset performed extensive empirical investigation using different state art convolutional neural networks cnn architectures krizhevsky simonyan zisserman szegedy began checking extent systems already trained data could directly tested icwt obtaining results much better chance systems perform accurately enough case perhaps one could expect hence followed recent trend based general idea transfer learning schwarz oquab pasquale chatfield simonyan zisserman pretrained networks adapted data hand indeed results confirm effectiveness strategies obtaining substantial improvements impressive methods quite provide close perfect accuracy one would wish hence proceeded taking closer look results starting question whether missing gap could imputed lack data indeed cnns known need massive amount data work often used improve results discuss detail later paper investigating latter question highlighted differences icwt datasets imagenet russakovsky generally identified clear differences object recognition task robotics respect scenarios typically considered learning vision along way analysis allowed test invariance properties considered deep learning giulia pasquale works quantify merits categorization also identification description discussion empirical findings concluded critical review main venue improvements pure machine learning perspective also taking extensive advantage robotic platform indeed bridging gap performance appears exciting avenue future multidisciplinary research next subsection discuss several related works rest paper organized follows sec introduces icwt dataset acquisition setting sec review deep learning methods considered empirical analysis reported sec categorization task sec object identification sec concludes study review possible directions improvement visual recognition robotics deep learning robotics deep learning methods receiving growing attention robotics adopted variety problems object recognition schwarz pinto held pasquale place recognition mapping object affordances nguyen grasping redmon angelova pinto gupta levine tactile perception baishya limit discussion work object recognition relevant work described paper schwarz authors demonstrate transfer learning deep convolutional neural networks cnns propose way include depth information camera main idea extract feature vector cnn train cascade support vector machines svms discriminate objects class identity position depth data encoded planar image using colorization scheme authors report performance increase washington benchmark lai work pinto shows robot use explorative actions like pushing poking objects autonomously extract training data train cnn paper employ less constrained setting schwarz cnns trained data acquired robot natural interaction undergo therefore challenging viewpoint transformations employ similar techniques transfer learning consider wider range architectures fine tuning techniques contrast pinto use human teacher see done object recognition icub robot perspective sec details acquisition setup gives control object trasformations clearly work could extended introducing using explorative actions similar ones pinto work held work pasquale similar work presented paper investigate invariance properties cnns learning examples focus however instance recognition whereas paper consider thus significantly extending pasquale problem object categorization addition perform detailed investigation various techniques systematic evaluation recognition performance specific object transformations datasets visual recognition robotics literature several datasets visual recognition robotics proposed coil nene aloi geusebroek washington lai kit kasper riverarubio bigbird singh rutgers amazon picking challenge dataset rennie one main characteristic datasets capture images object undergoes specific viewpoint transformations however datasets usually acquired strictly controlled turntable settings aiming provide accurate annotations information object pose image ground truth grasping manipulation rather substantial variations objects appearance consequence mainly contain changes point view visual transformations scaling background changes forth fundamental benchmark visual recognition tasks instance bakry use washington pascal xiang datasets order evaluate invariance deep learning methods analysis consequently focuses mainly viewpoint changes moreover since object typically positioned turntable subsequently rotated dataset show natural object transformations often occur practice see leitner review major limitations current datasets norb dataset lecun also acquired similar turntable setting one first released specifically support investigation invariance properties recognition methods fact images dataset artificially perturbed order increase variability cently borji motivated similar considerations presented dataset specifically authors aim create visual recognition benchmark beyond representing high number object instances provides also sufficient number images per object order study invariance properties deep learning methods also acquired turntable setting order collect annotations image terms object pose icwt dataset presented paper separates previous work objects captured undergoing natural transformations acquisition performed setting intended benchmark reproducing typical uncertainties faced visual recognition system robot task following discuss icwt dataset related acquisition setting detail icubworld dataset section present novel dataset visual recognition icubworld transformations icwt used empirical analysis work icwt latest release project whose goal benchmark improve visual recognition systems robotics icubworld datasets ciliberto fanello pasquale designed record prototypical visual experience robot humanoid icub metta performing tasks end devised implemented simple interaction application directly acquire images dataset robot cameras remarkable advantage collecting icubworld directly robot platform indeed resulting dataset offers natural testbed visual recognition robotics close possible real application particular ensures performance measured icubworld expected generalize well system deployed actual robot note aspect icubworld extremely relevant since visual biases make typically difficult generalize prediction performances across different datasets applications already previous work pinto torralba efros khosla hoffman rodner model shamir stamos tommasi shown empirically sec paper https giulia pasquale fig icwt categories category icwt report one example image one instance red categories appear also ilsvrc dataset black categories appear imagenet ilsvrc see supplementary material details currently acquire icubworld releases make extensive use robot physical capabilities manipulation exploration done current deep learning methods achieve already remarkable performance relying solely visual cues goal evaluate accuracy isolation robotic setting indeed exploiting robot body could provide dramatic advantages modeling recognition see instance montesano recently moldovan leitner dansereau would also prevent correctly assess contribution purely visual components possibly contribute icubworld indeed datasets visual recognition literature acquired directly robotic platform ramisa meger little oberlin one common reason scaling dataset size extremely expensive time consuming using single robot end appealing solution proposed million object challenge project oberlin involve multiple robots collect shared dataset visual experiences note used initial subset icwt pasquale paper present dataset first time entirety acquisition setup dataset overview data acquisition icwt followed protocol similar one fanello adopted previous icubworld releases ciliberto fanello pasquale human teacher shows object robot pronounces associated label annotate subsequent images robot exploits visual cues track collect images object human actor moves shows different poses work decided adopt object tracker using depth information pasquale place one exploiting detection independent motion scene ciliberto used collect previous icubworld releases fact shown pasquale allows extract precise bounding box around object interest developed application scale acquisition procedure hundreds objects collected multiple interactive sessions icwt available plan make also application publicly available order laboratories use protocol collect icwt largest icubworld release far comprising objects evenly organized categories typically found domestic environment fig reports sample image category icwt categories red figure also imagenet visual recognition challenge ilsvrc russakovsky found semantically visually similar classes among classification challenge remaining categories appear ilsvrc belong similar synset larger imagenet dataset deng provide qualitative intuition semantic variability within given category namely different visual appearance objects category fig shows sample image instances mug category refer reader supplementary material fig example images object instances icwt peculiarity icwt motivating name object shown multiple image sequences undergoes specific visual transformations rotations scaling background changes done test invariance visual models robotics done object recognition icub robot perspective table summary icubworld transformations dataset fig semantic variability sample images different object instances mug category provide qualitative intuition semantic variability icubworld see fig supplementary material examples categories obj per category days transformations rot rot scale bkg mix frames per session image change object orientation rotation applied fig third row background human moved object around robot keeping approximately distance scale pose object respect camera plane acquisition process background changes object appearance remains approximately fig fourth row mix human moved object freely front robot person would naturally showing new item child sequence nuisances combinations appear fig fifth row fig visual transformations excerpts sequences acquired one mug representing object undergoes specific visual transformations indeed since works study real nonsynthetic visual transformations goodfellow order evaluate invariance properties visual representations believe icwt could significant contribution direction knowledge icwt first dataset address invariancerelated questions robotics accounts much wider range visual transformations respect previous datasets object instance acquired different image sequences human supervisor performed different visual transformation object fig reports excerpts sequences contain respectively rotation human rotated object parallel camera image plane scale position object mantained see fig first row rotation similarly rotations object kept position scale however time human applied generic rotation object parallel image plane consequence different faces object shown camera fig second row scale human moved object towards cameras back thus changing object scale sequence composed approximately images acquired frames per second time interval except mix one lasted comprises images anticipated acquisition objects split multiple sessions performed different days acquisition location always little uncontrolled changes background across days lighting condition artificially controlled since wanted investigate role nuisance end acquired objects different times day different days lighting conditions slightly changing across objects within five sequences object acquired span minutes moreover repeated acquisition object two different days ended sequences per object containing visual transformations different light conditions adopted icub cameras resolution recorded centroid bounding box provided tracker frame details procedure see pasquale left right images acquired allow offline computation disparity map potentially improvement object localization segmentation tab summarizes main characteristics methods section review learning methods considered work particular briefly introduce principal concepts behind deep convolutional neural giulia pasquale networks cnns describe architectures used analysis algorithms adopted train apply refer interested reader goodfellow introduction cnns deep learning general deep convolutional neural networks deep convolutional neural networks cnns hierarchical models organized concatenation multiple processing layers structure allows mapping input signal images speech etc series subsequent representations progressively select features relevant considered task prototypical structure cnn see fig performs layer usually referred convolution layer following set operations convolutions local convolution respect bank learned linear filters spatial downsampling instance using strided convolutions non linearity sigmoid functions bishop recently rectifying linear units spatial pooling aggregate local responses signal single component instance taking maximum value observed cnn architecture characterized specific choice implementation operations filters locally processing input signal layer typically learned minimizing desired loss function overall classification accuracy supervised settings reconstruction error unsupervised ones spatial downsampling pooling operations make representation robust transformations input increasingly larger scales selectively retain features relevant task suppressing others recently common strategy image classification settings follow convolution layers set fully connected layers namely standard neural network last years however architectures fewer even one layer started preferred since achieve comparable better performance optimizing significantly fewer parameters szegedy classification settings final layer network typically softmax function maps cnn output individual class likelihood scores cnn https prediction obtained choosing class maximum score modular structure cnns allows training parameters simultaneously layers also known learning via lecun given large number parameters optimized order millions cnns typically need large amounts training data achieve good performance various forms data augmentation often needed artificially increase number training examples synthetically modifying images via geometrical illumination changes mitigate risk overfitting regularization techniques weights regularization dropout hinton srivastava recently batch normalization ioffe szegedy proved helpful work investigate performance modern cnns robotic setting icubworld analysis selected recent architectures achieving highest accuracy imagenet visual recognition challenge ilsvrc russakovsky used corresponding implementation trained ilsvrc publicly available within caffe jia framework summarize structures small variation alexnet model winner ilsvrc krizhevsky concatenates layers comprising parameters second classified ilsvrc simonyan zisserman maintains general structure alexnet layers comprising parameters relatively expensive train terms memory time large layers winner ilsvrc szegedy diverges previous architectures concatenates called inception modules uses one layer end reducing parameters number layers name short residual networks ilsvrc smaller version stacking layers parameters https http https https done object recognition icub robot perspective fig example convolutional neural network sec two knowledge transfer approaches considered work sec blue pipeline feature extraction case response one layers used feature vector shallow predictor like rlscs svms see sec trained new task red pipeline case network trained new task replacing final layer using original model see sec network image transfer learning techniques deep learning methods typically require large datasets successfully trained could principle prevent applicability problems training data scarce recent empirical evidence shown knowledge acquired training network largescale problem transferred new domain different approaches implement strategy proposed literature section review two method empirically assess experiments namely feature extraction feature extraction observed deeper layers cnn selectively respond specific properties visual scene typically characteristic specific objects object parts chatfield yosinski donahue zeiler fergus responses interpreted representations visual input responding features relevant task hand following intuition shown cnns trained datasets indeed used feature extractors smaller datasets leading remarkable performance schwarz oquab pasquale typically done training standard classifier support vector machine svm regularized least squares classifier rlsc bishop top feature vectors obtained one multiple layer responses strategy depicted fig blue pipeline interpreted changing last layer cnn ing new dataset keeping parameters network fixed implementation details empirical analysis work used feature extraction strategy transfer knowledge cnns trained imagenet icwt specific layers used experiments reported tab four architectures considered paper used rlsc gaussian kernel classifier features particular implemented nystrom approach considered rudi computationally appealing mid settings indeed allowed significantly speed experimental evaluation refer reader supplementary material details model selection image preprocessing protocols adopted work cnn trained dataset used learning process potentially smaller domains strategy known finetuning chatfield simonyan zisserman consists performing new training set initializing parameters network previously learned see fig red pipeline settings necessary adapt final layer new task changing number units order account new number classes discriminate potential advantage adapts parameters networks layers new problem rather final layers giulia pasquale table feature extraction layers four architectures considered work used notation adopted literature number identifies layer number label specifies type fully connected pooling layer model caffenet googlenet refer supplementary material work analysis using parameters discussion model selection protocols used experiments output layer table protocols caffenet googlenet base starting learning rate layers initialized original model layers learned scratch indicated using names caffe models row specifying starting learning rate used parameters refer reader caffe documentation results categorization section present empirical investigation deep learning methods robotic setting icubworld deep learning need knowledge transfer modern datasets visual recognition comprise extremely large number images million base ilsvrc challenge depicting objects wide range natural scenes extreme variability opens queslearned layers tion whether datasets icubworld resent smaller reality could interpreted simply dropout larger ones case deep solver sgd adam learning models trained imagenet would achieve decay policy polynomial exp decay high recognition performance icubworld well epochs without adaptation batch size result would lead appealing scenario robot simply download use viever flexibility comes price insual classifier trained prevolved training process indeed performance vious analysis shown computer vision network extremely dependent choice datasets essentially biased ultimately preventing many available learning algorithm generalize one domain recently used adapt network models learned pinto torralba efros model imagenet robotic tasks see eitel shamir khosla stamos pinto gupta redmon angelova tommasi hoffman rodner pasquale nguyen similarly recently observed conventional models trained imagenet perform poorly applied robotic settings goehring implementation details experiments formed caffenet googlenet address question evaluated four representative recent architectures shelf cnns ones reviewed sec task significantly faster image classification icwt experiments considered two main regimes restricted test set categories icwt network one updating one appear also ilsvrc challenge see sec layers keeping others fixed another fig reference compared results one aggressively adapting layers comprising average accuracy networks layers training dataset done corresponding categories imagenet dataset selecting different learning rates neurons test set icwt composed category layer size gradient steps iteration images object instances comprising refer two protocols transformations one day left camera unconservative adaptive report corresponding less differently specified always used camera parameters tab reference experiments experiments total images per catediscussed paper consider strategory imagenet downloaded images gies since observe significant performance corresponding categories refer supplementary differences following choices parameters caffenet adaptive conservative googlenet adaptive conservative done object recognition icub robot perspective accuracy icwt shelf icwt adapted imagenet random choice caffenet googlenet fig average classification accuracy networks trained ilsvrc tested icubworld dark blue imagenet gray test sets two datasets restricted shared categories see sec light blue reports classification accuracy networks transferred icubworld see sec details orange line shows recognition chance random classifier material corresponding synsets comprising average images per category fig reports average classification accuracy icwt dark blue imagenet gray immediately observed substantial drop performance testing icwt rather imagenet suggesting differences two datasets exist particular icubworld imagenet drop likely due biases imagenet see also tommasi hoffman rodner analysis bias outside scope work supplementary material provide evidence effect care point fair comparison restricted output networks classes considered experiment icwt original output vector provided networks refer supplementary material experiments reporting performance considering entire prediction knowledge transfer drop performance observed fig implies necessary train recognition system target domain however experiment also shows networks performed substantially better chance orange line suggests networks retained knowledge objects appearance therefore rather training novel architecture scratch ically requires large amounts training data time computational resources viable alternative transfer knowledge essentially adapting networks trained imagenet new setting see sec idea knowledge transfer recently received considerable attention deep learning community hoffman agrawal azizpour huh provides robust approach apply deep learning methods smaller datasets fig report classification performance achieved systems knowledge transfer applied light blue experiments followed protocol described sec rlsc predictors trained feature vectors extracted deeper layers cnns created separate training set icwt choosing instances category training keeping instance testing repeated experiments trials order allow instance category used test set fig reports average accuracy trials observe sharp improvement networks achieve remarkable classification accuracy range imagenet however corresponding performance still significantly higher although gap seems reduced recent architectures reasons gap following empirically address question showing observed behavior due fundamental differences robot vision image retrieval settings need data deep learning methods require large amounts data order trained effectively particularly true training network scratch valid also albeit smaller scale knowledge transfer techniques indeed mentioned sec common practice training cnn perform data augmentation namely artificially increase dataset size applying synthetic transformations original images rotations reflections crops illumination changes perspective robotic setting seems particularly favorable indeed data augmentation performed simply acquiring frames depicting object viewpoint illumination change naturally acquiring frames object particularly convenient robotics typically expensive gather instead images depicting different objects instances since robot needs directly observe section investigate impact giulia pasquale rlsc rlsc caffenet adapt caffenet cons googlenet adapt googlenet cons accuracy accuracy fine tuning caffenet googlenet fig recognition accuracy frames left accuracy caffenet googlenet models according conservative adaptive strategies see sec right accuracy rlsc classifiers trained features representations extracted architectures considered work see sec aspects robot vision taking icubworld testbed analysis note following use term instance refer specific object belonging given category frame denotes single image depicting scene object gain adding frames considered categorization task icwt compare performance learning models trained increasing number examples created training sets different size sampling respectively frames transformation sequence object category see supplementary material list categories used objects training validation test validation test sets contained images available corresponding instances order account statistical variability repeated experiments trials time leaving one different instance testing experiment trained tested one two available days icwt one camera fig reports average classification accuracy different learning models training examples provided surprisingly architecture achieve remarkably high accuracy already trained smallest training set show little improvement new data available finding contrast expectations since increasing dataset size seem key significant improvement performance secondary observations rlsc achieve comparable accuracy caffenet googlenet caffenet adapt caffenet cons googlenet adapt googlenet cons caffenet googlenet fig recognition accuracy instances number object instances available training left accuracy caffenet googlenet models according conservative adaptive strategies see sec right accuracy rlsc classifiers trained features extracted architectures considered work see sec confirm ilsvrc trends recent networks outperforming older versions features better googlenet using feature extraction rlsc note caffenet performs worse training data scarce high number parameters learned layers see sec supplementary material support findings supplementary material report results experiment performed using less example instances per category gain adding instances evaluated impact adding new object instances training recognition system consider process increasing semantic variability training set contrast increasing geometric variability showing frames known object instance considered categorization task sec created four training sets containing frames sampled increasing number instances per category namely dataset training data randomly available transformation sequences achieve identical total size images per category validation test sets contained frames remaining instance per category repeated experiments trials fig reports accuracy different models tested problem results show increasing semantic variability dramatically improves accuracy models similar margin around done object recognition icub robot perspective instances per category image retrieval imagenet robot vision icubworld frames per instance fig different training regimes robot vision image retrieval settings moreover observed trends would expect model achieve even higher accuracy object instances available secondary observations networks seems lead worse performance rlsc feature extraction caffenet googlenet also setting confirm ilsvrc results recent networks outperforming previous ones outperforming googlenet using feature extraction rlsc support findings supplementary material report results similar experiment discriminating categories robot vision image retrieval section considered main challenges associated robot vision robot vision image retrieval address visual recognition problem cast within two different training regimes identified two main causes difference depicted fig semantic variability namely amount different object instances available category amount frames per instance typically image retrieval settings large number images depicting different object instances category available instance images provided example ilsvrc training set comprises instances object categories image per instance provided opposite end spectrum robot vision settings easy gather large amounts example images single instance since robot move around observe object multiple points view however remarkable limitation total number categories instances experienced analysis shown limited availability semantic variation naturally occurs robotics applications dramatically affect recognition accuracy learning system moreover contrarily expectations observed limitation alleviated simply feeding training data frames network indeed classifiers trained datasets different size semantic variability achieved identical performance extremely problematic since robotics overhead collecting acquiring examples novel instance extremely expensive collecting views object comes low cost unfortunately adopt data augmentation strategies artificially increase semantic variability dataset done case variability findings appears clear bridging gap robot vision image retrieval needs deeper analysis end following deepen analysis performance cnns icwt particular focus concept invariance invariance sec saw models trained frames achieve comparable even better accuracy trained larger datasets results suggest corresponding networks could leverage robust data representation indeed section investigate extent models invariant visual transformations icwt indeed invariance robustness visual transformations scene highly desirable behavior designing training visual recognition system since dramatically reduce complexity learning problem anselmi increasing capability generalizing visual appearance object category limited number examples ideally one even appealing robotics applications usually robot required learn new objects fly recognize reliably unpredictable conditions test invariance cnns considered categorization problem introduced sec used object instances per category training validation testing however mix examples five available sequences rotation rotation scale giulia pasquale test mix test rot test scale test bkg mix bkg mix bkg mix bkg mix bkg mix bkg accuracy test rot caffenet googlenet fig generalization performance across different visual transformations learning models trained one transformations present icwt specified horizontal axis tested transformations accuracy reported separately plot tested transformation shaded areas represent improvement achieved including frames sequence instead subsampling background mix instead performed training validation using individual transformation tested learned model others repeated experiments trials considered two regimes one including images sequence another one subsampling images factor sec fig since previously observed including instances training set important including frames sec limit analysis one two available days icwt indeed setting aim study effect isolated viewpoint transformations without considering nuisances due example changes light setting conditions appear one day another experiment report accuracy approaches rely application rlsc top feature representations extracted using models see sec done isolate invariance properties networks trained imagenet note however performed experiments also networks previous tests observing similar results fig reports generalization capabilities models trained single transformation tested different one reference considered also training set obtained randomly sampling points training sets keeping similar size training set analogous used experiments sec size reduced images per category images per category subsampled regime report accuracy pled training sets bold line improvement achieved considering frames sequences shaded area color first confirm adding example frames transformation significantly improves networks invariance even transformation notice overall models exhibit invariance transformations included training set worth noticing cases best performance obtained training testing performed transformation transformations included training set demonstrates enriching dataset include transformations degrade performance moreover training mix achieves good performance tested every specific transformation suggests showing object robot natural way transformations appearing random combinations instead systematically collecting sequences comprising individual transformations feasible approach obtain predictors invariant transformations interesting note rather different models trained imagenet exhibit invariance pattern particular performance drops substantially testing transformation present training set expected transformations involving rotations object quite surprising affine ones namely scale rotation background convolutional structure cnn invariant design learned training imagenet investigate fact sec study invariance cnn models context object identification rather categorization done object recognition icub robot perspective knowledge transfer categorization identification sec seen knowledge acquired imagenet transferred icubworld domain successfully tackle categorization tasks natural question whether strategy would similarly favorable object identification indeed object identification settings semantic variability one instance per class categorization observed semantic variability extremely useful learning process sec simply adding images given instance almost redundant sec addressed question considering object identification task icwt compared cnn models trained increasing number examples setting similar one used categorization object identification problem chose object instances book flower glass hairbrush hairclip categories appear ilsvrc dataset created four training sets containing respectively rlsc fine tuning accuracy results object identification far work focused visual categorization problems namely assign given object instance corresponding category section move instead task object identification instance recognition consists labeling given image according object instance appearing problem indeed relevant robotics settings since reliable instance recognition skills would provide system finer control tasks manipulation grasping etc also considering domestic scenario robot could asked use specific object rather object category icub bring phone instead phone therefore parallel experiments categorization work assessed performance deep learning methods task object identification robotics note recently problem approached methods based keypoints extraction template matching lowe philbin collet crowley zisserman collet muja however recently observed appraoches relying holistic visual representations perform typically better settings icubworld ciliberto following section focus cnn architectures considered sec caffenet adapt caffenet cons googlenet adapt googlenet cons caffenet googlenet fig recognition accuracy frames identification left accuracy caffenet googlenet models according conservative adaptive strategies see sec right accuracy rlsc classifiers trained features representations extracted architectures considered work see sec images per object sampled randomly transformation sequences rot rot scale bkg see sec dataset images retained model selection images mix sequence used test classification accuracy methods considered categorization experiment images single day used experiments fig reports average accuracy cnn architectures trained growing number examples results stark contrast counterpart categorization sec clearly notice adding training data dramatically improves recognition capabilities networks suggests access views instance allows learning systems create nuanced model object apparently richer model provide advantage categorization settings extremely useful recognize object new images line observations notice also setting adaptive strategy significantly outperforms competitors suggesting representation learned categorization imagenet albeit already beneficial task improved adapt object identification setting findings confirm extend recent similar results instance retrieval literature babenko gordo invariance sec seen accuracy object identification system greatly improved provided examples instance observation seems contradict assumption cnn giulia pasquale googlenet caffenet test rot test rot test scale accuracy test mix bkg mix bkg mix bkg mix bkg mix test bkg fig generalization performance across different visual transformations identification rlsc trained one transformations present icwt specified horizontal axis tested transformations accuracy reported separately plot tested transformation designed already invariant transformations addressed question considering learning scenario similar one adopted sec investigate invariance categorization settings focused object identification specifically considered identification task restricted training set contain images single transformation sequence available icubworld sec resulting models tested separately remaining transformations assess ability learning systems generalize new viewpoints similarly sec considered one day evaluated learning methods based feature extraction plus rlsc order evaluate viewpoint invariance networks fig reports average accuracy models trained single transformation using respectively frames per object tested another notice performance varies remarkably one tested transformation another suggesting representations considered experiments invariant transformations observed icwt interestingly comparing two training regimes large small training set notice remarkable improvement observed sec adding frames confirmed extended also setting system trained single transformation time expected recent networks significantly outperform previous models achieving highest accuracy large margin similarly observed categorization setting fig reports quite low performance generalizing across transformations following discuss possible strategies address problem improving invariance section consider different strategies improve invariance properties cnns identification settings analysis motivated observation fig adaptive finetuning performs significantly better conservative feature extraction based strategies indeed suggests adapting representation cnns identification task helpful recognition possibly network actually learning transformations needs invariant end investigate whether cnn icubworld domain indeed invariance properties network identification task performed preliminary set experiments related question pasquale considered smaller set transformations strategies focused caffenet architecture consider learning setting experiments reported fig evaluate learning models first one following datasets improve invariance icubworld identification icwt dataset contains images instances object categories icwt used previous experiments namely oven glove squeezer sprayer body lotion soda bottle performed adaptive strategy identification task training validation sets obtained following protocol sec icubworld categorization icwt cat dataset icw performed categorization task objects done object recognition icub robot perspective accuracy icwt cat icwt icwt imnet imnet test rot test rot test scale accuracy test mix bkg test mix mix icwt cat icwt icwt imnet imnet bkg test rot mix bkg test rot mix bkg test scale mix bkg mix bkg mix bkg mix bkg mix test bkg test bkg fig experiment setting fig using different image representations provided caffenet top orange googlenet bottom blue network models according different strategies see sec considered adaptive strategy used train validation sets obtained following protocol sec icubworld imagenet icwt imnet dataset contains images categories sampled icwt imagenet note icwt categories appear ilsvrc synsets larger imagenet dataset see supplementary material list corresponding synsets images synset used performed categorization task dataset conceived test possibility cnn directly learn relation imagenet examples originally trained icwt images imagenet imnet dataset contains imagenet synsets corresponding object categories cnns later trained tested invariance namely book flower glass hairbrush hairclip performed categorization task dataset allows network focus data available task hand different images robot observe fig reports accuracy rlsc classifiers trained features extracted cnns datasets described training performed separately transformation similarly ous section dataset containing example images per object instance noticed identification task particularly advantageous leading dramatic improvement model trained features particular comparing results fig see icwt trained examples per object performs par method trained images per object moreover performance icwt general stable across different viewpoint transformations suggesting preliminary could indeed allowed cnns become partially invariant transformations icubworld interestingly cnns provide similar improvements identification cases even worse performance based classifier conclude note related categorization indeed following experiments could wonder whether similar strategy could improve invariance also categorization setting sec however empirical evidence showed performing datasets described performance cnns change significantly categorization task refer supplementary material analysis specific setting hypothesize due two possible reasons networks trained ilsvrc already highly optimized categorization gain adapting visual representation negative effect limited semantic variability icubworld respect imagenet may overcome potential benefits increasing invariance viewpoint transformations motivates proposed directions future investigation discuss following section discussion future work work studied application modern deep learning methods visual recognition tasks robotics challenged deep learning methods object recognition task specifically designed represent prototypical visual recognition problem real robotics application experiments show deep learning leads remarkable performance visual tasks object classification instance recognition proper adoption knowledge transfer strategies particular mixing deep learning conventional shallow classifiers based kernel methods plays fundamental role leverage visual representation learned datasets achieve high performance robotic domain however substantial gap still exists performance obtained two domains analysis shows performance gap due scarce semantic variability characterizes robotic domain due intrinsic cost acquiring training samples yet adoption robotic systems real applications requires push requirements achieve performance approaches indeed robotics applications failures due object potentially lead dramatically critical harmful consequences standard image retrieval scenarios therefore error rate systems need close possible zero considered production deployment environments shared humans section consider possible directions future research aimed mitigating impact differences performing visual recognition robotics goal discussion consider together picture present opinion robot vision problem could addressed improving invariance experiments sec show models tested work mostly locally invariant namely representation robust small changes local invariance main interest computer vision past lowe giulia pasquale mikolajczyk schmid current research invariance instead focused designing learning systems able learn encode representations robust dramatic global transformations object anselmi clearly robust visual representation viewpoint changes small deformations background variations less corresponding model trained dataset imagenet biased specific training conditions research invariant representations could therefore lead robust robotics systems need knowledge transfer also allow train classifier trained examples novel object reliably recognize different poses learning work followed typical approach categorization similarity relation assumed among different classes however visual recognition settings object categories often number visual features common wheels windows legs handles incorporating information similarities across object classes within learning model could significantly improve overall accuracy recognition system especially presence scarce training data particular literature learning proposed several approaches model relations across tasks object categories directly enforce learning problem available crammer singer micchelli pontil evgeniou joachims fergus lozano sindhwani learn unknown argyriou jacob minh sindhwani dinuzzo sindhwani ciliberto augmenting semantic variability simplest method increasing semantic variability share data acquired parallel different robotic systems indeed similar strategy used learn coordination robotic setting training networks using data acquired parallel several robots levine similarly million object challenge oberlin aims acquire large dataset object models sharing data acquired different laboratories owning baxter along similar direction plan extend icubworld help community icub robot time writing research groups world beyond expanding dataset http done object recognition icub robot perspective direction semantic variability would also allow collect multiple acquisitions object different conditions order extend analysis presented work nuisances like changes light setting indeed another critical aspect started consider supplementary material work alternative complementary approach data augmentation method often adopted image retrieval settings increase viewpoint also illumination variability training applying synthetic transformations image dataset rotating flipping cropping changing hue value visual augmentation cope semantic variability much challenging problem although recent work inverse graphics mansinghka kulkarni starting point direction exploiting contextual information work visual recognition considers frames independently rarely uses contextual information robot typically exposed continuous stream visual information information highly correlated exploiting correlations done many ways ranging trivial solutions like temporal averaging classification results complex ones rely scene reconstruction object tracking song integrating information robotics problem integrating informationwith rgb recently investigated within deep learning framework schwarz redmon angelova results although promising leverage representations learned datasets images future challenge determine best encode process information within cnn also acquire datasets directly training rgbdbased systems learning one way reduce cost supervision implement explorative strategies robot autonomously interacts environment collect training samples training instances scenario could extracted autonomously detecting invariances data correspond physical entities coherent motion pattern wang gupta salency cues strategies specific robotic domain could devised integrating multiple sensory modalities sinapov higy repertoire explorative actions designed specifically extract cues like grasping pushing objects montesano fitzpatrick hgman pinto conclusions paper described systematic study benefits deep learning methods robot vision tests devised prototypical vision task humanoid robot interaction exploited obtain realistic supervision train object recognition system presented icwt dataset investigation performance deep learning methods applied task results confirm deep learning remarkable step forward however still lot needs done reach level robustness reliability required real applications identified specific challenges possible directions research bridge gap confident next years rich exciting progress robotics acknowledgements work described paper supported center brains minds machines cbmm funded nsf stc award firb project funded italian ministry education university research gratefully acknowledge nvidia corporation donation tesla gpu used research references agrawal girshick malik analyzing performance multilayer neural networks object recognition proceedings european conference computer vision eccv anselmi leibo rosasco mutch tacchetti poggio unsupervised learning invariant representations theoretical computer science doi url http anselmi rosasco poggio 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systems zeiler fergus visualizing understanding convolutional networks springer international publishing cham url http doi giulia pasquale done object recognition icub robot perspective supplementary material giulia pasquale carlo ciliberto francesca odone lorenzo rosasco lorenzo natale icubworld transformations fig reports one image object instance icwt dataset report also reference number associated synset imagenet russakovsky note categories icwt part ilsvrc belong least similar synsets larger imagenet dataset deng table preprocessing operations executed images feeding networks mean subtraction scaling crop extraction caffenet image mean image size grid center mirrored googlenet pixel grid center mirrored pixel shorter side model cnns sec observed direct application cnns icubworld leads poor performance section provide additional details experiment reported fig image preprocessing applying deep cnns setting first evaluated impact considering coarser finer regions around object images indeed image icwt annotated coordinates object centroid bounding box provided segmentation depth map see sec took advantage rough localization order avoid considering full images classification compared different strategies extract crop images specifically tried either extract square crop fixed radius using information centroid use bounding box provided routine segmenting depth map smallest rectangle enclosing object blob cases tried two sizes using radius square crop leaving margin pixels around depth bounding box since deep cnns require fixed sized input caffenet considered models different preprocessing strategies applied order feed image network reproduced operations specified reference page particular caffe model convenience report tab general consist subtraction mean image mean pixel value dataset model trained extraction grid crops eventually multiple scales required size usually final prediction computed aggregating predictions crops gain robustness however since case image fact already crop around object compared advantage grid approach considering central crop single scale case fig reports results sec fig obtained testing cnns icwt different described strategies noticed performing finer grid scale mirrored localization object image blue green provides better performance architectures also considering central crop provides little advantage even detrimental reasonable already localized object fact considering portion image finally chose apply cnns extracting square region icwt images considering central crop light green fig hence need scale image considered single scale performance reported instance fig dark blue strategy used experiments chose extract regions instead relying bounding box depth map since aimed evaluating following analysis also invariance properties geometric transformations scaling precise applying networks imagenet instead used strategy suggested architectures see tab categorization results fig reported accuracy cnns applied icwt imagenet however since task reduced categorization setting considered prediction networks limited corresponding labels rather full vector scores normally produced cnn trained ilsvrc completeness fig report resulting accuracy cnns taking prediction class achieving maximum score available even present considered test sets one icwt reduced imagenet noticed performance drops even chance icwt since real chance rather viewpoint biases section preliminary analysis reported sec discussing potential biases imagenet done object recognition icub robot perspective soda bottle cellphone mouse coffee mug pencil case perfume remote ring binder soap dispenser sunglasses wallet oven glove squeezer sprayer body lotion book flower glass hairbrush hair clip fig example images objects icubworld transformations giulia pasquale accuracy disp multi crop disp disp multi crop disp multi crop multi crop caffenet googlenet fig average classification accuracy networks tested icwt segmenting object according different strategies icwt shelf imagenet accuracy object instances category rows matrix temporal dimension frame index reported horizontal axis row predictions sequence represented vertical bars hite prediction correct red wrong black sequence terminated noted since acquisition operator moving object back forth front robot starting close slowly moving backward eventually robot sequences cnn manages recognize object appears relatively large scale white bars mostly concentrated first half rows qualitative observation confirms recent studies herranz specifically highlight problem scale bias cnns preventing models trained datasets imagenet generalize settings objects mostly appear small part image like second part sequences fig report sequences containing rotation case operator rotating object always maintaining face visible robot constant speed indeed noticed many sequences periodic time intervals cnn recognize object corresponding configurations category appears less frequently imagenet dataset example observed soap dispensers mugs imagenet mostly placed tables consequently never recognized model pre selection section provide details regarding parameter selection learning methods adopted work described sec caffenet googlenet fig average classification accuracy networks trained ilsvrc tested icwt dark blue imagenet gray test sets two datasets restricted shared categories see sec feature extraction rlsc specify design choices made implement steps described sec feeding images networks order extract representation applied preprocessing pipeline explained motivated sec archiventing models generalize well tested tecture layers considered output extract repreicubworld without applying knowledge transfer techniques sentations specified tab caffenet vggto end report series excerpts showing indicated either layers since networks predictions cnns considered one layer provide tested icwt sequences qualitative analysis allows output representations full image best choice deto better understand frames icubworld actually pending application resorted rudi harder recognize cnns compare nystrom subsampling approach used prototypical examples imagenet implement gaussian rlsc algorithm fact inwe report predictions obtained using googlenet volves determining hyperparameters critical number training examples subsample made similar observations architectures approximate kernel matrix fig prediction computed maximum score among classes present test set rather order evaluate best output layer caffenet full vector restricted analysis also assign reasonable value subset test set considered fig specifically considered one small one large categorization tasks best recognized categories two specific visual transforon icwt representative smallest larger task mations scale rotation expected run analysis compared rlcs accuracy varying factors fig report predictions sequences containing scale transformation separately fixed categorization problem training object instances per category validating category plot represent sequences done object recognition icub robot perspective cellphone mouse mug remote soap dispenser fig predictions googlenet image sequences containing scale transformation reported categories different plots sequences object instances belonging category represented matrix rows temporal dimension frame index reported horizontal axis row predictions sequence represented vertical bars hite prediction correct red wrong black sequence terminated cellphone mouse mug remote soap dispenser fig fig image sequences containing rotation caffenet googlenet caffenet googlenet accuracy accuracy fig classification accuracy reported training rlsc image representations extracted four considered architectures large categorization experiment performed training examples varied horizontal axis performance training day light colors another one dark colors reported instance considered sequences visual transformations one camera considered one day among fig similar fig small categorization experiment training examples varying tween two available icwt training tested instance left training set days large task considered frames per sequence leading training examples per class whereas small task subsampled frames per sequence leading examples per class similarly sec fig reports average accuracy respectively small large experiment performed two experiments using representations four considered architectures reported separately using either layer gray pink caffenet case increased logarithmically value horizontal axis starting size training set small experiment stopped whereas large experiment stopped values observed little performance improvements respect much longer training times performance day trained reported light whereas performance day reported dark colors first observed features consistently performed better hence decided use layer extracting representations caffenet images icwt also observed relatively small values could provide good accuracies therefore chose fixed min experiments worth noting performed similar empirical analysis one reported order assess best output layer use extract visual representations models sec previously finetuned subsets icwt case observed features providing best performance therefore used layer experiments sections explained considering specialized features extracted models adapted setting general features models provide better generalization performance icwt section provide details performed procedure choice two strategies adaptive conservative reported tab general protocol image preprocessing pipeline similar one reported sec square region around object centroid extracted either subtracted mean image caffenet pixel googlenet training set training rest processing performed within caffe specifically extracting random crop randomly mirrored horizontally passing network test phase prediction one image taken extracting central crop instead random one see sec training set shuffled starting finetuning process since observed similarity images within batch negatively affected convergence backpropagation algorithm network evaluated model performance validation set every epoch finally chose epoch achieving highest validation accuracy left batch size values specified caffe reported tab number iterations equivalently epochs fixed empirically observing general performance saturating epochs models conservative caffenet involves learning parameters layers giulia pasquale therefore takes time converge case fixed number epochs hyperparameters choice section report empirical analysis performed order understand effect relative importance hyperparameters involved deep architectures caffenet googlenet scenario model selection per open problem dealing deep networks setting issue even complicated fact fixed reference task optimize training ilsvrc instead plan span wide range tasks comprising small large training sets indeed study presented work aims comparing models trained different categories objects transformations forth considered therefore two experiments used perform parameter selection rlsc approach small large categorization tasks described sec caffenet googlenet varying values multiple hyperparameters following report analysis separately two architectures caffenet considered parameters appearing tab varied value following way base starting learning rate layers initialized parameters model tried learned layers layers scratch specific starting tried learn starting set including also iii finally also empirical rule every time included one layer learn scratch decreased starting layers factor hence iii dropout percentage dropout layers tried default caffe value solver algorithm used stochastic gradient descent used sgd solver bottou caffe decay policy decay rate learning rates tried either polynomial decay exponent step decay decreasing factor every epochs tried possible combinations values parameters missing kept value specified caffe reference models see also previous section observed dropout percentage small influence left default value also observed polynomial decay smaller slope consistently better accuracy critical parameters instead base numbers layers learned scratch fig report example accuracy obtained respectively learning left right case decreased base layers horizontal axis performance reported fig day training light gray different one dark gray tried values base learning rate small experiment continuous line large experiment chose one small medium large value dots interpreting results useful recall strategy learns scratch three layers done object recognition icub robot perspective starting adam small small large large base accuracy accuracy starting base fig classification accuracy provided caffenet according different strategies either learning scratch left right performance reported day training light gray different one dark gray tried multiple values base small training set continuous line one small medium large value dots large one small small large large base sgd base fig classification accuracy provided googlenet according different strategies either using adam solver left sgd right rest figure similar fig decay policy using sgd used polynomial decay exponent using adam maintained learning rate constant caffenet tried combinations left mentioned parameters default values caffe first observed differently caffenet overall architecture impact learning layers scratch small three tested strategies behaving similarly chose last one iii slightly better others also observed little benefit using higher dropout percentages one little critical aspect instead choice solver end fig report accuracy obtained respectively using adam solver left sgd right latter case applied polynomial decay smaller slope since observed consistently better performance reported fig day training light gray different one dark gray varied base horizontal axis trying values small experiment continuous line one small medium large value large experiment dots observed sgd solver robust different choices base adam provides slightly better accuracies values base small large experiment therefore opted googlenet performed googlenet similar analthis latter solver ysis one reported caffenet similar observations caffenet architecwe recall considered googlenet architecture ture led conclude learning slowly smaller composed main branch ending one layer called base preferable add frames two identical auxiliary branches ending two layers called training set since model less prone overfit training condition base equal gain perforwe refer szegedy detailed description mance days passing continuos line model considering structure explored corresponding dot larger base preferable following parameters instead training set smaller reported tab base varied caffenet therefore also googlenet identified adaptive stratlearned layers always learned egy base equal conservative strategy scratch starting equal regarding base equal auxiliary branches tried cut learn also scratch starting equal finally iii learn scratch experiments categorization also setting starting layers section complement experiments discussed dropout tried either default caffe values paper analysis results reported higher pershow behavior observed specific conditions centage solver tried either sgd adam kingma considered paper number object classes considsolvers caffe ered tasks holds also settings right much free parameters one learning last layer left indeed notice latter strategy robust scenario base continuos line range left right also observe add frames training set strategies benefit learning slowly pass continuos line corresponding dot gain smaller base lrs indeed largescale experiment learning small base achieves best performance contrary learning large base almost benefit seeing frames even overfits day training therefore since analysis aimed spanning size training set wide range chose two representative strategies providing best performance respectively small experiment learning small actually base call conservative strategy account settings learning large base call adaptive strategy account settings giulia pasquale rlsc caffenet adapt caffenet cons googlenet adapt googlenet cons fig recognition accuracy frames number object views available training experiment fig executed training example instances per category caffenet adapt caffenet cons googlenet adapt googlenet cons caffenet googlenet caffenet googlenet fig recognition accuracy instances number object instances available training experiment fig executed categorization problem improving invariance cnns categorization rlsc caffenet adapt caffenet cons googlenet adapt googlenet cons caffenet googlenet accuracy rlsc accuracy accuracy fine tuning fig recognition accuracy instances number object instances available training experiment fig executed categorization problem increasing training set size fig reports results experiment shown fig performed including object instances per category training sets sake completeness complement information provided sec precising considered categorization task comprising following categories cellphone mouse coffee mug pencil case perf ume remote ring binder soap dispenser sunglasses lower wallet glass hairbrush hair clip book results confirm setting adding frames without increasing semantic variability used viable strategy improve categorization accuracy remains definitely lower performance achieved using example instances fig increasing semantic variability fig report results experiment analogous one shown fig respectively object categories rather noticed observations reported sec apply also namely accuracy increases remarkably example instances per category made available without saturating supporting claim semantic variability indeed critical categorization even settings involve categories mentioned end sec verify whether approach adopted improve invariance cnns identification settings may also adopted improve performance categorization considered categorization task sec fig performed sampling frames per sequence applying rlsc top representations extracted cnns experiment compared performance task achieved applying rlsc top representations extracted cnns previously finetuned end sec evaluated impact different strategies specifically considered exactly models sec first three strategies namely icwt icwt cat icwt imnet last one imnet imagenet synsets corresponding categories later involved categorization task report results fig using notation fig anticipated none strategies beneficial considering categorization task indeed achieve improvement performance respect baseline generalization across days section test robustness considered visual recognition systems respect variations changes illumination background neither semantic geometric current release icwt object instance acquired image sequences two different days order naturally images kind variations sake brevity main paper report experiments related aspect however relevant training visual recognition given day testing second one possibly small contextual variation scene cause system experience degradation recognition accuracy experiments reported report loss difference average classification accuracy observed testing model day trained available day icwt particular consider models trained categorization experiment sec sec done object recognition icub robot perspective accuracy icwt cat icwt icwt imnet imnet test rot test rot test scale test bkg mix bkg mix mix bkg mix mix bkg mix mix bkg mix mix bkg mix accuracy test mix test mix icwt cat icwt icwt imnet imnet test rot test rot test scale test bkg mix bkg mix mix bkg mix mix bkg mix mix bkg mix mix bkg mix fig experimental setting fig using different image representations provided caffenet top orange googlenet bottom blue network models according different strategies see sec rlsc rlsc accuracy accuracy fine tuning caffenet adapt caffenet cons googlenet adapt googlenet cons caffenet googlenet caffenet adapt caffenet cons googlenet adapt googlenet cons caffenet googlenet fig categorization accuracy frames generalization across days drop performance difference test accuracy observed testing models trained categorization task reported sec day training different one fig categorization accuracy instances generalization across days drop performance difference test accuracy observed testing models trained categorization task reported sec day training different one identification sec section test models day different one used training took images object instances selected testing original experiment second available day categorization fig fig report loss performance observed categorization setting respectively case tested relevance training set size see fig semantic variability see fig surprisingly settings networks exhibit substantial drop performance order accuracy cases even around critically adaptive strategy appeared effective tested day also strategy leads biggest drop implies cnn overfitting variability training day therefore less capable generalizing days interestingly however using less aggressive strategies conservative feature extraction based classifier modern networks googlenet seem general quite robust identification dramatic loss performance observed adaptive categorization could principle worrisome recall identification setting strategy adopted sec improve invariance properties cnns preliminary dataset interestingly however fig notice identification setting adaptive strategy seems giulia pasquale accuracy fine tuning rlsc caffenet adapt caffenet cons googlenet adapt googlenet cons caffenet googlenet fig identification accuracy frames generalization across days drop performance difference test accuracy observed testing models trained identification task reported sec day training different one overfitting training day albeit experiencing drop performance generalizing another day loss small par strategies considered work indeed noticed methods exhibit performance drop around accuracy
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age information network preemptive servers roy yates mar winlab department electrical computer engineering rutgers university ryates source submits status updates network delivery destination monitor updates follow route series network nodes node queue supporting preemption service characterize average age information input output node route induced updates passing poisson arrivals line network preemptive memoryless servers show average age accumulates successive network nodes ntroduction need timely knowledge system state remote monitoring applications led development analysis status update age metrics randomly arriving updates queued service facility early work showed generally source limit offered load particular updating must balance infrequent updates overly frequent updates induce queueing delays observation prompted study age lossy systems discard updates avoid building queues include served lcfs queue preemption either waiting service packet management mechanisms restrict number queued packets discard waiting packets become stale however contributions consider single hop communication systems recently effort examine age multihop network settings particular work closely related lgfs multihop networks studied update transmission times network links exponentially distributed sample path arguments used show preemptive lgfs policy results smaller age processes nodes network causal policy however structural results facilitate explicit calculation age work consider preemptive lcfs servers special case lgfs discipline multihop line network shown figure poisson arrivals memoryless preemptive servers use stochastic hybrid systems shs model age processes network derive simple expression average age node work supported nsf award presented ieee infocom age information workshop work motivated simple question regarding line network nodes service rates network traffic depends qualitatively service rates example packet dropping occur primarily node traffic pass quickly relatively little dropping node hand substantial dropping node little dropping node perspective average age monitor better dropping occur earlier later network reasonable hypothesis wastes resources faster downstream server fact see theorem average age monitor insensitive ordering servers work network model described section followed section iii summary results stochastic hybrid systems shs age analysis basis age analysis section apply shs tandem queue use model describes occupancy server network analysis may instructive shown section preemptive service facilitates analysis using fake updates technique reduces discrete state space one state simulation experiments provided section conclusions appear section vii ystem odel model updating process source submits update packets rate poisson process network depicted figure updates follow route nodes network monitor node update exponential service time independent arrival process service times nodes however forgo queueing network node preemptive server upon arrival server update immediately goes service update currently service preempted discarded useful model time update spends position network interface dominated waiting transmission example congested wireless network service time would dominated mac access delay transmission time packet negligible case would feasible desirable source monitor fig line network model node rate preemptive server update packet nominal update service preempted recent arrival starting time source submits status updates successive times update submitted time delivered time dependent random variables moreover preemption line network lossy updates discarded time recent received update max thus status update age refer simply age system performance given average status update age network figure denote average age monitor case node simple preemptive queue also called preemption service queue using graphical approach shown time average approaches prior work network modeled single service facility key analytical steps involve system time interarrival time delivered packets challenge computation correlation however network nodes graphical method fails queueing updates network several reasons line network lossy updates discarded preempted even calculation effective arrival rate updates monitor challenging departure process node even node memoryless arrivals node subsequent nodes fresh instead aged passage prior nodes age packet arriving node may correlated service times preceding nodes interarrival time update may correlated age updates reach monitor may subject survivor bias lucky enough service times avoid preempted iii tochastic ybrid ystems complexity lossy queueing process line network take different approach average age analysis following model system stochastic hybrid system shs hybrid state shs model discrete typically describes markov state queueing system row vector continuous captures evolution collection processes general shs powerful modeling framework many variations work use simplified form shs aoi analysis introduced piecewise linear process interest completeness summarize basics simplified shs details found references therein graphical representation markov chain state node transition directed edge transition rate note kronecker delta function ensures transition occurs state state define respective sets incoming outgoing transitions transition transition reset mapping induce discontinuous jumps continuous state aoi analysis employ linear mapping form xal transition causes system jump discrete state resets continuous state xal moreover discrete state continuous state evolves according using piecewise linear shs aoi elements binary see ones correspond certain relevant components grow unit rate state zeros mark components irrelevant state age process need tracked tracking age process transition reset maps binary linear mappings depend specific network system indexing scheme updates system transition rates correspond transition rates associated markov chain discrete state differences unlike ordinary markov chain shs may include discrete state unchanged reset occurs continuous state furthermore given pair states may multiple transitions discrete state jumps transition maps different sufficient average age analysis define vector functions note denotes discrete markov state probabilities similarly measures correlation age process occupancy discrete state fundamental assumption age analysis markov chain ergodic otherwise age analysis makes little sense assumption state probability vector always converges unique stationary vector satisfying shown obeys system first order differential equations vql fig shs markov chain line network nodes transition rates maps links shown table xal depending reset maps differential equation may may stable however stable converges limit case follows lim lim following convention age monitor average age process interest following theorem provides simple way calculate average age ergodic queueing system theorem theorem markov chain ergodic stationary distribution find solution differential equation stable average age aoi shs given age tandem ueue reemption use theorem evaluate age monitor line network figure intermediate nodes example demonstrate use theorem straightforward way evaluate average age network network set discrete states indicates node serving update packet also convenient refer states continuous state age monitor update service node age update node idle irrelevant set table table transitions arkov chain igure particular state node idle hold state otherwise node serving update state increases unit rate state follows state set follows age monitor grows unit rate hand states node serving update whose age growing unit rate similarly states node serving update whose age growing unit rate markov chain occupancy network shown figure edges labeled transition table lists state transition pair transition rate transition mapping xal matrix facilitate using theorem vql describe transitions note age monitor changes transitions update completes service node delivered monitor corresponding table transitions idle network fresh update arrives node age monitor unchanged arrival fresh irrelevant state state fresh update arrives preempts update service node age monitor unchanged arrival fresh irrelevant state state update node completes service moves node age monitor unchanged becomes irrelevant state update node update previously node state update node completes service delivered monitor system moves state age monitor becomes addition since irrelevant state arrival fresh update node induces state transition age unchanged node update fresh node age unchanged update node completes service delivered monitor system moves state age monitor becomes addition unchanged update node remains place since irrelevant state update node completes service moves node preempting update service node age unchanged becomes irrelevant state update node update previously node fresh update arrives node preempting update service node age unchanged new update node fresh unchanged update node stays place use theorem find average age first find stationary probabilities straightforward verify second need find satisfying defining follows table since three components twelve equations however irrelevant state follows similarly irrelevant state irrelevant state follows thus left eight equations algebra follows preemptive servers verifies average age monitor indeed insensitive ordering servers moreover comparing one see simple pattern ith node contributes age monitor verify observation nnode network however shs method employed nodes tracks state node network thus requires solution equations next section show extend simple pattern nodes using shs generates equations ode ine etwork fake pdates unusual feature server node tracking state node actually essential arrival goes immediately service whether update service node busy encodes age update service update completes service time enters service node whether preempts update may service node avoid tracking state node update departs node create fake update node timestamp age update departed new update node arrives node preempts fake update fake update causes delay arriving real update fake update depart node service node age update preempt node fake updates introduced shs derivation average age queue shs single discrete state section show extend fake updates approach prove following theorem theorem poisson arrivals rate line network servers service rates average age monitor transition marks arrival fresh update node continuous state set components unchanged transition update departs node arrives node node node fake update begins service transition update departs node delivered monitor age monitor reset age component whether representing real fake update ages unit rate thus following notation simplified defining applying theorem transitions given table obtain follows implying addition follows implying finally component implies implies claim follows theorem proof using fake updates markov chain singleton state space trivial stationary distribution shown figure transition link transitions shown table table omit entry since note fig shs markov chain line network nodes transition rates maps links shown table xal table table transitions arkov chain igure umerical xperiments theorem provides simple characterization average age preemptive line network examine age sample paths order get better sense age fluctuates across sequence preemptive servers focus line network nodes experiments let denote instantaneous age output node time start figure depicts representative sample paths sample paths result updates rate poisson process arriving node network service rates sample paths demonstrate average age grows predictably preemption thins update arrival process successive nodes thinning higher average age successive nodes consequence age growing updates also see evidence survivor bias occasionally updates pass quickly network enabling age monitor reset relatively low value figure examine reverse ordering servers downstream servers become progressively faster sample paths system updates dropped first node age processes nodes mimic age node small additional lag sample path evidence survivor bias however apparent figures general trend thinning updates induces larger age fluctuations nodes line see convergence average age examine running average age age age fig fifty arrivals three node line network arrival rate service rates fig sample paths running sample average arrival rate service rates bundles sample paths converging ages support theorem moreover server network memoryless preemptive server method fake updates greatly simplify age computation caution however fake updates appear useful servers support preemption service example preemptions occur waiting discrete state must track whether server busy must distinguish real updates fake updates cases system state grow exponentially number network nodes age fig fifty arrivals three node line network arrival rate service rates node figure plot sample paths node sample path updates arrive node rate poisson process nodes service rates sample path delivery time final received plot update monitor theorem expect see lim figure see bundles running average sample paths clustered around ages consistent conclusion moreover increasing variation apparently slower convergence running average age successive servers also consistent larger fluctuations age successive servers sample paths figures vii onclusions work use shs approach analyze age line network exponential servers two node example section apparent shs employed analyze variety queues simple networks described finite markov chains eferences kaul yates gruteser status often one update proc ieee infocom mini conference kam kompella ephremides effect message transmission diversity status age proc ieee int symp info theory pages june kaul yates gruteser status updates queues conf information sciences systems ciss march yates kaul status updating multiple sources proc ieee int symp info theory july costa codreanu ephremides age information packet management proc ieee int symp info theory pages june kam kompella nguyen wieselthier ephremides age information packet deadline proc ieee int symp info theory pages bedewy sun shroff information updates multihop networks proc ieee int symp info theory pages june bedewy sun shroff age information multihop networks corr talak karaman modiano minimizing wireless networks annual allerton conference communication control computing allerton pages oct yates kaul age information status updating multiple sources corr submitted ieee transactions information theory hespanha modelling analysis stochastic hybrid systems iee theory applications teel subbaraman sferlazza stability analysis stochastic hybrid systems survey automatica
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bulletin ieee technical committee learning technology volume number decision trees helpdesk advisor graphs gezerlis kalles use decision trees build helpdesk agent reference network facilitate advising junior less experienced staff better address telecommunication customer fault reports reports generate field measurements remote measurements coupled location data client attributes fused statistics produce models support provided beyond decision support models help identify staff act advisors based quality consistency predictability dealing complex troubleshooting reports advisor staff models used guide less experienced staff decision making thus advocate deployment simple mechanism exploits availability staff sound track record helpdesk act dormant tutors index customer relationship management decision trees knowledge flow graph introduction ustomer satisfaction key factor making clients loyal service provider telecommunications closely linked time takes fix problem way fault request handled customers considered loyal brand usually cost less serve big telecommunication organizations adopted standard operating procedures sops key performance indicators kpis help guide policies servicing customer fault complaints sops guidelines followed engineers help decide solve particular problem sops usually combinations rules decision trees constitute integral part staff training kpis numeric indices attempt capture aspects service quality perceived customer organization usually expressed terms average request handling speed percentage problems solved remotely repeated complaint rates though kpi may straightforward define measure relating soft attributes service provision elusive task example able conclude job time obvious target maintenance engineer ability draw organizational resources possible particular task though relating cost also straightforward manuscript received june revised september spyros gezerlis hellenic telecommunications agency greece deutsche telekom germany sgkezerlis dimitris kalles hellenic open university greece phone kalles linking measures customer satisfaction difficult exercise reason composite quality indices notoriously difficult define subject painstaking secrecy simplifications sops kpis usually deployed two levels support remote phone support handles incoming complaints calls service field technicians work landline network decision deploy support already providing remote support sometimes function remote support technician experience workload mental situation agility either approach judged based average cost incurs company also functions time element cost captured operational expense opex formulation one assumes complaint remotely resolved costs less resolved faster still keeps customer experience satisfactory level opexos standing opex site support opexrs remote resolution service typical values details business confidential complexity making decisions time pressure acceptable consistency organizational policies customer support cost containment overstated training key improvement paper puts forward unconventional approach based identification key personnel temporary act advisors peers discreet fashion contribution article first use decision trees identify inconsistencies cost incurred dealing customer complaints interpret inconsistencies measure ability company deliver customer experience complaint troubleshooting use inconsistencies rank helpdesk agents troubleshoot customer complaints generates knowledge hierarchy dynamically updated based identifying personnel advise need advised work serves bridge formality logic inherent expressing sops inherent vagueness regards best way resolve particular customer complaint also advances state practice customer relationship management crm genre little done recently regards customer complaints mainly focuses knowledge value client instead complaint resolution bulletin ieee technical committee learning technology volume number decision trees advisor flow graphs case study selected small group remote support agents technicians operated supervision one recorded measurements attributes internet complaints instances instance corresponds internet iptv internet protocol complaint record synchronization integrated access device central office routers addition metrics agents also recorded whether handled complaint helpdesk level referred site support one aims maximizing number issues resolved remotely without resort site support though makes obvious sense financial point view trivial problem resolution generates recurring complaints may may dealt agent next time thus raising costs seldom case expensive site support training set use build decision tree using algorithm looks like one shown table table snapshot training set agent product area internet athens internet patras internet athens profile sync max dist state state class variable showing type support product refers service client bought service complaint lodged area refers city fault occured profile refers nominal speed connection client actual capacity line service capacity example lowcost client might buy service even though line might accommodate much larger speeds sync refers actual synchronization speed client line max refers theoretical maximum attainable synchronization client line based signal strength attenuation variety physical characteristics dist refers copper cable length central circuit infrastructure measured training set made helpdesk reports recorded period weeks automatically logging data collected distributed corporate information systems helpdesk agents integrated data set generated combining data together decision trees experiment treat individual agents attributes used rweka analyzing data building models running delivered accuracy predicting whether complaint would resolved remote level fig shows snapshot decision tree dark agent attribute appears nodes near tree root clearly avoided agnostic sop dropping attribute along area attribute agents service requests central location may better resolved knowing singularity aspects actual physical network resulted accuracy fig decision tree using data labels agent decision node use data sets individual agents develop decision trees capture agent treats customer complaint well measure treatment uniformity measurement based partitioned data sets crosstested partition overall complaints data set groups according complaints assigned agents examine whether problem resolution practices one agent apply customer complaints handled another agent one hand approach allows see agent decision model compatible agents models hand however also allows consider model differences compared theoretical maximum practical maximum calculated error agent model table correctly classified instances results results among available agents shown table therein entry main diagonal refers testing accuracy note numbers even close suggesting inherently hard problem relative standard deviation rsd indicating small differences agent reclassifies training data generated concurring fig agent attribute seems dominate differences problem dealt entry describes experiment data agent used build decision tree subsequently tested data agent bulletin ieee technical committee learning technology volume number decision trees captures differences misclassifying customer complaints requiring site support remote resolution service opesx opesx rsd along main diagonal stands raises question agents fare differently asked classify data trained also highlighting substantial cost differences agents used classify data used agents might seem agent may unfairly penalized handling majority serious complaints requesting maintenance note complaints assigned agents randomly example pick experienced agents deal frustrated customers result one expects average hard easy problems uniformly allocated moreover performance review doubt also focus specific complaints dealt technique point mainly serves highlight inconsistencies dealing customer problems actual cost maintenance approach result question two particular agents performed different classification similar set complaints relatively less importance incorporating cost aspect decision tree classifier allowed build models show scale new data sets stress traditional techniques provide rather crude clues note traditional agent monitoring usually consists least indices per agent table reports conventional cost index agent using actual values two real kpi indices names well formula calculate composite cost index withheld due commercial secrecy kpi indices bear relation agentcost table finally reviews experiments ordered increasing ability tell agents apart terms performance table comparison technique resolution table average misclassification cost table traditional kpis cost evaluation agenti table iii effectiveness complaint resolution also taking account recurring complaints observe rsd mere clearly suggests traditional indices grossly hide differences conventional misclassification cost matrix actually note cost incurred carry maintenance activity choose ignore cost activity model correctly predicts class label maintenance activity site instance correctly classified model assign cost though incurs cost opexos resolve instead reserve cost opexos instances model classifies site though class label data set reads remote agents cost perspective shown table iii rsd cost using traditional kpis correctly classified instances average misclassification cost advisor flow graph collaborative training since agents models seem better suited classifying hereto instances might reasonable ask agents serve advisors colleagues though organizational level training involves review past cases well helpdesk tactics use one communication skills example helpdesk data analysis models serve training agent addresses troubleshooting report makes sense offer alternative approach problem crucial present advice hint recommendation followed since actual classification problem difficult advice sounds firm might easily resisted grounds counterexample seen occur capture picture agent help colleague draw advisor flow graph source nodes place agents better models whereas agents seem act increased misclassification costs placed destination nodes table iii used build advisor flow graph shown fig using equation corresponding adjacency weight matrix semantics advisor flow graph straightforward larger edge weights reflecting larger quality differences corresponding models according equation clarity purposes drawn graph three levels denote top level nodes nodes whereas bottom level nodes nodes example agent consult agents deciding whereas agent might also consult agent others unavailable window describing alternative action justification could suffice bulletin ieee technical committee learning technology volume number probably tackled basis simply put contribution article use decision trees classification mechanism knowledge transfer mechanism helpdesk technicians may easy build advisor graph deploying monitoring evolves actually improves quality service key part agenda close loop monitoring acting via acknowledgment work carried first author hellenic telecommunications organization ote greece acknowledge granting permission use individual technicians data data confidential property ote fig advisor flow graph showing flow information iii conclusions future work identify helpdesk agents capable advising colleagues using decision tree models agents deal customer complaints decision trees pinpoint agent differences help build graph capture possible advice flows agents advice flows essentially standard data mining activities used recommendation suitable decision making also reflection advisor graph dynamic knowledge hierarchy captures implicit reputation enjoyed members community practice even though knowledge may implicit volatile relevant leader selection via clustering performed optimization problems leaders help balance quality diversity case refers organizational goals effectiveness efficiency advisor graph gracefully scale number agents calculation carried subsequently selecting performed selecting available source nodes particular destination node also scale number attributes recorded complaint unnatural enhance record attributes agent similarities may detected exploited across variety criteria note however recording data agent age sex education background etc ran contrary law organizational regulations context case study limited data collection domain knowledge volatility issues arise advisor graph changes time due variability troubleshooting activities helpdesk agents expect application one might decide analyze agent groups larger time window establishing right combination group size time window likely difficult problem practical purposed references gustafsson johnson roos effects customer satisfaction relationship commitment dimensions triggers customer retention journal marketing ganesh arnold reynolds understanding customer base service providers examination differences switchers stayers journal marketing hwang jung suh ltv model customer segmentation based customer value case study wireless telecommunication industry expert systems applications quinlan programs machine learning morgan kaufmann publishers muggleton raedt inductive logic programming theory methods journal logic programming supplement reichheld one number need grow harvard business review december ngai xiu chau application data mining techniques customer relationship management literature review classification expert systems applications part khodakarami chan exploring role customer relationship management crm systems customer knowledge creation information management tseng knowledge management capability customer relationship management service quality journal enterprise information management soltani navimipour customer relationship management mechanisms systematic review state art literature recommendations future research computers human behavior hadzilacos kalles pierrakeas xenos small data sets revealing big differences advances artificial intelligence antoniou potamias spyropoulos plexousakis eds lncs turney classification empirical evaluation hybrid genetic decision tree induction algorithm journal artificial intelligence research tamargo garca falappa simari revision informant credibility orders artificial intelligence moubayed petrovski mccall leaders selection particle swarm optimisation intelligent data engineering automated learning yin wang eds lncs
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elicitation preferences single peaked trees apr palash dey indian institute science bangalore palash neeldhara misra indian institute technology gandhinagar mail april abstract multiagent systems often set agents preference ordering set items one would like know preference orderings various tasks example data analysis preference aggregation voting etc however often large number items makes impractical ask agents complete preference ordering scenarios usually elicit agents preferences asking hopefully small number comparison queries asking agent compare two items prior works preference elicitation focus unrestricted domain domain single peaked preferences show preferences single peaked domain elicited much less number queries compared unrestricted domain extend line research study preference elicitation single peaked preferences trees strict superset domain single peaked preferences show query complexity crucially depends number leaves path cover number distance path underlying single peaked tree whereas natural parameters like maximum degree diameter pathwidth play direct role determining query complexity investigate query complexity finding weak condorcet winner preferences single peaked tree show task much less query complexity preference elicitation observe number leaves underlying single peaked tree path cover number tree influence query complexity problem introduction multiagent systems often scenarios agents arrive consensus choosing multiple options typically agents preferences set items problem aggregating preferences suitable manner one problems social choice theory many ways expressing preferences set alternatives one comprehensive ways specify complete ranking set alternatives however one downsides model fact expensive solicit preference large number alternatives many agents involved since asking agents provide complete rankings impractical popular notion one elicitation ask agents simple comparison queries prefer alternative naturally gives rise problem preference elicitation hope recover complete ranking possibly relevant part ranking based small number queries paradigm voting popular general methodology aggregating preferences one devises voting rules mapping collection preferences refer votes either winning alternative consensus ranking keeping line terminology used voting refer alternatives candidates collection votes termed preference profile context fixed voting rule may also want query voters point determining winner aggregate ranking case may yet another refinement setting prior information agents likely vote may want determine voters query first able quickly rule large number alternatives explored goal elicit preferences prior structure one demonstrate scenarios imperative ask agent almost many queries would required determine arbitrary ranking however recent times considerable interest voting profiles endowed additional structure motivation first several application scenarios commonly considered rare votes adhoc demonstrating patterns whatsoever example notion preferences soon discuss length forms basis several studies analytical political sciences work eliciting preferences demonstrate structure argues notion also reasonable restriction applications outside domain political elections second motivation studying restricted preferences somewhat technical compelling understand structured preferences received considerable attention social choice theorists must first take brief detour foundational ideas shaped landscape voting rules understand today turns axiomatic approach social choice involves defining certain properties formally capture quality voting rule example would want voting rule informally speaking dictatorship would essentially mean discards one voter input unfortunately series cornerstone results establish impossible devise voting rules respect simplest desirable properties indeed classic works arrow gibbardsatterthwaite show straight forward way simultaneously deal properties like voting paradoxes nondictatorship unanimity etc refer elaborate discussion making matter worse many classical voting rules turn computationally intractable brings second reason structured preferences important consideration notion mentioned earlier excellent illustration refer reader section formal definition introduced captures essence structure political elections also turns extremely conducive many natural theoretical considerations begin one devise voting rules nice respect several properties preferences structurally elegant view winner determination since always admit weak condorcet winner candidate defeated candidate pairwise election thus working around condorcet paradox otherwise prominent concern general scenario landmark contribution show several computational problems intractable general setting become polynomially solvable consider preferences natural question point problem elicitation becomes easier get away fewer queries taking advantage structure provided peakedness turns answer affirmative shown detailed study definition involves ordering candidates called harmonious ordering authors work shows queries suffice assuming either harmonious ordering given one votes known dey misra show query complexity bound domain single crossing profiles large number voters return theme structural restrictions preferences turns single peaked preference domain subsequently generalized single peakedness trees roughly speaking profiles single peaked every path class continues exhibit many desirable properties single peaked domains example always exists weak condorcet winner many voting rules intractable unrestricted domain polynomial time computable underlying single peaked tree nice note class profiles single peaked trees substantially general class single peaked preferences note latter special case since path particular tree work addresses issue elicitation profiles trees seen significant generalization results detail specifics contributions parameter upper bound lower bound path width maximum degree path cover number number leaves distance path diameter log observation log observation log theorem log corollary log theorem log observation log even log corollary log even corollary log corollary log theorem log theorem log even corollary table summary query complexity bounds preference elicitation motivation contributions study query complexity preference elicitation preference profile single peaked tree provide tight connections various parameters underlying tree query complexity preference elicitation broad goal provide suitable generalization preference elicitation single peaked profiles profiles single peaked trees therefore consider various ways quantifying closeness tree path reflect measures might factor query complexity algorithm actively exploiting underlying tree structure summarize results preference elicitation table readers note parameters except diameter chosen small constants typically zero one two tree consideration path observe cases number leaves path cover number dependence parameter transparent recover results special case cases clear perspective provides additional mileage results matching lower bounds terms technique strategy scoop paths tree use known algorithms efficiently elicit preference parts trees paths efficiently merge information across board aspect algorithm varies depending parameter considered lower bounds typically come trees provide large degrees freedom reordering candidates typically trees many long paths stars arguments often subtle intuitive study query complexity finding weak condorcet winner preference profile single peaked tree able show weak condorcet winner found far fewer queries corresponding elicitation problem particular establish weak condorcet winner found using many queries profiles single peaked trees theorem also show bound best hope theorem also consider problem special case single peaked profiles showed queries necessary determine aggregate ranking show log queries suffice interested one weak condorcet winners moreover show bound tight condition algorithm interleave queries different voters theorem algorithm indeed satisfies condition finally expressing query complexity determining weak condorcet winner terms measure closeness path show algorithm query complexity log path cover number theorem number leaves corollary elaborate specific contributions preference elicitation design novel algorithms preference elicitation profiles single peaked tree leaves query complexity log corollary moreover prove exists tree leaves preference elicitation algorithm profiles single peaked tree query complexity log theorem show similar results parameter path cover number underlying tree theorem corollary provide preference elicitation algorithm query complexity log single peaked profiles trees made path deleting nodes theorem show query complexity upper bound tight constant factors theorem results show query complexity tightly depends number leaves path cover number distance path underlying tree show exists tree pathwidth one log corollary maximum degree corollary diameter corollary preference elicitation algorithm single peaked profiles tree query complexity log results show query complexity preference elicitation directly depend parameters next study query complexity finding weak condorcet winner profiles single peaked trees following results show weak condorcet winner found using many queries profiles single peaked trees theorem better query complexity preference elicitation moreover prove bound tight sense algorithm finding weak condorcet winner profiles single peaked stars query complexity theorem hand find weak condorcet winner using log many queries single peaked profiles theorem moreover show bound tight condition algorithm interleave queries different voters theorem algorithm indeed satisfies condition arbitrary underlying single peaked tree provide algorithm finding weak condorcet winner query complexity log path cover number theorem number leaves corollary summarize remark results generalize earlier works query complexity preference elicitation believe revisiting preference elicitation problem context profiles single peaked trees timely work also provides fresh algorithmic structural insights domain preferences single peaked trees related work already mentioned work addressing question eliciting preferences profiles closest predecessor work conitzer sandholm addressed computational hardness querying minimally winner determination also prove one would need make log queries even decide winner many commonly used voting rules matches trivial log upper bound preference elicitation unrestricted domain based sorting ding lin study preference elicitation partial information setting show interesting properties call deciding set queries boutilier provide empirical study preference elicitation probabilistic preference model devise several novel heuristics often work well practice rest paper organized follows introduce basic terminologies formally define problems section present results eliciting preference profile finding weak condorcet winner section section respectively conclude future works section preliminaries positive integer denote set suppose set voters preference alternatively called vote complete order set candidates denote set preferences tuple preferences voters called profile mentioned otherwise use denote number candidates number voters respectively let complete order subset candidates denote restriction order subset candidates denote restriction profile say candidate placed ith position preference preferred exactly candidates given voters profile candidate called condorcet winner every candidate strict majority voters prefer every candidate called weak condorcet winner voters profile exist candidate strict majority voters prefer every order denote reverse order preference set candidates called single peaked respect order every candidates whenever either candidate first position profile called single peaked respect order single peaked respect every notice profile single peaked respect order also single peaked respect order given path vertex another vertex tree define order induced path given tree set nodes set candidates profile called single peaked tree single peaked order induced every path tree every two candidates profile single peaked respect order candidates induced unique path call tree underlying single peaked tree known always exists weakly condorcet winner profile single peaked tree trees following definitions pertaining structural aspects trees useful pathwidth minimum width path decomposition set disjoint paths said cover tree every path cover number cardinality smallest set disjoint paths cover distance tree path smallest number nodes whose removal makes tree path diameter tree number edges longest path also list definitions subclasses trees special types trees see also figure tree star center vertex every vertex neighbor vertex tree subdivision star constructed replacing edge star path subdivision star called balanced exists integer distance every leaf node center tree caterpillar central path every vertex distance one tree complete binary tree rooted every nonleaf node exactly two children exists integer called height tree every leaf node distance either root node figure depicting classes trees path star balanced subdivision star caterpillar problem definitions known results suppose profile voters candidates pair distinct candidates voter introduce function query output function true voter prefers candidate candidate false otherwise formally state two problems consider paper definition preference elicitation given tree oracle access function query profile single peaked find definition weak condorcet winner given tree oracle access function query profile single peaked find weak condorcet winner suppose set candidates say algorithm makes queries exactly distinct tuples calls query query call number queries made algorithm query complexity state known results appeal later first observation employs sorting algorithm like merge sort elicit every vote log queries second follows merge subroutine merge sort observation preference elicitation algorithm query complexity log observation suppose form partition ranking candidates polynomial time algorithm finds given query complexity theorem preference elicitation algorithm query complexity single peaked profiles state weak condorcet winner problem asks eliciting point determining weak condercet winner recall least one winner guaranteed exist profiles trees definition weak condorcet winner given tree oracle access function query profile single peaked find weak condorcet winner results preference elicitation section present results preference elicitation profiles single peaked trees recall would like generalize theorem way profiles single peaked trees since usual single peaked profiles viewed profiles single peaked respect path propose following measures much tree resembles path leaves recall tree least two leaves paths trees exactly two leaves consider class trees leaves show algorithm query complexity log path cover consider notion path cover number tree smallest number disjoint paths tree partitioned clearly path cover number path one trees covered paths show algorithm query complexity log distance paths let size smallest set vertices whose removal makes tree path tree path said set simply empty set trees distance path sense vertex deletion provide algorithm query complexity log pathwidth maximum degree finally note paths also trees pathwidth one maximum degree two perspectives turn less useful particular trees parameters constant show elicitation hard would arbitrary profile therefore easy algorithm observation actually best hope first three perspectives employ seemingly capture appropriate aspect paths carry forward trees query complexities obtain tight matching lower bounds cases also considering structural parameters natural wonder class trees incomparable paths effective elicitation attempt direction consider trees bounded diameter however find useful examples show exist trees diameter two hard elicit general profiles remark point parameters polynomially computable trees making algorithmic results viable also parameters pathwidth maximum degree diameter show lower bounds trees parameters large trees pathwidth log maximum degree diameter roughly speaking also rules possibility getting good inverse dependence concrete example motivated algorithm paths diameter one might wonder algorithm query complexity possibility particular ruled ready discuss results table next show structural result trees namely tree leaves partitioned paths idea fix nonleaf node root iteratively find path low depth nodes depth node distance root leaf node disjoint paths chosen far formalize idea lemma let tree leaves polynomial time algorithm partitions disjoint paths every path proof first make tree rooted arbitrary nonleaf node partition tree paths iteratively follows initially every node tree unmarked set paths empty let path root node leaf node put mark nodes generally ith iteration pick unmarked node closest root breaking ties arbitrarily add path leaf node subtree rooted mark nodes since unmarked claim every node unmarked indeed otherwise suppose node already marked exists two paths one including another avoiding since currently unmarked already marked contradicts fact tree hence every node unmarked continue leaf nodes marked return since leaves every iteration least one leaf node marked since every contains leaf node algorithm runs iterations notice since algorithm always picks path consisting unmarked vertices set paths pairwise disjoint claim forms partition indeed otherwise must node remains unmarked end claim nodes subtree rooted unmarked includes least one leaf node contradicts fact algorithm terminates leaf nodes marked using lemma fact path account two leaves path cover number tree number leaves factor two lemma suppose path cover number tree leaves proof inequality follows fact path involve two leaves exists paths covering leaf nodes inequality follows fact lemma tree leaves partitioned paths algorithmic results present main algorithmic results begin generalizing result theorem single peaked profiles trees whose path cover number idea partition tree disjoint paths use algorithm theorem paths obtain order candidates path partition finally merge suborders intelligently formalize idea follows theorem preference elicitation algorithm query complexity log profiles single peaked trees path cover number proof since path cover number partition tree disjoint paths show elicit preference single peaked tree making log queries turn proves statement first find preference ordering restricted using theorem making queries every step needs queries since forms partition next merge orders obtain complete preference using standard divide conquer approach merging makes log queries thus query complexity algorithms log log towards results leaves show path cover tree leaves least lower bound easy since every path account two leaves upper bound comes careful partitioning nodes following maximal paths leaves root using careful marking scheme defer details argument full version using fact tree leaves partitioned paths use algorithm theorem obtain following bound terms leaves initially orders merge arbitrarily pair orders pairs one singleton odd renaming suppose pairings follows let define every odd integer merge get every using observation number queries algorithm makes iteration since forms partition end first iteration orders merge get algorithm repeats step log times obtain query complexity iteration corollary preference elicitation algorithm query complexity log profiles single peaked trees leaves finally given subset vertices whose removal makes given tree path elicitation algorithm makes log determine ordering among candidates path queries determine ordering among rest log queries using observation finally merge using observation leads following theorem preference elicitation algorithm query complexity log profiles single peaked trees distance path proof let smallest set nodes tree subgraph removal nodes path preference make log queries find using observation make queries find using theorem finally make queries find merging using observation gives overall query complexity log lower bounds turn lower bounds first result based counting argument showing query complexity terms number leaves given corollary tight constant factors indeed let consider subdivision star leaves let denote distance center total vertices one show candidates written level order candidates distance star ordered arbitrarily within ordering tells number possible preferences single peaked tree least obtain lower bound using decision tree argument wherein able show always possible oracle answering comparison queries answer way total space possibilities decreases factor half since tree must entertain least leaves account possibilities obtain claimed lower bound theorem let balanced subdivision star leaves preference elicitation algorithm single peaked profiles query complexity log proof suppose number candidates integer let center denote shortest path distance two nodes consider partition set candidates claim preference set candidates single peaked tree arbitrary order candidates every indeed consider path tree let candidate closest among candidates clearly single peaked respect path peak every thus number possible preferences single peaked tree least let preference elicitation algorithm single peaked profiles tree describe oracle answer queries algorithm makes every voter oracle maintains set possible preferences voter consistent answers queries algorithm already made voter beginning every voter argued whenever oracle receives query two candidates computes numbers orders prefers respectively oracle compute integers since oracle infinite computational power oracle answers voter prefers candidate updates set accordingly hence whenever oracle queried voter size set decreases factor two hand must correctness algorithm singleton set algorithm terminates every voter otherwise exists voter whose corresponding singleton exist two possible preferences single peaked tree consistent answers oracle given thus algorithm fails output preference voter correctly hence every voter must queried least log log log times since path cover number subdivided star leaves least also following corollary exists tree path cover number preference elicitation algorithm single peaked profiles query complexity log mimicking level order argument generic tree leaves using connection path cover leaves obtain lower bounds functions given useful subsequent results theorem let arbitrary tree leaves path cover number preference elicitation algorithm single peaked profiles query complexity log log proof let set leaves choose arbitrary nonleaf node root denote shortest path distance two candidates tree let maximum distance node partition candidates claim preference set candidates single peaked tree arbitrary order candidates every arbitrary order candidates indeed otherwise consider path tree let candidate closest among candidates clearly single peaked respect path peak number possible preferences single peaked tree least using oracle proof theorem deduce preference elicitation algorithm profiles single peaked tree needs make log queries bound respect path cover number follows lemma following results obtained simply applying theorem particular graphs instance use fact stars leaves pathwidth one obtain first part corollary appealing complete binary trees leaves pathwidth log second part examples also work context maximum degree diameter use stars caterpillars central path length next consider parameter pathwidth underlying single peaked tree immediately get following result preference elicitation trees pathwidths one log theorem fact pathwidths star complete binary tree one log respectively corollary exist two trees pathwidths one log respectively preference elicitation algorithm single peaked profiles respectively query complexity log corollary exist two trees maximum degree respectively preference elicitation algorithm single peaked profiles respectively query complexity log proof using theorem know preference elicitation algorithm profiles single peaked complete binary tree query complexity log since complete binary tree leaves result follows fact maximum degree node three binary tree case follows immediately theorem applied stars corollary exists two trees diameters respectively preference elicitation algorithm profiles single peaked respectively query complexity log proof cases follow theorem applied star caterpillar graphs central path length respectively final result follows theorem applied caterpillar graphs central path length shows bound theorem tight theorem integers exists tree distance path preference elicitation algorithm profiles single peaked query complexity log proof consider caterpillar graph length central path exists caterpillar graph since consider order set candidates order candidates single peaked order candidates clearly single peaked tree elicitation algorithm needs make queries involving candidates elicit due log queries elicit due sorting lower bound every voter proves statement weak condorcet winner show find weak condorcet winner profiles single peaked trees using fewer queries number queries needed find profile first note condorcet winner guaranteed exist found using queries pit arbitrary pair candidates use queries determine defeats push winning candidate forward repeat procedure clearly requiring rounds profile single peaked respect tree odd number voters condorcet winner procedure described would work otherwise simply find condorcet winner among first voters shown winner one weak condorcet winners overall profile therefore following upper bound begin following general observation observation let profile condorcet winner guaranteed exist find condorcet winner making queries proof two candidates find whether defeats simply asking voters compare takes queries algorithms maintains set candidates potential condorcet winners initialize iteration pick two candidates remove defeat vice versa using query complexity singleton iterations set singleton contain condorcet winner since find candidate condorcet winner every iteration thus size set decreases least one every iteration gives query complexity bound using observation develop weak condorcet winner algorithm query complexity profiles single peaked trees theorem weak condorcet winner algorithm query complexity single peaked profiles trees special case single peaked profiles even better take advantage fact median candidate guaranteed weak condorcet winner make log queries per vote find candidates placed first position votes using algorithm find median candidate show following profile single peaked path weak condorcet winner algorithm query complexity log shown let define frequency candidate number votes placed first position know median candidate according single peaked ordering candidates along frequencies defined weak condorcet winner single peaked profiles theorem weak condorcet winner algorithm query complexity log single peaked profiles path proof let profile single peaked respect ordering candidates find every voter candidate voter places first position using log queries using algorithm return median candidate proof let profile single peaked tree odd integer know exists condorcet winner since two candidates tie always exists least one weak condorcet winner every single peaked profile trees hence odd integer use observation find weak condorcet winner condorcet winner hence let assume even integer notice also single peaked odd number voters thus condorcet winner use observation find condorcet winner output weak condorcet winner claim weak condorcet winner indeed otherwise exists candidate defeats since even integer must defeat margin least two since pairwise margins even integers also defeats margin least one contradicts fact condorcet winner next result uses theorem paths path cover eliminating case even number voters idea setting aside one voter used theorem theorem let tree path cover number algorithm weak condorcet winner profiles single peaked query complexity log recalling number leaves bounds path cover number following consequence corollary let tree leaves algorithm weak condorcet winner profiles single peaked query complexity log state lower bounds pertaining weak condorcet winner first show algorithm single peaked profiles stars query complexity showing bound theorem tight theorem weak condorcet winner algorithm single peaked profiles stars must query complexity proof let star center vertex design oracle force weak condorcet winner algorithm single peaked profiles make queries every voter oracle maintains set marked candidates placed first position preference suppose oracle receives query compare two candidates voter order voter follows answers oracle already provided queries voter oracle answers accordingly otherwise answers unmarked marks otherwise oracle answers marks notice oracle marks one unmarked candidate every time queried claim must least votes queried least times exists votes least candidates unmarked scenario exists constant every least two candidates unmarked least votes algorithm outputs put first position least votes second position rest votes makes unique condorcet winner algorithm output put first position least votes second position rest votes makes unique condorcet winner hence algorithm fails output correctly cases contradicting correctness algorithm also resulting profile single peaked center first case second case therefore algorithm must query complexity concluding result uses intricate adversary argument shows query complexity weak condorcet winner single peaked profiles theorem essentially optimal provided queries different voters interleaved case algorithm theorem weak condorcet winner algorithm single peaked profiles interleave queries different voters query complexity log proof let profile single peaked respect ordering candidates oracle maintains two indices every voter candidate placed first position preference voter still consistent answers provided oracle till single peaked respect algorithm initializes one every voter oracle answers query way maximizes new value specifically suppose oracle receives query compare candidates voter ordering follows applying transitivity answers queries already made far voter oracle answers accordingly otherwise oracle answers follows oracle answers preferred else oracle answers preferred otherwise oracle answers preferred changes oracle answers preferred changes hence whenever oracle answers query voter value voter decreases factor two suppose election instance odd number voters let set voters claim first voters must queried log times suppose consider first voter queried less log times exist least two candidates placed first position vote oracle fixes candidates first positions votes queried till queried least votes way voters places candidate left first positions voters places candidate right algorithm outputs condorcet winner oracle makes unique condorcet winner placing top position unique median case algorithm output condorcet winner oracle makes unique condorcet winner placing top position unique median case hence algorithm fails output correctly cases thereby contradicting correctness algorithm theorem following result arbitrary tree theorem let tree path cover number algorithm weak condorcet winner profiles single peaked query complexity log proof let input profile disjoint paths cover tree number voters even remove arbitrary voter algorithm outputs condorcet winner rest votes correctness step follows proof theorem hence assume without loss generality odd number voters algorithm proceeds two stages first stage find condorcet winner profile every using theorem query complexity stage log log second stage find condorcet winner profile using theorem output query complexity second stage log hence overall query complexity algorithm log log log conclusions future work show algorithms preference elicitation profiles single peaked trees moreover prove query complexity algorithms optimal constant factors also show need query fewer number times preference elicitation want find weak condorcet winner work assume partial information preferences however many scenarios may knowledge profile interesting future direction research study partial information helps reduce query complexity preference elicitation acknowledgement palash dey wishes gratefully acknowledge support google india providing special fellowship carrying doctoral work references kenneth arrow difficulty concept social welfare journal political economy pages felix brandt markus brill edith hemaspaandra lane hemaspaandra bypassing combinatorial protections algorithms electorates journal artificial intelligence research jair pages felix brandt vincent conitzer ulle endriss lang ariel procaccia handbook computational social choice duncan black rationale group journal political economy pages vincent conitzer eliciting preferences using comparison queries journal artificial intelligence research jair thomas cormen introduction algorithms mit press vincent conitzer tuomas sandholm vote elicitation complexity strategyproofness eighteenth national conference artificial intelligence aaai pages vincent conitzer tuomas sandholm communication complexity common voting rules proceedings acm conference electronic commerce pages acm gabrielle demange orders tree mathematical social sciences ning ding fangzhen lin voting partial information questions ask proceedings international conference autonomous agents systems aamas pages international foundation autonomous agents multiagent systems palash dey neeldhara misra preference elicitation single crossing domain ijcai proceedings international joint conference artificial intelligence ijcai allan gibbard manipulation voting schemes general result econometrica journal econometric society pages katherine heinrich path decomposition matematiche mel vin hinich michael munger analytical politics cambridge university press john hopcroft jeffrey david ullman alfred vaino aho data structures algorithms volume boston usa tyler craig boutilier robust approximation incremental elicitation voting protocols ijcai proceedings international joint conference artificial intelligence ijcai pages tyler craig boutilier vote elicitation probabilistic preference models empirical estimation cost tradeoffs algorithmic decision theory pages springer andreu michael dennis whinston jerry green microeconomic theory volume oxford university press new york hervi moulin axioms cooperative decision making number cambridge university press dominik peters edith elkind preferences nice trees mark allen satterthwaite arrow conditions existence correspondence theorems voting procedures social welfare functions journal economic theory michael trick recognizing preferences tree mathematical social sciences lan hau chan edith elkind multiwinner elections preferences tree proceedings international joint conference artificial intelligence ijcai pages aaai press
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scheduling two agents single machine parameterized analysis danny dvir sep nimrod gerhard ben gurion university negev israel hermelin dvirs berlin germany weizmann institute science israel eindhoven university technology netherlands gwoegi abstract scheduling theory old area combinatorial optimization whereas much younger area parameterized complexity recently gained attention community aim bring two areas closer together studying parameterized complexity class scheduling problems analysis focuses case number jobs belonging second agent considerably smaller number jobs belonging first agent thus considered fixed parameter study variety combinations scheduling criteria two agents combination pinpoint parameterized complexity respect parameter scheduling criteria analyze include total weighted completion time total weighted number tardy jobs total weighted number jobs analysis draws borderline tractable intractable variants problems preliminary version work appeared proceedings international symposium parameterized exact computation ipec introduction scheduling area operations research provides fertile grounds several combinatorial problems typical scheduling problem given set jobs scheduled set machines arranged according specific machine setting objective determine schedule minimizes predefined scheduling criterion makespan total weighted completion time total weighted tardiness schedule various machine settings including single machine setting parallel machines scheduling problem may addition various attributes constraints refer reader extensive introduction area scheduling detailed survey classical results many scheduling problems typically hard problems include multitude parameters many proofs exploit fact parameters arbitrary large theory however many practical settings one parameters actually quite small example number different items processed shop might limited resulting scheduling instance limited number different processing times appear limited set planned delivery dates resulting scheduling instance limited number different due dates another example therefore natural ask whether scheduling problems become tractable parameters assumed comparatively small practice luckily framework answering questions recently developed computer science community theory parameterized complexity parameterized complexity facilitates analysis computational problems terms various instance parameters may independent total input length way problem instances analyzed according total input length also according additional numerical parameter may encode aspects input problem considered tractable algorithm optimally solves instance time allowed arbitrary computable function independent exponent required independent example considered tractable parameterized setting way model scenarios certain problem parameters typically much smaller total input length yet may small enough considered constant parameterized complexity enjoyed tremendous success since first developments early exemplified various textbooks subjects however currently papers attack scheduling problems parameterized perspective rather disappointing since scheduling problems seem particulary adequate parameterized analyses one scheduling problems lack algorithms finding optimal solutions several applications approximate solutions result big revenue losses gives strong motivation computing exact solutions even computing solutions requires lot resources secondly argued scheduling problems typically abundance natural problem parameters comparatively small practice thus algorithms whose running times grow exponentially parameters alone quite useful practical purposes aim paper help close gap research parameterized complexity area scheduling initiate parameterized analysis problems occurring setting scheduling contemporary area nowadays cutting edge scheduling research focus basic case two agents jobs processed single machine furthermore parameterized analysis focuses scenario second agent significantly smaller number jobs number jobs belonging agent thus taken parameter denoted throughout paper preform extensive parameterized analysis several scheduling problems respect parameter providing clear picture applicability parameterized algorithmics problems contribution work investigate variety combinations objective functions agent combination consider problem agent bound objective function goal determine whether exists schedule meets bounds simultaneously objective functions consider see section formal definitions total weighted completion time jobs weights goal minimize sum weighted completion time entire job set agent total weighted number tardy jobs jobs weights objective minimize total weighted number jobs terminate total weighted number jit jobs jobs weights goal maximize total weighted number jobs terminate precisely due date consider several combinations scheduling criteria corresponding problem npcomplete combination determine whether corresponding scheduling problem becomes tractable respect also subtleties consider unit weight case unit case paper organized according first agent scheduling criteria thus section deals case scheduling criterion agent total weighted completion time section focuses problems first agent criterion total weighted number tardy jobs section concerned problems first agent criterion total weighted number jit jobs related work set scheduling problems first introduced baker smith agnetis different combinations scheduling criteria baker smith focus analyzing problem finding schedule minimizes weighted sum two criteria agnetis focus analyzing problem minimizing first agent criterion keeping value second agent criterion greater given bound following two fundamental papers numerous researchers studied different combinations scheduling problems see detailed surveys problems appear framinan recent book agnetis give detail results directly related work appropriate sections remainder paper preliminaries section introduce notation terminology used throughout paper particular provide concrete definitions problems study well brief introduction theory parameterized complexity scheduling notation problem definitions problems considered paper input consists two sets jobs processed non preemptively single machine first set belongs agent second set belongs agent assume practical purposes one think much smaller let positive integer denoting processing time job moreover relevant let two positive integers representing due date weight job respectively schedule set disjoint time intervals processed single machine note represents completion time case one job agents jobs due dates use set lmax maxj represents time interval job denote lateness job early job otherwise tardy accordingly set set early jobs belongs agent set set tardy jobs belong agent also use denote set quality schedule measured two different criteria one per agent focus problems either one two agents criteria may either one following three possibilities weighted sum completion times denoted weighted number tardy jobs job indicator variable said tardy indicates whether number tardy jobs agent tardy use binary denotes weighted weighted number jit jobs job said use binary indicator variable indicates whether denotes weighted number jobs agent note first two criteria minimization criteria latter maximization criterion possible combination criteria consider decision problem given two positive integer bounds one agent need find exists job schedule bounds met case scheduling criterion sum weighted completion times weighted number tardy jobs bound regarded weighted number jobs refer job schedule exists feasible job schedule using standard three field notation scheduling denote set problems sometimes consider special cases problems use middle field denote restrictions input example problem problem scheduling criterion first agent weighted sum completion times second agent unweighed sum completion time agent jobs unit processing times basic concepts parameterized complexity theory main objective parameterized complexity theory analyze tractability problems respect input parameters necessarily related total input size thus problem instances measured terms input size also terms additional parameter context problem said tractable tractable fpt algorithm solves instance size parameter time function arbitrary computable exponential function long depends exponent independent reader referred excellent texts subject information parameterized complexity considered tractable even considered tractable note definition might allow quite large sufficiently smaller drastically outperforms common algorithms example moreover running time say moderate constants exponent quite fast practice case parameterized complexity provides convenient form analyzing complexity problem respect size given parameter context parameter instance always number jobs agent thus consider setting agent significantly jobs schedule nevertheless still wish meet agents criteria note problem already constant values parameter tractable algorithm problem imply thus context show one problems consider already agent constant number jobs excludes possibility problem tractable algorithm assumption showing hardness results use classical partition problem often used context scheduling problems definition partition problem given set positive integers encoded binary determine whether partitioned two sets weighted sum completion times section study problem three scheduling criteria discussed section first show three corresponding problems unlikely admit algorithm since even agent single job show theorem motivates study four special cases show case jobs agent unit weight problem becomes fpt criteria agent either weighted sum completion times weighted number tardy jobs however criteria agent number jobs problem remains intractable case well also provide fpt algorithm case criteria weighted sum completion times agent jobs unit processing times intractability general problem respect fact single agent problem solvable log time see smith gives hope least one problems tractable small unfortunately following theorem show case three criteria even second agent single job unit weight show via reduction partition problem see definition theorem problem either proof provide reduction partition problem defined given instance partition problem construct following scheduling instance agent jobs agent single job set moreover set case also set bound total weighted completion time agent set bound agent depends scheduling criterion sum completion times set weighted number tardy jobs set weighted number jit jobs set suppose partitioned two sets construct schedule first schedule agent jobs corresponding elements arbitrary order followed job followed jobs agents corresponding elements arbitrary order observe job completes time units bound agent met three criteria see first agent bound met well observe exclude precisely total weighted completion time agent jobs adding job increases completion time first agent jobs correspond elements unit thus contributes precisely total weighted completion time agent jobs first agent bound total weighted completion time met well direction suppose feasible schedule possible option let note set agent jobs scheduled job let denote agent remaining jobs since agent bound satisfied three possible criteria must moreover note total weighted completion time agent jobs since agent bound also met must agent jobs get since sum processing times thus setting yields solution partition instance fpt algorithm problem stark contrast result theorem next show although see agnetis problem much easier handle present fpt algorithm problem parameter using powerful result lenstra concerning mixed integer linear programs begin need following lemma easily derived using simple interchange argument see also agnetis prove argument less general case agents jobs unit weights lemma feasible solution problem exists feasible solution jobs agent scheduled order according shortest processing time spt rule consider problem due lemma assume without loss generality jobs agent numbered according spt rule allowing additional multiplicative factor running time algorithm focus reduced subproblem ordering second agent jobs predefined given subproblem renumber second agent jobs according ordering determine whether feasible schedule actually possible given subproblem need figure possible interleave two ordered sets jobs together way satisfies agents bounds towards aim formalize given subproblem mixed integer linear program milp number integer variables complete proof using celebrated result lenstra states determining whether given milp feasible solution tractable respect number integer variables formulate given subproblem milp define integer variable job ing number jobs belonging agent scheduled therefore add constraint moreover add constraint lemma bound total completion time first agent jobs formulated following constraint proof observe first term side constraint precisely total completion time first agent jobs job second agent scheduled add second agent jobs according intended meaning variables jobs agent scheduled prior presumed schedule causes increase completion time jobs belonging first agent thus adding agent jobs causes additional increase total completion time first agent jobs encoding second agent bound bit involved specifically introduce variable would like equal contribution first agent jobs completion time note intended meaning variable precisely pxj however encode directly linear constraint therefore introduce additional real valued variables corresponding denoted yij ensured adding constraint yij yij variables used provide steps contribution first agent jobs depicted fig accordingly add constraints yij naturally set furthermore add constraint yij equal intended meaning yij fig contribution jobs agent scheduled prior job increases completion time variable yij intended capture step contribution lemma bound total weighted completion time second agent jobs formulated following constraint proof observe first term side constraint total weighted completion time second agent ordered jobs assuming jobs agent scheduled argue second term contribution agent jobs suffices show contribution agent jobs completion time know contribution pxj since concerned feasible solutions constraints variables yin met yij max pxj summarize due lemma solve problem solving milp formulations integer variables correctness formulations follows lemmas analysis using lenstra result therefore obtain following theorem theorem fpt algorithm tractable respect problem determining whether exists schedule agents meet respective bounds total weighted completion times shown even jobs agents unit processing times complement result well result given theorem showing problem fpt respect jobs first agent unit processing times following lemma crucial construction fpt algorithm proven simple pairwise interchange argument lemma feasible schedule problem feasible schedule problem first agent jobs scheduled weight order assume without loss generality accordingly based lemma restrict search feasible schedule schedules job scheduled proof theorem allowing additional multiplicative factor running time algorithm focus reduced subproblem ordering second agent jobs also predefined given ordering renumber second agent jobs according ordering follows prove subproblem fpt respect proof uses ideas proof theorem thus briefly presented well define variables time represents number first agent jobs scheduled accordingly include constraint following constraint well bound weighted sum completion times second agent jobs expressed first term side weighted sum completion times second agent jobs scheduled one beginning schedule second term side corresponds contribution first agent jobs weighted sum completion times second agent jobs bound total weighted completion time first agent jobs need introduce set variables yjn variable meant encode contribution total weighted completion time first agent jobs note precisely use variables yij encode steps sum done proof theorem accordingly include set constraints moreover include set constraints also following set constraints yij definition finally encode bound total weighted completion time first agent jobs jwj first term side weighted sum completion time first agent jobs scheduled one beginning schedule second term side corresponds contribution second agent jobs weighted sum completion times first agent jobs theorem respect problem tractable problem leung proved next prove general problem problem fpt respect proof depends following lemma recall definitions section lemma feasible solution instance problem instance exists feasible solution jobs agent scheduled order decreasing order according spt rule jobs scheduled according edd rule iii jobs scheduled last arbitrary order consider problem define set subproblems corresponding possible ways partition set condition holds due lemma subproblem reduces instance lmax problem includes jobs agent early jobs agent reduced subproblem need find possible schedule jobs jobs indeed early completed later due date thus fact lmax problem solvable time see yuan leads following theorem theorem problem solvable time intractability consider instance problem problem feasible solution instance single job agent scheduled jit mode time interval thus instance equivalent instance problem single interval commonly denoted problem known following corollary corollary see adiri lee liman thus problem even weighted number tardy jobs section study variants problem form variants scheduling criteria agent weighted number tardy jobs note already single agent problem even due dates equal resulting dating back karp seminal paper therefore variant problem jobs first agent weights hard corollary problem due theorem restrict analysis unweighed problem provide algorithm case criteria agent also weighted number tardy jobs algorithm extended case jobs agent may arbitrary weights agents jobs unit processing times contrary criteria agent weighted number jit jobs show problem intractable already highly restrictive special cases know whether problem tractable criteria agent total weighted completion time case show algorithm algorithm question whether problem problem tractable remains open question nevertheless show problem solved much slower still time leads conclusion problem belongs parameterized class see formal definition solvable polynomial time upper bounded constant begin following easy lemma lemma feasible solution instance instance exists feasible solution problem jobs scheduled first followed jobs scheduled last arbitrary order iii jobs scheduled according edd rule order following lemma renumber jobs according edd rule furthermore divide original problem instances represent different processing order jobs consider given instance assume jobs numbered according processing order instance consider possible partitions subsets possible sets integers note pairs given pair looking restricted schedule satisfies following three conditions early jobs among jobs job scheduled right early jobs within iii note instance corresponding particular processing order feasible schedule iff restricted schedule corresponding pair exists show compute restricted schedule exists log time yield following theorem theorem problem solvable log time consider pair note goal find schedule early jobs translates solving disjoint instances single agent problem set jobs instance solved time slight modification classical algorithm moore gives total log time instances furthermore moore algorithm computes schedule minimum makespan final completion time amongst schedules tardy jobs thus composing schedules single schedule agents scheduling final job scheduled gives schedule minimizes schedules satisfy properties thus time determine whether exists restricted schedule corresponding theorem holds mention algorithm slightly improved jobs agent unit weights case know optimal order jobs order according shortest processing time spt rule thus try possible orderings reducing time complexity algorithm factor corollary fpt algorithm next show problem solvable log time problem fpt respect proof depends following lemma lemma feasible solution instance problem instance exists feasible solution jobs scheduled first according edd rule order followed jobs scheduled last arbitrary order consider instance problem define set instances corresponding possible ways partition set feasibility condition instance holds instance due lemma instances lmax problem need find feasible schedule agent subject scheduling set jobs early agnetis showed lmax problem solvable log log log time thus obtain following theorem fpt algorithm problem solvable log time problem even case unit processing time next complement result well theorem showing problem fpt respect first observe lemma holds well thus create instances instance instance set may vary given instance instances may assume furthermore due lemma instances fact instance lmax problem need find feasible schedule agent subject scheduling set jobs early latter problem solvable max log log time thus get theorem log time intractability problem solvable problem following observation made section conclude instance problem equivalent instance problem known see lee thus following corollary corollary problem weighted number jobs next consider problems form problems criteria agent total weighted number jit jobs recall job scheduled jit mode scheduled precisely time interval show either problem tractable also show problem tractable first agent jobs unweighed general case first agent jobs weighted left open general lemma problem problem known next prove problem fpt respect usual begin lemma feasible schedule problem jobs scheduled due date edd order moreover feasible solution problem feasible solution jobs set scheduled last arbitrary order following lemma assume without loss generality jobs indexed according edd rule show fixed ordering jobs agents determine polynomial time whether exists feasible schedule relative order agent jobs exactly ordering since orderings jobs agent imply problem fpt respect consider fixed ordering jobs agent convenience purposes let two dummy jobs agent given feasible schedule consider two jobs agent scheduled jit mode jobs agent scheduled obviously otherwise scheduled jit mode let denote number jobs maximal consecutive subsequence jobs total processing time greater following lemma holds lemma suppose feasible schedule scheduled jit mode jobs agent scheduled exactly jobs belonging agent scheduled prior exists feasible schedule sequence jobs scheduled right completion time job jobs scheduled prior completion job pair jobs let denote minimum contribution job sequence total weighted completion time agent feasible schedule schedule exists define according lemma compute using following formula let denote minimum total weighted completion time first jobs agent among partial schedules job set last job scheduled set schedule exists note definition completion time feasible partial schedule exactly time value computed using following recursion considers possible ways appending partial schedules fewer jobs agent one less jit job agent min otherwise algorithm computes possible values using recursion given equation computes preprocessing step values base cases recursion given report exists feasible schedule instance restricted relative fixed ordering jobs agent iff correctness algorithm immediate discussion let analyze time complexity ways order jobs agent order compute values values computed straightforwardly time using values equation values also computed time finally using dynamic programming along equation computing values requires time thus fixed ordering spend total time together gives time algorithm theorem problem solvable time problem next show modify ideas used section apply problem begin following analog lemma lemma feasible schedule instance problem jobs scheduled earliest due date edd order moreover feasible solution instance feasible solution jobs set scheduled last arbitrary order following lemma assume jobs numbered according edd rule create instances problem according possible candidate sets lmax holds instance fact instance problem includes jobs agent early jobs agent reduced instance need determine whether exists schedule job indeed early completed later due date problem polynomial time begin renumbering jobs section add two dummy jobs show solve furthermore let denote length maximal consecutive subsequence jobs total processing time greater lemma holds well assume scheduled jit mode agent jobs scheduled right completion jobs scheduled prior completion since jobs agent allowed late partial schedule includes jobs consecutive jit jobs early jobs agent scheduled early jobs agent scheduled two feasible partial schedule following two conditions holds condition condition let represent maximum total weighted number jit jobs among partial schedules job set jobs early last scheduled job note opposed section value represents criteria agent value computed following recursion considers possible ways appending partial schedules fewer early jobs agent one less jit job agent conditions hold max otherwise base cases recursion given algorithm reports exists feasible solution instance problem iff set running time algorithm bounded using similar analysis one given section thus obtain theorem known problem problem solvable time problem solvable weights either one two agents equal show problem even case unit processing time general problem arbitrary processing time fpt respect theorem problem proof given instance partition problem see definition construct following instance set moreover set note since jobs problem set recall due date one completed jit mode thus total gain agents restricted suppose partitioned two sets job schedule time interval schedule remaining jobs arbitrary order job fact scheduled jit mode implies thus feasible schedule constructed instance direction suppose exists schedule agent since total gain objective function agents restricted means setting solution next show obtain problem fpt respect end convenient view jobs time intervals job let time interval required scheduled jit mode scheduled within time interval thus pair jobs simultaneously scheduled jit mode iff means goal translates finding set pairwise disjoint time intervals total weight set intervals related jobs agent met bound try possible candidates subsets agent jobs pairwise disjoint time intervals subset compute subset agent jobs time intervals intersect time interval job subset intervals set compute maximum weight pairwise disjoint report found feasible schedule total weight jobs least subset jobs found possible candidate report instance feasible schedule correctness algorithm follows fact compute optimal set possible candidate candidates candidate computing set done log log log time moreover set computed log time using thus total algorithm runs log time theorem problem solvable log time conclusions open problems paper initiated parameterized analysis problems parameter studied number jobs belonging second agent considered three possible scheduling criteria total weighted completion time total weighted number tardy jobs total weighted number jit jobs possible combinations criteria agent analysis shows parameter indeed provides various positive results different settings summarized table several directions directly extend work first one find different parameters number different processing times input number different due dates many parameterizations make perfect sense practical applications second one consider scheduling criteria considered paper maximal lateness list three important questions left open directly work determine parameterized complexity variants unit weights unit processing times etc determine parameterized complexity problem problem running times algorithms presented sections improved specifically purely combinatorial algorithms problems avoiding ilps altogether hard fpt hard fpt fpt hard general cor open hard even cor hard general cor fpt hard even cor fpt open general fpt fpt fpt table table depicting results presented paper rows correspond different scheduling criteria agent columns associated different criteria agent acknowledgments research leading results received funding people programme marie curie actions european union seventh framework programme rea grant agreement number israel science foundation grant nimrod talmon supported postdoctoral fellowship algo references adiri bruno frostig rinnooy kan single machine scheduling single breakdown acta informatica agnetis billaut gawiejnowicz pacciarelli soukhal multiagent scheduling models algorithms imprint springer agnetis mirchandani pacciarelli pacifici scheduling problems two competing agents operations research baker smith model machine scheduling journal scheduling bodlaender fellows precedence constrained scheduling operations research letters brucker scheduling algorithms springer science cygan fomin kowalik lokshtanov marx pilipczuk pilipczuk parameterized algorithms springer downey fellows parameterized complexity springer fellows mccartin parametric complexity schedules minimize tardy tasks theoretical computer science flum grohe parameterized complexity theory eatcs series texts theoretical computer science springer garey johnson computers intractability guide theory gupta lee joseph leung efficient algorithms interval graphs graphs networks karp reducibility among combinatorial problems raymond miller james thatcher editors complexity computer computations pages kovalyov oulamara soukhal scheduling unbounded serial batching machine combinatorial optimization pages springer lee machine scheduling availability constraint journal global optimization lee liman single machine scheduling scheduled maintenance acta informatica lee choi leung pinedo approximation algorithms scheduling minimize total weighted completion time information processing letters lenstra integer programming fixed number variables mathematics operations research leung handbook scheduling algorithms models performance analysis springer leung pinedo wan competitive scheduling applications operations research mnich wiese scheduling meets tractability mathematical programming moore job one machine sequencing algorithm minimizing number late jobs management science mor mosheiov single machine batch scheduling two competing agents minimize total flowtime european journal operational research cheng yuan note complexity problem scheduling single machine journal combinatorial optimization niedermeier invitation algorithms oxford lecture series mathematics applications oxford univerity press oron shabtay steiner single machine scheduling two competing agents equal job processing times european journal operational research framinan common framework taxonomy multicriteria scheduling problems interfering competing jobs scheduling problems european journal operational research pinedo scheduling theory algorithms systems springer science business media shabtay dover kaspi scheduling involving criterion international journal production research smith various optimizers production naval research logistics van bevern mnich niedermeier weller interval scheduling colorful independent sets journal scheduling van bevern niedermeier parameterized complexity view scheduling jobs machines small looseness small slack corr yin cheng cheng wang scheduling two competing agents unrelated parallel machines omega yuan shang feng note scheduling two families jobs journal scheduling
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constant approximation capacitated violation may shi abstract study capacitated problem existing approximation algorithms violate either capacities upper bound number open facilities using natural relaxation problem one hope get violation factor soda introduced novel beyond limit gave approximation algorithm opens facilities use configuration soda give approximation capacitated problem seemingly harder configuration violate capacities result settles problem far algorithms concerned introduction capacitated problem ckm given set facilities together capacities set clients metric number asked open facilities give assignment connecting client one open facilities number open facilities bigger cardinality constraint facility connected clients capacity constraint goal minimize sum connection costs without capacity constraint famous problem first approximation algorithm given charikar guaranteeing solution within times cost optimal solution approximation ratio improved series papers current best ratio due byrka obtained improving part algorithm given svensson hand true constant approximation ckm known constantfactor results violate either cardinality capacity constraint aardal gave algorithm finds solution ckm opening facilities violating cardinality constraint factor guha gave algorithm approximation ratio relaxed uniform ckm department computer science university chicago demirci department computer science engineering university buffalo shil supported part nsf grant capacities connecting clients facility thus violating capacity constraint gave algorithm uniform ckm capacity violation improving algorithm capacities chuzhoy rabani gave ckm violating capacities factor using mixture schema lagrangian relaxations algorithm slightly relaxed version problem called soft ckm one allowed open multiple collocated copies facility ckm definition gave sometimes referred hard ckm opposed version recently byrka gave algorithm hard ckm keeping capacity violation factor algorithms ckm use basic relaxation problem known unbounded integrality gap even allowed violate either capacity cardinality constraint sense results considered reaching limits basic relaxation terms restricting violation factor order beyond limits introduced novel called rectangle presented approximation algorithm soft uniform ckm opening facilities later generalized author ckm introduced even stronger relaxation called configuration recently independently work paper byrka used configuration give similar algorithm uniform ckm violating capacities result paper use configuration give algorithm hard ckm respects cardinality constraint connects clients open facility running time algorithm thus result settled ckm problem view algorithms either violation violation sufficient constant approximation ckm known results ckm problem suggested designing algorithms capacity violation satisfying cardinality constraint harder designing algorithms cardinality violation note example best known cardinality violation factor ckm among algorithms using basic relaxation factor matches smallest possible cardinality violation factor dictated gap instance contrast best capacityviolation factor due gap instance basic largest known gap eliminates algorithms capacity violation smaller furthermore argue algorithms based basic configuration violation converted violation suggesting allowing capacity violation restrictive allowing cardinality violation suppose algorithm ckm violates capacity constraint factor based basic relaxation given solution basic given ckm instance construct new instance scaling factor scaling capacities factor valid solution allow connections fractional thus fractional capacities cause issues easy see valid solution soft capacities solution violates capacity constraint factor solution violates cardinality constraint factor thus considering new instance conclude algorithms based basic relaxation violating cardinality constraint gives power argument made algorithms based configuration one show valid solution configuration yields valid solution configuration however reduction direction work due constraint scaling variables factor yield valid solution techniques algorithm uses configuration introduced framework creates clustering facilities considered violation setting flexible sense one much freedom distribute extra facilities violation setting facility provide extra capacity however extra capacities restricted locations facilities particular need one level clustering form groups group contains fractional open facility groups facilities benefit extra capacities given scaling algorithm constructs distributions local solutions using dependent rounding procedure select local solution distribution solution formed concatenation local solutions small cost initial solution may contain facilities remove facilities bound cost incurred due removal open facilities remove facility guaranteed close group containing open facilities extra capacities provided facilities compensate capacity removed facility organization remaining part paper organized follows sections describe configuration introduced clustering procedure respectively section show construct distributions local solutions finally section show obtain final solution combining distributions constructed basic configuration section give configuration ckm start following basic relaxation min basic indicates whether facility open indicates whether client connected facility constraint cardinality constraint assuring number open facilities constraint says every client must fully connected facilities constraint requires facility open order connect clients constraint capacity constraint well known basic unbounded integrality gap even allowed violate cardinality constraint capacity constraint factor gap instance setting facility capacity metric consists isolated groups facilities clients collocated words distances within group distances groups nonzero integral solution instance group one open facility therefore even connect client group open facilities groups hand fractional solution basic relaxation opens facilities group serves demand group using facilities group note gap instance disappears allow capacity violation order overcome gap setting introduced novel ckm called configuration formally state let fix set facilities let sufficiently large integers let stands subset size convenience also treat set holds every let zsb indicate event indicate event number open set open facilities exactly facilities indicates event open every event happen notice always client indicates keep variables notational purposes every connected integral solution variables event variables following constraints valid help understand constraints good think zsb zsb zsb zsb constraint says open constraint says zsb exactly one exactly one constraint says connected exactly one constraint definition variables constraint holds mentioned earlier constraint says zsb connected facility constraint capacity constraint constraint least open facilities says configuration obtained basic adding variables constraints every since exponentially many subsets know solve efficiently however note polynomially many variables fixed given fractional solution basic relaxation construct values variables check feasibility constraints polynomial time rounding algorithm either constructs integral solution desired properties outputs set constraints infeasible latter case find constraint configuration satisfy run ellipsoid similar instance given show gap still unbounded cardinality constraint violated instead capacity constraint less let group facilities clients method rounding algorithm iterative way see notations fix solution basic cost every let define dav connectionp every every denote dip value solution dav note dav set dav facilities clients shall let simply use let let denote minimum distance simply use moving demands set open facilities decided optimum connection assignment clients facilities computed solving minimum cost problem due integrality matching polytope may allow connections fractional good fractional assignment good integral assignment use following framework design analyze rounding algorithm initially one unit demand client course algorithm move demands fractionally within moving units demand incurs cost end demands moved facility units demand open facility positive amount demand goal bound total moving cost number open facilities representatives black components groups algorithm starts bundling facilities together process creates bigger bigger clusters end nicely formed network sufficiently big clusters facilities see figure illustration clustering representatives bundles initial moving demands first phase use standard approach facility location problems partition facilities bundles bundle associated center called representative set representatives bundle total opening least let initially repeat following process becomes empty select client smallest dav add remove clients thus removed use variants index representatives variants index general clients family voronoi diagram centers let every initially location add closest subset use denote union voronoi regions centers lemma following statements hold max dav dav exists dav dav yuv every representatives facilities bundles black components black components forest groups groups forest figure clustering procedure first phase figure partition bundles centered set representatives second phase figure partition family black components construct rooted forest third phase figure partition family groups formed contracting group single node proof first consider property assume dav dav add remove clients satisfying must property consider iteration removed representative added iteration satisfy property consider property property since dav dav implying yuv due constraint consider property property client dav dav since added next lemma shows moving demands facilities corresponding representative cost much lemma every duv proof property every thus since forms partition get following corollary corollary initial moving demands corollary move demands first every move units demand moving cost step exactly step demands every units demand every move units demand moving cost step thus initial moving demands set representatives representative xuv units demand black components second phase employ construction partition set representatives family black components rooted forest many good properties example black component far away parent root black component contains total opening simplicity say total opening representative yuv total opening bundle forest large degree algorithm requires forest degree property guaranteed using representation describe framework run classic kruskal algorithm find minimum spanning tree mst metric color edges mst black grey white kruskal algorithm maintain set emst edges added mst far partition initially emst edge distance two endpoints sort edges ascending order lengths breaking ties arbitrarily pair order partition add edge emst merge two partitions containing respectively color edges emst every say weight yuv every representative weight least property subset representatives say big weight least say small otherwise edge emst consider iteration kruskal algorithm edge added mst iteration merged partition containing partition containing new partition small call black edge small big call grey edge directed similarly small big grey edge directed big say white edge treat black white edges undirected edges grey edges directed edges define black component mst maximal set vertices connected black edges let set black components thus indeed forms partition contract black edges mst remove white edges resulting graph forest trees black components edge directed grey edge later lemma show grey edges directed towards roots trees every component define shortest distance representative representative component forest may many make forest binary use representation trees specific every component sort according order add directed edge first child directed edge every two adjacent children ordering child appearing later ordering child appearing earlier let new forest naturally defines new relationship components lemma satisfy following properties every spanning tree representatives every edge spanning tree every root component every component every root component either component parent component parent every component two children rest section dedicated proof lemma first prove properties original forest show black edges representatives considered edges kruskal algorithm assume otherwise consider first edge considered iteration connected yet add minimum spanning tree since black component gray white either case new partition formed adding weight implies edges added later mst black moreover contradicts fact black component therefore black edges length implying property focus tree initial forest small black component black edges representatives added mst edge first edge added mst grey edge directed black component white small black since black component thus grey edge therefore growth tree kruskal algorithm follows first grey edge added two black components one big small define root big component time add new small black component current via grey edge directed process white edges incident may added tree rooted tree grey edges edges directed towards root property holds moreover length grey edge parent stronger property since property root big black component suppose contains two representatives singleton consider last black edge added make since black edge small therefore proving property move prove properties lemma final forest used representation obtain thus property holds property independent forest thus still holds component root root thus properties maintained since sorted children component according values constructing binary tree property holds every component parent forest parent initial forest either child former case latter case due property path connecting representative representative internal vertices representatives edges length moreover representatives due properties thus thus property holds finishes proof lemma groups third phase apply simple greedy algorithm forest partition set black components family groups group contains many black components connected contracting group forest set black components becomes forest set groups group total opening unless partition set groups using technique similar rooted tree construct apgroup black components follows initially let contain root component repeat following procedure choose component adjacent smallest add thus construction connected constructed group add remove black components broken many rooted trees apply procedure recursively rooted tree constructed partition set components every contract components single node rooted forest becomes rooted forest set groups naturally defines relationship following lemma uses properties way construct lemma following statements hold set groups rooted forest root group contains single root component root group group let parent distance representative group children proof root component property thus root group contains single root component exactly property constructing group tree terminating condition thus condition hold thus implying property property component thus root group terminating condition constructing implies total weight less right last black component added implying property consider property property easy see group constructed tree following property value component component let group parent let black components thus path components edges path length property moreover property implies representatives component path connected edgessof length thus find path representatives every edge path length property total representatives components contained well thus distance exactly property finally since forest binary every group contains components every group contains children implying property constructing local solutions section shall construct local solution distribution local solutions given set union black components local solution contains pair facilities open amount supply demand satisfied thus shall use supplies satisfy demands initial moving demands thus require two main properties need distribution satisfy expected size distribution big cost matching demands supplies small distinguish concentrated black components black components roughly speaking component concentrated fractional solution clients either almost fully served facilities almost fully served facilities shall construct distribution local solutions concentrated component require constraints satisfied return set separation oracle let vector satisfying constraints roughly speaking defines distribution local solutions local solution good big total demand satisfied small algorithm randomly selects distribution defined condition good fact concentrated guarantees total mass good local solutions distribution large therefore factors lose due conditioning small components construct single local solution instead distribution local solutions moreover construction union components instead individual component components comprise close fact move demands arbitrarily within without incurring much cost thus essentially treat distances representatives concerned two parameters facility distance capacity using simple argument optimum fractional local solution minimizes cost matching demands supplies almost integral contains fractionally open facilities fully opening two fractional facilities find integral local solution small number open facilities remaining part section organized follows first formally define concentrated black components explain importance definition define used measure cost satisfying demands using supplies construction local solutions concentrated components components stated theorem lemma respectively concentrated black components earth mover distance definition concentrated black component except choose parameter differently definition define every black component black component said concentrated otherwise large enough use denote set concentrated components denote set nonconcentrated components next lemma shows importance completeness paper include proof lemma proof let every property fact thus dav first inequality implies second inequality triangle inequality third one equalities simple manipulations notations recall total demand initial moving thus according lemma concentrated use charge cost moving units demand provided moving distance big compared gives freedom handling components concentrated amount demand moved must comparable guaranteed configuration order measure moving cost satisfying demands using supplies define earth mover distance definition earth mover distance given set demand vector supply vector earth mover distance defined emdv inf functions every every technical reason allow fraction supply unmatched shall use xuv denote amount demand initial moving set representatives use denote vector restricted coordinates distributions local solutions concentrated components section construct distributions components proving theorem let let assume constraints satisfied find distribution pairs every support moreover distribution satisfies support size emdj prove theorem first construct distribution satisfies properties modify obtain final distribution notice typical black component however root component containing single representative might large let assume deal case end section since constraints satisfied use variables satisfying constraints construct distribution pairs indicates set open facilities indicates clients connected facilities let pand section simplicity shall use shallpuse denote denote denote denote distribution defined follows initially let zsb increase zsb satisfying every every pair support every moreover either integral since distribution pairs hard see every every support size definition say pair good interested good pairs support show total probability good pairs distribution large let denote set pairs satisfying property denote set pairs satisfying property notice markov inequality proof following lemma uses elementary mathematical tools lemma proof idea use property concentrated get intuition consider case every either thus pairs support every thus assume towards contradiction sort pairs support according descending order define follows take first pair ordering total value pairs ordering plus greater define define fix client indicator variable event inequality comes fact every decreasing function assumption summing inequality exists number every function hard see minimized every every leading contradiction thus finishes proof lemma overall good second inequality used fact focus good pair support since since assumed large enough let set indicated every property satisfied set thus indeed forms distribution pairs moreover support size support thus property holds lemma let good notice since thus condition requires upper bounded threshold smaller thus property long simplicity use denote good denote scaling factor used define indeed shall lose factor later thus shall prove property term right lemma emdj depends follows every proof focus good pair defined every call demand vector supply vector since good thus satisfy demands emd satisfy demands two steps first step give colorspto supplies demands color correspondent client notice xuv every xuv units demand color every units supply color step match supply demand using following greedy rule unmatched demand color unmatched supply color match much possible cost step total cost moving supplies demands color lemma step max max units unmatched demand second step match remaining demand supply every move remaining supply step supplies demands match arbitrarily total cost max diam diam diameter notice since good expected cost first step similarly expected value first term lemma consider second term notice max also max max max min implying summing inequality clients max expected value second term lemma finishes proof lemma point may apply following operation repeatedly take pair shift pair increase thus assume property holds property may unsatisfied every support satisfy property focus support considering achieves emd find matching every emd emd shift sum far distribution pairs satisfies properties property replaced property missing property satisfy property shall apply following lemma massage distribution lemma given distribution pairs satisfying construct another distribution every pair every pair support max property requires probability pair happens large compared probability happens property requires every support property corresponds requiring even count size expected size still going proof lemma shall away pairs formally let define property satisfied markov inequality since thus every pair implying property every pair support thus property holds consider case case fractional number let distribution obtained conditioning pairs markov inequality every pair moreover since conditioned event pairs support modify obtain final distribution notice pair thus second inequality due every pair every pair let due inequality define implying max finally scale vector properties hold scaling factor every pair property overall holds finishes proof lemma lemma finish proof theorem case apply lemma obtain distribution property properties remain satisfied property also holds lost factor expected cost obtain final distribution initially let every pair every support apply following procedure increase otherwise take arbitrary set increase due property every pair support property implies increase using following operation take arbitrary pair support let set decrease increase eventually guarantee thus properties satisfied finishes proof theorem handle case properties root black component contains single representative yuv first find nearly integral solution open facilities close two facilities serving minimum amount demand spread demand among remaining facilities since least open facilities remaining increase amount demand open facility factor let may scale factor course algorithm consider following variables min setting obtain solution value duv lemma value duv fix optimum solution since two two fractional moreover yuv least facilities support shall reduce size support repeating following twice procedure consider support smallest value let scale factor every change still xuv value objective function scaled factor let let every yuv properties satisfied moreover emdj duv since value duv scaled factor let contains single pair probability properties satisfied satisfy properties manually add facilities probability case finishes proof theorem local solutions unions components section construct local solution union close black components lemma let set black components assume exists every find pair dyb dyb emdv proof shall use algorithm similar one used handling case section simplicity let effective capacity consider following variables min setting obtain valid solution objective value duv lemma lemma value fix optimum solution since two every polytope two fractional let let every properties satisfied prove property compute emdv move demands supplies cost xuv first term proved bound second term due fact optimum solution finishes proof lemma rounding algorithm section describe rounding algorithm start giving intuition behind algorithm concentrated component construct distribution local solutions using theorem shall construct partition representatives union nearby components set apply lemma construct local solution independently randomly choose local solution every distribution constructed move demands open facilities small cost property property however may open facilities even expectation noticing fractional solution opens facilities set extra number facilities come two places property theorem may open expectation facilities property lemma may open dyb dyb facilities reduce number open facilities shall shut remove alreadyopen facilities move demands satisfied facilities survived open facilities concentrated component responsible removing facilities expectation set responsible removing facilities lemma allows bound cost moving demands caused removal provided moving distance big respect capacity constraint factor allowed scale concentrated components components groups sets figure figure gives forest set groups denoted empty polygons figure gives set union components solid polygon supplies survived open facilities factor requirements satisfied forest structure groups fact group contains fractional opening property due forest structure property always enough open facilities locally support removing facilities order guarantee always open facilities need use dependent rounding procedure opening removing facilities many previous algorithms incorporate randomized rounding procedure random selections vertex points polytopes respecting marginal probabilities many cases randomized selection procedure derandomized since explicit linear objective shall optimize formally describe rounding algorithm every group use denote set construct partition follows root group add empty group add way empty construct partition except components also define consider set follows every add thus forms partition see figure definition section describe procedure opening set facilities whose cardinality may larger section define procedure remove removes one open facility wrap algorithm section constructing initial set open facilities section open set facilities whose cardinality may larger construct supply vector concatenation local solutions constructed easy construct local solutions components set components correspondent apply lemma obtain local solution add let every notice either contains single root black component contains black components group former case diameter issat property latter case let arbitrary representative representative property thus properties lemma satisfied concentrated components obtain distributions local solutions applying theorem every check constraints satisfied return separation plane fractional solution otherwise apply theorem component obtain distribution produce local solutions concentrated components shall use dependent rounding procedure respects marginal probabilities mentioned earlier shall define polytope procedure randomly selects vertex point polytope let expectation according distribution notational convenience shall use denote bbc dbe consider following polytope defined variables indicator vector local solutions indicates whether responsible removing one facility shall call remove later changing variables vertex point defined two laminar families tight constraints thus integral lemma integral proof avoid negative coefficients shall let focus variables consider set tight constraints define vertex point tight constraints define matroid base polytope laminar matroid every pair support thus constraint equivalent bsj true since constraint holds distribution transformation constraints easy see tight constraints also define laminar matroid thus classic matroid theory set tight constraints define integral solution thus integral set every lemma point polytope every consider pairs support thus total number variables proof notice expected size according distribution budget number open facilities open expected number facilities beyond budget easy seep constraints hold every due properties large enough contains root component thus constraint holds constraints hold point randomly select vertex point every every since integral every add let unique local solution every finishes definition initial let recall xuv demand initial moving every initial demand vector later shall remove facilities update satisfy following properties maintained rounding algorithm proceeds every property due properties property holds since xuv remove procedure section define procedure remove removes facilities updates procedure takes set input root black component let root group containing concentrated component let parent group group containing otherwise union components group let group let calling remove require following properties hold maintaining properties procedure remove remove exactly one open facility change increase factor every increase factor every moreover moving cost converting old new black component facility every emdv increased factor formally describe procedure remove first highlight key ideas assume root component choose arbitrary facility notice facilities shut send demands sent need increase supplies factor otherwise shall shut facility smallest value since least facilities satisfy units unsatisfied demands using facilities thus total amount demands moved comparable either case cost redistributing demands big root component shall shut facility smallest value formally describe procedure remove first consider case root component either component union components group case let arbitrary facility due property find let component contains shall shutdown matching achieves emdv due property total supply equals total demand every shall move units demand total amount demand moved exactly every receive units demand update new demand vector decrease every scale factor every property cost moving demands thus property holds remove change every scale factor new vector emdv increase scaled factor every thus properties satisfied properties trivially true moreover maintained properties consider case case shall remove facility run remove smallest value notice due property let remove facility consider function achieves emdv shall redistribute demands new demand remove change scale total cost redistributing demands procedure due property since properties satisfied properties maintained case root component handled similar way case property least facilities remove facility smallest using argument guarantee properties maintain properties obtaining final solution obtain final set facilities call remove procedures order consider group using order consider group already considered parent group root group contains single root component repeat following procedure twice facility call remove call remove group following let repeat following procedure twice facility call remove every call remove lemma procedure proof first show whenever call remove properties hold concentrated component called remove notice initially due constraint due order considering components property never removed facility calling remove thus property holds check call remove thus property holds property group initially consider first inequality due property constraint second due property may remove facility set call remove satisfying one following conditions concentrated component child group union components contains components case removed facilities due constraint remove facilities thus shall remove facilities thus property holds thus every call remove successful concentrated component called remove initially calling remove never removed facility thus number times call remove least initial value minus overall number facilities removing procedure first inequality due constraint since integer properties constraint final times initial every finally every thus capacity constraint violated factor set large enough remains bound expected cost solution done bounding cost transferring original final well cost matching final first focus transferring cost property call remove transferring cost black component notice scaled factor always cost union components quantity call remove twice thus contribution transferring cost concentrated component quantity might large however probability call remove otherwise property expected contribution transferring cost lemma thus overall expected transferring cost consider matching cost since maintained property matching cost bounded emdv due property quantity increased factor course removing facilities initial expectation quantity due properties found set facilities vector every set large enough cost matching vector vector thus obtained ckm violation references karen aardal pieter van den berg dion gijswijt shanfei approximation algorithms hard capacitated location problems european journal operational research mohit singh ola svensson algorithms capacitated facility location proceedings annual ieee symposium foundations computer science focs arya garg khandekar meyerson munagala pandit local search heuristic facility location problems proceedings annual acm symposium theory computing stoc pages new york usa acm jarosaw byrka krzysztof fleszar bartosz rybicki joachim spoerhase approximation algorithms hard capacitated problems proceedings annual symposium discrete algorithms soda jarosaw byrka thomas pensyl bartosz rybicki aravind srinivasan khoa trinh improved approximation positive correlation budgeted optimization proceedings annual symposium discrete algorithms soda jarosaw byrka bartosz rybicki sumedha uniyal approximation algorithm uniform capacitated problem capacity violation robert carr lisa fleischer vitus leung cynthia phillips strengthening integrality gaps capacitated network design covering problems proceedings eleventh annual symposium discrete algorithms soda pages philadelphia usa society industrial applied mathematics charikar guha improved combinatorial algorithms facility location problems proceedings annual ieee symposium foundations computer science pages charikar guha tardos shmoys approximation algorithm problem extended abstract proceedings annual acm symposium theory computing stoc pages new york usa acm julia chuzhoy yuval rabani approximating capacities soda pages sudipto guha approximation algorithms facility location problems phd thesis stanford usa jain mahdian saberi new greedy approach facility location problems proceedings annual acm symposium theory computing stoc pages new york usa acm jain vazirani approximation algorithms metric facility location problems using schema lagrangian relaxation acm shanfei improved approximation algorithm hard uniform capacitated problem approx proceedings international workshop combinatorial optimization problems international workshop randomization computation approx shi uniform capacitated beyond natural relaxation proceedings annual symposium discrete algorithms soda shi approximating capacitated open facilities proceedings annual symposium discrete algorithms soda pages shi ola svensson approximating via proceedings annual acm symposium theory computing stoc pages new york usa acm lin vitter minimum packing constraint violation extended abstract proceedings annual acm symposium theory computing stoc victoria british columbia canada pages shmoys tardos aardal approximation algorithms facility location problems extended abstract stoc proceedings annual acm symposium theory computing pages new york usa acm
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may remarks weak containment property groupoids semigroups claire abstract locally compact groupoid said weak containment property full coincide reduced one although known property strictly weaker amenability show two properties mild exactness assumption apply result get informations corresponding weak containment property semigroups introduction notion amenability locally compact groups takes many forms well understood see instance amenability introduced measured setting discrete group actions countable equivalence relations zimmer end seventies soon renault extended notion general measured groupoids locally compact groupoids followed studies example group actions detailed general study provided monograph particular long known every amenable groupoid weak containment property sense full reduced coincide yet time converse left open locally compact groupoid amenable weak containment property locally compact groups well known true due theorem hulanicki generally true transitive locally compact groupoid groupoid acting transitively set units proved recently weak containment property action exact discrete group compact space implies amenability action leading believe positive answer question full generality however willett exhibited nice simple example groupoid weak containment property without amenable paper show nevertheless answer question quite often positive suffices groupoid satisfies weak form exactness call inner exactness see definition covers many examples every minimal locally compact groupoid invariant closed subsets unit space empty set whole set units inner exact particular locally compact groups inner exact every continuous action mathematics subject classification primary secondary key words phrases groupoids semigroups weak containment amenability claire exact locally compact group locally compact space corresponding transformation groupoid inner exact however long known exist groupoids inner exact first example non hausdorff locally compact groupoid given skandalis later order provide counterexamples conjecture groupoids higson lafforgue skandalis exhibited hausdorff locally compact groupoids fact bundles groups inner exact mentioned example willet kind relations weak containment amenability partially clarified paper follows see theorem theorem let inner exact locally compact groupoid weak containment property measurewise amenable corollary let locally compact groupoid two following conditions equivalent measurewise amenable every full crossed product coincide reduced crossed product measurewise amenability notion slightly weaker topological amenability many usual cases instance groupoids difference explained main part text see definition remark inner exact groupoid weak containment property equivalent amenability answers question raised willett previous theorem following generalization covers interesting examples theorem let locally compact groupoid equivalent inner exact groupoid weak containment property measurewise amenable established results section turn section case discrete semigroups limit ourself semigroups far case discrete groups namely inverse semigroups defined section groups give partial answers recurrent question concerning semigroups right definition amenability semigroup semigroups consider attached full reduced generalizing classical case groups three obvious candidates notion amenability natural wonder relations left amenability exists left invariant mean semigroup weak containment property full reduced semigroup nuclearity reduced semigroup problem addressed many papers see cite course three properties equivalent discrete group let consider first case inverse semigroup see section denote useful feature semigroup full reduced described remarks weak containment property via groupoid canonically associated consequence see case reduced nuclear weak containment property since amenable general fact stated example given willett reinterpreted setting inverse semigroups allows show general inverse semigroups see example answers question raised remark example clifford inverse semigroup inverse semigroup disjoint union groups set idempotents contained center gives example clifford semigroup weak containment property although groups amenable answers question raised observe notion left amenability interesting except zero element element every indeed inverse semigroup zero left amenable since dirac measure zero left invariant mean even zero left amenability imply weak containment property fortiori nuclearity see example however using theorem present class inverse semigroups conditions equivalent theorem let strongly inverse semigroup idempotent pure morphism exact group see definition weak containment property reduced nuclear theorem next consider case pair group containing unit pointed handy tool order study left inverse hull inverse semigroup nice properties propositions following xin define full full extends definition given nica ordered groups definition hand quotient reduced first observe left amenability always implies nuclearity see proposition true general weak containment property implies left amenability shown nica considered free group generators semigroup generated using uniqueness property cuntz algebra nica proved weak containment property although left amenable note generated isometries weak containment property equivalent uniqueness moreover extension algebra compact operators therefore nuclear whether weak containment property implies nuclearity old problem raised several authors instance laca raeburn remark recently xin using theorem give following partial answer theorem claire theorem let subsemigroup exact group weak containment property implies nuclear result follows fact satisfies assumptions theorem xin introduced independence property rephrased saying quotient map onto injective case occurs instance ordered groups nuclearity implies weak containment property thus order facilitate reading paper first section recall main facts groupoids used sequel fix notation emphasize locally compact spaces always hausdorff unless explicitly mentioned second countable locally compact groupoids always come equipped haar system hilbert spaces separable preliminaries groupoids assume reader familiar basic definitions groupoids details refer let recall notation terminology groupoid consists set subset called set units two maps called respectively range source maps composition law inverse map operations satisfy obvious rules facts composition law product associative elements act units see definition set usually denote set units locally compact groupoid groupoid equipped locally compact topology structure maps continuous topology induced topology induced assume range therefore source map open necessary condition existence haar system denote algebra continuous complex valued functions compact support definition let locally compact groupoid haar system family measures indexed set units satisfying following conditions support exactly support every continuity every function continuous invariance examples transformation groupoid let locally compact group acting continuously right locally compact space topological product space remarks weak containment property natural groupoid structure space units range source maps given respectively product given inverse denote groupoid haar system given left haar measure similarly one defines left action groupoid group bundle locally compact groupoid range source maps equal lemma one choose left haar measure group way forms haar system explicit example given section etale groupoids locally compact groupoid called range therefore source map local homeomorphism onto discrete open moreover family counting measures forms haar system see proposition groupoids associated actions generally partial actions define discrete groups partial transformation groupoid partial action discrete group locally compact space family partial homeomorphisms open subsets meaning extends topology induced product topology domain groupoid range source maps given respectively product defined inverse given already said sequel locally compact groupoids implicitly supposed hausdorff second countable equipped haar system three examples mentioned haar system representations locally compact groupoid let locally compact groupoid haar system set space involutive algebra respect following operations define norm max sup sup definition representation bounded operators hilbert space every claire example let denote image inverse map let defined representation generally let radon measure denote measure defined formula let image inverse map define operator formula representation called induced representation associated full completion respect norm sup runs representations reduced completion respect norm sup obviously identity map extends surjective homomorphism onto remark assume locally compact group let observe involution introduced usual one group theory denotes modular function usually involution defined map isomorphism involutive algebra involution introduced usual one involution full reduced defined canonically identified respectively classical full reduced group similarly full reduced transformation groupoid identified full crossed product reduced crossed product respectively complex valued functions vanishing infinity familiar result group theory relates bijective natural way representations full group unitary representations group similar result holds groupoids statement requires preparation remarks weak containment property let locally compact groupoid let radon measure set say equivalent case denote derivative groupoid equipped measure called measured groupoid definition unitary representation triple measure measurable field hilbert spaces fundamental sequence measurable vector fields iii measurable action isometries every isometric isomorphism identity map function measurable denote hilbert space square integrable sections define operator formula representation called integrated form simply crucial result due renault asserts every representation disintegrated fact groupoid assumed second countable needed theorem proposition let representation hilbert space unitary representation unitary equivalent integrated form say disintegrates example left regular representation measure given note well known easy see map defined formula isometric isomorphism implements unitary equivalence claire denote norm closure define seminorm sup ranges representations disintegrate denote obtained separation completion respect seminorma note canonical surjective homomorphisms onto onto play crucial role rest paper related reductions groupoid give details next section reductions groupoid let groupoid subset set subgroupoid called reduction reduced single element group called isotropy group let locally compact groupoid haar system let locally compact subset meaning locally compact groupoid whose haar system obtained restriction haar system let measure let support closed subset faithful representation see corollary instance follows canonically identified besides representations contains representations disintegrate follows canonical surjective map onto let consider general situation closed subset given set well known inclusion extends injective homomorphisms similarly restriction map extends surjective homomorphisms onto onto moreover sequence exact facts refer page section proposition detailed proof hand corresponding sequence respect reduced always exact shown groupoid skandalis appendix example another interesting class examples provided higson lafforgue skandalis authors consider residually finite group decreasing sequence finite index normal subgroups let alexandroff compactification set denote respect quotient homomorphism let quotient afor simplicity include notation since haar system always implicitely given remarks weak containment property equivalence relation equipped quotient topology natural structure hausdorff locally comb range source maps given pact groupoid group bundle space units equivalence class fibre bundle quotient group call groupoid basic result sequence exact whenever kazdhan property even exact crossed products definition actions groupoids refer let recall facts definition let locally compact space equipped homomorphism centre multiplier algebra sense exists approximate unit every given simplicity write instead let open subset view ideal denote closed linear span closed ideal fact see corollaire set whenever write instead instead denote quotient map set recall kak injective upper semicontinuous see upper field let two morphism morphism case factors morphism let locally compact spaces continuous map associated see definition let locally compact groupoid haar system action given structure isomorphism every isomorphism deduced factorization equipped action say claire let set space continuous sections compact support upper field defined respect following operations see proposition define norm max sup sup full crossed product enveloping banach obtained completion respect let define reduced crossed product let consider hilbert module defined completion space continuous compactly supported functions respect inner product set extends bounded operator adjoint acting hilbert way get representation reduced crossed product completion respect norm see remarks note locally compact subset natural structure moreover canonically isomorphic respectively let set since see action trivial moreover canonically isomorphic respectively denoted except authors consider instead explains formula exactly remarks weak containment property amenability weak containment reference section notion amenable locally compact groupoid many equivalent definitions recall two let recall notation given locally measure defined compact groupoid measure definition definitions say amenable exists net mxi family probability measures mxi continuous sense function dmxi continuous limi uniformly compact subsets say approximate invariant continuous mean note amenable locally compact subset groupoid amenable proposition proposition let locally compact groupoid haar system amenable exists net functions every limi uniformly compact subsets limi uniformly compact subsets also need notion measurewise amenability definition proposition let locally compact groupoid let measure say measured groupoid amenable exists net functions limi weak say measurewise amenable amenable measured groupoid every measure remark amenable groupoid measurewise amenable converse true groupoids countable orbits see theorem particular case groupoids locally compact groups theorem let locally compact groupoid consider following conditions measurewise amenable every measure canonical surjection onto injective claire nuclear every canonical surjection onto injective canonical surjection onto injective moreover isotropy groups discrete every equivalence proved theorem contained corollary well fact isotropy discrete proof see theorem proposition definition say action locally compact groupoid inner exact every closed ideal sequence exact say inner exact canonical action inner exact every invariant closed subset sequence exact term inner definitions aims highlight consider short sequences respect specific given action groupoidc possible definition exactness groupoid following one candidates considered definition say groupoid exact sense short every action inner exact notion studied kirchberg wassermann locally compact groups proved particular property equivalent exactness discrete group examples every minimal groupoid inner exact particular every locally compact group inner exact every groupoid inner exact let locally compact group acting right locally compact space transformation groupoid indeed let action acts straightforward check canonically isomorphic moreover identification functorial cthis notion used another context remarks weak containment property fact groupoid exact strong sense indeed acts amenably compact space therefore acts amenably see proposition acts amenably momentum map proper facts imply proof proof showing group acting amenably compact space see instance theorem details notion exactness groupoids given let partial action discrete group locally compact space associated groupoid whenever exact purpose paper give easier proof inner exactness let closed subset set partial action induces partial action domain domain acts restriction similarly one defines partial action partial transformation groupoids respectvely isomorphic observe acts partially moreover proposition groupoid canonically isomorphic reduced crossed product associated partial action make observation two partial actions finally conclude using fact proved theorem following sequence exact case groupoid group bundle considered next proposition proposition let groupoid group bundle locally compact space let following conditions inner exact every sequence exact iii continuous field fibres iii proof obvious let prove equivalence iii let canonical surjective map onto field lower sense lower semicontinuous every see instance theorem hand indeed set map extends continuously order define structure din non discrete case fact proved recent preprint brodzki cave claire continuity get note follows lemma function upper kernel theorem let locally compact groupoid two following conditions equivalent inner exact groupoid measurewise amenable proof theorem gives converse show every quasiinvariant measure canonical surjective map injective still using theorem let support set following diagram commutative two lines exact injective elementary diagram chasing shows injective recall canonically identified see section observe factorizes surjective follows injective following theorem converse proposition already known discrete isotropy since condition implies nuclear corollary let locally compact groupoid following conditions equivalent measurewise amenable every proof recalled theorem conversely immediately implies weak containment property holds following corollary immediate since amenability measurewise amenability coincide groupoid corollary let groupoid inner exact instance minimal weak containment property amenable notion topological equivalence locally compact groupoids refer definition corollary let locally compact groupoid equivalent inner exact groupoid weak containment property measurewise amenable remarks weak containment property proof assume weak containment property equivalent inner exact groupoid theorem groupoid weak containment property therefore measurewise amenable conclude use fact measurewise amenability preserved equivalence theorem willett considered constructed sequence normal subgroups finite index free group two generators groupoid weak containment property although amenable applications semigroup semigroups consider two kinds semigroups inverse semigroups group inverse semigroup semigroup every exists unique element references notion note groups inverse semigroups exactly one idempotent set idempotents plays crucial role abelian one defines equivalence relation exists idempotent quotient group called maximal group homomorphism image since every homomorphism group factors group trivial zero element frequent situation abuse notation also denote quotient map onto zero denote set zero set given set denote inverse semigroup partial bijections zero element application empty domain theorem proposition identifies inverse semigroup let group containing unit left inverse hull inverse generated injection unit namely observe map defined every element form let recall important properties proposition let subgroup application sending well defined satisfies proof every left reversible assertion lemma proved proposition claire recall inverse semigroup partial order defined follows exists idempotent see page instance definition inverse semigroup said kernel equivalently every element greater idempotent idempotent zero means definition let inverse semigroup morphism grading application group addition say idempotent pure morphism application group exists inverse semigroup called strongly note without zero strongly remark let map isomorphism indeed surjective also surjective assume since idempotent pure see idempotent therefore unit nica introduced toeplitz inverse semigroup inverse subsemigroup generated maps therefore definition say satisfies toeplitz condition give section another characterization toeplitz condition along examples proposition assume satisfies toeplitz condition let defined proposition image case greatest element proof see proposition lemma groupoid associated inverse semigroup let inverse semigroup recall construction associated groupoid described detail denote space maps whenever zero equipped topology induced product space space called spectrum locally compact totally disconnected note monoid unit element therefore compact semigroup acts follows domain open compact set define equivalence relation exists quotient respect equivalence relation equipped remarks weak containment property quotient topology range class source composition law given see details general hausdorff inverse semigroups interested like see quotient topology hausdorff proposition let strongly inverse semigroup let idempotent pure morphism partial action spectrum groupoid topologically isomorphic groupoid associated partial action particular hausdorff moreover compact unit proof result described partial action spectrum defined setting empty set depend choice shown lemma moreover theorem groupoid canonically isomorphic proposition let strongly inverse semigroup assume idempotent pure morphism image greatest element groupoid equivalent transformation groupoid action hausdorff locally compact space proof let since therefore closed subset moreover follows cocycle sending injective closed injectivity means map injective cocycle said closed closed since cocycle injective closed exists locally compact space endowed continuous action transformation groupoid equivalent unit compact equivalence given groupoid isomorphism onto reduction see theorem theorem corollary let group containing unit groupoid defined partial action compact space satisfies toeplitz condition equivalent transformation groupoid defined action locally compact space proof proposition idempotent pure morphism use proposition follows propositions ein proofs carried assuming without zero immediately extend setting ffor notion equivalence groupoids refer definition claire weak containment inverse semigroups let inverse semigroup let recall definition full reduced details see given set banach full defined enveloping algebra universal representations partial isometries left regular representation defined otherwise extension faithful reduced generated still denote extension left regular representation zero follows ideal preferable get rid set similarly denote canonical surjective homomorphism see onto canonically isomorphic shown respectively reason introduced nuclear injective note case definition say weak containment property equivalently isomorphism observe weak containment property groupoid property recall semigroup left amenable exists left invariant mean inverse semigroup zero course left amenable since dirac measure zero left invariant mean nuclear groupoid amenable therefore weak containment property converse following example shows left amenability even absence zero imply weak containment property also shows weak containment property strictly weaker nuclearity precisely authors consider definition also slightly different space require maps satisfy proof also works setting remarks weak containment property example let residually finite group decreasing sequence formally hls example whose notation keep let groupoid defined example view inverse semigroup following way product given isnthe smallest ofothe two elements identify set set idempotents product two idempotents given spectrum set homeomorphic compact space groupoid associated space equivalence classes pairs map sending class isomorphism topological groupoids onto maximal group homomorphism image finite group idempotent pure note zero zero let observe clifford semigroup since amenable semigroup left amenable result duncan namioka see proposition sequence one defined willett weak containment property nuclear moreover clifford semigroup every subgroup amenable although weak containment property hand realize finite index subgroup choose intersection kernel reduction map corresponding weak containment property observed remarks hence left amenability imply weak containment property general next theorem gives particular sufficient condition equivalence weak containment property nuclearity reduced theorem let strongly inverse semigroup let idempotent pure morphism assume amenable nuclear assume exact weak containment property equivalent nuclearity proof proposition groupoid associated partial action spectrum amenable amenable see therefore statement holds exact groupoid inner exact example corollary weak containment property amenable therefore nuclear since claire example let directed graph associated graph inverse semigroup see section let free group generated set edges idempotent pure morphism satisfying assumption proposition weak containment property see theorem although amenable cardinal weak containment semigroups embedded groups section consider discrete group contains unit denote left regular representation denote isometry given reduced generated isometries right definition full speculative universal generated elements vpq every big instance murphy proved commutative semigroup universal nuclear reasonable definition full introduced xin definition variant definition variant denoted adopt definition full sequel denote proposition defined let recall see lemma inverse semigroup canonically isomorphic inverse semigroup partial isometries vqn let also recall surjective homomorphism see lemma therefore following situation definition say weak containment property note weak containment property implies weak containment property definition let assume addition define partial order setting say ordered group every either equivalently every pair elements common upper bound least common upper bound see lemma toeplitz condition equivalent following property every exist vqn see instance proof proposition ordered semigroups semigroups satisfy toeplitz condition see xin also introduced important condition called independence definition describe note remarks weak containment property contained group condition equivalent injectivity see theorem satisfied ordered groups see lemma proposition let group assume satisfies independence condition nuclearity implies weak containment property proof assume nuclear since see groupoid amenable follows thus weak containment property proposition let group amenable nuclear left amenable amenable subgroup nuclear proof assume amenable proposition proposition groupoid amenable since isomorphic groupoid defined partial action follows nuclear well quotient suppose left amenable left reversible therefore amenable subgroup see propositions see nuclear apply first part proof theorem let exact group containing unit two following conditions equivalent nuclear satisfies independence condition proof weak containment property implies weak containment property also independence assertion follows corollaries converse proposition questions example groupoid satisfies weak containment property although amenable provided willett bundle groups opposite principal groupoids units elements source range satisfy weak containment property without amenable would also interesting know whether examples transformation groupoids discrete group acting locally compact space respect true exact groupoid weak containment property compact question seems quite difficult already raised boundary compact set equipped natural action answer positive indeed weak containment property implies sequence claire exact roe willett proved exactness property implies property thus exact find example pair nuclear weak containment property find characterization weak containment property locally compact groupoid added remark read paper kang found nice proof roewillett result thank allowing reproduce proof let discrete group let furstenberg boundary fix element denote canonical inclusion let recall see follows identity map idc extends unital contraction onto therefore exists conditional expectation onto let denote resp canonical homomorphism resp map known injective also injective indeed conditional expectation yields completely positive map onto identity map therefore injective assume groupoid weak containment property following commutative diagram first line exact sequence canonical map injective implies groupoid weak containment property commutativity diagram deduce also weak containment property since action furstenberg boundary minimal follows corollary groupoid amenable therefore exact remarks weak containment property references claire jean renault amenable groupoids volume monographies enseignement enseignement geneva foreword georges skandalis appendix germain claire dynamiques non commutatifs math claire amenability exactness dynamical systems trans amer math electronic claire exact groupoids preparation blanchard hopf bull soc math france jacek brodzki chris cave kang exactness locally compact groups arxiv madalina buneci associated transitive groupoids univ craiova ser mat alcides buss ruy exel ralf meyer reduced fell bundles inverse semgroups arxiv john crisp marcelo laca toeplitz algebras artin groups aust math duncan paterson inverse semigroups proc edinburgh math soc ruy exel inverse semigroups combinatorial bull braz math soc ruy exel starling charles amenable actions inverse semigroups arxiv ruy exel partial dynamical systems fell bundles applications arxiv harry furstenberg boundary theory stochastic processes homogeneous spaces harmonic analysis homogeneous spaces proc sympos pure vol xxvi amer math providence higson lafforgue skandalis counterexamples conjecture geom funct michel hilsum georges skandalis morphismes espaces feuilles kasparov une conjecture connes ann sci norm sup hulanicki means condition locally compact groups studia mehrdad kalantar matthew kennedy boundaries reduced discrete groups arxiv mahmood khoshkam georges skandalis regular representation groupoid applications inverse semigroups reine angew mahmood khoshkam georges skandalis crossed products groupoids inverse semigroups oper theory eberhard kirchberg simon wassermann operations continuous bundles math marcelo laca iain raeburn semigroup crossed products toeplitz algebras nonabelian groups funct landsman ramazan quantization poisson algebras associated lie algebroids groupoids analysis geometry physics boulder volume contemp pages amer math providence mark lawson inverse semigroups world scientific publishing river edge theory partial symmetries gall kasparov claire xin semigroup amenability semigroups funct xin nuclearity semigroup connection amenability adv xin partial transformation groupoids attached graphs semigroups arxiv masayoshi matsumura characterization amenability group actions operator theory david milan inverse semigroups amenability weak containment operator theory david milan benjamin steinberg inverse semigroup crossed products groups geometry dynamics paul muhly dana williams renault equivalence theorem groupoid crossed volume nyjm monographs albany state university new york university albany gerard murphy generated commuting isometries rocky mountain alexandru nica generated isometries operators operator theory alexandru nica groupoid construction actions certain inverse semigroups internat magnus dahler norling inverse semigroup associated left cancellative semigroups proc edinb math soc magnus dahler norling reduced inverse semigroup math alan paterson weak containment clifford semigroups proc roy soc edinburgh sect alan paterson amenability volume mathematical surveys monographs american mathematical society providence alan paterson groupoids inverse semigroups operator algebras volume progress mathematics boston boston birant ramazan quantification par des phd thesis university jean renault groupoid approach volume lecture notes mathematics springer berlin jean renault des produits operator theory jean renault ideal structure groupoid crossed product operator theory appendix georges skandalis jean renault sundar groupoids associated ore semigroup actions operator theory jean renault dana williams amenability groupoids arising partial semigroup actions topological higher rank graphs arxiv marc rieffel continuous fields coming group cocycles actions math john roe rufus willett ghostbusting property funct aidan sims dana williams renault equivalence theorem reduced groupoid operator theory rufus willett groupoid whose maximal reduced arxiv robert zimmer hyperfinite factors amenable ergodic actions invent remarks weak containment property robert zimmer von neumann algebra ergodic group action proc amer math robert zimmer amenable ergodic group actions application poisson boundaries random walks functional analysis cnrs umr cedex address
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may aggregation supports along lasso path pierre bellec abstract linear regression fixed design propose two procedures aggregate collection supports collection subset possible supports cardinality elements depend data procedures satisfy oracle inequalities assumption design matrix use procedures aggregate supports appear regularization path lasso order construct estimator mimics best lasso estimator restricted eigenvalue condition design matrix satisfied estimator achieves optimal prediction bounds finally discuss computational cost procedures introduction let two positive integers consider mean estimation problem unknown subgaussian vector exp exp noise level euclidean norm observe wish estimate design matrix size given may larger require model exists goal find estimator prediction loss small empirical loss defined accepted presentation conference learning theory colt supports along lasso path setting lasso known achieve good prediction performance tuning parameter define lasso estimate solution convex minimization problem argmin identity matrix size optimal choice tuning parameter log numerical constant restricted eigenvalue condition holds definition universal tuning parameter log leads good prediction performance bickel however columns correlated restricted eigenvalue condition satisfied question optimal choice tuning parameter still unanswered even noise level known empirical theoretical studies van geer lederer hebiri lederer dalalyan shown columns correlated lasso estimate tuning parameter substantially smaller universal parameter leads prediction performance substantially better lasso estimate universal parameter summarize papers raise following question problem selection tuning parameter find datal driven quantity prediction loss small high probability paper focus different problem namely problem lasso aggregation construct estimator mimics prediction performance best lasso estimator construct estimator high probability min constant small quantity goal achieve small prediction loss high probability goal select lasso estimate small prediction loss look estimator prediction performance almost good prediction performance lasso estimate estimator may different form parameter supports along lasso path motivation consider instead following let deterministic vectors goal mimic best approximation among well known literature aggregation problems estimator form integer suboptimal theorem rigollet tsybakov section juditsky proposition gerchinovitz thus optimal procedure valued discrete set optimal procedures problem valued convex hull set examples exponential weights procedures proposed leung barron dalalyan salmon procedure dai although lot progress made various aggregation problems knowledge previous work deals problem aggrel gation nonlinear estimators collection based sample setting present paper observation lasso estimates independent performed data used construct lasso estimators aggregate show aggregation nonlinear estimators form possible nonlinear estimators without assumption instance estimator achieves log given section denotes number nonzero coefficients max given design matrix call support subset cardinality denoted supp set indices given support denote square matrix size orthogonal projection linear span columns whose indices belong denote set subsets consider following problem problem aggregation collection supports let collection supports estimator valued construct estimator high probability min function takes small values supports along lasso path set family supports let emphasize cardinality elements depend data note support minimizes subject section construct estimator satisfies log nonempty supports literature aggregation problems one given collection estimators deterministic integer goal mimic best estimator collection tsybakov references therein novelty present paper consider aggregation collection estimators cardinality collection depends data main contributions present paper following section propose estimator satisfies oracle inequality log nonempty supports estimator noise level estimator solves explain corollary section used construct procedure aggregates nonlinear estimators form section devoted using result section construct estimator satisfies log computational complexity procedure sum complexity regularization path lasso complexity convex quadratic program proofs found appendix aggregation family supports throughout section let collection supports let real valued estimator let cardinality let supports supports define weights rigollet tsybakov note construction constant greater set subsets given support supports along lasso path least squares estimator linear span covariates indexed consider two estimators based first estimator defined follows define criterion log log support lower bound direct consequence upper bound proved rigollet tsybakov holds criterion nature aic bic variants massart define estimator log argmin estimator orthogonal projection onto linear span columns whose indices procedure close one studied massart define second estimator valued convex hull let cardinality let supports holds let define simplex follows define let log penq penq penalty inspired recent works procedure dai used derive sharp oracle inequalities aggregation linear estimators dai bellec supports along lasso path density estimators bellec penalty pushes towards another penalty points finally term log pushes coordinate size support large define estimator minimizer function defined argmin theorem let positive integers let let matrix size let collection subsets assume noise satisfies let real valued estimator let estimator defined satisfies probability greater exp min log furthermore estimator satisfies probability greater exp log min previously studied aggregation problems one given collection estimators deterministic integer goal construct estimator high probability min small error term increases tsybakov references therein theorem different nature several reasons first set random cardinality depend observed data second error term appears inside minimum depend cardinality estimator theorem set subsets previously studied exponential screening estimator rigollet tsybakov sparsity pattern aggregate rigollet tsybakov special case deterministic contains possible supports exponential number supports along lasso path supports computing sparsity pattern aggregate practice hard mcmc algorithm developed rigollet tsybakov compute approximate solution sparsity pattern aggregate knowledge theoretical guarantee mcmc algorithm converge good approximation polynomial time sparsity pattern aggregate satisfies sharp oracle inequality yields minimax rate balls assumption design matrix dai tsybakov construct estimator one solve optimization problem convex quadratic program size simplex constraint complexity computing polynomial cardinality thus small possible construct efficiently cardinality decreases prediction performance estimator becomes worse computing becomes easier problem construct set supports high probability exists support simultaneously bias size small construct set prediction loss estimator small note theorem needs assumption set design matrix following corollary perform aggregation family nonlinear estimators form set estimators family share design matrix matrix deterministic corollary let positive integers let let matrix size let collection subsets assume noise satisfies let family estimators valued cardinality family elements depend data let real valued estimator let define supp let estimator estimator satisfies probability greater exp min log using similar result readily obtained estimator supports along lasso path leading constant aggregation supports along lasso path let recall properties lasso path efron given observation exists positive integer finite sequence supp supp thus finite number supports lasso path section study estimator theorem special case supp aggregate supports appear lasso path theorem let positive integers let let matrix size assume noise satisfies let real valued estimator let let knots lasso path let supp family supports appear lasso path let estimator estimator satisfies probability greater exp min log lasso estimator using similar result readily obtained estimator leading constant computational complexity procedure theorem polynomial number knots lasso path discussed section rest section assume come back estimation noise level section interestingly theorem need assumption design matrix estimators good performance soon possibly unknown support loss small supports along lasso path prediction guarantees restricted eigenvalue condition goal section study prediction performance procedure defined theorem restricted eigenvalue condition design matrix definition condition satisfied min min following result reformulation bickel theorem theorem bickel let diagonal elements equal assume let assume condition satisfied let event probability greater lasso estimator tuning parameter log satisfies simultaneously log largest eigenvalue matrix thus restricted eigenvalue condition satisfied lasso estimator universal parameter log enjoys simultaneously norm order true sparsity prediction loss order log theorem direct consequence theorem bounds theorem let positive integers let let matrix size let subset assume let assume condition satisfied let knots lasso path let supp family supports appear lasso path let supports along lasso path estimator estimator satisfies probability greater exp log furthermore log using similar result readily obtained estimator different constants proof theorem theorem event probability greater log let event defined theorem using simple inequality log log bounds obtain holds event union bound event probability greater finally obtained integration procedure studied theorem aggregates supports along lasso path using procedure similar result holds estimator leading constant equal theorem following implications first fixed prediction performance estimator similar lasso universal tuning parameter multiplicative factor involves numerical constants quantity soon operator norm bounded constant estimator studied theorem enjoys best known prediction guarantees second theorem implies estimator satisfies prediction bound simultaneously confidence levels holds probability greater contrast lasso estimator universal parameter depends fixed confidence level lasso estimator universal parameter supports along lasso path satisfies prediction bound confidence level knowledge known whether lasso estimator universal parameter satisfies similar bound different confidence levels regard estimator studied theorem provides strict improvement compared lasso universal parameter third estimator theorem satisfies bound prediction bound expectation knowledge known whether lasso estimator universal parameter satisfies similar bound expectation assuming bound tight putting computational issues aside prediction performance procedure theorem substantially better performance lasso universal parameter soon bounded constant upper bound similar given belloni theorem remark namely belloni prove lasso estimator universal tuning parameter satisfies high probability sparsity true parameter constant depends sparse eigenvalues matrix belloni condition upper bound used instead prove results similar replaced smaller constant depends sparse eigenvalues computational complexity lasso path computing estimator theorem done two steps compute full lasso path let supp supp supports appear lasso path knots lasso path compute solution quadratic program defined step assume complexity computing negligible compared complexity step step time complexity step complexity convex quadratic program size number knots lasso path thus global cost computing estimator theorem polynomial exist efficient algorithms compute entire lasso path efron however mairal proved values regularization path lasso contains knots hence supports along lasso path design matrix observation exact computation full lasso path realizable polynomial time order fix computational issue mairal propose algorithm computes approximate regularization path lasso fixed algorithm guaranteed terminate less knots points approximate path duality gap smaller approximation algorithm used instead computing exact lasso path one may compute estimator collection supports appear approximate path computed algorithm mairal another solution avoid computational issues follows let positive integer instead computing lasso path one may consider grid tuning parameters aggregate supports corresponding lasso estimates advantage approach twofold first lasso estimate computed standard convex optimization solvers second time complexity procedure guaranteed polynomial corollary procedure satisfies probability greater min log oracle inequality strong however least one lasso estimates enjoys small prediction loss small norm prediction loss also small fully procedure using lasso section proposes fully procedure based squareroot lasso choice grid comes empirical theoretical observations correlated design matrix exists tuning parameter smaller universal parameter enjoys better prediction performance universal parameter van geer lederer hebiri lederer dalalyan let log universal parameter squareroot lasso belloni confidence level let conservatively small value tuning parameter let integer supports along lasso path consider geometric grid compute lasso estimators parameters possible perform computation simultaneously pham references therein let supp supports computed lasso estimators let variance estimated lasso universal parameter choice return estimator estimator estimator returned procedure enjoys theoretical guarantee min log probability greater similar guarantee leading constant obtained estimator using concluding remarks presented two procedures aggregates datadriven collection supports procedures satisfy oracle inequalities given theorem main result paper sections study situation collection supports appear along lasso path procedures may used datadriven collections well procedures allow one perform prediction performance computational cost contains supports procedures achieve optimal prediction guarantees assumption design matrix realized polynomial time hand cardinality small say polynomial possible compute estimators polynomial time view one look set following properties supports along lasso path set small estimators computed rapidly set contains support simultaneously small procedures enjoy good prediction performance natural choice collection supports appear along lasso path choice studied sections another natural choice aggregate supports several lasso estimators since lasso weak conditions design meinshausen definition corollary research investigate means construct collection two properties satisfied acknoledgements would like thank alexandre tsybakov helpful comments writing manuscript appendix proof theorem matrix define operator norm frobenius norm sup kakf respectively proof define event log define event event simultaneously supports log log log supports along lasso path used log lemma event min log obtain use fact event remains bound probability event denote complement event proceed union bound follows definition lemma log yields exp established exp proof close argument used massart giraud section recent reference model selection novelty present paper consider collection estimators proof let let notational simplicity definition rewritten log log log define event define log log supports along lasso path notation using simple inequality implies event log log using obtain log log define events event holds remains bound probability event fixed using log let fixed using log event probability greater log log thus proof union bound completes proof appendix technical lemmas lemma estimator let minimizer almost surely min max log log log defined supports along lasso path proof lemma let function convex differentiable rewritten log simple algebra log log summing last display equality applied get log log since least squares estimator linear span covariates rewritten thus log log log boyd vandenberghe section yields furthermore linear function simplex maximized vertex almost surely obtain supports along lasso path lemma let supports quantity defined satisfies probability greater exp proof lemma let almost surely clear satisfies chernoff bound yields exp clear apply concentration inequality get probability greater exp finally let rank matrix orthogonal projector hence applying concentration inequality hsu yields probability greater exp union bound completes proof references pierre bellec optimal bounds aggregation affine estimators submitted url http pierre bellec optimal exponential bounds aggregation density estimators accepted bernoulli url http alexandre belloni victor chernozhukov lie wang pivotal estimation via lasso nonparametric regression ann issn url http supports along lasso path peter bickel acov ritov alexandre tsybakov simultaneous analysis lasso dantzig selector ann issn url http lucien pascal massart gaussian model selection eur math soc jems issn url http stephen boyd lieven vandenberghe convex optimization cambridge university press dong dai philippe rigollet tong zhang deviation optimal learning using greedy ann issn url http dong dai philippe rigollet lucy xia tong zhang aggregation affine estimators electron url http arnak dalalyan joseph salmon sharp oracle inequalities aggregation affine estimators ann issn url http arnak dalalyan mohamed hebiri johannes lederer prediction performance lasso arxiv preprint bradley efron trevor hastie iain johnstone robert tibshirani least angle regression ann issn url http discussion rejoinder authors gerchinovitz prediction individual sequences prediction statistical framework links around sparse regression aggregation techniques phd thesis paris url https christophe giraud introduction statistics volume monographs statistics applied probability crc press boca raton isbn mohamed hebiri johannes lederer correlations influence lasso prediction ieee trans inform theory march url http daniel hsu sham kakade tong zhang tail inequality quadratic forms subgaussian random vectors electron commun issn url http juditsky rigollet tsybakov learning mirror averaging ann issn url http supports along lasso path gilbert leung andrew barron information theory mixing regressions ieee trans inform theory issn url http julien mairal bin complexity analysis lasso regularization path arxiv preprint nicolai meinshausen bin recovery sparse representations data ann issn url http pham laurent ghaoui arturo fernandez robust sketching multiple lasso problems arxiv preprint philippe rigollet alexandre tsybakov exponential screening optimal rates sparse estimation ann issn url http philippe rigollet alexandre tsybakov sparse estimation exponential weighting statist issn url http tsybakov aggregation minimax optimality estimation proceedings international congress mathematicians seoul appear sara van geer johannes lederer lasso correlated design improved oracle inequalities probability statistics back models processes volume inst math stat ims pages inst math beachwood url http
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submitted annals statistics arxiv unified treatment multiple testing prior knowledge using sep aaditya rina martin michael university california university significant literature studies ways employing prior knowledge improve power precision multiple testing procedures common forms prior knowledge may include priori beliefs hypotheses null modeled prior weights differing importances hypotheses modeled differing penalties false discoveries multiple arbitrary partitions hypotheses known possibly overlapping groups indicating dis similarity hypotheses knowledge independence positive arbitrary dependence hypotheses groups allowing aggressive conservative procedures present unified algorithmic framework called global null testing false discovery rate fdr control allows scientist incorporate four types prior knowledge simultaneously recovering wide variety common algorithms special cases introduction multiple hypothesis testing classical highly active research area dates back least initially unpublished manuscript tukey entitled problem multiple comparisons given large set null hypotheses multiple testing commonly deals deciding subset reject guaranteeing notion control number false rejections practical importance incorporate different forms prior knowledge existing multiple testing procedures prior knowledge yield improvements power precision also provide interpretable results focus paper methods control false discovery rate fdr test global null hypothesis incorporating number following considerations using prior weights using penalty weights partitioning hypotheses groups incorporating knowledge dependence structure within data including options estimating adapting unknown number nulls independence reshaping rejection thresholds preserve error control guarantees msc subject classifications primary secondary keywords phrases multiple testing false discovery rate prior knowledge simes adaptivity group fdr ramdas barber wainwright jordan presence arbitrary dependence challenge incorporate forms structure maintaining internal consistency coherence consonance among pattern rejections acceptances existing work managed simultaneously employ one two four considerations present general unified framework integrating four performing test controlling fdr framework allows scientists mix match techniques use multiple different forms prior knowledge simultaneously framework simplifies analysis existing procedures generalize conditions known work outline contributions outline rather presenting general version framework outset introduce framework gradually beginning simple weighting schemes introducing tools notation needed encompass complex schemes since types prior information considered paper already motivated applied settings focus conceptual mathematical aspects assumptions section formalize range settings studied terms assumptions underlying provide data multiple testing assumed jointly consider three settings independence positive dependence arbitrary dependence generalized bhy procedures discuss use reshaping functions introduced blanchard roquain guard arbitrary dependence use describe set procedures generalize classical procedure procedure thus refer generalized bhy develop general lemma core fdr control proofs first application show yields short proof fdr control generalized bhy procedures three settings dependence bhy section discuss two natural ways incorporate describe procedure uses prior penalty weights simultaneously using lemma show procedure controls weighted fdr three settings dependence immediately implies old new results algorithms studied genovese benjamini hochberg blanchard roquain adaptive prior section present adaptive procedures independent unified multiple testing prior knowledge improve power incorporating estimate proportion nulls first proposed storey propose novel adaptive procedure employ prior penalty weights well null proportion estimates using new lemma succinctly prove adaptive procedure controls weighted fdr independence grouped adaptive procedures section tackle problem controlling fdr single userdefined partition hypotheses may may formed elementwise particularly discuss one way achieve based simes first showing simes test studied hochberg liberman valid even positive dependence reshaped work arbitrary dependence using novel group lemma demonstrate use novel weighted simes tests conjunction adaptive procedures control weighted group fdr unifying multilayer grouped adaptive procedure section generalize previous section allow use multiple arbitrary partitions hypotheses possibly arranged relative process develop algorithmic framework called extending earlier work barber ramdas allows use following features extensions different prior penalty weights every group every partition adaptivity partitions whose groups known independent incomplete partitions contain leftover set overlapping groups within partition reshaping functions guard arbitrary dependence algorithmic framework guarantees internal consistency meaning rejected hypotheses groups always mutual agreement combine lemma novel proof techniques prove guarantees weighted group fdr control partitions simulataneously three forms dependence sometimes reprove existing results literature usually purposes illustrating novel proof technique cases label results propositions cases results new found elsewhere literature label theorems table summarizes contributions relative related work given space limitations list related references defer detailed discussions related work point paper settings discussed form incorporated structure prior work paper benjamini hochberg benjamini yekutieli proposition prior weights fdr genovese proposition penalty weights fdr benjamini hochberg finos salmaso proposition prior penalty weights fdr blanchard roquain null proportion adaptivity fdr prior penalty weighted null proportion adaptivity fdr storey proposition proposition theorem none global null simes benjamini yekutieli proposition prior weights global null hochberg liberman proposition single partition groups fdr barber ramdas proposition multiple arbitrary partitions fdr multiple possibly incomplete arbitrary partitions possibly overlapping groups prior penalty weights group null proportion adaptivity fdr barber ramdas theorem theorem table table summarizes contributions relative known results regarding procedures multiple testing traffic lights represent results proved independence easy positive dependence intermediate arbitrary dependence hardest top box indicates uniformity fdr bound proved inequality instead strict equality bottom box indicates adaptivity provided also constrained simes ramdas barber wainwright jordan none false discovery rate unified multiple testing prior knowledge background begin presenting background multiple testing fdr control overviewing common assumptions literature providing discussion procedure helpful introduce basic definitions notation pair vectors use notation mean orthant ordering definition nondecreasing sets functions set said nondecreasing implies say function nonincreasing implies finally order handle ratio often arises fdr control results adopt dotfraction notation dotfractions behave like fractions whenever denominator nonzero formally derive properties dotfractions detail appendix false discovery rate fdr standard multiple testing setup given set different denoted random vector corresponds different null hypothesis let denote subset true null hypotheses goal reject subset null words select subset retaining control number false discoveries consider algorithm based observed vector chooses subset hypotheses reject seminal work benjamini random subset denoted hochberg proposed measure performance algorithms quantity called false discovery rate fdr gave simple procedure provably control fdr defined terms false discovery proportion given number false null fdp hypotheses true incorrectly historical footnote procedure proposed eklund albeit heuristic without formal guarantees ramdas barber wainwright jordan total number discoveries meaning null hypotheses rejected fdr expectation random ratio fdr goal construct procedures control fdr target level guarantee fdr dependence assumptions assume marginal distribution null meaning stochastically dominated uniform distribution precisely index assume course uniformly distributed trivially satisfy condition use phrase uniformity describe situation null marginally exactly uniform phrase employed understood null marginally regarding assumptions joint distribution three possible kinds dependence considered paper independence positive dependence arbitrary dependence independent setting assumed mutually independent second possibility positive dependence formalized positive regression dependence subset condition prds short definition prds say vector satisfies prds null index nondecreasing set function nondecreasing original positive regression dependence assumption introduced lehmann well prds assumption first made benjamini yekutieli instead definition one prove conditions essentially equivalent prds condition holds trivially independent also allows amount positive dependence let consider simple example provide intuition let multivariate gaussian vector covariance matrix null components correspond gaussian variables zero mean letting cdf standard gaussian vector prds unified multiple testing prior knowledge every index entries covariance matrix see benjamini yekutieli additional examples type noted prds assumption closely related assumption multivariate total positivity order studied karlin rinott since implies prds results paper hold prds also immediately also hold generalized procedures begin discussing classical procedure given vector let denote associated vector order statistics given order statistics procedure target fdr level rejects smallest max convention set choosing empty value set rejections definition must contain exactly many hypotheses benjamini hochberg prove procedure controls fdr desired level assumption fdr rejection set mutually independent result shown hold also positive dependence setting benjamini yekutieli assumptions made joint distribution however conceivable procedures must guarded proclaiming discovery benjamini yekutieli originally proposed running procedure modified threshold order maintain fdr control arbitrary dependence introduce family generalizations modification refer generalized procedures simply procedures throughout paper order first need notion reshaping function introduced blanchard roquain definition reshaping fixed probability measure define reshaping function ramdas barber wainwright jordan fixing reshaping function generalized procedure target fdr level makes max rejections convention oif set empty rejection set given contains exactly many hypotheses noting probability measure thus requiring lowest fall lower strict threshold immediately see choice generalized procedure conservative procedure run target fdr level example see benjamini yekutieli original proposal running procedure modified threshold equivalent running procedure reshaping function summarize known properties procedures proposition procedure generalized procedures following properties independence uniformity method fdr positive dependence method fdr arbitrary dependence generalized procedure reshaping function yields fdr statement proven inequality benjamini hochberg benjamini yekutieli proved statement equality well proving statement statement specific reshaping function arbitrary reshaping function statement proven blanchard roquain role reshaping naturally universally optimal choice different choices base probability measure hence lead different behaviors associated procedures typical choice measure discrete distribution support choice sensible since number rejections always integer indeed unified multiple testing prior knowledge noted blanchard roquain continuous measure replaced discrete measure yield reshaping function strictly larger relevant integer range leading powerful procedure contrast discrete distributions considered past work paper also considers case continuous measures continuous measures interest context procedures considered sequel function corresponds total weight assigned subsets hypotheses opposed count total weight fractional one longer reduce case discrete measures many different examples reshaping functions connections formulations multiple testing methods found literature see blanchard roquain sarkar alternative derivation procedures alternate way motivate procedures useful designing algorithms later paper let fdp denote fdp procedure rejects less equal note fdp fdp approximation follows servation expect roughly many null order guard unex pected dependence structures would reshape thepdenominator fdp undercount rejections replacing choose largest threshold estimated fdp concretely define max fdp ramdas barber wainwright jordan reject hypotheses corresponding smaller hard verify threshold rule corresponds rejecting set hypotheses original rule procedures respectively lemma fdr control set elements let denote set elements leaves element vector use denote vector coordinate set zero course contain information reconstructed definition loop function said satisfy property loop null index refer loop situations vector represents threshold satisfies loop even though threshold may differ significantly either lie thresholds words perspective threshold might well instead develop intuition lemma follows note assumption null reformulated fixed course uniform inequality holds equality following lemma guarantees property continues hold certain random thresholds recall term nonincreasing interpreted coordinatewise respect orthant ordering def lemma lemma let null hypothesis nonincreasing function independent unified multiple testing prior knowledge furthermore additionally assume range satisfies loop uniformly distributed inequality replaced equality nonincreasing function prds respect constant function reshaping function arbitrary dependence constant functions reshaping functions arbitrary dependence proofs statement equality statement given appendix statement inequality recovered special case statement proved blanchard roquain also proved general statement one reshaping function present bound required proof novel group lemma lemma central succinct proofs results paper also extended allow multiple reshaping functions used corollary give first hint power lemma begin providing short proof procedure controls fdr proof proposition prove statement first definition fdr fdr ramdas barber wainwright jordan step follows applying lemma turning statement repeat argument incorporating reshaping function fdr step follows lemma order prove statement equality applying lemma straightforward see satisfies loop condition completeness statement proved general setting proposition note style arguments fdr control entire set rejected hypotheses proved considering individually new see example sarkar however past works either considered specific functions proved inequality version statement proof generalization arguments made special case procedure heesen janssen using prior penalty weights consider setting weights associated hypothesis prior work focused two natural interpretations priors nullity hypotheses penalties wrongly rejecting nulls accordingly introduce sequence positive penalty weights positive priorp weights along normalization conditions two weight sequences interpreted follows penalty weights encode unequal importances hypotheses hypotheses indicate interest scientist hypotheses hence count differently measuring number false total discoveries prior weights encode prior evidence null large implies prior belief hypothesis likely small indicates opposite weights might learned data collected older study different subjects treat weights constants straightforward extend results bayesian setting weights random null distributed super uniformly even conditioning weights unified multiple testing prior knowledge order model setting define modified criterion following benjamini hochberg called weighted fdr fdru order model setting hand one need alter definition fdr since weights simply allow hypotheses reject alter measure error present version procedures incorporate sets weights simultaneously procedures following blanchard roquain define weighted procedure bhuw follows set reject hypotheses threshold given max fdp fdp useful rewrite procedure follows let represent weights corresponding order statistics weighted define sum first weights define max reject hypotheses ocorresponding smallest weighted values bhuw procedure equivan lently described rejecting first corresponding total penalty weight wish use reshaping function guard arbitrary dependence among instead use doubly weighted procedure denoted byuw instead define max fdp fdp corresponds max ramdas barber wainwright jordan implicitly choice reshaping function considered fixed suppressed notation procedure setup following guarantees weighted fdr proposition procedures given following properties independence uniformity maxi bhuw cedure yields fdru positive dependence bhuw procedure yields fdru arbitrary dependence byuw procedure reshaping function yields fdru present proof let discuss special cases sets weights used blanchard roquain proved proposition also proposition inequality proof proposition mimics presented notation completeness penalty weights used independent one may set bhuw procedure recover procedure proposed benjamini hochberg proved proposition albeit inequality prior weights used genovese proposed priorweighted procedure controls unweighted fdr independence result recovered setting bhuw procedure invoking proposition weights equal one bhuw byuw reduce unweighted procedures proposition reduces proposition proof proposition first prove statements conciseness unite proofs two statements define function identity setting reshaping function setting note write weighted unified multiple testing prior knowledge fdr equation fdru recalling definition multiply divide obtain fdru inequality follows observation function nonincreasing null hypothesis apply lemma lemma next turn proof statement begin noting maxi range next show satisfies loop condition respect index holds independence uniformity inequality holds equality lemma prove loop first easy observe since strictly positive next also straightforward hypothesis already rejected procedure setting zero change order apply lemma last condition need verify assuming holds rejected also holds indeed rank within order statistics guaranteed since rejected setting number rejections must bounded since unchanged continues hold condition may infer condition also hold since side ramdas barber wainwright jordan null proportion adaptivity prior penalty weights recall original unweighted procedure provided fdr control level significantly smaller target leading possible loss power compared procedure fully utilized fdr budget known independent storey collaborators proposed simple method estimate number nulls incorporating estimate procedure leads procedure possibly higher power refer method storey adaptive procedure rest section assume independent demonstrate combine benefits adaptivity prior penalty weighting analyzing novel procedure storey adaptive method fix constant define quantity conservative estimate proportion nulls since analogy procedure procedure chooses max fdp fdp rejects smaller procedure comes following guarantee proposition independence procedure guarantees fdr storey coauthors establish result using martingale arguments blanchard roquain gave alternate proof using lemma prove proposition since follows special case theorem proceed discussion result need following notation define estimate nullproportion unified multiple testing prior knowledge obtained set zero estimated future reference note using simple properties binomials easy show estimate satisfies bound turns bound property estimator require prove fdr control fact bound satisfied several estimators examples see sarkar blanchard roquain propose empirically compare several estimates find storey method best power amongst considered alternatives adaptive procedure section demonstrate simultaneously incorporate prior weights penalty weights storey adaptive estimator introduce shorthand maxj define procedure follows define estimated null proportion estimate false discovery proportion fdp min choose threshold rejection max fdp reject hypotheses min words apply weighted bhuw procedure also incorporating weighted null proportion estimator procedure following guarantee theorem independence procedure controls fdru level ramdas barber wainwright jordan notice set weights unity recover exactly procedure theorem reduces proposition also set recover exactly bhuw procedure prepare proof theorem define analogous estimate weighted null proportion function observation crucial proof theorem important property required proof following inequality establish need sophisticated result weighted binomials present bound recovered special case lemma setting weights one lemma inverse binomial lemma given vector constant bernoulli variables bernoulli weighted sum satisfies since lower bound follows jensen inequality include bound provide context upper bound detailed proof found appendix order see required property follows lemma define since applying lemma guarantees simple algebra leads property ready prove theorem unified multiple testing prior knowledge proof theorem definition fdru min fdru min min min step uses fact construction procedure step follows multiplying dividing whenever numerator one term moreover since min fdru min iii step iii follows since conditioning fully determines step follows elementary bound min noting function nonincreasing hence applying lemma ensures fdru inequality follows property concluding proof ramdas barber wainwright jordan combining one layer groups prior penalty weights null proportion adaptivity demonstrate incorporate prior structural knowledge hypotheses separated partition known groups procedures introduced far next section generalize ideas handle multiple arbitrary partitions consisting groups provided meaning corresponding test whether group entirely null one may trivially run bhy procedure control group fdr section tackle setting formed combining using simes procedure would like control fdr possibly presence prior penalty weights main technical tool lemma lemma analogy elementwise lemma lemma formal setup suppose partitioned hypotheses groups size access weights group level prior weights penalty weights simes group may formed using weighted procedure weights discussed separately later summarize data takes form group weights group weights wish select subset groups sbgrp proportion null groups groups consisting entirely null hypotheses high define set null groups hgrp group contains null hypotheses weighted group fdr given grp fdrgrp bgrp order test group rejection need compute pggrp order true need ensure unified multiple testing prior knowledge whenever pggrp super uniformly distributed whenever hgrp null group many statistics proposed combine elementwise pare independent options include fisher rosenthal gaussian cdf originally proposed stouffer bonferroni correction known powerful also works arbitrary dependence proposal proposal fixed setting arbitrary either built combining individual hypotheses within group constructed new independent data matter pvalues constructed one may simply apply bhy procedure appropriately weighted reshaped adapted control fdr one example special interest closely related bhy procedure setting group formed elementwise using simes procedure next discuss generalized simes tests global null simes proposed improvement bonferroni procedure global null testing level first calculate simes using reshaping function simes min reject hgn simes connection bhy procedure quite transparent note simes bhy procedure makes least one rejection level well known simes pvalue simes really bonafide result recover special case proposition simesw test global null simes test extended hochberg liberman incorporate prior weights independence define weighted hypothesis calculate generalized simesw group simesw min henceforth tilde used signified reshaping function calculating simes within single group continue use notation without tilde comparing across multiple groups ramdas barber wainwright jordan global null hypothesis group hypothesis consists entirely nulls rejected level simesw general setting individual within group may arbitrarily dependent instead consider reshaped weighted simes given rsimesw min reshaping function recall definition following result states weighted reshaped simes really bonafide proposition global null hypothesis weighted simes following properties independence uniformity maxi simesw exactly uniformly distributed positive dependence simesw distributed arbitrary dependence reshaped simes rsimesw distributed statement first proven hochberg liberman statement unit weights hommel statements straightforward consequences properties weighted procedures completeness prove proposition proof proposition first consider simes without reshaping setting positive dependence examining definition bhuw procedure prior weights uniform penalty weights see weighted simes simesw minimum threshold passes bhw procedure simesw min bhw makes least one rejection level words equivalence simesw recalling number rejections bhu procedure target fdr level note global null run bhuw level discovery made bhuw unified multiple testing prior knowledge false discovery hence fdp equal one whenever least one discovery zero otherwise fdr previous result proposition proves fdr bhuw procedure hence global null simesw independence uniformity proposition also implies statement holds equality next turn setting arbitrary dependence use reshaped weighted simes rsimesw setting similarly see rsimesw min byw makes least one rejection level compare generalized procedure uses reshaping function use reshaped simes test applying proposition show rsimesw equal fdr corresponding procedure therefore proposition implies quantity demonstrate employ weighted simes control group fdr first independence groups clarity later general assumptions purpose following lemma prove useful lemma analogy superuniformity lemma present following lemma contains analogous bounds settings independent positively dependent base case group constructed simes pvalue setting arbitrarily dependent base case group constructed fisher long valid null lemma lemma let hgrp group let pag denote group pag let denote remaining nonincreasing function base pvalues independent simesw pag ramdas barber wainwright jordan nonincreasing function base pvalues positively dependent simesw pag base arbitrarily dependent constant reshaping function function pag valid group function property pag superuniform whenever null group example may take pag rsimesw pag reshaped weighted simes formed using reshaping function let set possibly overlapping null groups meaning agk represent corresponding simes nonincreasing function base positively dependent simes proof lemma relies lemma found appendix remark thatsstatement different statement applied null group indeed statement arguments simes procedure simes original base desired statement bootstrapped apply root entire tree null groups internal node stores simes calculated children group fdr control single partition tools developed far place simple construct procedure controls fdr group level given single partition hypotheses groups first form summary pggrp group run either procedure procedure resulting list pggrp according dependence assumptions first construct group group may choose use weighted simes collapse group single summary reducing overall problem standard multiple testing unified multiple testing prior knowledge problem avoid conflicting notation weights let denote weights individual hypotheses computing simes referring notation setup define compute simesw group simesw simesw pag pggrp simesw pag min smallest group choose use weighted rsimes computed reshaping function grp rsimesw pag min group subscript reshaping function indicates group free choose different functions though may little reason exercise freedom note definitions numerator uses instead depending normalize weights may sum weights groups sum weights within group need hence adjustment appropriate normalization argued earlier simesw pag bonafide superuniform group intersection hypothesis composed nulls whenever independent positively dependent arbitrary dependence reshaped rsimesw pag one guaranteed superuniform finally wish use simes free set pggrp pag function property valid group null hgrp pag superuniform includes reshaped simes special case pag rsimesw pag also encompasses fisher rosenthal group constructions principle could also computed raw data alter results proved long pggrp still superuniform null group hgrp ramdas barber wainwright jordan next state results fdr control group testing setting expected independence free use adaptivity arbitrary dependence obligated use reshaping independence null proportion estimate fixed naturally defined bgrp pggrp proposition fdr control recalling setup independence group procedure applied pggrp achieves fdrgrp positive dependence base bhuw procedure applied simes simes pag achieves fdrgrp arbitrary dependence group byuw procedure applied pggrp achieves fdrgrp provide explicit proof proposition since proofs fdrgrp control directly mimic corresponding proofs fdru control procedures applied base single alteration appropriate statements lemma invoked place corresponding statements lemma proof proposition follows special case proof theorem multilayer procedure introduced next section observed barber ramdas unweighted case view group fdr procedure described subsection interpolation weighted simes test weighted procedure seen considering two extremes taking one single group size recovers simes test global null across hypotheses taking instead groups size one recovers procedures testing individual hypotheses course algorithm controls fdr guarantee elementwise fdr hypotheses considered individually rather groups general multilayer method introduced next section gives type simultaneous guarantee various forms dependence fdr control internal consistency multiple partitions turn general case allow multiple partitions set hypotheses barber ramdas continuing support prior weights penalty weights adaptivity reshaping arbitrary dependence well generalizations overlapping groups unified multiple testing prior knowledge incomplete partitions presented framework specialize case single partition procedures discussed previous sections formal setup follows partitions interest mth partition many groups agm partitions groups within partition specified order order partitions described makes difference final output similarly ordering within partition groups also arbitrary section define null set mth partition containing groups entirely filled null hypotheses natural notion falsely discovered group group entirely composed true nulls algorithm incorrectly proclaimed least one discovery within group effectively rejected global null hypothesis within group group partition associated summary pgm assume derived base may generally calculated raw data another source without altering results interpretability arguably interest maintain internal consistency among rejected hypotheses groups single partition groups say set rejections individual level group level internally consistent reject groups containing least one rejected hypothesis reject hypothesis group also rejected require rejections internally consistent simultaneously partitions later extend notion two ways partitions leftover sets overlapping groups weak strong internal consistency notion internal consistency extension multilayer setting requirement coherence consonance defined classical paper gabriel explored depth fwer literature sonnemann finner romano even case one layer groups top individual hypotheses handling consistency controlling fdr trivial example sequential procedure ramdas barber wainwright jordan first rejecting groups target fdr level rejecting individual hypotheses within rejected groups level may neither succeed controlling fdr individual level due accounting selection bias succeed internally consistent groups may get selected first round none individuals within group may get selected second round method easily generalized partitions similarly parallel procedure choosing groups reject independently individuals reject may also fail internally consistent naive solution issue rejecting hypotheses whose groups rejected every may also fail control fdr levels see barber ramdas examples phenomenon algorithm guarantee fdr control hypothesis group levels holds simultaneously multiple arbitrary partitions hypotheses also incorporating weights adaptivity maintaining internal consistency rejected hypotheses introduce framework formally framework algorithmic framework presented section generalizes framework barber ramdas roughly speaking algorithm multivariate extension classical procedures based following sequence steps select hypotheses layer whose smaller initial threshold reject elementary hypothesis contained selected group every layer layer reject group hypothesis contains rejected elementary hypothesis estimate layer lower initial thresholds layer repeat steps desired level partitions use name extended algorithm noting six important ways paper adds flexibility earlier framework weights partition associated two sets many positive weights one group partition penalties priors used take differing group sizes account unified multiple testing prior knowledge reshaping within across layers arbitarily dependent use reshape thresholds layer possibly different layers simes used form group level may place weights base purpose calculating weighted simes additionally within group pgm simesw pam arbitrarily dependent would choose reshaping functions potentially different functions chosen group comput ing pgm rsimesw pam adaptivity partition whose known independent independence groups necessarily within groups incorporate storey adaptivity every layer constant define weighted null proportion estimator partition pgm use adaptivity one layer may improve power layers since groups discovered one layer allows individual hypotheses discovered hence groups discovered layers incomplete partitions allow partitions let partition leftover set represent elements partition belong group partition gives additional flexibility user may want assign hypotheses groups important note another set counted calculating grouplevel fdr layer meaning discoveries within leftover set alter fdr layer hypotheses leftover set internal consistency constraints imposed layer overlapping groups allow groups partition overlap elementary hypothesis need part single let denote set groups belongs two natural extensions internal consistency follows ramdas barber wainwright jordan weak internal consistency reject every partition following holds either least one rejected group containing leftover set meaning belong group layer strong internal consistency reject every partition following holds either every group contains rejected remark two notions internal consistency handled framework monotone notion internal consistency still works meaning decreasing increase number rejections levels arbitrary group new algorithm longer necessarily uses simes group layers words layer arbitrary formed combining elementwise way mentioned might constructed raw data depending constructed appropriately use adaptivity independence simes used reshaping dependence choose use fisher rosenthal combination methods needed reject subset hypotheses subset groups partition exact choice rejection sets defined next subsection given sbm define weighted fdr mth partition fdr rejections made internally consistent satisfy fdr simultaneously describe situation allowing layers two layers may useful imagine write pvalues rectangular grid within row interpretable meaning say hypotheses corresponding point space similarly within column different meaning say hypotheses corresponding point time addition controlling overall fdr may also want control spatial fdr using row partition group row temporal unified multiple testing prior knowledge fdr using column partition group column even fdr using rectangular partition group block refer figure visualization see barber ramdas discussion numerical simulations well neuroscience application fig consider hypotheses written grid four partitions mentary rows columns blocks top row underlying truth leftmost panel showing three panels showing groups partition hence identified bottom row example set discoveries leftmost panel showing rejections three panels showing groups correspondingly rejected correct rejections black false rejections false discovery proportions partition respectively deriving algorithm run algorithm need search rejection thresholds layer thresholds parametrized weighted discovery counts layer reader cautioned necessarily integer instead real number corresponding loosely total rejected penalty weight weights set equal indeed corresponds number groups layer rejected given first perform initial screening layer separately init min ramdas barber wainwright jordan choose use reshaping would instead define init min null proportion estimate defined choose use adaptivity simply set equations used either setting remark obliged one option group level set pgm simesw convenience recompute notation present section recall simesw min smallest choose use reshaped simes would instead compute simesw min reshaping function chosen group layer also recall groups layer independent may use adaptivity setting otherwise would set define total set rejections either sbweak init wish enforce weak internal consistency alternately sbstrong init strong internal consistency interpret expressions see weak consistency hypothesis rejected every layer either leftover set belongs least one groups rejected rewrite condition init sbweak either sbm unified multiple testing prior knowledge strong consistency hypothesis rejected every layer either leftover set belongs rejected belong groups rewrite init sbstrong either sbm finally redefine set groups layer rejected order enforce internal consistency init sbm sbm sbweak sbstrong desired examining either definitions readers may verify appropriate weak strong notion internal consistency satisfied course definitions depend initial choice vector since would like make large number discoveries would like use large possible time controlling false discovery rates sbm fdp define set feasible vectors input parameters suppress implicit dependence particular penalty weights equal one consistency condition defining feasible equivalent requiring numbers rejections layer vector elementwise analogous procedure generalized procedure find value least many rejections threshold condition viewed generalization setting condition described blanchard roquain also worthy note algorithm barber ramdas derived terms thresholds instead number rejections ramdas barber wainwright jordan corresponding feasibleity condition fdp type estimate fdp indeed fdp avoid simplicity associating comparing derivation one barber ramdas see viewpoint viewpoint equivalent lead algorithms however dealing arbitrary dependence proofs vastly simpler terms terms explaining switch choice notation paper procedures choose largest feasible thresholds given max choice defines algorithm algorithm rejects hybb potheses defined rejections layer given sbm defined next summarize theoretical guarantees theoretical guarantees following proposition states actually maximum corner set feasible vectors defined equaproposition let set feasible vectors tion let maximum feasible vector defined equation barber ramdas proved result setting original algorithm completeness prove general setting appendix vector feasible perspective selfb also feasible perspective consistency captured fdr control specifically next theorem proves find set guarantees simultaneous control fdru partitions theorem procedure finds definition satisfies following properties simultaneously base independent group given employing adaptivity defining pgm simesw pam guarantees fdru unified multiple testing prior knowledge base group given pgm fdru simesw pam arbitrarily dependent constructed arbitrarily assumption pgm superuniform null meaning valid using reshaping group guarantees fdru setting part additionally groups layer independent independent pam using reshaping adaptivity layer guarantees fdru remark difference parts two results guarantee use adaptivity set madapt layers use adaptivity set remaining layers fdr control maintained across layers long madapt independence statement independent condition fails madapt fdr control layers fact affected proof given appendix integrates various ingredients required handle weights null proportions groups detailed earlier sections proof includes new ideas independent interest specific handling overlapping groups dependent one application statement base independent overlapping groups group formed using fisher rosenthal combinations base recently katsevich sabatti prove case fdr controlled even without using reshaping albeit constant factor larger target level practice accurate side information group structures rejected hypotheses likely respect may significantly improve precision achieving lower fdr theoretical bound without affecting power much however inaccurate side information may significantly lower power since would additional misguided constraints meet aspects explored simulations barber ramdas special cases setting single partition recovers algorithms discussed paper finest partition groups containing one hypothesis reduces exactly generalized bhy procedures discussed section course include variants bhyuw ramdas barber wainwright jordan described section employ adaptivity recover method weighted variant section instantiate coarsest partitions single group containing hypotheses recover exactly generalized simes test weighted variants section instantiated single partition groups hypotheses recover exactly test controls discussed end section theorems propositions paper essentially deduced special cases theorem algorithm fdr control input possibly incomplete partitions possibly overlapping groups indices vector base group pgm group layers target fdr levels sets prior weights penalty weights thresholds adaptive null proportion estimation reshaping functions desired initialize set definition repeat update mth vector defining sbm equation using weak strong consistency desired let max end vectors unchanged one full cycle output adaptive vector efficient implementation although one employ bruteforce grid search find algorithm presented algorithm able find vector efficiently using style procedure strict generalization algorithm name barber ramdas code procedure publicly available https following proposition provides correctness guarantee algorithm proposition output algorithm maximum feasible corner defined equations result proved barber ramdas setting original algorithm take integer values unified multiple testing prior knowledge algorithm slightly subtle due presence penalty weights proof proposition general setting given appendix discussion paper provides simple proofs many existing multiple testing also generalizing conditions global null test simes weighted simes test hochberg liberman fdr controlling procedures benjamini hochberg benjamini yekutieli weighted fdr controlling procedure benjamini hochberg weighting procedure genovese adaptivity procedure storey also described several new procedures including weighted versions adaptive weighted simes test showed unify concepts grouped setting finally algorithm unifies generalizes procedures building original framework introduced barber ramdas procedures analyzed generalized fully cover huge literature fdr controlling procedures example procedures paper procedures much work also done alternative styles procedures like methods example benjamini liu propose procedures control fdr independence later procedures benjamini liu romano shaikh provably control fdr arbitrary dependence gavrilov extending adaptive control independence adaptive procedures analyzed benjamini independence blanchard roquain dependence different methods incorporating weights procedures also studied different notion weighted simes proposed benjamini hochberg benjamini bogomolov also propose ways take single partition groups account yekutieli discusses hierarchical testing lemmas lemma grouped setting lemma used quickly prove fdr control many procedures may useful tool exploring potential extensions multiple testing procedures broader settings acknowledgments thank wenyu chen helping implement new algorithm etienne roquain clarifying point procedures thank eugene katsevich aditya guntuboyina sujayam saha larry wasserman fanny yang relevant ramdas barber wainwright jordan sions authors also thankful audience members statistics departments stanford wharton davis louis workshop higher order asymptotics inference nips workshop adaptive data analysis whose questions partly shaped work work supported part office naval research grant number air force office scientific resesarch grant number nsf award alfred sloan fellowship appendix proof lemma statement lemma follows directly blanchard roquain independently reproved barber ramdas statement inequality equality follows special case since independence special case positive dependence distribution null change conditioning independent set pvalues statement proved also blanchard roquain prove statements statement prove first part lemma assumptions function satisfies property respect index uniformly distributed independent remaining since ignore possibility following calculations since satisfies loop condition seen separately considering happens numerator side zero one since determines immediately follows last step follows since range uniformly diso tributed independent therefore unified multiple testing prior knowledge concludes proof lemma independence uniformity statement let probability measure chosen definition reshaping function definition let drawn independently let qmprobability measure corresponding distribution therefore last step holds lemma appendix proof lemma bound follows immediately jensen inequality since plower split argument upper bound three cases case integer weights first suppose weights integers case binomial number weights equal simple calculation ramdas barber wainwright jordan shows binomial binomial case one weight suppose exactly one weights without loss generality take let binomial bernoulli note inequality holds since simplifying get inequality uses fact calculated case unified multiple testing prior knowledge case general case suppose least two weights let let min inequality follows simple calculation using assumption define new vector weights defining calculation proves marginalizing note least one fewer weight repeating process inductively see reduce case one weight case case proves lemma appendix proof group lemma proof lemma straightforward applying lemma precisely define augmented vector pag define function since pag assumed superuniform since hgrp null group means pag superuniform index null augmented vector applying lemma place index yields desired bound lemma simply special case lemma since independence special case positive dependence conditioning independent set change distribution pag lemma proof strategy reduce statement form lemma becomes applicable note simply take approach proof lemma define augmented vector simesw pag know vector positively whether prds entry simesw pag aim mind let number discoveries made bhw procedure run within group level using connection simes test ramdas barber wainwright jordan procedure may write since bhw procedure level ith rejected hence may conclude feg defined feg taking expectations sides applying lemma immediately proves lemma specifically know function also assumed therefore feg also given lemma proved proof lemma follows exactly argument except last equation replaced lemma invoked place lemma appendix proof proposition definition thus definition nondecreasing function immediately implies sbm sbm therefore layer second inequality holds observation definition set feasible vectors since holds proves feasible vector hence unified multiple testing prior knowledge appendix proof theorem able handle four cases theorem define function identity using reshaping theorem statements using reshaping theorem statements also let using adaptivity theorem statements let defined theorem statements adaptivity used fix partition since assumption assume without mention tion implies sbm null group must calculate fdp sbm init min first inequality follows definition feasible set init second follows since sbm sbm definition last init step uses definition sbm without reshaping theorem statements reshaping theorem statements multiplying numerator denominator term taking expectations sides deduce fdr min calculations place prove four statements theorem ramdas barber wainwright jordan theorem statement define function maps vector note nonincreasing function since nonincreasing function definition procedure nondecreasing function returning see min fdr note event pgm allows inequality returning condition group fdr equality holds function outside group last inequality holds lemma unified multiple testing prior knowledge finally observe independence different groups partition implies indicator variables phm independent bernoulli probabilities success thus consequence lemma prove analogous property plugging back bounds fdr finally obtain fdru theorem statement proof statement follows steps without need condition since use adaptivity define function gmm nonincreasing function since nonincreasing function returning proof statement calculate fdr pgm lemma know therefore fdr desired theorem statement turn proving method reshap ing define define constant ggm returning calculate fdr ramdas barber wainwright jordan pgm since pgm lemma know therefore assumed superuniform null group fdr theorem statement proof part combines calculations part adaptivity used part reshaping used defined define part note longer constant nonetheless proceeding part calculate fdr next condition outside group fdr last step holds since function finally apply lemma show conditional expected values course subtlety must condition note fixing function regarded function remaining unknowns pam still value constant pgm tgm pam indeed superuniform since due independence pam distribution changed therefore apply lemma place treated constant random vector pam pgm see fdr therefore unified multiple testing prior knowledge finally need bound proof part see indicator variables phm independent since phm thm pam sets pam assumed independent furthermore since pam assumed valid superunih form means variable phm bernoulli chance success therefore bound calculated proof part holds well fdr concludes proof four parts theorem appendix proof proposition first introduce notation let vector sth pass algorithm prove induction initialization suppose show consider layer pass algorithm stage vectors update applying induction also inner loop assuming prove definition algorithm max since since sbm nondecreasing function definition side expression turn definition feasible vector therefore feasible set must induction true desired ramdas barber wainwright jordan suppose algorithm stabilizes full passes completing mth layer last pass algo rithm vectors however since gorithm stops sth pass means using observation definition see means definition proposition induction also know iteration completes proof appendix properties dotfractions section verify dotfractions satisfy many properties ordinary fractions thus notation safely used throughout proofs main results following property shown hold assuming dotfractions appearing equation inequality well defined hence throughout assume various properties used dotfractions expression may use properties never side note observe paper always use comparing two fractions prove first bound reduces must since otherwise would undefined prove second bound reduces must since otherwise would undefined case comparing scalar prove must undefined trivially true sides equal zero unified multiple testing prior knowledge adding numerators prove reduces must otherwise dotfractions undefined sides equal zero multiplying fractions prove reduces either otherwise would undefined sides equal zero cancelling nonzero factors see simply apply noting assumption multiplying scalar see reduces must undefined sides zero properties carry fractions dotfractions settings familiar manipulations fractions may longer correct example relatedly add fractions usual way may equal fails implicitly would assuming order make two denominators may fail references rina foygel barber aaditya ramdas multilayer false discovery rate control grouped hypotheses journal royal statistical society series statistical methodology yoav benjamini marina bogomolov selective inference multiple families hypotheses journal royal statistical society series statistical methodology ramdas barber wainwright jordan yoav benjamini yosef hochberg controlling false discovery rate practical powerful approach multiple testing journal royal statistical society series statistical methodology yoav benjamini yosef hochberg multiple hypotheses testing weights scandinavian journal statistics yoav benjamini wei liu multiple test procedure controls false discovery rate tel aviv tel aviv university yoav benjamini wei liu multiple hypotheses testing procedure controls false discovery rate independence journal statistical planning inference yoav benjamini daniel yekutieli control false discovery rate multiple testing dependency annals statistics yoav benjamini abba krieger daniel yekutieli adaptive linear procedures control false discovery rate biometrika gilles blanchard etienne roquain two simple sufficient conditions fdr control electronic journal statistics gilles blanchard roquain adaptive false discovery rate control independence dependence journal machine learning research dec gunnar eklund massignifikansproblemet unpublished seminar papers uppsala university institute statistics volume gunnar eklund paul seeger massignifikansanalys statistisk tidskrift livio finos luigi salmaso methods using weights journal statistical planning inference ruben gabriel simultaneous test theory multiple comparisons annals mathematical statistics pages yulia gavrilov yoav benjamini sanat sarkar adaptive procedure proven fdr control independence annals statistics pages christopher genovese kathryn roeder larry wasserman false discovery control weighting biometrika nicholas heard patrick choosing methods combining arxiv preprint philipp heesen arnold janssen dynamic adaptive multiple tests finite sample fdr control journal statistical planning inference yosef hochberg uri liberman extended simes test statistics probability letters hommel tests overall hypothesis arbitrary dependence structures biometrische zeitschrift james hongyu zhao harrison zhou false discovery rate control groups journal american statistical association samuel karlin yosef rinott classes orderings measures related correlation inequalities multivariate totally positive distributions journal multivariate analysis eugene katsevich chiara sabatti multilayer knockoff filter controlled variable selection multiple resolutions arxiv preprint erich leo lehmann concepts dependence annals mathematical statistics pages morgenstern berechnung des maximalen signifikanzniveaus des testes unified multiple testing prior knowledge wennk untern gegebenen tests zur ablehnung metrika joseph romano azeem shaikh stepdown control false discovery proportion optimality pages institute mathematical statistics joseph romano azeem shaikh michael wolf consonance closure method multiple testing international journal biostatistics das maximale signifikanzniveau des tests lehneh wennk untern gegebenen tests zur ablehnung metrika ludger random variables maximum sums advances applied probability pages sanat sarkar methods controlling false discovery rate indian journal statistics series pages sanat sarkar stepup procedures controlling fdr journal statistical planning inference paul seeger note method analysis significances masse technometrics john simes improved bonferroni procedure multiple tests significance biometrika eckart sonnemann allgemeine multipler testprobleme bern institut mathematische statistik und versicherungslehre eckart sonnemann allgemeine multipler testprobleme edv medizin und biologie english version original article minor corrections finner general solutions multiple testing problems biom eckart sonnemann helmut finner multiple testprobleme multiple hypotheses testing pages springer john storey direct approach false discovery rates journal royal statistical society series statistical methodology john storey jonathan taylor david siegmund strong control conservative point estimation simultaneous conservative consistency false discovery rates unified approach journal royal statistical society series statistical methodology samuel stouffer edward suchman leland devinney shirley star robin williams american soldier adjustment army life studies social psychology world war vol john wilder tukey problem multiple comparisons introduction parts princeton university vladimir vovk combining via averaging arxiv preprint daniel yekutieli hierarchical false discovery methodology journal american statistical association aaditya ramdas departments statistics eecs university california berkeley aramdas rina foygel barber department statistics university chicago rina martin wainwright departments statistics eecs university california berkeley wainwrig michael jordan departments statistics eecs university california berkeley jordan
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cooperative robust estimation local performance guarantees mar zamani paper considers problem cooperative estimation linear uncertain plant observed network communicating sensors take novel approach treating filtering problem view point local sensors network interconnections accounted via uncertain signals modelling estimation performance nodes information communicated nodes treated true plant information subject perturbations node endowed certain believes perturbations filter design proposed distributed filter achieves suboptimal consensus performance furthermore local performance estimator also assessed given additional constraints performance nodes conditions shown useful tuning desired estimation performance sensor network ntroduction research cooperative filtering estimation networked systems gained much momentum past decade aiming developing efficient estimation algorithms large assemblies networked sensors mentioned references reflect common trend literature main objective accomplish globally optimal suboptimal estimation performance network usually performance individual sensors considered problems observation motivates question relationship estimation performance individual filters within distributed estimation network performance overall network paper considers problem within specific framework distributed consensus estimation approach also targets global convergence problem however decoupling global solution rather define local objective function terms uncertain signals capturing performance nodes leads decoupling distributed filter implicitly maintaining meaningful connection network way local objective function abstracts dependence nodes eliminating need consider exact models raw measurements nevertheless convergence local filter dependent rest network provide conditions render convergence network filters furthermore asserting conditions individual performances nodes guaranteed performance work supported australian research council discovery projects funding scheme project work done first author school engineering unsw canberra canberra australia zamani dst group australia ugrinovskii school engineering unsw canberra canberra australia email ugrinovskii individual filters established conditions express neighbours maintain certain level accuracy local filter also guarantees nominated performance establish relationship analyze distributed filter network consisting estimators solving auxiliary optimal filtering problem every node sense approach bears resemblance decentralized control controller constructed regulate local subsystem mentioned auxiliary filtering problem originates shown yield interconnected filters exchange information network nodes although parameters filter could computed online decentralized fashion new element paper compared neighboring information interpreted node node considered agnostic amount energy error true state plant neighbours estimates state contrast consider model node perceives relationship energy neighbours error accuracy filter give detailed discussion idea later paper note technically model adds constraint energy error inputs arising auxiliary minimum energy problems constraint form integral quadratic constraint previously used robust decentralized control problems filtering problems see however unlike problems parameters constraints used play role tunable parameters adjusted according desired local global performance also serve indicators sensitivity individual filters neighbours performance main result paper sufficient conditions network parameters ensure performance network consisting proposed minimum energy filters mentioned global disturbance attenuation guaranteed conditions also certain local properties node filters established show conditions admit form convex semidefinite program enables constructing filter network yielding suboptimal disturbance attenuation notation euclidean space vectors euclidean norm positive semidefinite matrix kakx denotes lebesgue space signals diag denotes block diagonal matrix diagonal blocks kronecker product matrices smallest eigenvalue symmetric matrix roblem ormulation reliminaries plant distributed estimator consider linear system respectively state unknown modeling disturbance input latter assumed integrable matrices known however initial state unknown considered part uncertainty system main objective paper determine conditions plant state estimated network filters using plant measurement indicates measurement taken node network measurement rpi imperfect subject measurement disturbance taking values rmi also belongs space assumption coefficients measured output matrices matching dimensions rpi rpi addition direct measurements plant node receives information nodes network form cij wij fij estimate state neighbouring node signal values rmij represents communication errors uncertainty communication channel assume gij fij network graph describing communications filtering nodes assumed directed node edge sets denoted respectively neighborhood node set nodes send information node denoted cardinality denoted laplace matrix network graph denoted following many papers distributed estimation consider class interconnected filters processing direct measurements neighbours information cij means observer form kij cij wij estimation problem paper determine coefficients kij ensure convergence network trajectories plant also guarantee acceptable attenuation detrimental effects disturbances estimation error formally properties formulated follows given positive semidefinite matrix rnn collection positive semidefinite matrices constants wish determine collection filters form guarantee following properties absence disturbances estimation error filter converges zero asymptotically presence disturbances network filters attains type disturbance attenuation property kvi norm provided neighbours node contribute sufficient effort quantitatively defined later assist also guaranteed node kei kvi constant determined later properties formalize desired attributes distributed filter want achieve particular property specifies desired global disturbance attenuation performance across sensor network using network decoupled filter equations note decoupled equations governing gains filters provided later furthermore property articulates desired local disturbance attenuation provided sufficient contribution neighbours sufficient contribution condition quantitatively defined later paper remark properties jointly generalize property consensus introduced also see example let laplacian matrix graph obtained network graph reversing edges choice results left hand side equal tor thepweighted disagreement cost nodes generally letting diag reduces property strong robust synchronization introduced addition property describes attenuation properties individual node filters including property analysis constitutes main difference problem posed previous work area distributed estimation representation neighboring information make performance analysis individual filters possible let introduce mismatch disturbancefree information contained signal cij corresponding true version information wij rpij psfrag replacements signals information received sensor represented cij wij fij kij fij wki system signals play role exogenous disturbances signal wki represents output used agent neighbour interpretation allowed construct minimum energy filters form property initial condition arbitrary integrable disturbances arbitrary kvi kei equation regarded additional measurement plant affected disturbances treating signals purpose filter derivation disturbances additional effect decoupling node neighbours indeed consider error dynamics filter node kij wij fij positive constant matrices whose existence determined certain lmi conditions also zij zij given matrices associated confidence node performance node condition provides type bound energy filter estimation errors node expressed terms energy disturbances affecting node similar property important difference former condition includes energy signals depend neighbours accuracy also according level disturbance attenuation stated nodes goal revisit design filters obtain possibly sharper property least local filters filters provide means assessing local performance sensitivity neighbours errors owing relation viewpoint node error dynamics network seen interconnection two systems representing respectively error dynamics errors dynamics rest system see figure motivated propose following condition formally capture sensitivity node accuracy neighbours filters every exist positive definite symmetric matrices constants dij fki fig subsystems representation error dynamics system kei dij generalized form property condition reflects neighbours accuracy influences local disturbance attenuation property every node therefore follows use condition establish quantitative characteristic neighbours effort mentioned demonstrate role vividly take example scalar suppose holds provided satisfied small since according energy bounded suggests node tolerate relatively small mismatch inputs able guarantee however small energy accomplished corresponding neighbour suggests eigenvalues may indicative sensitivity local filters fidelity neighbours estimates section show matrices computed jointly attenuation levels provides means performance tuning local filters similarly constants dij describe bound estimation error energy node prepared tolerate neighbour response hypothetically estimating perfectly known plant utmost precision indeed hypothetical case condition reduces bound energy mismatch disturbance signal neighbour distributed estimation problem position present formal definition distributed estimation problem described section problem determine collection filters form matrices rpij constants following conditions hold given positive semidefinite matrix rnn network filters achieves properties found following implication holds found signals satisfy filter guarantees satisfaction condition satisfied stress global performance properties proposed distributed filter proved without using condition iqc used guarantee certain local performance node subject acceptable performance neighbours latter development analogous iqcs used derivation decentralized robust controllers quantify uncertainty arising system interconnections see however different decentralized controllers references aim maintain coupling filters ensure cooperation iii istributed minimum energy filtering local performance guarantees section main results presented explained section goal obtain converging sense distributed filter provides global estimation performance described item problem also characterize quantitatively connection local properties filters sensitivity estimation accuracy neighbours solve problem first introduce auxiliary robust minimum energy filtering problem involving modified version standard minimum energy cost cost tional depending signals affecting measurements cij available node follows kyi kxt kcij wij kri compared standard minimum energy cost functional includes additional weighted penalty tracking error node see last term shown inclusion term enforces guaranteed performance filter minimum energy estimate sought weight matrix term regarded parameters filter process selecting matrices optimize proposed however different cost include direct quadratic penalty instead derivation local filters impose constraint mismatch signals modifications auxiliary robust minimumenergy filtering problem consists determining set unknowns compatible measurements communications cij minimizing energy cost subject constraint inf inf inf denotes class vector signals obtained stacking satisfying originated minimum energy filtering least square fitting problem lead likely trajectory compatible data node cij subscripts highlight trajectory consistent data collected interval node definition end point trajectory estimate state given measurement data cij solve constrained optimization problem apply method fact since cost depends requires solve family minimum energy filtering problems replaced arbitrary signal take fixed point mapping generated family minimum energy filtering problems due lack space omit details proceed assuming fixed point let vector otherwise define kxt kyi kcir wir fixed consider unconstrained optimization problem inf optimization problem standard optimal tracking problem fixed terminal condition unique solution condition establish relationship problem constrained inner optimization problem let holds also convenience define vector whose jth component dij otherwise lemma every corresponding set nonempty value inner optimization problem finite inf sup lemma lower bound value problem follows inf inf inf sup esign consider following optimization problem inf inf inf solution problem involves differential riccati equation wij gij lemma given fixed suppose differential riccati equation symmetric nonsingular solution interval following filter computes recursively minimizer optimization problem interval wij cij wij value optimization problem finite given kcij kyi let holds dre bounded positive definite solution lemma following theorem summarizes discussion theorem given constants matrices suppose set empty condition holds filter computes recursively process satisfies condition compared distributed minimum energy filter obtained family suboptimal minimum energy filters node parametrized able apply theorem necessary method computing least one vector every node next section present algorithm accomplishes task addition algorithm obtains matrices constants consistent found thus providing complete solution problem robust distributed estimator algorithm compute solution problem utilizes collection linear matrix inequalities lmis including condition following matrix inequalities wij symmetric matrix composed follows diagonal blocks defined gij wij pii also blocks wij wji wji wij wji lmis represent linear constraint variables since represents disturbance attenuation level distributed filter suitable set filter parameters interest minimizes variable numerically achieved solving convex optimization problem sup subject let value supremum theorem let pair stabilizable given positive semidefinite weighting matrix rnn suppose feasible collection matrices scalars satisfy constraints convex optimization problem riccati equation positive definite bounded solution furthermore corresponding filtering algorithm verifies claims problem table olutions problem simulation simulation simulation indicates robustness estimators respect accuracy neighbours improved moderately increasing node minj minj conclusions paper proposed distributed filtering algorithm utilizing filtering approach design constituent filters algorithm employs decoupled computation individual filter coefficients achieved considering estimation error neighbouring agents additional exogenous disturbances weighted according nodes confidence neighbours estimates conditions obtained proposed filter provides guaranteed internal stability desired disturbance attenuation network error dynamics addition local filter guarantees certain disturbance attenuation assisted neighbours also provided simulation example confirms convergence proposed filter case system undetectable pairs nodes tuning filter discussed reduce dependence local filters neighbours accurate estimates theorem shows solving sdp problem allows determine suboptimal well local disturbance attenuation levels characterize local performance node filters see well matrices condition consistent performance sensitivity performance obtained local filters neighbours accuracy assessed using eigenvalues explained section process illustrated example presented next llustrative xample eferences section simulated network five sensor nodes considered estimate plant plant state matrix input matrix matrix corresponds one regimes controlled chua electronic circuit considered network consists five nodes connectivity described set directed edges matrices taken note none pairs observable detectable also following communication matrices taken wij also let fij system two distributed filter designs compared filters designed achieve suboptimal consensus performance selected first optimization problem solved parameters next additional constraint imposed computed levels local attenuation minimum eigenvalues computed matrices property guaranteed theorem shown table one see first case filters nodes much larger constants substantially smaller values eigenvalues matrices together features indicate filters significantly sensitive accuracy neighbours unexpected given pairs detectable second distributed kalman filter embedded consensus filters proc ieee cdc ecc subbotin smith design distributed decentralized estimators formations fixed stochastic communication topologies automatica nelson freeman decentralized filtering system proc american contr pages louis ugrinovskii distributed robust filtering consensus estimates automatica ugrinovskii langbort distributed estimation uncertain systems via dissipativity theory iet control theory mortensen recursive nonlinear filtering opt theory hijab minimum energy estimation phd dissertation berkeley zamani ugrinovskii distributed filtering proc ieee cdc los angeles ugrinovskii orsi decentralized robust control uncertain markov jump parameter systems via output feedback automatica moheimani savkin petersen robust filtering prediction smoothing observability uncertain systems ieee transactions circuits systems part fundamental theory applications clark holton first look graph theory world scientific singapore ugrinovskii synchronization parameter varying systems via relative consensus application synchronization uncertain bilinear systems automatica ugrinovskii distributed robust estimation randomly switching networks using consensus automatica petersen ugrinovskii savkin robust control design using methods william mceneaney filtering nonlinear systems syst contr athans falb optimal control introduction theory applications
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published conference paper iclr fast onvolutional ets ith fbfft gpu erformance valuation apr nicolas vasilache jeff johnson michael mathieu soumith chintala serkan piantino yann lecun facebook research broadway new york usa ntv jhj myrhev soumith spiantino yann bstract examine performance profile convolutional neural network cnn training current generation nvidia graphics processing units gpus introduce two new fast fourier transform convolution implementations one based nvidia cufft library another based facebook authored fft implementation fbfft provides significant speedups cufft whole cnns convolution implementations available open source faster nvidia cudnn implementation many common convolutional layers synthetic kernel configuration discuss different performance regimes convolutions comparing areas straightforward time domain convolutions outperform fourier frequency domain convolutions details algorithmic applications nvidia gpu hardware specifics implementation fbfft also provided ntroduction deep convolutional neural networks cnns emerged one promising techniques tackle large scale learning problems whether image face recognition audio speech processing natural language understanding convolutional layer within networks provides useful properties translation equivariance activations limiting factor use convolutional nets large data sets recently computational expense krizhevsky demonstrated training large cnns millions weights massive data sets tractable graphics processing units gpus properly put use since renewed interest cnns insufflated fresh breath various frameworks implementations including torch collobert theano bergstra krizhevsky caffe jia many frameworks based around codes nvidia gpus using cuda garland discuss contributions convolution performance gpus namely using fast fourier transform fft implementations within torch framework summarize theory behind training convolutional layers time frequency domain section detail implementations first based nvidia cufft cublas libraries section evaluate relative performance nvidia cudnn library chetlur different configurations section significantly outperform cudnn time domain convolution implementations wide range problem sizes second implementation motivated limitations using black box library cufft application domain describe reaction implemented opensource implementation batched fft batched fft called facebook fft fbfft achieves speedup cufft sizes interest application domain implementation achieves gpu efficiency ratios certain cases describe ongoing effort improve performance solution based algorithmic tiling section conclude implementation released part fbcuda fbcunn opensource libraries http published conference paper iclr onvolution discrete convolution used cnns quickly summarize implementation formulation mirroring mathieu forward propagation fprop inputs set input feature planes different filter kernel weights producing output feature planes input output feature part minibatch feature planes reduced summed pointwise bprop gradient loss respect outputs convolved kernels reduction finally kernel weights updated using gradient loss respect weights accgrad reduction purposes paper use set symbols interchangeably refer size input plane matrix size filter kernel matrix size output planes size implement convolution per matlab terminology input zero padding input mirror padding around margins input optionally popular convolution implementation unroll data computation form large matrix multiplication chellapilla strategy followed many implementors since matrix multiplication linear algebra primitive available virtually platform possible provide instances direct calculation faster matrix unrolling large krizhevsky challenging provide implementation faster small subset possible convolution problems introducing strides form convolution performing convolution every offset popular way reduce computational cost expense precision memory accesses required similar fewer reuse opportunities hand convolution theorem convolution two discrete signals performed lower asymptotic complexity performing multiplication frequency domain applied forward pass becomes denotes complex conjugation pointwise product discrete fourier basis used largest two components convolved linearity dft allows one perform sum fourier domain desired applying fft yields log procedure lieu original similar transformations apply two passes call frequency domain convolution contrast time domain convolution via direct computation torch practice forward pass hence extended input size output size even bigger performance section published conference paper iclr strided convolutions via fft implemented efficiently obtain good performance brosch tam consider paper fft onvolution mplementation section discuss implementation strategies using nvidia cufft libraries efficiency fft onvolution etails described general formulation three types convolutions section borrow torch naming convention input weight output gradoutput gradinput gradweight stored singleprecision floating point tensors layout stored memory using socalled bdhw format explicit expression input image row data innermost varying dimension table describes operations fft computation forward pass using operators cgemm matrix multiplication similar implementations follow two passes prefix denotes gradients suffix denotes frequency domain tensors rest suffix denotes transposed tensors table implementation detail forward propagation input output eif eiff inf eiff inf eif cgemm eif outf outf exact tensor dimensions also given taking advantage hermitian symmetry property dft inputs store half complex entries remaining obtained complex conjugation results array sizes also perform interpolation serves multiple purposes first necessary handle boundary second required interpolate operands fourier finally padding impact fft algorithm used practice well floating point operation count operations section following conversion frequency domain perform transpositions prepare tensors cgemm matrix multiplication library calls transposition converts bdhw layout hwbd transposition currently implemented using cgeam routine also considering transposition routine cgemm library calls performed transposed tensors frequency domain casting operation cgemm call allows benefit heavily tuned cublas routine eventually transpose result back bdhw format perform inverse fft point resulting real tensor always case typically tensors published conference paper iclr clipped appropriate final size fprop bprop accgrad fft esign pace discuss implementation aspects explored multiple factors influence computational efficiency ffts transform size prime factor decomposition whether batched iterated single transforms applied deep learning domain commonplace deal small sizes undesirable properties efficiency drop order cufft implements ffts ubiquitous algorithm cooley tukey takes advantage trigonometric equalities recursively decompose reuse computations discussed supplement decomposition built specialized kernels fixed sizes correspond prime factor decomposition cufft implements specialized building blocks radix sizes sizes use efficient kernels exploiting conjugate symmetry property admit prime factor decomposition using radices expensive bluestein algorithm used bluestein results used time domain fact image kernel perform fft larger size may handled efficiently exploiting efficient larger sizes balanced extra cost introduced subsequent transposition matrix multiplication steps table last case one best tradeoff easily guessed cufft also batched mode optimizations multiple ffts size performed blas esign pace cublas library also comes different implementations batched single operation modes choice implementation options larger batches small matrices cublascgemmbatched library call smaller batches larger matrices multiple cublascgemm calls host intermediate batch matrix sizes devices compute capability higher support dynamic parallelism allows cuda kernels launch kernels beneficial many launches small matrices note discussion applies multiplications transposition matrix size either number matrices vendor libraries usually optimized throughput latency expect efficient larger sizes along critical dimensions image size batch case multiple kernel case due build system limitations able experiment dynamic parallelism strategy leave future work system level use cuda streams buffering cuda resources intermediate buffers remove synchronization points across convolutions mindful memory consumption address keep one single buffered copy type tensor involved behavior tailored bulk synchronous execution layers gpu adapted multiple asynchronous convolutions gpu buffers automatically expanded required reused much possible autotuning combine implementation simple autotuning strategy devise strategy selection mechanism runs problem size caches fastest strategy dozen later reuse autotuning strategy explores different possible fourier basis sizes decomposed powers cufft efficient implementation words fft dimension size explore sizes input size power search space reduced single point addition fourier basis sizes weigh various cublas calls asynchronous modes http published conference paper iclr fft onvolution erformance erformance versus dnn configurations compare cufft convolution results nvidia cudnn library chetlur contains one fastest general purpose convolution methods gpu using matrix unrolling decent performance many problem sizes thanks heavy autotuning cublas codes different problems strong baseline reason image cnns date part used square input images filters though rectangular filters valid problems notably text cnns collobert thus restrict problem domain much space used practice areas perhaps large small due current engineering concerns evaluate cudnn convolution table table configuration elements evaluated dimension sizes evaluated minibatch size input filters output filters kernel output figures performance summaries cufft convolution versus cudnn nvidia tesla averaged across three passes problem size corresponds minibatch size multiplied number input output planes one pass reduction dimension many possible combinations may map problem size cudnn performance varies greater degree cufft across passes due asymmetry convolution sizes pass fact larger convolution kernel seen gradient accumulation essentially free fourier domain averaging three passes together provides proxy overall performance corresponds output deeper layers image cnns output size decrease increase depth corresponds moving upper right lower left graph black areas chart due failed cufft runs due memory pressure undetermined potential cufft issues fft convolutions make large kernel sizes inexpensive make performance three passes roughly equal table hand penalizes smaller kernels compared cudnn kernels figure cufft performance poor compared cudnn overhead multiple kernel launches streaming memory multiple times input size often outweigh algorithmic advantage fft however largest problem sizes convolution via fft still advantageous top speed faster cudnn kernels figure show increasing dominance fft strategy top speed faster tendency confirmed larger kernel sizes maximum speedup cudnn parameterized output rather input size implied valid choice published conference paper iclr output size output size problem size problem size figure kernel output size figure kernel problem size output size problem size figure kernel output size figure kernel problem size problem size speedup output size figure kernel figure kernel published conference paper iclr cnn erformance table show performance real cnns alexnet krizhevsky overfeat fast sermanet comparing cudnn kernels torch first layer uses cudnn cufft runs strided layers use cufft timings include convolutional layers network table alexnet overfeat fast performance network kernel fprop bprop accgrad total alexnet cufft cudnn overfeat fast cufft cudnn table shows performance cudnn cufft convolution implementation representative layer sizes assuming data present gpu speedups range cudnn unsurprisingly larger smaller contribute reduced efficiency fft surprisingly experience noticeable speedups small kernels long input tensor remains small size optimal fft sizes autotuning finds reported columns note padding found autotuner column trillion equivalent reductions per second floating point achieved implementation nvidia tesla cuda number represents throughput kernel needs achieve order match implementation computed metric compare relative efficiency across problem padding sizes cases time domain convolution would need exceed peak order match throughput fbfft mplementation section presumes familiarity gpu architecture refer supplement details designing libraries multiple objectives must balanced memory tradeoffs maximizing locality without sacrificing much parallelism good instruction mix register usage mapping strategy computation data memories compute elements key principle design set leaf kernels performance reduce larger problem combination kernels data loop tiling irigoin triolet recursive decompositions gunnels since vendors sustain high performance large class application domains exist parameter configurations carefully tuned approach significantly outperforms libraries shin common deep learning use convolutional layers consist many batched small convolutions tiny relative dsp hpc standards put regime fall outside highly tuned regime feature dimensions often smaller gpu warp sizes often fit exclusively registers rather shared memory smem sensitive latencies determined possible obtain better efficiency existing batched cufft mode cnns imitations fft cufft library black box explicitly embedded input output arrays consequence one may need allocate duplicate larger memory different fftw compatibility padding mode transforms published conference paper iclr table representative layer performance layer fprop bprop accgrad fprop bprop accgrad fprop bprop accgrad fprop bprop accgrad fprop bprop accgrad params params params params params cudnn cufft speedup region copy data tensors padded tensors memory consumption spurious copies affect latency significantly instead devised implementation batched fft fft sizes reaches efficiency occupancy also implemented ifft kernel based fft kernel implementation use clipping conditionally load value reading within bounds constant otherwise approach used automatic code generation tools halide relies aggressive properties cuda compiler allows efficient control flow rather using explicit loop prologues epilogues mechanism require additional memory allocation particularly desirable latency sensitive mode additionally since cufft cublas closed source impossible take advantage algorithmic simplifications may available instance forward pass computation shown table result first cufft call form fbfft return form two innermost data dimensions transposed allows remove full data transposition fft kernels another optimization yet explore eliminating bit reversal portions fft ifft done performing fft decimation frequency dif ifft decimation time dit discussed supplement warp level fft fft size batched fft power two sizes view single warp small distributed system lockstep collective communication capabilities program fashion valiant implement dif enforce following invariants steps warp thread originally loads one real element input vector locally computes one complex twiddle factor root unity step warp threads exchange data another thread warp parallel produce new value published conference paper iclr warp threads exchange twiddle factors another thread warp parallel produce new value two exchanges written one instruction steps fft computed stored distributed bit reversed manner within register across warp sizes bit reversal implemented single warp shuffle either load twiddle factors device memory compute sincosf function subsequently swap within registers greatly reduces reliance either memory bandwidth special functional unit expense additional registers decision explicitly loading twiddle factors device memory computing tradeoff arithmetic intensity memory bandwidth sizes arithmetic pipeline bottleneck loading twiddle factors memory two special sizes results performance increase respectively discussion applies fft independent fft within larger fft fourier transform separable implemented sets multiple fft transpositions sets fft first set comprises ffts second set comprises ffts hermitian symmetry following standard techniques lyons pack real ffts single complex fft extra term quantity makes computation bring performance lowering occupancy chose dimension kernels size introduce additional control flow handle border case results additional performance implement transposition smem across warps following ruetsch micikevicius data already resident registers main concerns limiting smem usage keep occupancy high limiting using vector instructions avoid saturating unit lsu fft fft size size building block extend strategy larger sizes use single warp approach compute full fft main difference computation distributed across multiple registers across threads warp fourier coefficients twiddle factors registers per thread perform full fft per warp performance cufft wins happens register usage limits occupancy much outperform cufft hard register limit similarly fft still well within application domain following modifications handle multiple registers per thread hermitian symmetry allows perform half computation tradeoff adding divergence performing less work benefits reduced computations dominate divergence losses take advantage trigonometric symmetries twiddle factor distribution compute fraction roots unity needed fft distributed register register copies twiddle factor across warp across registers requires different implementation managed implement fully within registers bit reversal occurs across registers across warps bits represent register bits represent warp without sophisticated implementation results indirect addressing registers costly implement simple bit reversal smem occupancy bottleneck fft fft case intermediate transpose becomes significantly expensive experimented various strategies keep occupancy high including partial transpositions within warp use minimal amounts smem iscussion report relative performance implementation fbfft compared cufft various batch input sizes interest number batches consider depends dimension published conference paper iclr cnn layers well parallelization strategy may involved typical sizes interest fbfft faster tried million batches larger sizes gains stabilize around efficiency goes memory used speedup various sizes batches fbfft speedup number batches fbifft speedup speedup various sizes batches number batches figure fft ifft cufft speedup various sizes batches number batches fbifft speedup fbfft speedup speedup various sizes batches number batches figure fft ifft cufft figure shows performance case numbers exercise implicit zerocopy padding expect additional gains incorporate fft convolution implementation outperforms cufft cases interest dramatically smaller batch sizes small batch sizes also correspond latency sensitive regime figures cufft based implementation performs quite worse cudnn achieve efficiency occupancy size batch size reported nvvp figure shows performance case relative performance gains sizes modest case even losing cufft size small batch sizes magnitude relative gains various batch sizes drops faster case looking performance fft obtain speedup cufft batches ratio obtained batches coupled tiling strategy section emphasize sizes interest actually depend input batch sizes vary whole spectrum interfaced fbfft convolution module ran experiments kernels different convolution passes inputs sizes problem size used swapping fft implementation observed overall mean speedup standard deviation geometric mean minimum speedup despite sometimes performing computations unexpected two computations perform number flops accounting hermitian symmetry plus fact efficiency cufft increases fbfft remains high almost constant published conference paper iclr fbfft interpolate power experiments exercise padding lower memory footprints fbfft compared cufft yet reflect additional optimizations tiling bit twiddling elision urrent imitations uture work current implementation fbfft heavily relies shuffle instructions spite good efficiency utilize available memory bandwidth due load store instructions kernel competing shuffle instructions unit lsu consequence first bottleneck number instructions issued lsu instance kepler capability throughput floating point operations per cycle throughput shuffles future investigate release faster implementations become available temporary memory overhead requirements common issue performing convolutions fourier domain first implementation introduced following memory buffers support implementation input output weight tensors store buffer frequency array buffer complex transpose buffers store fourier representation generally limited weight tensor independent size global memory pressure introduce reuse buffers layer pass opportunity reuse fft results hidden layer reducing cost forward ffts asynchronously precompute ffts weight tensors gradients better fill gpu utilization pipeline using cufft additionally pad input weight output tensors explicitly best performing common fft size using cufft additional temporary memory reserved cufftplan fbfft padding implicit temporary memory buffer needed reach size hand fbfft supports square convolutions whose size power consequence much padding could occur adversely affect performance memory consumption tiling strategy describe next good way circumvent problem additionally recently developed transposed batched cgemm permits removal complex transposed buffer problem tool like maxas lavin could valuable fbfft provides gains cufft sizes tiling strategy input used exploit advantage kernel significantly smaller input decompose large convolution several smaller ones simplicity consider convolution single input plane trivially extended let input size kernel size write vector formed contiguous elements let definition convolution convolution input size computed convolutions inputs size cost convolution goes log log log formula see optimal order get complexity log strategy allows speed forward backward propagation tiling also used reduce memory cost temporary storage running tiles parallel tiles run parallel need scratch space potential expense parallelism efficiency gradient accumulation reuse strategy since involves larger convolution input size kernel size however similar formula published conference paper iclr optimizations planned well since much data already available registers shared memory implementing transpose via recursive decomposition pointwise multiplications fourier domain especially tiling rather small matrix multiplication routines integrated rest convolution kernel code might win cublas prevent need multiple cuda kernel launches associated overhead finally mentioned earlier bit reversal portions eliminated fft using dif ifft using dit onclusion summarize achieve significant gains cnns using ffts cufft convolution implementation achieving speedups cudnn common sizes reaction cufft cublas limitations context specific application domain developed fft implementation fbfft suited deep learning problem sizes large batches small feature planes fbfft faster cufft transforms problems interest convolution faster cufft well mean sizes wish exploit given new efficient primitive size convolution continuing work bit twiddling transposition pointwise multiplication optimizations continuing work tiling make computational advantage size apply larger convolution problems allow reduced training time use ever larger deeper cnns acknowledgments would like thank julien demouth nvidia suggested improvements still possible virtue current implementation lsu rather memorybound eferences bergstra james breuleux olivier bastien lamblin pascal pascanu razvan desjardins guillaume turian joseph david bengio yoshua theano cpu gpu math expression compiler proceedings python scientific computing conference scipy june oral presentation bluestein leo linear filtering approach computation discrete fourier transform audio electroacoustics ieee transactions december issn brosch tom tam roger efficient training convolutional deep belief networks frequency domain application images neural computation doi url http burrus sidney fast fourier transforms url http chellapilla kumar puri sidd simard patrice high performance convolutional neural networks document processing lorette guy tenth international workshop frontiers published conference paper iclr handwriting recognition baule france october rennes suvisoft url https http chetlur sharan woolley cliff vandermersch philippe cohen jonathan tran john catanzaro bryan shelhamer evan cudnn efficient primitives deep learning corr url http collobert kavukcuoglu farabet environment machine learning biglearn nips workshop collobert ronan weston jason bottou karlen michael kavukcuoglu koray kuksa pavel natural language processing almost scratch mach learn november issn url http cooley james tukey john algorithm machine calculation complex fourier series mathematics computation garland michael grand scott nickolls john anderson joshua hardwick jim morton scott phillips everett zhang yao volkov vasily parallel computing experiences cuda ieee micro july issn doi url http giles mike course cuda programming nvidia gpus lecture url http gunnels john henry greg van geijn robert family matrix multiplication algorithms proceedings international conference computational iccs london isbn url http irigoin triolet supernode partitioning proceedings acm sigplansigact symposium principles programming languages popl new york usa acm isbn doi url http jia yangqing shelhamer evan donahue jeff karayev sergey long jonathan girshick ross guadarrama sergio darrell trevor caffe convolutional architecture fast feature embedding arxiv preprint krizhevsky alex url https krizhevsky alex sutskever ilya hinton geoffrey imagenet classification deep convolutional neural networks pereira burges bottou weinberger eds advances neural information processing systems curran associates url http lavin andrew maxdnn efficient convolution kernel deep learning maxwell gpus corr url http lyons richard understanding digital signal processing longman publishing boston usa edition isbn mathieu henaff mikael lecun yann fast training convolutional networks ffts corr url http jonathan barnes connelly adams andrew paris sylvain durand amarasinghe saman halide language compiler optimizing parallelism locality recomputation image processing pipelines acm sigplan conference programming language design implementation pldi seattle usa june doi url http published conference paper iclr ruetsch greg micikevicius paulius optimizing matrix transpose cuda technical report nvidia january sermanet pierre eigen david zhang xiang mathieu michael fergus rob lecun yann overfeat integrated recognition localization detection using convolutional networks international conference learning representations iclr cbls april url http shin jaewook hall mary chame jacqueline chen chun fischer paul hovland paul speeding autotuning specialization proceedings acm international conference supercomputing ics new york usa acm isbn valiant leslie bridging model parallel computation commun acm august issn doi url http volkov better performance lower occupancy gpu technology conference url http published conference paper iclr upplement fft onvolution erformance reakdown show breakdown cufft convolution performance steps indicated table timings add reported performance previous table report additional copies needed also enforce force extra synchronizations isolate contribution operation abstracting details fft ifft take significant amount compute resources address section table cufft convolution performance breakdown layer fprop bprop accgrad fprop bprop accgrad fprop bprop accgrad fprop bprop accgrad fprop bprop accgrad fft trans fft trans cgemm trans ifft particular case ffts take runtime due wasteful interpolation kernel tensor minimal size compute fft input array without interpolation loss cases tiling strategy developing see section result large additional performance gains fft ecimation ime requency fourier transform projects functions onto harmonic orthogonal basis discrete fourier transform vector vector wnkj wnj unity traditional algorithm recursively decomposes computation odd even part wnk wnk published conference paper iclr figure dit output ordered left dif input ordered right burrus decomposition called decimation time dit alternate decomposition performs decimation frequency dif wnkj wnkj power decimations recursively decompose perfectly balanced tree take advantage symmetry properties roots unity dataflow graph fft butterfly shape good way visualizing computations symmetry dit dif order operations applied whether input output order shuffled figure gpu rogramming variety references available describe cuda nvidia various gpu architectures garland discuss detail implementation fbfft much depends upon specifics kepler gpu architecture nvidia gpus execute code granularity warp defined set threads existing architectures thread assigned lane within warp threads execute simt single instruction multiple thread fashion meaning warp atomic unit execution holds single program counter thus execute single instruction time across threads collections warps brought together blocks ctas together share region fast shared memory resident chip blocks exchange data via much slower global memory resident gpu host cpu address space individual threads within warp free take divergent paths since single present branch execution serialized threads participating branch question disabled words threads take divergent code paths would obtain computational efficiency divergent code paths hard avoid nvidia instruction set means reduce cost giles one predicated instructions used small branches warp threads execute parts branch threads side effects block threads access register file registers per thread kepler registers allocated statically cuda compiler important performance factor writing cuda kernels data kept registers much possible avoid communications published conference paper iclr registers cuda addressable possible declare static array within registers operate elements limitation addressing performed using statically determined constants compiler translate accesses register number known compile time indirect addressing also supported results copies local region within global memory essentially constitutes register spilling even presence caches using local memory usually comes performance consequence design kernels using aggressive inlining template parameters unrolling directives make register accesses statically addressable kepler architecture introduced specialized shuffle instructions exchange data registers within warp synchronously avoids shared global memory interestingly shuffle instructions allow dynamic indexing array held registers long array distributed cyclic fashion across registers thread within warp float arr simulates linear array float realarr arr holds elements lane holds element arr holds elements lane holds element arr holds elements lane holds element example warp threads read value held realarr float val arr must statically known dynamic many warps run parallel switched gpu hardware cycle enough parallelism available measured occupancy gpu first approximation long latency operations hidden thanks fast context switching registers shared memory come finite quantities gpu compute multiprocessor limited resources partitioned compiler hardware amongst computations level cuda kernel increased usage registers shared memory reduce gpu occupancy limits ability hide long latency operations reduced occupancy necessarily result performance loss volkov often performance tradeoffs increasing decreasing threads per block shared memory per block registers per thread hard discover problem one many reasons designing implementation aims efficient problem difficult bleeding edge cases little local memory consumption helps performance instance restricting number registers per thread increase occupancy
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compositional abstraction networks control systems dissipativity approach dec majid murat abstract paper propose compositional scheme construction abstractions networks control systems using interconnection matrix joint properties subsystems abstractions proposed framework abstraction control system possibly lower dimension used substitution original system controller design process moreover provide procedure constructing abstractions class nonlinear control systems using bounds slope system nonlinearities illustrate proposed results network linear control systems constructing abstraction compositional way without requiring condition number gains subsystems use abstraction substitute synthesize controller enforcing certain linear temporal logic specification example particularly elucidates effectiveness compositional reasoning systems introduction modern applications power networks biological networks internet congestion control manufacturing systems networked systems inherently difficult analyze control rather tackling network whole approach severely restricts capability existing techniques deal many numbers subsystems one develop compositional schemes provide certifications main structural properties subsystems interconnections past years several results compositional abstractions control systems early results include compositional abstractions control systems useful verification rather synthesis results employ exact notions abstractions based simulation relations simulation maps constructive methodologies exist rather restricted classes control systems contrast exact notions compositional approximate abstractions introduced recently useful controller synthesis examples include compositional construction finite abstractions linear nonlinear control systems infinite abstractions nonlinear control systems class stochastic hybrid systems zrmng works abstraction finite infinite possibly lower dimension used substitution original system controller design process proposed results zrmng use type conditions facilitate compositional construction abstractions resulting type requirements intrinsically condition spectral radius interconnection matrix general depends size graph violated deteriorated number subsystems grows work propose novel compositional framework construction infinite abstractions networks control systems using dissipativity theory first adapt notion storage function dissipativity theory quantify joint properties control subsystems abstractions given network control subsystems storage functions propose conditions based interconnection matrix joint properties subsystems abstractions guaranteeing network abstractions quantitatively approximate behaviours network concrete subsystems proposed compositionality conditions enjoy specific interconnection structures provide compositional abstractions control systems without requiring zamani arcak condition number gains subsystems illustrate point example section furthermore provide geometric approach construction abstractions class nonlinear control systems corresponding storage functions using bounds slope system nonlinearities related work compositional construction infinite abstractions networks control systems also proposed type conditions used facilitate compositional construction abstractions use conditions type requirements inherently condition spectral radius interconnection matrix general depends size graph dissatisfied number subsystems grows hand necessarily case broader conditions fact compositionality requirements may condition number gains subsystems interconnection matrix enjoys properties section although results provide constructive procedures determine abstractions linear control systems propose techniques construction abstractions class nonlinear control systems using bounds slope systems nonlinearities results assume internal input output space dimensions component network equal corresponding ones abstraction case paper interconnection matrix permutation one one paper general interconnection matrix recent results establish stability stabilizability networks control systems compositionally using dissipativity properties components hand results provide construction abstractions networks control systems compositionally using abstractions components joint properties control systems notation sets nonnegative integer real numbers denoted respectively symbols footnoted subscripts restrict usual way denotes positive real numbers symbol denotes vector space real matrices rows columns symbols denote vector elements one zero vector identity zero matrices respectively closed open intervals denoted respectively symbols denote corresponding intervals given vectors rni use denote concatenated vector given vector kxk denotes euclidean norm note given iff given symmetric matrix denote maximum minimum eigenvalues denote diag block diagonal matrix diagonal matrix entries given function simply use denote given function essential supremum denoted ess sup continuous function said belong class strictly increasing said belong class continuous function said belong class fixed map belongs class respect fixed nonzero map decreasing respect control systems class control systems studied paper formalized following definition definition control system tuple state external input internal input external output internal output spaces respectively compositional abstraction networks control systems dissipativity approach subsets sets measurable functions time open intervals respectively continuous map satisfying following lipschitz assumption every compact set exists constant zkx external output map internal output map locally absolutely continuous curve state trajectory exist input trajectories satisfying almost call tuple trajectory consisting state trajectory output trajectories input trajectories satisfies also denote state reached time inputs initial condition state uniquely determined due assumptions also denote corresponding external internal output value respectively call external output trajectory internal output trajectory external input trajectory internal input trajectory mainly used interconnection purposes remain available interconnection see definition later detailed information remark control system internal inputs outputs definition control systems definition reduces tuple map becomes correspondingly equation describing evolution system trajectories reduces storage simulation functions first introduce notion storage functions adapted notion storage functions dissipativity theory notion storage functions characterizes correlation inputs outputs single control system proposed notion storage functions characterizes joint correlation inputs outputs two different control systems case two control systems internal inputs outputs notion storage functions recovers one incremental storage functions introduced definition let two control systems external output space dimension continuously differentiable function called storage function exist matrices appropriate dimensions symmetric matrix appropriate dimension conformal block partitions one zamani arcak one obtains use notation exists storage function control system possibly called abstraction several key differences notion storage function corresponding one simulation function definition definition requires internal signals live spaces respectively necessarily case moreover choice input satisfying depends whereas definition also depends internal input finally point definition positive definite matrix simulation function definition also storage function definition rest conformal block partitions zero recall notion simulation functions introduced modifications definition let two control systems continuously differentiable function called simulation function exist one use notation exists simulation function let point differences definition definition sake brevity simply assume every exists holds whereas definition authors use interface function feed input enforcing function definition assumed identity furthermore frame decay condition dissipative form definition decay condition given implication form note notions storage functions definition simulation functions definition comparable general former defined control systems internal inputs outputs latter defined control systems without internal inputs outputs one readily verify notions coincide control systems without internal inputs outputs next theorem shows importance existence simulation function quantifying error output behaviours ones abstraction theorem let suppose simulation function exist function exists following inequality holds proof theorem similar one theorem zrmng omitted due lack space let illustrate importance existence simulation function correspondingly inequality simple example assume given control system interested compositional abstraction networks control systems dissipativity approach computing control input keep output always inside safe set instead one compute control input abstraction keeping output always inside potentially easier due lower dimension existence simulation function hence inequality imply exists control input always inside inf note one choose initial conditions minimize first term hence smaller error satisfaction desired property remark note satisfy triangle inequality one divide coefficients appearing right hand side factor get less conservative upper bound remark note one given interface function maps every input satisfied similar definition input realizing section show map constructed class readily given nonlinear control systems compositionality result section analyze networks control systems show construct abstractions together corresponding simulation functions using storage functions subsystems definition network control systems based notion interconnected systems described interconnected control systems define interconnected control system following definition consider control subsystems rni rmi rpi static matrix appropriate dimension defining coupling subsystems interconnected control system denoted follows functions internal variables constrained interconnection control subsystems illustrated schematically figure figure interconnection control subsystems zamani arcak composing simulation functions storage functions assume given control subsystems rni rmi rpi together corresponding abstractions storage functions use denote corresponding functions matrices corresponding conformal block partitions appearing definition next theorem one main results paper provides compositional approach construction abstractions networks control systems corresponding simulation functions theorem consider interconnected control system induced control subsystems coupling matrix suppose control subsystem admits abstraction corresponding storage function exist matrix appropriate dimension matrix equality satisfied diag diag diag simulation function interconnected control system coupling matrix proof first show inequality holds function one gets function defined max defining function one obtains satisfying inequality show inequality holds well consider exists rmi consequently compositional abstraction networks control systems dissipativity approach vector satisfying pair subsystems internal inputs given derive following inequality using conditions definition matrices inequality rewritten define functions min max construction readily satisfies inequality hence conclude simulation function figure illustrates schematically result theorem remark let assume ipi control subsystem assumptions analytical feasibility conditions matrix inequality derived special interconnection matrices including negative positive feedback interconnection zamani arcak figure compositionality results provided conditions satisfied skew symmetric interconnection negative feedback cyclic interconnection finally extension cactus graphs provided details chapter abstraction synthesis class nonlinear control systems section concentrate specific class nonlinear control systems quadratic storage functions first part formally define specific class nonlinear control systems deal section second part assume abstraction given provide conditions storage function third part shown geometrically construct abstraction together storage function finally discuss feasibility key condition based results section hold class nonlinear control systems class nonlinear control systems considered section given satisfies use tuple refer class control systems form remark linear including zero function zero matrix one remove push term hence tuple representing class control systems reduces linear one therefore every time use tuple implicitly implies nonlinear nonzero compositional abstraction networks control systems dissipativity approach similar shown without loss generality assume class nonlinear control systems one define new function satisfies rewrite aef remark simplicity derivations restrict systems single nonlinearity however would straightforward obtain analogous results systems multiple nonlinearities satisfies furthermore proposed results also extended systems multivariable nonlinearities satisfying multivariable sector property along lines context observer design note class nonlinear control systems remark used widely model many physical systems including active magnetic bearing flexible joint robot fuel cell power generators underwater vehicles quadratic storage functions consider quadratic storage function form matrices appropriate dimensions order show storage function abstraction concrete system require following key assumption assumption let assume constant exist appropriate dimensions matrix matrices equality inequality hold denote zero matrices appropriate dimensions next rather straightforward result provides necessary sufficient geometric condition existence matrix appearing condition lemma given condition satisfied matrix zamani arcak note feasibility characterization lmi involved discussed details end section linear variables note matrix inequality bilinear variables fix constant however assuming square invertible matrix introducing new variables multiplying sides denote zero matrices appropriate dimension one obtains matrix inequality fixed lmi linear matrix inequality variables constant remark note one combine compositionality condition simultaneous search quadratic storage functions subsystems form particular assume given control subsystems one consider matrices lmi decision variables instead fixed without loss generality solve combined feasibility problems although combined feasibility problem may huge large networks solving directly may intractable one use alternating direction method multipliers admm solve feasibility problem distributed fashion along lines proposed provide one main results section showing conditions storage function theorem let suppose assumption holds exist matrices hold function defined storage function providing proof point always exist matrices satisfying implying naturally better simplest abstraction therefore one seek small possible elaborate construction satisfying details next subsection compositional abstraction networks control systems dissipativity approach proof holds implying inequality proceed showing inequality readily verified holds holds note given choose via following linear interface function appropriate dimension matrix using equations definition interface function get using obtain following expression slope restriction one obtains constant depending takes values interval using expression reduces using young inequality help inequality one gets following upper bound positive constant zamani arcak using computed upper bound inequality satisfied functions matrix next result shows conditions actually necessary storage function provided structure interface function matrices appropriate dimension theorem let suppose defined storage function interface given equations hold proof since storage function exists function follows holds implies let consider inputs since inequality reduces using results lemma lemma zrmng inequality implies existence function holds given interface function using obtain since positive definite derive holds hence follows remains show hold first assume since using first inequality one gets since assumption one obtains implies let consider inputs therefore inequality reduces choosing one readily verify obtain implies hence holds free design parameter interface function using results remark note matrix minimize function hence reduce upper bound proposition choose minimizing given error output behaviors choice compositional abstraction networks control systems dissipativity approach far extracted various conditions original system matrices abstraction matrices ones appearing conditions ensure storage function corresponding interface function refining control signal designed one apparently requirements enforce condition matrix example one select making abstract system fully actuated hand one ask existence storage function additionally require controllable behaviors absence internal inputs concrete system preserved abstraction refer interested readers subsection section details mean preservation controllable behaviors next theorem requires condition order guarantee preservation controllable behaviors theorem let suppose exist matrices satisfying matrix given assumed satisfy matrix every trajectory exists trajectory hold proof let trajectory going show trajectory use derive agt use equations definition derive agt agt showing trajectory follows concludes proof remark note previous result establishes absence internal inputs definition refer interested readers details properties controllability systems surjective smooth map zamani arcak construction abstractions provide several straightforward sufficient necessary geometric conditions matrices appearing definition storage function corresponding interface function proposed geometric conditions facilitate constructions matrices first recall lemma providing necessary sufficient conditions existence matrices appearing condition lemma consider matrices exist matrices satisfying give necessary sufficient conditions existence matrices appearing conditions respectively lemma given resp exists matrix satisfying resp resp matrix appropriate dimension lemma given exist matrices satisfying lemmas provide necessary sufficient conditions resulting construction matrices together matrices appearing definition interface function matrices computed next lemma provides necessary sufficient condition existence matrix appearing condition lemma given exists matrix satisfying matrix appropriate dimension although condition readily satisfied choosing one preferably aim finding nonzero smooth later satisfaction compositionality condition already mentioned choice matrix free one also construct ensuring preservation controllable behaviors extra conditions given lemma recalled next provides necessary sufficient conditions existence satisfying lemma consider matrices injective let exists matrix satisfying matrices appropriate dimensions ker similar lemma give necessary sufficient conditions existence satisfying lemma consider matrices injective let exists matrix satisfying ker note conditions resp complete characterization mainly matrices together matrices result compositional abstraction networks control systems dissipativity approach construction matrices chosen freely appropriate dimensions resp computed summarize construction abstraction storage function corresponding interface function table table construction corresponding storage function interface function given satisfying compute matrices pick injective lowest rank satisfying resp compute compute compute compute compute satisfying compute satisfying rather nonzero choose freely resp appearing compute feasibility lmi subsection discuss sufficient necessary feasibility conditions lmi restrictive case positive constant denotes zero matrix appropriate dimension convert feasibility conditions restricted version lmi ones two dual control problems feasibility restricted dual one designing controller rendering linear system strictly positive real spr duality linear assignment control problem section restricted version lmi reduces virtue lemma conditions mean linear control system enforced spr disturbance output control law therefore feasibility restricted version lmi dual feasibility control problem system enforced spr control law using schur complement one readily verify restricted version lmi equivalent zamani arcak means dual system input output enforced strictly less control law section refer interested readers theorem deriving sufficient necessary feasibility conditions restricted version lmi looking corresponding dual control problems namely enforcing spr assigning linear note context observer design control feasibility dual control problems investigated several physical problems example consider linear control system satisfying matrix assume laplacian matrix undirected graph complete graph following block diagonal structure diag partition taking values rni introducing ini ini ini satisfying one readily verify coupling matrix given goal aggregate taking values rni governed satisfies one readily verify conditions satisfied ini ini ini denotes zero matrices appropriate dimensions moreover satisfies conditions hence function storage function satisfying condition condition ini ini input rni given via interface function note computed compositional abstraction networks control systems dissipativity approach look coupling matrix satisfying condition follows ldiag diag note existence satisfying graph laplacian means subgraphs form equitable partition full graph although restricts choice partition general complete graph partition equitable choosing using matrix reduces condition reduces always holds without restrictions size graph order show inequality used always true laplacian matrices undirected graphs sake simulation fix let synthesize controller via abstraction enforce specification defined ltl formula requires output trajectory closed loop system evolves inside set avoids sets indicated blue boxes figure visits indicated red boxed figure infinitely often use scots synthesize controller enforce synthesis process restricted abstract inputs given set initial states since obtain output trajectory illustrated figure note would possible synthesize controller using scots original system without intermediate approximation remark result highlights advantage type conditions proposed storage function example also satisfies requirements simulation function defined however resulting type involves spectral hence tion reduces using results one readily verify number components increases quality approximation deteriorates unless interface gain increasing desirable results high amplitude inputs spectral radius square matrix denoted defined max eigenvalues zamani arcak figure specification closed loop output trajectory sets given conclusion paper proposed first time notion storage function relating concrete control system abstraction quantifying joint correlation notion adapted one storage function dissipativity theory given network control subsystems together corresponding abstractions storage functions provide compositional conditions network abstractions approximate original network approximation error quantified compositionally using storage functions subsystems finally provide procedure construction abstractions together corresponding storage functions class nonlinear control systems using bounds slope system nonlinearities one main advantages proposed results based condition comparison existing ones based type condition former enjoy specific interconnection matrix provide compositional conditions section acknowledgments authors would like thank mahmoud khaled simulation section references aamo arcak fossen kokotovic global output tracking control class systems monotonic velocities international journal control arcak pedersen varigonda adaptive observer design fuel cell hydrogen estimation proceedings american control conference pages june arcak kokotovic control systems nonlinearities ieee transactions automatic control july arcak kokotovic feasibility conditions circle criterion designs systems control letters arcak meissen packard networks dissipative systems springerbriefs electrical computer engineering springer international publishing baier katoen principles model checking mit press april das kumar new bounds spectral radius graphs discrete mathematics april fan arcak observer design systems multivariable monotone nonlinearities systems control letters compositional abstraction networks control systems dissipativity approach frehse compositional verification hybrid systems using simulation relations phd thesis radboud universiteit nijmegen girard pappas hierarchical control system design using approximate simulation automatica godsil royle algebraic graph theory graduate texts mathematics springer new york isidori nonlinear control systems communications control engineering london edition kerber van der schaft compositional analysis linear control systems proc acm int conf hybrid systems computation control pages lin sontag wang smooth converse lyapunov theorem robust stability siam journal control optimization meissen lessard arcak packard compositional performance certification interconnected systems using admm automatica pappas lafferriere sastry hierarchically consistent control systems ieee transactions automatic control june pola pepe benedetto symbolic models networks control systems ieee transactions automatic control rungger zamani compositional construction approximate abstractions proceedings international conference hybrid systems computation control pages acm new york usa april rungger zamani compositional construction approximate abstractions interconnected control systems ieee transactions control network systems rungger zamani scots tool synthesis symbolic controllers proceedings international conference hybrid systems computation control pages acm new york usa april scholtz monitors distributed wave controllers electromechanical disturbances power systems phd thesis massachusetts institute technology september sontag mathematical control theory volume new york edition stan sepulchre analysis interconnected oscillators dissipativity theory ieee transactions automatic control february tazaki imura bisimilar finite abstractions interconnected systems proceedings international conference hybrid systems computation control hscc pages tabuada pappas lima compositional abstractions hybrid control systems discrete event dynamic systems willems dissipative dynamical systems part general theory archive rational mechanics analysis yakubovich solution certain matrix inequalities automatic control theory doklady akademii nauk sssr young classes summable functions fourier series proceedings royal society london mathematical physical engineering sciences zrmng zamani rungger mohajerin esfahani approximations stochastic hybrid systems compositional approach ieee transactions automatic control forthcoming department electrical computer engineering technical university munich munich germany address zamani url http department electrical engineering computer sciences university california berkeley usa address arcak url http
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truncated random measures feb trevor campbell jonathan huggins jonathan tamara broderick abstract completely random measures crms normalizations rich source bayesian nonparametric priors examples include beta gamma dirichlet processes paper detail two major classes sequential crm representations superposition organize novel existing sequential representations used simulation posterior inference two classes constituent representations subsume existing ones previously developed hoc manner specific processes since complete crm used explicitly computation sequential representations often truncated tractability provide truncation error analyses type sequential representation well normalized versions thereby generalizing improving upon existing truncation error bounds literature analyze computational complexity sequential representations conjunction error bounds allows directly compare representations discuss relative efficiency include numerous applications theoretical results normalized crms demonstrating results enable straightforward representation analysis crms previously available bayesian nonparametric context introduction background crms truncation process sequential representations series representations superposition representations truncation analysis series representations superposition representations stochastic mapping hyperpriors normalized truncation analysis series representations superposition representations hyperpriors simulation computational complexity summary results discussion acknowledgments date february first authorship shared jointly campbell huggins campbell huggins broderick appendix additional examples beta process beta prime process appendix technical lemmas appendix proofs sequential representation results correctness proof expected number rejections behavior appendix proofs crm truncation bounds protobound series representation truncation superposition representation truncation stochastic mapping truncation truncation hyperpriors appendix proofs normalized truncation bounds normalized series representation truncation normalized superposition representation truncation truncation hyperpriors references introduction many data sets view data points exhibiting collection underlying traits instance document new york times might touch number topics themes individual genetic data might product populations ancestors belonged user activity social network might dictated varied personal interests traits directly observed common approach model trait frequency rate broader population airoldi inferential goal learn rates well data point exhibits trait since traits unknown priori cardinality also typically unknown data set grows larger reasonably expect number traits increase well cases example expect uncover topics read documents ancestral populations examine individuals genetic data unique interests observe individuals social network bayesian nonparametric bnp priors provide flexible principled approach creating models number exhibited traits random grow without bound may learned part inferential procedure generating countable infinity potential individual data point exhibits finitely models enable growth number observed traits size data set practice however impossible store countable infinity random variables memory learn distribution countable infinity variables finite time conjugate priors likelihoods developed orbanz theoretically circumvent infinite representation altogether perform exact bayesian posterior inference broderick however priors likelihoods often single piece within complex generative model ultimately approximate posterior inference scheme markov chain truncated random measures monte carlo mcmc variational bayes required approximation schemes often necessitate full explicit representation latent variables one option approximate prior related finitedimensional prior replace infinite collection random traits finite subset likely traits first enumerate countable infinity traits full model write paired trait topic document rate frequency discrete measure captures sequence indexed pairs random bayesian model random measure many cases distribution defined specifying sequence simple familiar distributions known sequential representation given sequential representation natural way choose subset traits keep first traits discard rest resulting approximate measure approach called sequential representations shown exist completely random measures crms kingman ferguson klass large class nonparametric priors includes popular models beta process hjort kim gamma process ferguson klass kingman brix titsias james numerous sequential representations crms developed literature ferguson klass bondesson james broderick crm priors often paired likelihood bernoulli process thibaux jordan negative binomial process zhou broderick poisson likelihood process titsias likelihood process determines much trait expressed data point sequential representations also exist normalized completely random measures ncrms perman perman james pitman regazzini lijoi provide random distributions traits dirichlet process ferguson sethuraman ncrms typically paired likelihood assigns data point single trait using ncrm discrete distribution since crms many possible sequential representations method required determining use application hand representation selected choosing truncation level main contributions enable principled selection representation truncation level using approximation error provide comprehensive characterization different types sequential representations crms filling many gaps literature sequential representations along way classify representations two major groups series representations constructed transforming homogeneous poisson point process superposition representations superposition infinitely many poisson also possible truncate sequential representation random level removing atoms weights less specified threshold argiento muliere tardella though approach straightforward use posterior inference algorithms ncrms sometimes referred normalized random measures independent increments nrmis lijoi regazzini james campbell huggins broderick point processes finite rate measures also introduce two novel sequential representations crms provide theoretical guarantees approximation error induced truncating sequential representations give error function prior process likelihood process level truncation truncation error bounds crms studied previously past work focused specific combinations crm priors particular sethuraman ishwaran james ishwaran zarepour blei jordan paisley generalized roy roychowdhury kulis processes current work give much general results bounding truncation error results fill large gaps analysis truncation error often measured terms total variation distance data distributions induced full truncated priors provide first analysis truncation error sequential representations beta process bernoulli likelihood thibaux jordan beta process negative binomial likelihood zhou broderick normalization generalized gamma process brix process generalized inverse gamma lijoi lijoi discrete likelihood moreover even truncation results already exist literature ishwaran james paisley roychowdhury kulis improve error bounds factor two reduction arises use point process machinery crms circumventing total variation bound used originally ishwaran james upon modern truncation analyses built obtain truncation error guarantees bounding probability data drawn full model use feature available truncated model thinking terms probability provides intuitive interpretation bounds communicated practitioners used guide choice truncation level remainder paper organized follows section provide background material crms establish notation first main theoretical section section describe seven different sequential crm representations including four series representations three superposition representations two novel next provide general theoretical analysis truncation error series superposition representations section provide analogous theory normalized versions representation section via infinite extension trick gumbel maddison determine complexity simulating representation section section summarize results table provide advice select sequential representations practice proofs results developed paper provided appendices truncated random measures background crms truncation consider poisson point process rate measure min process generates apcountable infinity values almost surely finite sum bnp trait model interpret rate frequency trait typically paired parameter associated trait topic document shared interest social network assume throughout space distribution constructing measure placing mass atom location results completely random measure crm kingman shorthand write crm completely random measure generated described crm trait distribution left implicit notation effect results possible deterministic components crm kingman considered brevity components added assuming purely atomic analysis modified without undue effort crm prior typically combined likelihood generates trait counts data point let proper probability mass function support collection conditionally independent observations given distributed according likelihood process xnk xnk independently across across desideratum expresses finite number traits encoded assumption since trait counts typically latent full generative model specification indep define observed data conditional density respect measure space instance sequence represents topic rates document corpus might capture many words document generated topic might observed collection words document since sequence countably infinite may difficult simulate perform posterior inference model one approximation scheme define truncation since finite truncation used exact simulation posterior error arises using full crm quantify error consider propagation support may also considered long holds results except representation still hold since rely upon behavior campbell huggins broderick bayesian model define analogous definitions indep standard approach measuring distance use metric marginal densities respect measure final observations ishwaran james paisley kpn bounds kpn also bounds probability contains feature truncation sections interpretation may easier digest since depend observation model instead framed terms underlying traits practitioner trying estimate process illustrate practical application theoretical developments work provide number examples throughout involving gamma process brix denoted discount parameter scale parameter mass parameter rate measure setting yields undiscounted gamma process ferguson klass kingman titsias gamma process often paired poisson likelihood throughout present work use rate parametrization gamma distribution match gamma process parametrization density given gam appendix provides additional example applications theoretical results two crms beta process teh broderick bernoulli negative binomial likelihood beta prime process bpp broderick likelihood sequential representations sequential representations heart study truncated crms provide iterative method terminated point yield finite approximation infinite process choice termination point determines accuracy approximation thus natural first step providing coherent treatment truncation analysis sequential representations past work two major classes sequential representation truncated random measures used series representations form superposition pck resentations form inner sum atoms crm section examines four series representations ferguson klass bondesson three superposition representations two novel broderick james show sequential representations specific crms fit seven general representations finally discuss stochastic mapping procedure useful obtaining new representations transformation others proofs results section may found appendix series representations series representations arise transformation homogeneous poisson point process tend somewhat difficult analyze due dependence atoms also tend produce simple representations small truncation error sections throughout paper let exp ordered jumps homogeneous poisson process let measure satisfying basic conditions let ferguson klass define inf inverse tail measuer say representation write ferguson klass showed implies crm representation analogous inverse cdf method generating arbitrary random variable uniform random variable homogenous poisson process playing role uniform random variable also optimal sequential representation sense sequence generates elegant general approach simulating representation difficult inverting function analytically intractable except cases example gamma process exponential integral function abramowitz stegun representation thus neither inverse computed closed form one must resort numerical approximations bondesson bondesson say bondesson representation write density theorem shows bondesson representations constructed large albeit restricted class crm rate measures offer novel proof theorem appendix using induction strategy introduced banjevic campbell huggins broderick similar proof ideas also used prove truncation error bounds sequential representations section use slight abuse notation brevity measure absolutely continuous respect lebesgue measure density respect lebesgue measure theorem bondesson representation bondesson let rate measure satisfying nonincreasing density implies crm example bondesson representation following representation gamma process described bondesson banjevic since obtain exp thus follows theorem exp condition fails hold apply theorem thinning using nomenclature say thinning representation write probability measure absolutely continuous respect showed implies crm note probability decreasing thus representation generates atoms effect removed increasingly frequently becomes inefficient example thinning representation let gam thinning representation gam rejection using nomenclature say rejection representation write measure satisfying poissp unif showed implies crm representation similar thinning representation except sequence generated poisson process rather allows causing frequency generating ineffective atoms decay assuming appropriately chosen representation thus constructed efficient truncated random measures thinning representation calculate efficiency terms expected number rejections number identically zero proposition expected number rejections remark written densities respect lebesgue measure integral proposition rewritten example rejection representation following consider call crm lomax process lomp related lomax distribution use method analytically since thus rejection representation unif unlike thinning construction given example finite number rates set zero almost surely particular expected number rejections constant example rejection representation case use instead use method analytically since rejection representation unif expected number rejections representation efficient large extremely inefficient small superposition representations superposition representations arise infinite sum crms finite rate measure tend easier analyze series representations decouple atoms summed crms produce representations larger truncation error sections throughout let measure satisfying basic conditions let decoupled bondesson say decoupled bondesson representation write density vki indep vki poiss tki gam novel superposition representation though special cases already known paisley roychowdhury kulis theorem shows decoupled bondesson representation applies class crms bondesson representation section campbell huggins broderick theorem decoupled bondesson representation let specified theorem fixed implies crm proof theorem appendix generalizes arguments paisley roychowdhury kulis free parameter controls number atoms generated outer sum index principled selection made trading computational complexity section truncation error section example decoupled bondesson representation arguments paralleling made example show representation roychowdhury kulis follows directly application theorem exp bondesson representation setting theorem apply condition fails hold broderick james let say representation write indep poiss indep broderick james showed implies crm rate measure likelihood selected conjugate exponential family noting rate sampled mixture exponential family distributions indep zki zki indep zki categorical example representation gamma process values found using integration parts standard gamma distribution integral zki sampled gamma distribution inspection log log indep zki gam say representation write density truncated random measures poiss vki ukik ukij vki indep ukij beta novel superposition representation although previously developed special case beta process broderick name representation arises fact exhibits types behavior broderick mild conditions show theorem appendix theorem shows conditions implies crm proof appendix relies notion stochastic mapping lemma powerful technique transforming one crm another note lemma case deterministic function via mapping may recovered setting lemma crm stochastic mapping let crm probability kernel crm theorem representation let rate measure satisfying let density rate measure beta process implies crm example representation choose gam using change variable follows immediately theorem gam best knowledge authors representation gamma process novel campbell huggins broderick truncation analysis sequential representations developed section shares common structural outer infinite responsible generating countably infinite number atoms crm section terminate outer sums finite truncation level resulting truncated crm possessing finite number atoms develop upper bounds error induced truncation procedure truncated crm error bounds section rely lemma tightening factor two bound ishwaran james proposition shows bound lemma tight without assumptions data likelihood lemma crm protobound let crm truncation indep indep denoting marginal densities respectively kpn supp supp proposition protobound tightness borel exists likelihood lemma tight factor proof results section including lemma proposition found appendix provided truncation results use generative model lemma summarized table section throughout section given likelihood model define notational brevity asymptotic behavior truncation error bounds specified tilde notation lim series representations series representations viewed functional standard poisson point process sequence random variables distribution particular may write form jumps homogeneous poisson point process measurable function every truncated crm takes form lemma appendix generalization arbitrary discrete random measures theory applies equally well measurable space use truncated random measures theorem provides general truncation error bound series representations form specifies range guarantees bound decays theorem series representation truncation error error approximating series representation truncation satisfies kpn indep gam indep furthermore remark alternate form sometimes easier use practice found applying standard geometric series formula yields simplified upper bound derived noting bound usually gives asymptotics main task using theorem develop truncation error bound series representation evaluating integrand definition thus next evaluate integrand provide expressions truncation error bound four series representations outlined section throughout remainder section defined theorem cdf representation representation inf evaluate bound use transformation variables fact conclude recent work representation developed monte carlo estimates error truncated random measure moments known inverse arbel contrast result provides explicit bound truncation error bound require knowing often challenging aspect applying representation campbell huggins broderick example truncation lomp poisson likelihood recall example lomax process lomp crm rate measure log using log since integral upper bounded log log log log replacing approximation setting two terms equal obtain product logarithm function xex thus using fact conclude exp bondesson representation representation writing expectation explicitly integral measure using transformation variables log given bondesson representation definition example truncation bondesson representation let since log truncated random measures second equality follows using power series exponential integral abramowitz stegun chapter thus thinning representation representation distribution since lemma event using fact analytic bounds thinning representation specific processes tend opaque notationally cumbersome simply compare truncation error section representations numerical approximation rejection representation assume use representation simulate poissp rejection representation satisfies inf using techniques thinning representations example truncation poisson likelihood using fact log arguing example see integral upper bounded log log log replacing approximation setting two terms equal obtain defined conclude campbell huggins broderick example truncation poisson likelihood integral upper bounded setting two terms equal solving obtain superposition representations superposition representations truncated crm takes form let denote tail measure superposition property poisson point processes kingman tail measure crm rate measure independent crm following result provides general truncation error bound superposition representations specifies range guarantees bound decays theorem superposition representation truncation error error approximating superposition representation crm truncation satisfies kpn furthermore remark series representations alternate form sometimes easier use found applying standard geometric series formula truncated random measures simplified upper bound derived noting bound usually gives asymptotics main task using theorem develop truncation error bound superposition representation determining tail measure following provide tail measure three superposition representations outlined section decoupled bondesson representation point process superposition average atoms generated independent weights indep indep form gam therefore tail measure density bound decoupled bondesson representation therefore expressed example decoupled bondesson representation truncation using fact equivalent factor bound roychowdhury kulis representation constructive derivation representation broderick proof theorem immediately yields therefore representation truncation error bound expressed using formula example representation truncation standard gamma integral yields sum telescoping canceling terms campbell huggins broderick asymptotic result follows lemma analyze case use hospital rule take limit integral lim log log canceling terms telescopic sum yields log log asymptotic result follows application lemma representation point process superposition erage atoms generated independent weights form indep indep beta therefore tail measure density random variable truncation error bound may expressed example representation truncation let random variable density place using final equality follows ishwaran james theorem thus case follows lemma applied stochastic mapping show truncation bounds developed elsewhere paper applied crm representations transformed using lemma crm denote transformation object defined respect corresponding object denoted tilde example place use make function notation proposition one applies stochastic mapping crm one usually also wants change likelihood thus also changes proof proposition may found appendix truncated random measures proposition truncation error stochastic mapping consider representation crm truncation error bound likelihood stochastic mapping probability kernel truncation error bound hyperpriors practice prior distributions often placed hyperparameters crm rate measure conclude investigation crm truncation error showing bounds developed section modified account use hyperpriors note make dependence hyperparameters explicit notation proposition proposition crm truncation error hyperprior given hyperparameters consider representation crm let given series representation superposition representation error approximating truncation satisfies kpn example decoupled bondesson representation truncation standard choice hyperprior mass gamma distribution gam combining proposition example normalized truncation analysis section provide truncation error bounds normalized crms ncrms examples include dirichlet process ferguson normalized gamma process brix james pitman lijoi normalized process kingman lijoi given crm define corresponding ncrm via measurable subset likewise given truncated crm define normalization via note simulation algorithm used simply normalizing result depend particular representation crm thus applies equally representations section first step analysis ncrm truncations define approximation error manner similar crm truncations since normalized distributions thus observations generated generated definition crms developments section theoretical results section rely general upper bound provided lemma proposition shows bound lemma tight without assumptions data likelihood campbell huggins broderick lemma ncrm protobound let crm let truncation let normalizations respectively indep indep kpn supp marginal densities respectively proposition protobound tightness borel exists likelihood lemma tight factor analysis crms section relied heavily poisson process stucture rates unfortunately rates possess structure thus lack many useful independence properties rates must sum one likewise sampling depend atoms independently randomly selects single atom based rates rather using basic definitions random quantities derive error bound decouple atoms using technique extreme value theory gumbel random variable location scale denoted gumbel defined cumulative distribution function corresponding density interesting property gumbel distribution one perturbs logprobabilities finite discrete distribution gumbel random variables arg max resulting set sample discrete distribution gumbel maddison technique invariant normalization arg max invariant corresponding constant shift space present purposes develop infinite extension result lemma infinite sampling collection positive numbers let gumbel random variables arg log exists unique distribution arg max log categorical proof result along others section may found appendix utility lemma allows construction without problematic coupling underlying crm atoms due normalization rather dealing directly rates perturb gumbel random variables characterize distribution maximum rate process combination distribution lemma yields key proof technique used develop truncation bounds theorems results presented section summarized table section truncated random measures series representations following result provides general truncation error bound normalized series representations specifies range guarantees decays use general series representation notation distribution measurable function every theorem normalized series representation truncation error bound error approximating series representation ncrm truncation satisfies kpn gam furthermore example dirichlet process dirichlet process concentration normalized gamma process example exp section therefore antiderivative log using antiderivative evaluate integrals formula writing expectation gam explicitly making change variables log last equality found multiplying dividing integrand making change variables log therefore truncation error bounded kpn bound example exponential decay reproduces earlier truncation error bound rates due ishwaran james ishwaran zarepour however techniques used past work generalize beyond dirichlet process developed apply ncrm superposition representations following result provides general truncation error bound normalized superposition representations specifies range guarantees decays rely property truncation tail mutually independent crms expressed tail measure denoted campbell huggins broderick theorem truncation error bound normalized superposition representations error approximating superposition representation ncrm truncation satisfies kpn furthermore bound applied using tail measures derived earlier section example dirichlet process example view dirichlet process concentration normalized gamma process first lemma integral exponential exp exp example shows provides tail measure decoupled bondesson representation log substituting result using fubini theorem swap order integration summation evaluating integral making substitution yields noting therefore kpn find tightest bound one minimize respect given example normalized gamma process lemma integral exponential exp exp truncated random measures standard gamma integral yields multiplying previous two displays integrating yields used change variables find upper incomplete gamma function therefore kpn setting expression immediately yields bound normalized process kingman multiplying eqs integrating yields log obtain bound splitting integral intervals bounding section separately obtain bound via transformation therefore asymptotically min kpn log max truncation studied previously argiento threshold weights unnormalized crm beyond fixed level prior normalization develop error bounds method truncation results directly comparable present work due different methods truncation sequential representation termination versus weight thresholding hyperpriors crm case place priors hyperparameters ncrm rate measure conclude investigation ncrm truncation error showing bounds developed section modified account hyperpriors note make dependence hyperparameters explicit notation proposition proposition ncrm truncation error hyperprior given hyperparameters consider representation crm let normalization let given series representation superposition representation error approximating truncation satisfies kpn campbell huggins broderick example dirichlet process place lomax prior lomp combining proposition example yields kpn simulation computational complexity sequential representations section generated different finite sequence distributions resulting different expected computational cost truncation level thus truncation level appropriate parameter compare error bounds different representations require characterization computational cost investigate mean complexity representation number random variables sampled function truncation level representations section begin series representations value series representation generates single trait rate composed transformation random variables thus series representations work satisfy constant inspection representation remaining series representations superposition representations hand generate poisson random variable determine number atoms value generate atoms therefore mean simulation complexity takes form constants might depend value decoupled bondesson representation since atom requires generating three values vki tki representation since atom requires generating three values zki note since decreasing sequence representation since atom requires vki beta generating random variables therefore summary results table summarizes truncation simulation cost results applied beta normalized gamma beta prime processes results bondesson representation well decoupled bondesson representations previously known reproduced results results table novel best authors knowledge interesting note bounds expected costs within representation classes often form aside constants across classes however vary significantly indicating chosen sequential representation process influence truncation error process fig shows comparison truncation error bounds vary expected computational cost simulation normalized gamma process poisson likelihood observations results shown thinning rejection representations computed approximation formula others use truncated random measures table asymptotic error bounds simulation cost summary error bounds presented constant varies models bernoulli obe odds bernoulli poi poisson random measure lomp poi bpp poi obe asymptotic error bound bpp see poi obe bpp bpp poi obe bpp poi obe bpp poi obe complexity log min bpp expressions examples sections note bondesson decoupled bondesson representations exist representations provide bounds examples shown normalized gamma process leave numerical approximation results theorems open problem similar figures processes particular beta bernoulli provided appendix note bounds presented improved factor two versus comparable past results literature due reliance lemmas rather earlier bound found ishwaran james fig top row shows results process representations except thinning capture exponential truncation error decay due fact thinning representation generates increasingly many atoms weight expected number atoms outer index representation decays representation lowest truncation error expected representation generates nonincreasing sequence weights must efficient arbel based figure appendix processes appears bondesson representation typically provides best tradeoff simplicity campbell huggins broderick mean computational cost error error error error mean computational cost mean computational cost mean computational cost error error mean computational cost mean computational cost figure truncation error bounds representations normalized process left column unnormalized process right column normalized process row displays results different value discount parameter efficiency used whenever conditions theorem satisfied technical conditions satisfied rejection representation good alternative ease theoretical analysis concern decoupled bondesson representation provides comparable efficiency analytical simplicity superposition representation bottom two rows fig show results process representation options limited technical conditions bondesson decoupled bondesson representations satisfied rejection representation often best choice due simplicity competitive performance representation however one must take care check efficiency beforehand using proposition given particular choice example choice present work makes rejection representation inefficient fig beta bernoulli fig processes efficient process fig yields reasonable results representation good choice truncation truncated random measures bound approaches exponential decay process larger representation good alternative based results fig appears single dominant representation situations provided representation intractable often however guideline rejection bondesson representations tend good choices processes rejection representations good choices processes discussion investigated sequential representations truncation error bounds simulation algorithms normalized completely random measures past work development analysis tools occurred hoc basis results present paper contrast provide comprehensive characterization analysis different types sequential crm representations available practitioner however number remaining open questions limitations first work consider influence observed data analyses assume priori perspective truncation typically performed data incorporated via posterior inference variational inference mixture blei jordan latent feature model however analysis posteriori truncation studied past work well ishwaran james gelfand kottas ishwaran zarepour language crms observations introduce component posterior process unobserved traits drawn possibly normalized ordinary component crm ishwaran zarepour broderick anticipate property makes observations reasonably simple include truncation tools provided present paper used directly unobserved ordinary component component may treated exactly addition important open questions regarding sequential representations developed work unknown whether generalized versions bondesson decoupled bondesson representations developed larger classes rate measures representation provide partial answer decoupled bondesson case regarding representations one might expect use conjugate exponential family crms broderick would yield expression truncation bound cases provided paper indeed case integrals evaluated exactly expression found however unable identify general expression applicable conjugate exponential family crms based examples provided conjecture expression exists finally fundamental connections representations left largely unexplored work open area research although progress made connecting decoupled bondesson representations hierarchies generalized beta processes roy sec final remark one primary uses sequential representations past work development posterior inference procedures paisley blei jordan present work mean computational cost error campbell huggins broderick error error mean computational cost mean computational cost figure truncation error bounds process provides guidance truncated representations best paired inference methods leave open direction future research require theoretical empirical investigation acknowledgments authors thank anonymous reviewers thoughtful comments led substantial improvements improvements presentation content paper authors supported office naval research muri grant huggins supported government awarded dod air force office scientific research national defense science engineering graduate ndseg fellowship cfr campbell broderick supported darpa award appendix additional examples section provide example applications theory beta process beta prime process beta process beta process teh broderick discount parameter concentration parameter mass parameter crm rate measure setting yields standard beta process hjort thibaux jordan beta process often paired bernoulli likelihood negative binomial likelihood failures bern negbinom note bernoulli likelihood negative binomial likelihood truncated random measures bondesson representation beta thus follows theorem beta case bondesson representation equivalent representation since exp beta representation used teh equivalent bondesson representation obtain truncation bound bernoulli likelihood case argue example result generalizes applies theorem apply directly since however representation obtained using trick paisley let poiss beta thus crm rate measure generated according theorem crm rate density given hence thinning representation let beta thinning representation beta rejection representation obtain rejection representation let rate measure log thus apply method analytically since constructed use campbell huggins broderick construct rejection representation unif expected number rejections ordinary hypergeometric function parenthetical superscripts indicate partial derivatives quantity monotonically diverges obtain truncation bound bernoulli likelihood case consider settings separately setting two terms equal solving obtain conclude log log log setting two terms equal solving conclude defined decoupled bondesson representations decoupled bondesson representation paisley extended broderick setting broderick construction fact trivial representation truncated random measures decoupled bondesson representation special case bernoulli likelihood case truncation bound decoupled bondesson representation power law representation arguments example result generalizes paisley applies representation representation beta process refer reader broderick details note standard beta integral yields hence zki almost surely beta demonstrating construction due thibaux jordan special case representation obtain truncation bound bernoulli likelihood case first consider setting using lemma simplify sum asymptotic result follows lemmas use lemma arrive digamma function asymptotic result follows lemma also bound truncation error case negative binomial likelihood fixed number failures assuming integration parts yields sum telescoping canceling terms campbell huggins broderick asymptotic result follows lemmas analyze case use hospital rule take limit yielding lim computing error bound canceling terms telescoping sum asymptotic result follows application lemma stochastic mapping transform gamma process beta process applying stochastic mapping gam gam using lemma yields applying result bondesson representation yields gam exp unlike bondesson representation applies hyperpriors consider truncating bondesson representation beta process hyperprior mass parameter standard choice hyperprior gamma distribution gam combining proposition truncation bound beta prime process beta prime process bpp broderick discount parameter concentration parameter mass parameter crm rate measure beta prime process often paired odds bernoulli likelihood case truncation results odds bernoulli likelihood mean computational cost error error error truncated random measures mean computational cost mean computational cost figure truncation error bounds beta bernoulli process bondesson representation nonincreasing thus follows theorem bpp truncation bound first upper bound follows jensen inequality thus error bound process thinning representation let thinning representation bpp rejection representation take rate measure lomp rejection representation bpp unif expected number rejections constant digamma function since log representation remains efficient even fairly large values obtain truncation campbell huggins broderick bound use approach example log log log log setting two terms equal solving conclude product log function defined similarly example case instead use since rejection representation unif expected number rejections representation efficient large extremely inefficient small following approach example integral upper bounded setting two terms equal solving obtain decoupled bondesson representation follows theorem arguments bondesson case bpp using trivial bound calculations analogous case obtain upper bound exp truncated random measures representation transform gamma process beta prime process bpp applying stochastic mapping using lemma yields bpp applying result representation example yields novel representation gam gam implies bpp using trivial bound calculations analogous case obtain upper bound asymptotic simplification representation process thus error bound also case appendix technical lemmas lemma uniformly bounded sequence random varip ables proof without loss generality assume hypothesis exists follows markov inequality lemma measure exists proof without loss generality let suppose implication hold exists let sequence continuity hence atomic contradiction lemma absolutely continuous measure continuous satisfy min lim campbell huggins broderick proof min implies fact since otherwise contradiction since continuous bounded range exists finite assume particular thus contradiction thus lemma assume twice continuously differentiable function following properties exists increasing sequence lim sup sup constant increasing sequence proof taylor expansion yields assumptions ensure lim lim hence lim lim lemma gautschi thus lemma digamma function truncated random measures proof analyzing right hand side yields induction supposing result true demonstrates desired result next proceed induction using recurrence relation abramowitz stegun chapter right hand side evaluates supposing result true demonstrating result lemma dxn proof campbell huggins broderick digamma function using lemma asymptotic expansion abramowitz stegun chapter obtain since xdn using lemma previous two displays yields result lemma log proof integration parts standard gamma integral taking limit via hospital rule yields lim log lemma standard asymptotic equivalence properties appendix proofs sequential representation results correctness proof theorem first show density since nond decreasing exists almost everywhere hence furthermore final equality follows assumed behavior since partition random variables independent suffices show measurable set complement random variable correct characteristic function define family random measures conditioning truncated random measures note two terms left hand side independent thus write characteristic function complement multiplying sides making change variable yields eiav characteristic function random variable density differentiating sides respect rearranging yields conclude exp using integration parts definition rewrite eiav iav eiav iav eiav campbell huggins broderick final equality follows assumed behavior combining previous two displays setting concludes proof exp exp exp proof theorem already shown density let pck crm rate measure law using product distribution formula log let cdf derived preceding arguments conclude rate measure log cdf rewritten combining previous two displays conclude proof theorem since representation case vki almost surely previously shown broderick simply apply stochastic mapping result lemma density truncated random measures proof expected number rejections proof proposition equalities follow definition integrating applying campbell theorem behavior formalize sense representations fact produce behavior let poiss znk analyze number features observations number features appearing times observations power law analysis beta process broderick use bernoulli likelihood process however bernoulli process applicable whereas general replacing bernoulli process poisson likelihood process natural choice since znk bern asymptotically since thus bernoulli poisson likelihood processes behave asymptotically relevant asymptotic analysis therefore able show crms representations beta process broderick call types power law behavior condition tails heavy theorem assume continuous density exists constant depending almost surely order prove theorem require number additional definitions lemmas approach follows broderick reader encouraged consult details discussion power law campbell huggins broderick behavior crms throughout section lemma crm rate measure let homogeneous poisson point process rate define furthermore let let follows campbell theorem kingman first lemma characterizes power law behavior slowly varying function satisfies lemma broderick proposition slowly varying function ctd transferring power law behavior trivial since next lemma justifies transferring power law behavior truncated random measures lemma broderick lemmas satisfies furthermore max final lemma confirms asymptotic behaviors almost surely expectations lemma assume satisfies slowly varying functions almost surely proof theorem combining three lemmas result follows soon show satisfies constant may change line line begin rewriting using change variable since integrable continuous function combining upper bound yields since integrable dominated convergence limit lim exists finite moreover since continuous density exists hence thus campbell huggins broderick hence sufficiently small thus sufficiently small shows holds appendix proofs crm truncation bounds protobound lemma protobound let two discrete random measures let collection random measures generated supp supp let collection random variables generated via define analogously finally define supp supp almost surely joint distribution kpy marginal densities proof lemma direct application lemma crms crm truncation technical condition satisfied weights sampled independently atom proof lemma direct application lemma ncrms normalization crm distribution crm normalization truncation technical condition satisfied conditioning supp equivalent normalization proof lemma begin expanding conditioning denoted conditioning brevity kpy conditioning truncated random measures first term arises fact function equal distribution assumption substituting back kpy finally fubini theorem kpy proof propositions proof applies results since borel exists measurable mapping random variable atomic measure define let conditional exists txn since least one atom lacks using definitions facts observe txn txn tzn txn tzn campbell huggins broderick ptz ptx since txn finite sum one random variables follows thus lemma exists define family likelihoods unif let conditioned probability least implies inequalities equalities hence conclude kpy series representation truncation recall section series representation generally form exp jumps homogeneous poisson point process measurable function distribution every note lemma fact repeatedly prove useful proofs section proof theorem based following lemma lemma hypotheses theorem supp supp exp proof let supp supp use proof strategy banjevic induction since multiplying sides making change variable yields differentiating sides respect rearranging yields truncated random measures since lemma solve conclude exp exp use inductive hypothesis gam trivially holds inductive hypothesis holds using tower property exp follows setting proof theorem main result follows combining lemmas applying jensen inequality using monotone convergence upper bound follows immediately fact integral fix follows lim supp supp lemma rfollows continuous mapping theorem conclude hence theorem representation truncation error conclusions theorem hold proof theorem first make change variables obtain campbell huggins broderick finally use fact monotone convergence theorem thinning representation truncation error conclusions theorem hold proof theorem since using monotone convergence theorem rejection representation truncation error conclusions theorem hold proof theorem since eliminate innermost integral truncated random measures making change variable reasoning analogously proof theorem obtain theorem bondesson representation truncation error hypotheses theorem satisfied conclusions hold proof theorem proved using theorem take alternative approach using direct poisson process arguments lemma gam supp supp exp proof lemma measure distribution crm define supp supp tvk prove exp tve set obtain desired result proof proceeds induction event supp supp equal supp thus supp turn equivalent probability thinning crm times process atoms probability crm atoms since poisson process measure atoms probability exp exp second equality follows change variables probability inductive hypothesis gam campbell huggins broderick using tower property condition fact tvk since exp desired result follows setting first combine lemmas apply jensen inequality bounds fact follows arguments proof theorem superposition representation truncation proof theorem begin lemma note supp supp supp supp generating view point process representing atoms bernoulli trial contained thinned success probability generate atom failed times thus remaining process thinned therefore event equivalent event thinned atoms using fact poisson process measure atoms probability formula supp supp since nonnegative error bound lies interval show first note splitting individual summed components combining results eqs yields lim truncated random measures stochastic mapping truncation proof proposition let notational brevity define event supp supp corresponding transformed event supp supp bern truncation hyperpriors proof proposition repeating proof lemma appendix except additional use tower property condition hyperparameters additional use fubini theorem swap integration expectation jensen inequality kpy supp supp appendix proofs normalized truncation bounds proof lemma first demonstrate arg max note arg max log arg max exp log exists due monotonicity exp similarly existence either proves existence since gumbel exp log exp log exp therefore exp log sufficient demonstrate exp log finally since positive sequence converging finite number elements greater set exp log thus arg max exp log arg max exp log campbell huggins broderick right hand side exists computes maximum finite nonempty set numbers note arg max guaranteed single element since log purely diffuse distribution existence uniqueness arg max demonstrated compute distribution first note log exp exp arg max log log log last integral since integrand gumbel density normalized series representation truncation proof theorem first apply lemma kpy supp next jensen inequality supp supp remaining part proof quantifies probability sampling generates atom support equal support since normalization use trick based lemma rates perturb gumbel random variables quantify probability max occurs within atoms first split sequential representation truncation tail using form next define maximum gumbel perturbed rates max log truncated random measures gumbel since homogeneous poisson process condition value equivalent conditioning event poisson process exactly atoms atom since depend ordering max log max log unif therefore maximum collection independent random variables compute cdf simply taking product cdfs random variables using standard techniques transformation independent random variables dudv log log dudv defining function next define analogous maximum tail process rates sup log conditioning sup log homogeneous poisson process note since poisson point process log using poisson process stochastic mapping rate measure dudv therefore equal probability poisson point process atoms position greater since poisson process measure atoms probability dudv campbell huggins broderick noticing integrand gumbel density use fubini theorem swap integrals evaluate dudv taking derivative yields density respect lebesgue measure therefore using trick lemma substitute results original bound yielding kpy supp using substitution fact bound simple consequence fact supp asymptotic result follows fact supp normalized superposition representation truncation proof theorem initial technique proof theorem yields kpy supp remaining part proof quantifies probability sampling generates atom support since following developments similar focus discussion reintroduce tail quantities necessary first transform rates stochastic mapping log gumbel resulting new poisson point process rate measure probability points poisson point process less value equal probability atoms defining supremum points process combined basic properties poisson point processes exp dwdt truncated random measures using fubini theorem swap integrals evaluate inner integral analytically noting integrand gumbel density exp exp take derivative respect obtain density respect lebesgue measure derivation holds true replacing therefore using trick lemma substitute results original bound yielding kpy supp using substitution fact dominated swap integration differentiation fact bound lies simple consequence fact supp fact error bound asymptotically approaches consequence monotone convergence theorem applied decreasing sequence functions truncation hyperpriors proof proposition repeating proof lemma appendix except additional use tower property condition hyperparameters additional use fubini theorem swap integration expectation jensen inequality kpy supp references abramowitz stegun eds handbook mathematical 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artificial intelligence statistics paisley zaas woods ginsburg carin construction beta international conference machine learning perman order statistics jumps normalised stochastic processes applications perman pitman yor sampling poisson point processes probability theory related fields pitman lecture series regazzini lijoi distributional results means normalized random measures independent annals statistics series representations infinitely divisible random annals probability campbell huggins broderick series representations processes perspective point resnick mikosch eds processes theory applications chapter boston roy scheme generalized beta indian buffer processes hierarchies arxiv preprint roychowdhury kulis gamma processes variational international conference artificial intelligence statistics sethuraman constructive definition dirichlet statistica sinica teh indian buffet processes advances neural information processing systems teh ghahramani construction indian buffet 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pure known results applications open problems feb juan uwe fabrizio zanello abstract note presents discussion algebraic combinatorial aspects theory pure various instances pure appear described several open problems deserve investigation also presented dedicated david eisenbud occasion birthday introduction pure fascinating objects arise several mathematical areas subject extensive research yet knowledge pure limited goal note survey known results motivate investigations pointing connections various interesting problems much material paper drawn recent monograph authors mats boij rosa cited article bmmnz often quote results giving name author year result published also hint history ideas development theory pure interest pure already indicated definition one hand pure defined vector whose entries record number monomials fixed degree order ideal generated monomials degree pure hilbert function finitedimensional graded algebra level socle concentrated one degree monomial relations extensive literature monomial ideals extensive literature level algebras pure form bridge two theories outline work pure point view sequences broad array applications occur many settings largely combinatorial ones section review basic results theory pure focus qualitative aspects shape instance cases pure osequences known unimodal first weakly increasing peak reached weakly decreasing however pure may fail unimodal even arbitrarily many include discussion recent results conditions force unimodality connections various combinatorial problems subject section face vectors pure simplicial complexes examples pure particular existence certain block designs steiner systems related pure special case existence finite projective planes equivalent existence particular pure work paper done first author sponsored national security agency grant number simons foundation grant work paper done second author sponsored national security agency grant number simons foundation grant juan migliore uwe nagel fabrizio zanello another challenging problem stanley conjecture matroid complex pure discuss recent progress however conjecture remains open general section describe results enumeration pure focus asymptotic properties particular follows number variables large almost pure unimodal conclude note collection open problems mentioned earlier sections monomial level algebras finite nonempty set monic monomials indeterminates called monomial order ideal whenever monomial dividing defined vector counting number monomials degree monomial order ideal called pure maximal monomials partial ordering given divisibility degree pure pure monomial order ideal reasons clear shortly call socle degree type pure number maximal monomials notice think indeterminates polynomial ring field question whether given sequence pure depend choice thus many results matter choose however situations choosing nice field allows use special algebraic tools say something pure case make whatever additional assumptions need let infinite field consider standard graded artinian usually monomial ideal without loss generality assume contain nonzero linear forms define codimension let consider action monomials contraction mean action generated yrar monomial ideal define inverse system annr one check consists monomials identifying viewed monomial order ideal recalling standard graded algebra hilbert function defined dim observe defined order ideal coincides hilbert function furthermore pure monomial order ideal level algebra socle annihilator homogeneous maximal ideal concentrated one degree called socle degree necessarily degree maximal monomials dimension socle space equal type pure see details inverse systems thus study pure boils study possible hilbert functions artinian monomial level algebras cases rule candidates pure showing even level algebra hilbert function often need use structure monomial algebras pure known results applications open problems basic tool macaulay theorem determine sequence even determine hilbert function artinian algebra refer details macaulay theorem recall statement let positive integers exist uniquely determined integers called expansion set macaulay theorem following theorem macaulay let standard graded algebra hilbert function possibly infinite sequence integers satisfies growth condition theorem every value thus sequences occur hilbert function standard graded algebra example sequence pure even growth degree degree big similarly pure hilbert function level algebra see finally hilbert function level algebra pure monomial level algebra hilbert function fact first nonunimodal level hilbert function discovered codimension see third author boyle shown fact smallest possible hilbert function challenge determine additional conditions imposed requiring hilbert function artinian level monomial algebra first result due originally stanley subsequent proofs given watanabe ikeda reid roberts roitman herzog popescu lindsey concerns monomial complete intersections note though equivalent property proven earlier bruijn van ebbenhorst tengbergen kruyswijk result requires introduce notion weak strong lefschetz properties consequence result pure type perhaps critical alternative proofs could given influence study slp wlp general related topics commutative algebra hardly overstated result throughout paper call sequence unimodal nondecreasing degree nonincreasing past degree call strictly unimodal strictly increasing degree possibly constant range strictly decreasing reaches zero zero past point unimodality central concept combinatorics combinatorial commutative algebra also branches mathematics see instance classical surveys stanley brenti theorem see sources let characteristic zero let artinian monomial complete intersection xar juan migliore uwe nagel fabrizio zanello let general linear form positive integers homomorphism induced multiplication maximal rank particular true consequence pure type strictly unimodal arbitrary artinian homogeneous ideal maximal rank property called strong lefschetz property slp case called weak lefschetz property wlp consequence wlp range injective short exact sequence thus first difference hilbert function standard graded algebra range say differentiable range also follows sequence wlp holds peak hilbert function reached hilbert function nonincreasing therefore whole hilbert function unimodal remark simple examples show theorem may fail positive characteristic turns question positive characteristics monomial complete intersection weak strong lefschetz property leads unexpected connections problem determining number certain plane partitions lozenge tilings families lattice paths see one early important results pure due hibi theorem hibi let pure socle degree whenever following two important consequences flawless first half nondecreasing latter result later improved following algebraic hausel theorem hausel let monomial artinian level algebra socle degree field characteristic zero general linear form induced multiplication injection particular field sequence first half differentiable following additional results differentiability theorem bmmnz every finite differentiable first half pure converse hausel theorem particular finite differentiable pure truncation nondecreasing pure socle degree differentiable pure known results applications open problems turns best possible result direction proposition bmmnz exist nondecreasing pure socle degree differentiable example illustrate preceding result example observe first pure since truncation polynomial ring four variables pure order ideal arises using monomials degree also pure since order ideal generated monomial abcd four new variables work polynomial ring eight variables consider pure order ideal generated monomials degree resulting since first difference constructed desired example putting aside class nondecreasing pure turn question unimodality three factors whether nonunimodal example exist codimension number variables socle degree type easy application macaulay theorem gives standard graded algebra codimension two unimodal hilbert function fact hard show level monomial hilbert function strictly unimodal see iarrobino hence interesting questions arise codimension begin results involving socle degree also bring codimension follows hibi result flawlessness pure socle degree unimodal next case socle degree already necessarily unimodal thanks example example observe pure since truncation polynomial ring five variables also pure since corresponds maximal monomial abcd hence reasoning consider one copy eleven copies hvectors pure twelve different rings work tensor product rings follows nonunimodal pure socle degree natural question smallest codimension nonunimodal pure exist socle degree remains open following results however theorem boyle pure socle degree three variables unimodal pure socle degree four variables unimodal four variables exist nonunimodal pure socle degrees boij third author gave nonunimodal pure codimension socle degree smallest known example codimension follows juan migliore uwe nagel fabrizio zanello boyle result codimension open cases socle degrees notice codimension previous theorem leaves open socle degrees turn questions involving type course follows immediately theorem pure type unimodal else deduced pure using wlp collection results obtained showed sense limits wlp study pure theorem bmmnz field characteristic zero following hold monomial artinian level algebra type three variables wlp thus pure type codimension differentiable reaches peak possibly constant nonincreasing reaches zero fix two positive integers monomial artinian level algebras codimension type possess wlp least one following true iii proof surprisingly long intricate main point cases able show artinian monomial level algebras exist wlp notice theorems require characteristic zero statements injectivity surjectivity require property characteristic indeed great deal research carried see happens positive characteristic refer overview results consequences shape pure indeed characteristic free noted since hilbert function monomial ideal depend characteristic one studying artinian level monomial algebras fixed type theorem serious limitation usefulness wlp however study pure osequences determine combinations codimension type socle degree force unimodality still conceivable wlp play useful role trivial example know monomial complete intersections codimension possess wlp thanks theorem known except codimension noted whether complete intersections property knowledge special case monomial ideals enough say complete intersection hilbert functions unimodal perhaps similar phenomenon allow wlp continue play role study pure first approach using philosophy obtained cook second author lifted monomial ideal ideal reduced set points one variable showed general artinian reduction wlp concluded hilbert function original monomial algebra unimodal regardless whether wlp reason mention useful tool studying wlp introduced first second authors harima watanabe whose study continued brenner kaid study syzygy bundle use theorem see used show complete intersection field characteristic zero wlp idea restrict general line say one defined general linear form key restricted ideal codimension hence pure known results applications open problems matrix minimal syzygies two consecutive degrees idea height complete intersections information obtained considering module syzygies sheafifying obtain syzygy bundle applying theorem general line defined course introduces questions semistability syzygy bundle mostly omit general setting following result summarizes idea nicely least codimension three theorem let artinian homogeneous ideal whose syzygy bundle semistable restriction splits general line wlp restriction splits general line wlp applications approach higher codimension found instance returning unimodality questions tools wlp also needed following theorems boyle relied decomposition results analysis complete intersection hilbert functions use wlp theorem boyle codimension pure type strictly unimodal codimension pure type strictly unimodal natural question whether pure type arbitrary codimension unimodal also fixed codimension interesting problem determine types force unimodality record smallest known nonunimodal example codimension type given finally one ask nonunimodal pure answer nonunimodal theorem bmmnz integers exists pure variables nonunimodal exactly maxima course price theorem paid large socle degree type fact even face vectors simplicial complexes much smaller subset pure nonunimodal arbitrarily many peaks see result considerably extends theorem even though unlike arbitrary pure number variables becomes necessarily large number peaks increases good topic build bridge algebraic combinatorial sides theory pure interval conjecture interval property introduced third author conjectured existence set hilbert functions level suitably juan migliore uwe nagel fabrizio zanello symmetric way gorenstein algebras namely says two necessarily finite sequences class integer sequences coincide entries one say index positive integer sequences also given level gorenstein hilbert functions nearly impossible characterize appears natural property one strongest structural results might hope achieve set sequences example proved holds gorenstein hilbert functions socle degree since gorenstein hilbert functions symmetric means fixed gorenstein ranges minimum possible value say latter maximum allowed polynomial ring variables achieved compressed gorenstein algebras see papers iarrobino third author works notice existence hilbert functions especially helpful view fact codimensions value known fact gorenstein hilbert functions highly nonunimodal asymptotically lim proved three authors solving longstanding conjecture stanley see also broad generalizations interval property still wide open today level gorenstein algebras combinatorial direction bmmnz also conjectured pure name icp pure simplicial complexes topic discussion next section pure icp proved number special cases importantly known socle degree number variables theorem bmmnz icp holds set pure socle degree thanks result stokes third author recently developed new approach leading proof stanley matroid conjecture see next section icp remains open instances three variables must pointed recently disproved four variable case constantinescu varbaro see remark found following counterexample example consider pure order ideals generated yzt pure osequences however exhaustive computer search sets three monomials degree four variables reveals sequence pure contrary icp worth remarking however level provide counterexample arbitrary level algebras indeed level algebra four variables whose inverse system generated two pure known results applications open problems sums sixth powers six general linear forms sixth power one general linear form pure still wide open little progress made far general time still unclear exact scope interval property also use areas combinatorial algebra even enumerative combinatorics well known hold set hilbert functions graded algebras see macaulay theorem arbitrary simplicial complexes theorem complexes bmmnz instead fails quite dramatically example matroid conjecturally another subset pure see next section refer bmmnz details stanley third author recently looked context posets class ranked posets generalizing young lattice integer partitions lattice even though fails general might reasonable property conjecture instance natural class differential posets pure combinatorics much motivation study pure comes combinatorics section give overview side theory order put context definition pure given introduction quickly recall notion posets order ideals introduction theory refer chapter stanley new edition poset short partially ordered set set equipped binary relation reflexive antisymmetric implies transitive implies order ideal poset subset closed respect thus monomial order ideals finite order ideals poset monomials polynomial ring binary relation divisibility notice ranked poset rank monomial degree therefore macaulay exactly possible rank functions order ideals since every hilbert function satisfying macaulay theorem achieved monomial algebra similarly seen pure rank function monomial order ideal whose generators antichain maximal monomials degree another fundamental class order ideals contained boolean algebra poset subsets ordered inclusion identifying integer vertex order ideals usually called simplicial complexes vertices elements simplicial complex dubbed faces maximal faces facets dimension face cardinality minus dimension largest dimensions faces notice identify variable simplicial complexes also coincide order ideals generated squarefree monomials particular define pure simplicial complexes whose facets dimension clearly rank vectors called pure special subset pure generated squarefree monomials juan migliore uwe nagel fabrizio zanello example simplicial complex order ideal generated thus pure complex dimension whose pure equivalently pure generated two squarefree monomials similarly macaulay theorem arbitrary know characterization class pure thanks classical theorem see however analogously things become dramatically complicated hopeless say comes attempting characterization pure last section bmmnz begun study pure still little known today beyond known arbitrary pure besides obvious intrinsic importance simplicial complexes central object algebraic combinatorics combinatorial algebra topology name subjects pure also carry fascinating applications connections finite geometries design theory want focus next portion section follow discussion probably first observed characterization pure simplicial complexes would imply instance steiner systems special case classification finite projective planes one major open problems geometry steiner system set together collection called blocks every contained exactly one block steiner systems special family block designs refer reader two texts find truly vast amount information combinatorial designs instance steiner triple system sts steiner system dubbed steiner quadruple system order system since dealing maximal sets cardinality case clear identify element variable existence steiner systems similarly block designs equivalent existence certain pure example let consider sts order constructing design tantamount determining family squarefree degree monomials say squarefree degree monomial divides exactly one clearly since squarefree degree monomials exist words exists pure notice also exists pure exists pure since seven degree monomials total degree divisors needs exactly three linear divisors must squarefree easy see sts order pure indeed exist using monomial order ideal generated pure known results applications open problems simple numerical observations show necessary condition sts order exist congruent modulo classical result kirkman also sufficient words pure pure congruent modulo different challenging problem classification steiner systems isomorphism even existence particular systems sometimes brings story nontrivial amount interesting algebra illustration mention case steiner systems intimately connected first sporadic finite simple groups ever discovered called mathieu groups see mathieu five groups denoted respectively fact arise automorphism groups steiner systems transformations systems preserve blocks exists one sts order called fano plane see figure reasons clear minute words seven monomials previous example isomorphism possible set generators pure order ideal pure also exists unique sts however reasonable believe number nonisomorphic sts increases extremely quickly large instance nonisomorphic sts order order similarly possible orders steiner quadruple systems known nicely characterized since obvious necessary conditions turn also sufficient exists order congruent modulo proved hanani words reasoning since four possible given pure pure congruent modulo however increase cardinality blocks things become obscure due high complexity computing combinatorial designs large vertex set well lack general theory juan migliore uwe nagel fabrizio zanello already steiner quintuple systems trivial necessary conditions congruent modulo longer sufficient example steiner quintuple system exists see smallest value existence currently open words unknown whether pure perhaps family examples steiner systems finite projective planes deserve special mention recall projective plane collection points lines two lines intersect exactly one point two points lie exactly one line one also assumes exist four points three collinear order avoid uninteresting pathological situations projective plane finite easily seen number points equal number lines number must form integer order plane projective plane order line contains exactly points duality point intersection exactly lines words finite projective planes steiner systems reader may want consult introduction area thus example steiner system fano plane unique smallest possible projective plane similarly argued earlier terms pure one show projective plane order exists pure pure major open problem geometry asks classification finite projective planes even possible values may assume conjecturally always power prime standard algebraic exercise using finite field theory construct projective plane order best general necessary condition known today still following theorem theorem order projective plane congruent modulo sum two squares thus instance consequence theorem projective plane order exists words pure however already ruling existence projective planes order required major computational effort see lam case still open notice least certain values number nonisomorphic projective planes order large general classification seems entirely reach smallest exists one nonisomorphic projective plane four possible cases already known veblen second important application pure want discuss brings attention special class simplicial complexes called matroid complexes pure known results applications open problems matroids ubiquitous mathematics often show surprising ways see algebraic theory matroids began seminal paper stanley introduced pure finite matroid naturally identified pure simplicial complex restriction subset also pure complex one associates given simplicial complex following squarefree monomial ideal field called ideal complex quotient algebra ring standard fact combinatorial commutative algebra see ring matroid complex level although course positive except degenerate cases thus level since artinian reduction however even though presented monomials notice artinian reductions general far monomial require taking quotients general enough linear forms following spectacularly simple conjecture stanley predicts matroid complex nonetheless find artinian monomial level algebra conjecture matroid pure problem characterizing matroid appears hopeless conjecture motivated much algebraic work done matroids past years see highly nonexhaustive list main approach conjecture given certain class matroids explicitly produce pure monomial order ideals matroid pure recently stokes third author introduced abstract approach conjecture main idea inspired latest progress pure made bmmnz particular proof interval conjecture icp socle degree try reduce stanley conjecture much possible one properties pure thus avoiding explicit construction monomial ideal matroid approach already led proof conjecture matroid complexes dimension case focus large portion thesis simply followed lines theorem matroid pure osequences generally following first concrete still tentative general approach conjecture see assuming conjecture holds matroid complexes whose deletions respect vertex cones may difficult show techniques paper conjecture true general following two natural still bold assumptions juan migliore uwe nagel fabrizio zanello matroid differentiable long nondecreasing fact incidentally would hausel swartz proved exist first half matroid carry way suppose shifted sum two pure differentiable long nondecreasing also pure enumerations pure seen pure arise several areas yet properties well understood important questions addressed even classification available example one would like estimate number pure given codimension socle degree happens asymptotically moreover seen pure far unimodal want nevertheless one may ask odds pure unimodal order discuss questions let denote sets pure differentiable respectively codimension socle degree recall given two functions one says asymptotic writes limr limits taken approaching infinity consider integrating passing provides differentiable thus since finite differentiable pure theorem following inclusions results linusson see imply large cardinalities asymptotically equal follows large codimensions almost pure precisely one theorem bmmnz fix positive integer large almost socle degree differentiable namely since pure hilbert functions level algebras immediately get following consequence corollary bmmnz fix positive integer let set level hilbert functions codimension socle degree large almost level sequences pure unimodal outcome changes drastically fix additional parameter socle type since monomial degree divisible distinct variables observe pure known results applications open problems proposition bmmnz let set pure codimension socle degree type bound sharp note contrast result analogous restriction level hilbert functions example gorenstein algebras type admit positive socle degree codimension fact asymptotically number hilbert functions known recall socle degree symmetric differentiable degree initially stanley iarrobino see hoped hilbert functions gorenstein algebras although true see first counterexample almost true fact differentiable extended symmetric sequence result moreover every hilbert function gorenstein algebra see obviously first half hilbert function gorenstein algebra taken together follows number gorenstein hilbert functions negligible theorem bmmnz fix positive integer let set gorenstein hilbert functions codimension socle degree let set codimension socle degree large almost gorenstein hilbert functions precisely returning pure would interesting determine least find good estimate number pure codimension socle degree type seems difficult problem however simplest case easy combinatorial answer fact pure type hilbert function complete intersection algebra whose inverse system monomial form yrar may assume partition since easy see distinct partitions lead different hilbert functions arrive following result proposition bmmnz number partitions integer exactly parts consider slightly different asymptotic enumeration question fix positive integers since number monomials dividing monomials degree finite finitely many pure socle degree type denote set determining exactly seems reach however one may hope least find order large first interesting case namely settled theorem bmmnz let denote number pure socle degree type lim proof gives information consider pure hibi theorem fact monomials degree divisible distinct quadratic monomials follows see two juan migliore uwe nagel fabrizio zanello functions fact asymptotically equal notice fixed possible values fall one following three regions illustrated figure region region region iii iii using superscripts denote sets pure region shown bmmnz iii lim lim arguments use crucial way interval property pure socle degree unfortunately property fails general nevertheless would interesting extend results socle degree lim open problems section collect interesting problems remain open area pure discussed previous sections related problems addressed largest type pure unimodal independently codimension socle degree even proving pure type unimodal codimension would interesting fixed codimension largest type pure unimodal fixed codimension largest socle degree pure unimodal smallest codimension exists nonunimodal pure socle degree determine asymptotically number pure socle degree type large order magnitude first example nonunimodal pure due stanley fact cohenmacaulay simplicial complex hence particular pure pure known results applications open problems smallest number variables number vertices complex allowing existence nonunimodal pure tahat recently determined sharp lower bound nonunimodal socle degree socle degree stanley original example produced examples larger socle degree low stanley problem year imu volume mathematics frontiers perspectives asks among things matroid unimodal even matroid notice matroid much smaller subset vectors major recent breakthrough problem see huh huh katz consequence work hence unimodality known see lenz huh representable matroids notice also general matroid would imply matroid proved dawson lenz acknowledgements wish thank david cook richard stanley helpful comments third author also thanks bierbrauer interesting discussion connections group theory steiner systems thank referee careful reading paper references nonpure shellability subspace arrangements complexity formal power series algebraic combinatorics series formelles combinatoire dimacs series discrete mathematics theoretical computer science volume boij migliore nagel zanello shape pure mem amer math soc vii boij zanello level algebras bad properties proc amer math soc boyle unimodality pure thesis university notre dame brenner kaid syzygy bundles weak lefschetz property illinois math brenti unimodal sequences algebra combinatorics geometry update jerusalem combinatorics contemporary mathematics bruck ryser nonexistence certain finite projective planes canadian math bruns herzog rings rev cambridge studies advanced mathematics cambridge university press cambridge chari matroid inequalities discrete math chari two decompositions topological combinatorics applications matroid complexes trans amer math soc chen guo jin liu trivariate monomial complete intersections plane partitions commut algebra cho iarrobino hilbert functions level algebras algebra chowla ryser combinatorial problems canadian math colbourn dinitz eds handbook combinatorial designs crc press boca raton constantinescu varbaro matroid complexes preprint juan migliore uwe nagel fabrizio zanello cook lefschetz properties monomial complete intersections positive characteristic algebra cook nagel hyperplane sections subtlety lefschetz properties pure appl algebra cook nagel weak lefschetz property monomial ideals lozenges illinois math appear cook nagel enumerations deciding weak lefschetz property preprint dawson collection sets related tutte polynomial matroid graph theory singapore lecture notes vol springer berlin bruijn van ebbenhorst tengbergen kruyswijk set divisors number nieuw arch wiskunde loera kemper klee small matroid complexes electron combin dembowski finite geometries reprint original classics mathematics springerverlag xii laksov compressed algebras conference complete intersections acireale lecture notes mathematics vol springer berlin geramita inverse systems fat points waring problem secant varieties veronese varieties parametric spaces gorenstein ideals queen papers pure applied mathematics curves seminar queen vol geramita harima migliore shin hilbert function level algebra mem amer math soc stokes zanello pure matroid ann comb appear hanani quadruple systems canad math harima characterization hilbert functions gorenstein artin algebras weak stanley property proc amer math soc harima migliore nagel watanabe weak strong lefschetz properties artinian algebra hausel quaternionic geometry matroids cent eur math hausel sturmfels toric varieties doc math herzog popescu strong lefschetz property simple extensions preprint arxiv hibi said pure combin theory ser huh milnor numbers projective hypersurfaces chromatic polynomial graphs amer math soc huh matroids logarithmic concavity preprint huh katz characteristic polynomials bergman fan matroids math ann iarrobino compressed algebras artin algebras given socle degrees maximal length trans amer math soc iarrobino kanev power sums gorenstein algebras determinantal loci springer lecture notes mathematics springer heidelberg kirkman problem combinatorics cambridge dublin math lam search finite projective plane order amer math monthly lenz realizable matroid complex strictly combinatorics probability computing appear zanello monomial complete intersections weak lefschetz property plane partitions discrete math lindner rodger design theory crc press boca raton lindsey class hilbert series strong lefschetz property proc amer math soc linusson number combinatorica pure known results applications open problems macaulay properties enumeration theory modular systems proc london math soc mathieu sur des fonctions plusieurs sur les former sur les substitutions qui les laissent invariables math pures appl liouville mathieu sur fonction cinq fois transitive liouville journ xviii merino chip firing game matroid complexes discrete models combinatorics computation geometry discrete math theor comput sci maison inform math discrete mimd paris migliore nagel monomial ideals almost complete intersections weak lefschetz property trans amer math soc migliore nagel reduced arithmetically gorenstein schemes simplicial polytopes maximal betti numbers adv math migliore nagel tour weak strong lefschetz properties preprint migliore nagel zanello lower bound second entry gorenstein conjecture stanley proc amer math soc migliore nagel zanello bounds asymptotic minimal growth gorenstein hilbert functions algebra neel neudauer matroids known math mag generalized permutohedra cotransversal matroids pure preprint okonek schneider spindler vector bundles complex projective corrected reprint edition appendix gelfand modern classics basel basel pottonen exists steiner system combin theory ser oxley matroid theory oxford university press oxford pastine zanello two unfortunate properties pure preparation proudfoot matroid complex unpublished note reid roberts roitman complete intersections hilbert functions canad math bull schweig lattice path matroid electron combin sekiguchi upper bound dilworth number rees number noetherian local rings hilbert function adv math stanley complexes higher combinatorics aigner reidel dordrecht boston stanley hilbert functions graded algebras adv math stanley unimodal sequences algebra combinatorics geometry ann new york acad sci stanley combinatorics commutative algebra second progress mathematics boston boston stanley positivity problems conjectures algebraic combinatorics mathematics frontiers perspectives amer math providence stanley enumerative combinatorics vol second cambridge university press cambridge stanley zanello rank function differential poset electron combin stokes matroids arithmetic degree squarefree strongly stable ideals thesis university kentucky swartz matroid complexes combin theory ser swartz finite buildings higher connectivity combin theory ser tahat nonunimodality thesis michigan technological university preparation juan migliore uwe nagel fabrizio zanello veblen wedderburn geometries trans amer math soc watanabe dilworth number artinian rings finite posets rank function commutative algebra combinatorics advanced studies pure math vol kinokuniya north holland amsterdam white theory matroids encyclopedia mathematics applications cambridge univ press cambridge white matroids applications encyclopedia mathematics applications cambridge univ press cambridge zanello extending idea compressed algebra arbitrary algebra zanello extending idea compressed algebra arbitrary cases nonexistence algebra zanello codimension level algebra zanello interval conjectures level hilbert functions algebra department mathematics university notre dame notre dame usa address department mathematics university kentucky patterson office tower lexington usa address department mathematical sciences michigan technological university houghton usa address zanello
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optimal power allocation imperfect hardware analysis untrusted relaying sep networks ali kuhestani student member ieee abbas mohammadi senior member ieee wong fellow ieee phee lep yeoh member ieee majid moradikia muhammad khandaker member ieee abstract taking variety realistic hardware imperfections consideration propose optimal power allocation opa strategy maximize instantaneous secrecy rate cooperative wireless network comprised source destination untrusted relay assume either source destination equipped multiple antennas lsma system rest equipped single antenna prevent untrusted relay intercepting source message destination sends intended jamming noise relay referred cooperative jamming dbcj given system model novel closedform expressions presented high ratio snr regime ergodic secrecy rate esr secrecy outage probability sop improve secrecy performance system optimizing associated hardware design results reveal beneficially distributing tolerable hardware imperfections across transmission reception front ends node system secrecy rate may improved engineering insight equally sharing total imperfections relay transmitter receiver provides best secrecy performance numerical results illustrate proposed opa together appropriate hardware design significantly increases secrecy rate kuhestani mohammadi electrical engineering department amirkabir university technology tehran iran wong khandaker department electronic electrical engineering university college london london yeoh school electrical information engineering university sydney nsw australia moradikia electrical engineering department shiraz university technology shiraz iran index terms physical layer security untrusted relay hardware imperfections optimal power allocation hardware design ntroduction security wireless communication networks conventionally implemented physical layer using key based cryptography complement highly complex schemes wireless transmitters also validated physical layer exploiting dynamic characteristics associated communication links physical layer security pls promising paradigm safeguarding wireless communication networks without incurring additional security overhead massive mimo systems key enabling technology wireless communication networks provide significant performance gains terms spectral efficiency energy efficiency new technology employs coherent processing across arrays hundreds even thousands base station antennas supports tens hundreds mobile terminals additional advantage massive mimo inherently secure traditional mimo systems antenna array exploited transmitter precisely aim narrow directional information beam towards intended receiver received ratio snr several orders magnitude higher incoherent passive eavesdropper however security benefits severely hampered cooperative networks intended receivers may also potential eavesdroppers context pls cooperative jamming involves transmission additional jamming signals degrade received snr potential eavesdropper applied source intended receiver node set nodes source destination source relay beamform jamming noise orthogonal spatial dimension desired signal recently several works considered interesting scenario untrusted relaying cooperative jamming performed intended receiver referred cooperative jamming dbcj real life untrusted relay may collaborate provide reliable communication several practical scenarios may include untrusted relay nodes heterogenous wireless networks intermediate nodes may used assist transmission networks important protect confidentiality information untrustworthy relay concurrently relying increase reliability communication thanks dbcj strategy positive secrecy rate still attained untrusted relay networks recent years several works focused performance analysis power allocation security enhancement untrusted relaying networks specific authors proposed optimal power allocation opa strategy maximize instantaneous secrecy rate relaying network relaying scenario considered exploiting direct link sourcebased artificial noise injection scheme proposed hinder untrusted relay intercepting confidential message power allocation strategy also proposed optimally determine information jamming signal powers transmitted source opa problem imperfect channel state information investigated notably aforementioned works considered perfect hardware communication network practice hardware equipments experience detrimental impacts phase noise imbalance amplifier quantization errors converters mixers filters oscillators imperfections distorts signals way hardware imperfections unavoidable severity imperfections depends quality hardware used transceivers behavior component modeled detail purpose designing compensation algorithms even compensation remain residual transceiver imperfections problem challenging especially high rate systems networks exploiting inexpensive equipments although contributions security based wireless networks assumed perfect transceiver hardware investigated impact particular imperfections imbalance phase noise presence external eavesdropper paper goes beyond investigations considering residual hardware imperfections physical layer security design paper take account opa untrusted relay network nodes suffer hardware imperfections either source destination equipped multiple antennas lsma nodes equipped single antenna dbcj protocol operated first phase destination perfectly removes jamming signal via self interference cancelation second phase system model main contributions paper summarized follows inspired first present generalized system model transceiver hardware imperfections secure transmission network based calculate received instantaneous sndr relay destination formulate opa source destination maximizes instantaneous secrecy rate untrusted relaying accordingly novel solutions derived exact opa addition new simple solutions derived opa high snr regime according opa solutions novel compact expressions derived ergodic secrecy rate esr secrecy outage probability sop high snr regime applied arbitrary channel fading distributions gain insights new expressions presented rayleigh fading channels asymptotic results highlight presence secrecy rate ceiling basically different prefect hardware case highlight ceiling phenomenon independent fading characteristic two hops provide new insights hardware design secure communications end cost constraint transceiver hardware node formulate hardware design problem aforementioned network maximize secrecy rate results reveal secrecy rate improved optimally distributing level hardware imperfections transmit receive front ends node notation use bold lower case letters denote vectors denote identity matrix zeros matrix respectively denote euclidean norm conjugate transpose transpose operators respectively stands expectation random variable denotes probability denote probability density function pdf cumulative distribution function cdf respectively denotes circularly symmetric complex gaussian mean variance diag stands main diagonal elements matrix exponential integral max max stands maximum value ignal ystem epresentation system model shown fig system model consideration wireless network one source one destination one untrusted relay equipped one fig secure transmission presence transceiver imperfections first second figures show downlink uplink transmissions respectively relay acts helper eavesdropper solid lines represent first phase transmission dashed line represents second phase transmission antenna equipped lsma denoted respectively corresponds downlink uplink scenarios cellular system base station equipped lsma mobile user relay equipped single antenna scenario many antennas whereas single antenna opposite applies scenario nodes operate mode accordingly receive transmitted signal transmitting jamming signal hence direct link unavailable also assume channels satisfy reciprocity theorem transmission complex gaussian channel denoted hsr ins hrd respectively transmission denoted hsr hrd respectively consider slow fading channel coefficients vary independently one frame another frame change within one frame additive white noise receiver represented complex gaussian variable variance define snrs per link hence average snrs per branch given represents transmit snr network maximum ratio transmission mrt beamforming maximal ratio combining mrc processing applied node improve overall system performance let represent ratio snrs lsma consider scenario scenario two special cases taken account paper observed numerical results analysis satisfactory even moderate values dbcj technique applied degrade received signal untrusted relay decipher desired information whole transmission performed based tdma based protocol message transmission divided two phases broadcast phase relaying phase consider total transmit power budget power allocation factor transmit powers respectively first phase transmits intended signal power concurrently jams gaussian noise confuse untrusted relay power simplicity transmit power set accordingly second phase transmission simply broadcasts amplified version received signal power order consider residual transceiver imperfections conventional compensation algorithms applied node generalized system model taken account imperfection transmission reception segments denoted respectively introduced distortion noises experimental results many theoretical investigations verified distortion noises gaussian distributions due central limit theorem key property variance distortion noise antenna proportional signal power antenna accordingly scenario diag srns khsr furthermore kst diag khrd diag scenario imperfections cases also given krt khrd design parameters kit kir characterize level imperfections transmitter receiver hardware respectively parameters interpreted error vector magnitudes evms evm determines quality transceivers defined ratio average distortion magnitude average signal magnitude since evm measures joint effect different hardware imperfections compensation algorithms measured directly practice lte evm requirements range kit kir smaller values needed achieve higher spectral efficiencies remark interference analysis paper supports scenario large enough number interfering signals typical wireless environments gaussian assumption interference valid applying central limit theorem signal representation let denote unit power information signal jamming signal respectively according combined impact hardware imperfections generalized channel model received signal scenarios expressed respectively yrul yrdl wst hsr hrd hsr hrd hsr khsr hrd khrd represent mrt transmit weight vectors respectively observe propagated distortion noises noise treated interference untusted relay potential eavesdropper result engineering insight beneficially forward hardware imperfections make system secure instead injecting artificial noise friendly jammer relay amplifies received signal first phase amplification factor krr kst krr note scenario scenario received signal scenarios jamming signal cancelation respectively given hrd hdr hrd wsh hsr hrd ghrdnr hsr hrd hrd noise information signal distortion noise hsr hrd ghrd hsr hrd hrd hrd hdr hrd noise information signal distortion noise according algebraic manipulations sndr given krr kst krr krr based manipulations sndr calculated krr krt krt kst kst analysis assume evms perfectly known consider estimation errors future work krt krr krt krr krr krt define krr total imperfection level respectively remark perfect hardware received snrs perfect hardware derived setting level imperfections nodes zero derived sndrs section reduce special case follows seen mathematical structure derived sndrs complicated compared perfect hardware case since terms manifest denominator propose opa solution general scenario imperfect hardware generalization done section iii main contribution work scenario simplified krr krr krt krt krr moreover scenario obtain based conclude although intercept probability reduced increasing imperfection secrecy rate also degraded therefore great interest intelligently distribute tolerable hardware imperfections across transmission reception radio frequency front ends nodes improve secrecy rate network hardware design approach analyzed section main contribution paper iii ptimal ower llocation section proceeds analyze optimal power allocation problem aim maximizing instantaneous secrecy rate extending results opa solved perfect hardware investigate power allocation factor presence hardware imperfections instantaneous secrecy rate evaluated substituting find since goal distribute power optimally secrecy rate achievable instantaneous secrecy rate reformulated given log monotonically increasing maximization equivalent maximization therefore opa factor obtained solving following constrained optimization problem arg max lemma function section based lemma following corollary corollary function one maximum despite constant functions proposition quasiconcave function feasible set optimal point given proof derivative given seen leads two solutions easy examine feasible solution practical values kit kir derived according corollary find quasiconcave function feasible set make analysis tractable provide new compact expressions opa high snr regime case expressions simplified case scenario expressions changed substituting opa solutions high snr regime expressed following tractable forms rgodic ecrecy ate section derive esr proposed secure transmission scheme case scenarios since straightforward obtain expression exact esr scenarios exact esr includes double integral expressions due complicated structures therefore proceed first deriving new analytical expressions esr high snr regime applied arbitrary channel fading distributions based new expressions derived esr rayleigh fading channels despite prior works literature investigated esr based perfect hardware assumption various untrusted relaying networks take account hardware imperfections new results section generalize recent results esr useful secrecy metric representing rate average secure transmission rate achievable esr given following proceed evaluate parts case scenarios towards goal present following useful lemma lemma positive constants cdf new derived proof start definition cdf follows last probability equals one otherwise equals downlink scenario plugging obtain find terms deterministic constants leads secrecy rate ceiling high snr regime based lemma part given last equation follows integration parts expression straightforwardly evaluated channel fading distribution either directly simple numerical integration furthermore based part constant value conclude amount information leakage independent transmit snr position relay depends evms network nodes replacing compact esr expression achieved channel distribution case rayleigh fading due fact exponential applying part expressed solution substituting esr expression becomes conclude esr exclusively characterized level imperfections nodes function transmit snr channel gain uplink scenario substituting yields observe first hop snr contributes secrecy rate performance following similar procedure scenario esr performance case arbitrary fading distribution expressed case rayleigh fading esr expression given observed esr entirely determined average channel gain first hop transmit snr level imperfections network nodes also find increasing number antennas impact esr large ecrecy utage robability section similar esr results general expressions first presented sop applied channel distribution presence transceiver imperfections high snr regime based derive novel expressions sop rayleigh fading channels sop denoted pso criterion determines fraction fading realizations secrecy rate supported accordingly overall sop defined probability system instantaneous secrecy rate able support target transmission rate pso following focus case respectively downlink scenario substituting based sop definition obtain psodl log last equation follows using lemma worth pointing sop expressions allows straightforward evaluation sop channel fading distribution simple numerical integration conclude sop always target transmission rates threshold depends evms nodes interestingly event holds channel fading distribution network topology transmit snr therefore explained section secrecy rates never achieved due secrecy rate ceiling furthermore conclude target transmission rates smaller threshold pso approaches zero increasing snr similar perfect hardware whereas sop always equals one target transmission rates larger threshold result fundamentally different perfect hardware case sop goes zero increasing snr target transmission rate rayleigh fading channels exponential therefore new simple sop expression presence transceiver hardware imperfection given psodl exp note results section generalize results derived case untrusted relaying perfect hardware uplink scenario similar case sop obtained substituting using sop definition yields pso sop derived exp psoul special case rayleigh fading channels observed numerical results expressions sufficiently tight medium high transmit snrs ardware esign section aim gain insights design hardware maximize secrecy rate untrusted relaying networks depending fixed cost network node show segments transmission reception front ends designed specially given maximum tolerable hardware imperfection node derive new analytical results characterizing hardware imperfections distributed transmission segment reception segment node maximize secrecy rate therefore find kit kir maximize secrecy rate kit kir kitot mathematically speaking goal solve following optimization problem krt krr arg max krt krr krtot tot based instantaneous secrecy rate increasing function transmit snr since aim achieve high transmission rates consider asymptotic snr regime solve hardware design problem observed numerical studies results high snr analysis applied successfully finite snrs asymptotic snr regime random distributions asymptotic received sndrs respectively given substituting secrecy rate ceiling given conclusions insights concluded first secrecy rate ceiling event appears asymptotic snr regime significantly limits performance system event different perfect hardware case esr increases increasing snr note ceiling effect independent fading distribution following focus scenarios separately conclude hardware design overall network downlink scenario following proceed solve optimization problem independently discussing hardware design follows proposition suppose krt krr krtot hence secrecy rate ceiling maximized krt krr tot proof please see appendix tot proposition suppose thus secrecy rate ceiling maximized tot tot tot tot tot proof please see appendix uplink scenario similar scenario two propositions provided follows proposition suppose krt krr krtot thus secrecy rate ceiling maximized krt krr tot proof please see appendix tot proposition suppose hence secrecy rate ceiling monotonically decreasing function proof case negative feasible set based propositions provide following corollary conclusion analysis provides new insights system design corollary consider cooperative network one multiple antennas node communicates single antenna node via single antenna untrusted relay let assume predefined cost assigned node maximize secrecy rate network following considerations taken account according propositions total cost relay node divided half transmission reception front ends better apply level imperfections every transceiver chain instead utilizing mix transceiver chains according proposition design multiple antennas node designers persuaded use hardware reception front end hardware exact opa simulation exact opa simulation approximate opa using approximate opa using epa epa secrecy rate ceiling using ergodic secrecy rate perfect hardware imperfect hardware transmit snr fig ergodic secrecy rate versus transmit snr exact derived expressions perfect imperfect transceiver hardwares number antennas source destination set imperfect case secrecy rate ceiling observed transmission front end hardware imperfections reception end multiple antennas node close zero according proposition design single antenna node quality requirements transmission end obeys observed numerical examples obtain tot typical values evms vii umerical esults iscussions section numerical results provided verify accuracy derived closedform expressions section lsma lsma respectively also cases multiple antennas multiple antennas compare esr performance exact esr simulations opa numerically evaluated finite numbers antennas using bisection method addition equal power allocation epa plotted benchmark furthermore concepts secrecy rate ceiling practical hardware insights section numerically presented numerical evaluations ergodic secrecy rate case case exact opa simulation exact opa simulation approximate opa using approximate opa using transmit snr fig ergodic secrecy rate versus transmit snr exact derived expressions different levels hardware imperfections number antennas source destination set transmission links nodes modeled rayleigh fading channel average channel gains specified moreover lsma number antennas set number antennas set fig depicts esr versus transmit snr cases perfect krt krr kst imperfect cases number antennas set observed figure simulation exact opa evaluated using bisection method good agreement derived high snr solutions perfect imperfect hardwares contrast perfect hardware figure shows esr ceiling phenomenon occurs imperfect hardware reveals performance limits realistic networks high snr regime figure also reveals hardware imperfections low impact low snrs significant high snr regime furthermore observed proposed opa increases secrecy rate floor approximately scenarios respectively compared epa fig examine accuracy derived solutions considering seen numerical theoretical curves opa epa secrecy outage probability opa opa imperfect exact opa simulation perfect exact opa simulation imperfect approximate opa using perfect approximate opa using imperfect epa perfect epa transmit snr fig secrecy outage probability versus transmit snr transmission different target transmission rates perfect imperfect hardware good agreement across snr regimes moreover observed increasing level hardware imperfections achievable secrecy rate degraded approximately high snr regime figs show sop function transmit snr lsma based scenarios respectively different target secrecy rates theoretical curves plotted derived analytical expressions marker symbols generated simulations observed figures negligible performance loss caused transceiver hardware imperfections low target secrecy rate increasing target secrecy rate substantial performance loss revealed interestingly network imperfect hardware always outage secure communications transmit snr exactly predicted analytical results section reason target secrecy rate derived thresholds mentioned sop system always equals one thresholds also seen figures despite opa technique sop curves imperfect hardware perfect hardware slope thus opa secrecy outage probability epa opa opa imperfect exact opa simulation perfect exact opa simulation perfect approximate opa using perfect approximate opa using imperfect epa perfect epa transmit snr fig secrecy outage probability versus transmit snr transmission different target transmission rates perfect imperfect hardware hardware imperfections lead snr offset unveiled curve shifting right sop performance epa technique approaches saturation value high snr regime imperfect hardware observation reveals secrecy performance advantage proposed opa scheme compared epa finally provide figs illustrate insights designing practical systems presented section simulation assume total hardware imperfection tot node equals krtot based propositions maximize secrecy rate design transmission reception front ends krt krr lsma based proposition obtain lsma based proposition design hardwares defining hardware imperfection vector krt krr consider following four different hardware design schemes design designed randomly example design designed optimally based propositions designed randomly design designed randomly designed optimally lsma ergodic secrecy rate opa design opa design opa design opa design epa design transmit snr fig ergodic secrecy rate versus transmit snr lsma various imperfection distributions transmission tot tot reception ends considered lsma design designed optimally lsma lsma results depict hardware design based propositions provides higher esr performance compared case random hardware design design cases optimizing one node designs furthermore show analysis presented section based high snr analysis utilized auspiciously medium snrs addition seen figures mentioned different hardware designs esr performance close together low snr regime difference esr performance designs large high snr regime finally understand figure proposed opa together design significantly outperforms scenario epa design viii onclusion physical transceivers inseparable segments traditional new emerging wireless networks literature works considered impact ergodic secrecy rate opa design opa design opa design opa design epa design transmit snr fig ergodic secrecy rate versus transmit snr lsma various imperfection distributions transmission tot tot reception ends considered hardware imperfections security based transmissions little understood regarding impact untrusted relaying networks paper taking hardware imperfections consideration proposed optimal power allocation opa strategy maximize instantaneous secrecy rate cooperative wireless network comprised source destination untrusted relay based opa solutions new expressions derived ergodic secrecy rate esr secrecy outage probability sop rayleigh fading channels expressions effectively characterize impact hardware imperfections manifest existence secrecy rate ceiling enhanced increasing snr improving fading conditions also illustrate hardware imperfections low impact low snrs significant high snr regime issue reveals hardware imperfections taken account developing high rate systems networks improve secrecy performance network finally presented hardware design approach numerical results depict optimally distributing hardware imperfections transmission reception segments improve secrecy performance ppendix let take derivative krt using chain rule partial derivations follows using obtain substituting krr krtot krt obtain tot krt krtot krt krt krtot krt krt tot krt krtot krt krt krtot krt krt substituting tedious manipulations yields krtot tot tot expression shows concave function krt feasible set krt single solution tot ppendix following similar approach proposition evaluate let substitute tot compute following derivations tot tot tot tot tot tot krt expression obtained similar changing krt substituting manipulations obtain tot tot tot tot tot straightforward see concave function feasible set single solution simply calculated ppendix write using yields considering krt krtot krr one obtain krtot krr substituting solving yields krr tot acknowledgements authors would like thank lajos hanzo helpful comments improve paper eferences mukherjee fakoorian huang swindlehurst principles security multiuser wireless networks survey ieee commun surveys tutorials vol chen gerstacker chen survey techniques physical layer security ieee commun surveys doi yang wang geraci elkashlan yuan renzo safeguarding wireless communication networks using physical layer security ieee commun vol apr rusek persson lau larsson marzetta edfors tufvesson scaling mimo opportunities challenges large arrays ieee sig proc vol larsson tufvesson edfors marzetta massive mimo next generation wireless systems ieee commun vol marzetta noncooperative cellular wireless unlimited numbers antennas ieee trans wireless vol kapetanovic zheng rusek physical layer security massive mimo overview passive eavesdropping active attacks ieee commun vol jun yener secure communication using untrusted relay case cooperative jamming proc ieee globecom new orleans khandaker wong masked beamforming presence eavesdroppers ieee trans inf forens vol khandaker wong robust secrecy beamforming presence eavesdroppers ieee wireless commun vol liu petropulu poor joint jamming wireless physical layer security destination assiatance proc asilomar conference signals systems computer pacific grove usa huang swindlehurst cooperative jamming secure communications mimo relay networks ieee sel areas vol huang mukherjee swindlehurst secure communication via untrusted relay fading channels ieee trans signal vol may sun zhang niu performance study systems untrustworthy relay nodes ieee trans veh vol wang teh secure cooperative relaying networks untrustworthy relay nodes ieee globecom washington mabrouk tourki hamdi secure cooperative 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constrained deep transfer feature learning applications yue qiang ecse department rensselaer polytechnic institute street troy usa sep jiq abstract feature learning deep models achieved impressive results data representation classification various vision tasks deep feature learning however typically requires large amount training data may feasible application domains transfer learning one approaches alleviate problem transferring data source domain target domain existing transfer learning methods typically perform transfer learning often ignore specific properties transferred data must satisfy address issues introduce constrained deep transfer feature learning method perform simultaneous transfer learning feature learning performing transfer learning progressively improving feature space iteratively order better narrow gap target domain source domain effective transfer data source domain target domain furthermore propose exploit target domain knowledge incorporate prior knowledge constraint transfer learning ensure transferred data satisfies certain properties target domain demonstrate effectiveness proposed constrained deep transfer feature learning method apply thermal feature learning eye detection transferring visible domain also applied proposed method facial expression recognition second application experimental results demonstrate effectiveness proposed method applications introduction feature learning deep models active research area recent research demonstrated feature learning methods effective features learnt representation classification input data many computer vision tasks including face recognition object detection scene classification feature second feature space step feature learning transferred target data step constrained transfer learning first feature space step feature learning transferred target data step constrained transfer learning source data source domain transferred data target data raw data space intermediate domains target domain figure framework proposed constrained deep transfer feature learning method algorithm iteratively performs transfer learning feature learning deeper feature spaces learning deep model however typically requires large amount training data hence feature learning domains scarce data feasible transfer learning one approaches address problem help feature learning target domain transferring data knowledge datarich source domain transfer learning compensate lack data target domain also benefit tasks target domain experience gained source domain transfer learning techniques usually involve instance transfer feature transfer model parameter transfer relational knowledge transfer transfer learning techniques used natural language processing document classification etc however typical transfer learning techniques usually perform transfer fixed shallow feature space fixed feature space may effectively fill semantic gap target source domains another issue transfer learning purely data driven without adequately considering certain inherent properties target data tackle issues propose constrained deep transfer feature learning method perform simultaneous transfer learning feature learning exploiting knowledge target source domains general framework shown figure specifically propose iteratively perform constrained transfer learning feature learning several increasingly deeper feature spaces gradually transfer knowledge source domain target domain learn features target domain furthermore propose exploit target domain knowledge incorporate prior knowledge constraint transfer learning ensure transferred data satisfies certain properties target domain principles proposed framework following merits first progressive transfer allows creation several intermediate pseudo domains bridge gap source target domains knowledge transfer continues intermediate domains gradually approach target domain second feature learning performed level knowledge transfer happens also performed progressively higher level knowledge transfer continues hence knowledge transfer feature learning intertwine step improving feature learning knowledge transfer finally imposing constraints transferred data ensure transferred data possess certain desired characteristics also semantically meaningful remaining part paper organized follows section reviews related work section discusses restricted boltzmann machine deep boltzmann machine models section introduces proposed method section shows experimental results section concludes paper related work recently increasing number works concentrate learning good representations data deep models salakhutdinov build hierarchical deep model learn features object recognition handwritten character recognition similar proposed method works perform feature learning based multimodal data different domains example ngiam use deep autoencoder learn common features audio video data deep multimodal dbm constructed learn shared features images texts however contrast existing works learn shared representations across domains work focuses transferring knowledge one domain help feature learning another domain multimodal feature learning methods usually require lot paired training data specifically handle case limited data target domain rich data source domain transfer learning refers learning methods leverage knowledge source domain help learning target domain transferred knowledge includes training instances features model parameters relational knowledge example triadaboost proposed transfer instances within framework figure rbm data one domain rbm transfer learning dbm model adaboost feature spaces found minimize differences also work combines transfer learning feature learning deep features learned dbm model using data source domain selected document classification target domain different work first learns deep features transfers learned features among multimodal data work preforms transfer feature learning multiple levels deep architecture importantly add target domain knowledge constrain transfer learning feature learning procedure background introduce proposed method first review restricted boltzmann machine rbm deep boltzmann machine dbm models rbm undirected probabilistic graphical model figure captures joint probability binary input data data target domain multiple binary hidden nodes exp energy function exp partition function parameters conditional probabilities follows denotes sigmoid function given training data parameters learned maximizing log likelihood stochastic gradient ascend algorithm approximated contrastive divergence algorithm dbm model shown figure assume two layer model thereafter consists one layer visible nodes multiple layers binary hidden nodes represents probability visible nodes similar equation new energy function ignoring bias terms table notations represent target data source data pseudo target data transferred source domain necessary use superscript indicate pairwise data corresponding data domains data one domain data notations pairwise target source data data unpaired target data unpaired source data algorithm constrained deep transfer feature learning data pairwise source target data datap unpaired target data datau unpaired source data datas result learned features stage layer constrained transfer learning learn joint probability target source data constraint transfer unpaired source data sli sampling pseudo target data datau datau unpaired pseudo target data datau etn parameters different layers dbm learned manner jointly fine tuned feature learning using rbm dbm hidden nodes top layer inferred input data using equation techniques considered new feature representations transfer learning input data concatenated data source target domains rbm figure learn joint probability distribution transfer learning performed using joint probability constrained deep transfer feature learning general framework proposed constrained deep transfer feature learning method motivated following intuitions first straightforward think directly applying dbm model illustrated section learn deep features target domain however dbm learning usually requires lot training data variations limited data target domain capture joint distribution target source data bridge source target domains could transfer additional large amount source data target domain pseudo data target domain feature learning second significant domain differences one time transfer may effective stated therefore propose perform transfer feature learning progressively different levels feature spaces generate intermediate pseudo target spaces gradually approach final target space third underlying mechanism produces data target data must follow certain properties want preserve transferred data hence constrain transfer learning ensure satisfaction properties therefore propose constrained deep transfer feature learning method table shows notations general framework shown figure algorithm first stage perform transfer learning feature learning iteratively transfer learning step capture joint distribution feature learning learn features target pseudo target data project data learned feature space constrained transfer feature learning end joint stage jointly features based deeply transferred pseudo target data real target data target source data using rbm figure bridge transfer additional large amount source data pseudo target data sampling conditional distribution addition impose constraint learning third feature learning step transferred pseudo target data combined original target data shaded area figure feature learning using rbm figure pseudo target data real target data projected learned feature space new feature space next level transferred pseudo target data considered source data constrained transfer feature learning next iteration finally similar dbm model figure learned features parameters jointly based deeply transferred pseudo target data real target data iterative transfer learning feature learning converge deep feature learning shown converging transfer learning also converges due gradually reduced gaps target domain source domain iterations following subsections discuss step details constrained transfer learning one layer section first discuss learn joint probability target source data semisupervised manner constraints discuss generate pseudo target data sampling learning joint probability knowledge transfer visible thermal work use rbm model illustrated section learn joint probability target source data shown figure model defines joint probability exp energy function similar format equation recall rbm usually learned contrastive divergence algorithm however standard learning rbm model requires large amount pairwise source target data limited application tackle problem propose semisupervised learning method specifically learn model parameters maximizing log likelihood pairwise source target data datap unpaired target data datau unpaired source data datas arg max data log datau log two parameters balance different terms equation equation marginal distributions data one domain summing missing data domain following algorithm solve optimization problem equation using gradient ascent algorithm gradient model parameters log likelihood term calculated data data model thermal thermal mean eye figure constrained deep transfer feature learning thermal eye detection pairwise visible thermal facial images thermal mean eye generate samples following algorithm estimate model dependent expectations gibbs update model starting samples calculated using data dependent probabilities data datap datau datas data log datau visible represent data dependent model dependent expectations algorithm approximately calculate data dependent expectation need sample unknown variables data dependent probabilities pdata note learning setting data dependent probabilities pdata differ data type specifically pdatap pdataut pdataus pairwise data could directly sample using equation however unpaired data intractable thus use gibbs sampling generate similarly transfer learning constraints every domain data represent observations underlying latent objects often either physical biological process latent objects produce observed data target domain example shown figure case eye detection thermal images transferring knowledge visible domain intensities eyes measure temperature near eye skin surface eye temperature determined blood flow eye skin surface hence intensity distribution near eye thermal image mainly determined underlying latent vascular structure distribution example tear dust area usually high temperature due blood flow artery beneath figure hand pupil area eye lash region temperature expected low lack blood flow unique facial anatomy structures eye regions lead unique distinct target data pattern universal across subjects result ensure physically semantically meaningful transfer propose impose certain constraints transfer learning order preserve target data properties certain unique shape appearance characteristics based intuitions illustrated modify parameter learning problem equation adding constraint arg max data first term log likelihood function defined equation second term cost function ensure transferred data satisfies certain properties solve optimization problem equation depending type could still use gradient ascent algorithm derived gradient first term parameters section need calculate gradient second term parameters detailed calculated discussed section pseudo target data generation sampling given learned joint probability could transfer large amount additional source data datau target domain generate pseudo target data done sampling datau using gibbs sampling method iteratively calls equation feature learning one layer iteration combining transferred pseudo target data real target data goal feature learning learn rbm model figure captures variations target pseudo target data uses hidden nodes new features formally rbm defined equation parameter learning formulated arg max datat datau first term second term represent log likelihood real target data datat datap datau pseudo target data dataet parameter balances two terms learn features apply standard contrastive divergence algorithm difference gradient calculated based terms equation given learned model target pseudo target data new representation calculated using equation next iteration learned feature space constrained transfer learning feature learning continue interact convergence joint feature constrained transfer feature learning stage initially learned features addition deeply transferred pseudo target data top feature layer denoted joint feature need project pseudo data back original space repeatedly calling equation parameters get deeply transferred pseudo data original space based pseudo target data real target data apply standard algorithm deep boltzmann machine model fix point equation used estimate expectation persistent markov chain used estimate model dependent expectation experimental results eye detection thermal images transferring visible domain demonstrate proposed framework constrained deep transfer feature learning applied eye detection thermal images comparing facial analysis visible domain thermal facial analysis sensitive eye localizations however limited works eye detection thermal images furthermore existing thermal eye detection techniques typically use visible image features suboptimal since thermal visible images formed based different principles example thermal images contain limited texture gradient information due heat diffusion phenomenon visible image features usually focus encoding detailed information feature learning thermal images alleviate problem limited thermal training images make difficult leverage existing deep learning models address challenge propose apply proposed constrained transfer feature learning method perform thermal feature learning transferring eye data visible domain thermal domain compensate lack thermal data target domain refers thermal patches including eye patches background patches source domain refers visible eye patches note need transfer eye patches positive data since generate many negative data sampling background thermal eye detection learned thermal features using proposed method train svm classifier search eye scanning window manner implementation details databases use four databases including visible thermal facial behavior database vtfb mahnob laughter database natural visible thermal facial expression database nvie facial recognition technology feret database vtfb synchronized visible thermal videos flir thermal camera subjects additional thermal videos subjects spontaneous facial expressions arbitrary head poses mahnob laughter database provides thermal facial sequences subjects moderate head poses neutral happy facial expressions nvie database contains thermal facial sequences subjects spontaneous posed expressions feret database provides visible facial images different head poses moderate expressions training use synchronized thermal visible images first subjects vtfb images pairwise data use thermal images ditional subjects vtfb images visible images feret images unpaired data test method thermal images remaining subjects vtfb images mahnob database images nvie database images method use two layer dbm hidden nodes learn features rbm models used learn joint distributions hidden nodes respectively experiments height width image patches distance patches normalized background patches least distance away eyes augment training data rotating resizing images following learn rbms hidden nodes raw visible thermal image patches respectively treat values hidden layers preprocessed data speed learning procedure matlab implementation takes hours train full model two layers constraint single core machine impose target domain constraint discussed section propose impose constraint transferred thermal eyes must satisfy general appearance pattern shown fig thermal eye achieve goal propose capture target data pattern using thermal mean eye obtained averaging existing target data basic idea individual thermal eye may vary person unique eye structure averaging thermal mean eye capture commonality canceling differences commonality resulted shared underlying eye structure figure shows mean eye example apparently capture unique pattern thermal eye brighter inner eye corner darker eye center true across subjects different conditions transferring visible eyes thermal domain help thermal feature learning transfer learning step equation becomes arg max nsu iqj denotes positive eye data second term enforces mean transferred pseudo thermal eye close given mean eye ically source data datas htiqj represents expected transferred thermal eye pseudo mean transferred pseudo thermal eyes close given mean eye assume gradient second constraint term objective function equation denoted calculated follows nsu nsu iqj iqj datas evaluation criterion eye detection error fined error max represent detected ground truth eye locations subscript denotes left right eyes regard error successful detection evaluation criteria follows methods fair comparison evaluation proposed method section evaluate proposed constrained deep transfer feature learning method vtfb database note constrained transfer learning step discussed section joint probability thermal visible eyes learned different approaches learning pairwise data using standard algorithm proposed semisupervised learning paired unpaired data discussed section learning pairwise data constraints second term equation learning data constraints full model experiments compare four variations study properties proposed method evaluation constrained transfer learning method layer one first evaluate constrained transfer learning method layer one evaluate learned joint probabilities different learning strategies based two criteria including reconstruction error log likelihood pairwise data reconstruction error refers average pixel difference transferred pseudo thermal eyes sampled ground truth thermal eyes calculate testing set using method seen table semisupervised learning constraint improve performances standard learning method combining together full model achieves best performance table evaluation different constrained transfer learning methods layer one methods pairwise data pairwise constraint semi constraint reconstruction error log likelihood evaluation thermal eye detector constrained deep transfer features subsection evaluate eye detection performance based features learned using proposed method dbm two iterations algorithm constrained transfer learning step layer deep model use four different learning strategies different methods compare proposed method different strategies baseline method baseline method learns features using standard deep boltzmann machine model based thermal data without knowledge transfer shown table even pairwise data proposed transfer feature learning method improves baseline learning constraint boost performances combining together constrained deep transferred features learned data increase detection rate comparing baseline also implement two baselines using dbm visible training data refers dbm feature learning based visible data visible features learned visible data thermal data model parameters thermal feature learning initialized parameters visible features seen table inferior proposed method table eye detection rates using different methods methods dbm baseline pairwise data pairwise constraint semi constraint dbm visible baseline dbm visible finetune baseline eye detection rate learning different amount training data vary training data train proposed method features classifier baseline method eye detection rates shown figure specifically keep reducing number training subjects subjects pairwise data subjects unpaired data comparing proposed method always learns better features baseline comparing constraints important improve performances different sets data dbm semi constraint eye detection rate different training data sets figure learning different amounts training data comparison features compare learned features proposed method standard image features scale invariant feature transform sift feature histogram oriented gradients hog feature fair comparison train linear svm training data experimental results differ due features shown figure learned features proposed method significantly outperforms designed features percentage images successful detections transfer different layers figure shows important perform constrained transfer learning multiple layers since boosts performance comparing one layer transfer addition transferring deeper feature space layer slightly better transferring shallow feature space layer sift features hog features proposed method constraint semi constraint eye detection rate error figure cumulative error distribution curves using eye detectors different features transfer layer transfer layer transfer layers different constrained deep transfer feature learning methods figure transfer different layers visualization figure show learned filters parameters proposed method filters lower level capture local dependencies represent small dots similar gabor filters filters higher level capture global variations frontal image source image target figure neutral left right facial expression recognition comparison existing thermal eye detectors databases best knowledge two works report eye detection results publicly available thermal face databases image patches represented haar wavelet coefficients two successive classifiers coarse fine constructed based features different levels details cascade manner information whole face region used support eye detection thermal facial images identifies face features svm used learn eye detector table illustrates detector outperforms method mahnob laughter database nvie database detector significantly better method different features including learned features using proposed constrained deep transfer feature learning method learned features using conventional dbm handcrafted hog lbp features addition vary number training subjects target domain observations first learned features significantly better features second settings transferring knowledge source domain learned features using proposed method outperform features learned dbm facial expression recognition rate learned filters layer learned filters layer figure learned filters different layers hog lbp dbm proposed method number training subjects target domain table comparison existing thermal eye detectors mahnob laughter database nvie database mahnob laughter database methods reported proposed method detection rate nvie database methods reported proposed method detection rate second application proposed constrained deep transfer feature learning method limited visible thermal domains extended domains simultaneous knowledge transfer deep feature learning section apply proposed framework facial expression recognition second application goal application learn effective feature representation classify neutral face view transferring data frontal view source view target unique facial structure view used constrain transferring learning experiments used database figure contains facial images varying head poses facial expressions illumination conditions training testing figure shows results experiments compared classification accuracies linear svm figure facial expression classification accuracies different features numbers training subjects target domain conclusion paper propose constrained deep transfer feature learning framework order perform feature learning target domain instead performing transfer learning fixed feature space propose simultaneously perform transfer feature learning iteratively increasingly higher level feature spaces order minimize semantic gap source target domains furthermore ensure transferred data semantically meaningful consistent underlying properties target domain incorporate target domain knowledge constraints transfer learning applied proposed framework thermal feature learning thermal eye detection transferring knowledge visible domain also applied facial expression recognition experiments demonstrate effectiveness proposed framework applications future would apply framework vision applications acknowledgements work described paper supported part national science foundation grant number nsf part army research office grant number aro references blitzer mcdonald pereira domain adaptation structural correspondence learning proceedings conference empirical methods natural language processing emnlp pages stroudsburg usa association computational linguistics chen flynn bowyer visible light face recognition comput vis image underst dai yang xue boosting transfer learning proceedings international conference machine learning icml pages mekyska beyond cognitive signals cognitive computation girshick donahue darrell malik rich feature hierarchies accurate object detection semantic segmentation proceedings ieee conference computer vision pattern recognition cvpr gross matthews cohn kanade baker image vision hinton training products experts minimizing contrastive divergence neural computation hinton osindero fast learning algorithm deep belief nets neural computation hinton salakhutdinov reducing dimensionality data neural networks science july martinez binefa pantic facial component detection thermal imagery computer vision pattern recognition workshops cvprw ieee computer society conference pages ngiam khosla kim nam lee multimodal deep learning internation conference machine learing icml pages pan yang survey transfer learning ieee transactions knowledge data engineering petridis martinez pantic mahnoblaughter database image vision computing journal phillips moon rizvi rauss feret evaluation methodology algorithms ieee trans pattern anal mach saenko kulis fritz darrell adapting visual category models new domains proceedings european conference computer vision part pages salakhutdinov hinton deep boltzmann machines proceedings international conference artificial intelligence statistics volume pages salakhutdinov murray quantitative analysis deep belief networks proceedings international conference machine learning pages salakhutdinov tenenbaum torralba learning models pattern analysis machine intelligence ieee transactions srivastava salakhutdinov multimodal learning deep boltzmann machines bartlett pereira burges bottou weinberger editors advances neural information processing systems pages taigman yang ranzato wolf deepface closing gap performance face verification june wang liu peng chen wang natural visible infrared facial expression database expression recognition emition inference ieee trans multimedia wang liu shen eye localization thermal infrared images pattern recognition zhang deep transfer learning via restricted boltzmann machine document classification machine learning applications workshops icmla international conference volume pages zhou lapedriza xiao torralba oliva learning deep features scene recognition using places database ghahramani welling cortes lawrence weinberger editors advances neural information processing systems pages curran associates
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prime ideals regular sequences symmetric polynomials sep neeraj kumar bstract let polynomial ring denote power sum symmetric polynomial xan consider following two questions describe subsets set polynomials generate prime ideal set polynomials regular sequence produce large families prime ideals exploiting serre criterion normality theorem help arithmetic considerations vanishing sums roots unity also deduce several results concerning regular sequences symmetric polynomials ntroduction let polynomial ring sequence elements ring called regular sequence ideal proper image nonzero divisor following notation macdonald let denote power sum symmetric polynomial complete symmetric polynomial elementary symmetric polynomial degree respectively let set positive integers given set denote set power sum symmetric polynomials paper discuss following two questions question let polynomial ring subsets ideal generated set polynomials prime ideal question let polynomial ring subsets set polynomials regular sequence similarly ask questions complete symmetric polynomials elementary symmetric polynomials obvious reasons whenever regular sequence generating prime ideal hpa regular sequence well specifically focus question problems fundamental nature interesting algebraic geometric point view general setting question mentioned eisenbud general extremely difficult prove given ideal polynomial prime chapter powerful methods known showing primeness ideal hochster method principal radical system serre criterion normality theorem use serre criterion normality paper study question began paper dimention zero case conca krattenthaler watanabe question highly key words regular sequence prime ideal symmetric polynomial mathematics subject classification primary secondary neeraj kumar beautiful conjecture conca krattenthaler watanabe states given positive integers gcd form regular sequence abc mod see conjecture necessary condition follows lemma partial evidence support sufficient condition see theorem similarly authors also formulated conjecture three complete symmetric polynomials form regular sequence see conjecture joint paper martino could provide evidence conjectures proving special instances employed technique serre criterion show primeness ideal instance shown ideal hpa prime see theorem also shown ideal prime see proposition way succeeded give families regular sequences help computer calculations proposed conjecture conjecture one main result paper theorem partially answers conjecture paper managed produce families prime ideals exploiting serre criterion help arithmetic considerations vanishing sums roots unity main results paper following let polynomial ring let hpa let suppose smallest prime factor factorization max prime ideal let polynomial ring let let hpa pma prime ideal section contains preliminary results section discuss problem whether two power sum symmetric polynomials generate prime ideal answer certain extent purely terms arithmetic conditions degree polynomials number indeterminates see theorem let hpa pma show prime ideal see theorem show hxna form regular sequence also show two complete symmetric polynomial form regular sequence similar results also hold power sum elementary symmetric polynomials see lemma proposition computer calculations cocoa suggest prime ideal obvious provide evidence example eneralities preliminary results let polynomial ring let power sum symmetric polynomial complete symmetric polynomial elementary symmetric polynomial degree respectively xai xia xia prime ideals regular sequences instance one symmetric polynomials following newton formula see equation respectively aha nen use following lemma prove smoothness symmetric polynomials lemma also use lemma example lemma technical lemma let polynomial ring symmetric polynomials one following iii pxai proof write polynomial involving taking partial derivative one obtains euler formula thus conclude taking partial derivative one obtains proceeding conclude claim iii obvious neeraj kumar given natural number consider roots unity field complex number may ask ourself natural numbers exist roots unity define equation vanishing sum power roots unity weight equation said vanishing sum roots unity weight instance set given let set weights exists vanishing sum roots unity prime factorization form qar theorem lam leung theorem weight set exactly given nqr poonen rubinstein classified minimal vanishing sums weight similar treatment mann one may also see remark note theorem given depends prime divisors multiplicity occur factorization theorem also shows vanishing sum roots unity must weight smallest prime divisor lemma let subset roots unity suppose smallest prime factor factorization let natural number max one proof claim follows remark rime ideals regular sequences begin section useful lemma states partial derivatives complete symmetric polynomial form regular sequence polynomial ring see lemma use lemma show complete symmetric polynomials partial derivatives smooth polynomials see lemma lemma recall irreducibility schur polynomial polynomial ring see theorem special case discuss smoothness schur polynomial see example discuss main results paper theorem theorem respectively also discuss case two complete symmetric polynomials generating prime ideal polynomial ring lemma let let polynomial ring let following holds iii form regular sequence form regular sequence form regular sequence proof let ideal generated partial derivatives prime ideals regular sequences aha thus clearly lemma lemma thus observe moreover proceeding similarly chain containment ideals therefore recall proposition consecutive complete symmetric polynomials form regular sequence thus conclude moreover complete intersection ideal claim obvious let ideal generated partial derivatives proof iii similar except fact time choose carefully reason want use fact know form regular sequence following lemma discuss smoothness symmetric polynomials lemma let let polynomial ring following holds smooth hence irreducible element smooth hence irreducible element iii smooth hence irreducible element proof claims obvious give elementary proof proof iii similar proof non constant polynomial homogeneous bezout theorem hypersurfaces intersects projective space since gives singular point hypersurface suffices prove hxna common zero claim follows lemma following lemma discuss smoothness partial derivatives complete symmetric polynomials lemma let let polynomial ring let smooth hence irreducible proof enough show smooth similar proof given lemma suffices show partial derivatives common zero set notations convenience instance let let ideal generated partial derivatives neeraj kumar also note observation one obtains taking partial derivatives summing get integer number irrelevant thus conclude also also know consider ideal thus get proceeding similarly obtain integer number irrelevant thus conclude also also know thus get also note proceeding similar way obtain following relations thus using similar argument lemma conclude moreover complete intersection ideal thus claim follows following macdonald schur polynomial defined det det prime ideals regular sequences partition integers schur polynomial following relation see equation det remark irreducibility schur polynomial discussed dvornicich zannier let given partition gcd schur polynomial irreducible see theorem recall special forms schur polynomial simply note irreducibility follow theorem lemma show irreducibility polynomials also smoothness obtained results partially extend domain partition irreducibility schur polynomial assuming schur polynomial irreducible one may ask whether true schur polynomial also smooth answer positive case computational evidence shows answer negative general however special case partition form schur polynomial turns smooth discuss proof following example example let polynomial ring let partition schur polynomial smooth proof similar proof given lemma suffices show partial derivatives common zero taking partial derivatives get lemma conclude see unless thus common zero also common zero complete intersection ideal thus claim follows record following convention follow onwards throughout paper conventions list symmetric polynomials fik respective degrees deg always assume unless otherwise specified following proposition see two complete symmetric polynomials always form regular sequence similar results also hold power sum elementary symmetric polynomials neeraj kumar proposition let let polynomial ring following holds form regular sequence form regular sequence iii form regular sequence proof prove lemma irreducible polynomial hence domain irreducible polynomial factored lower degree complete symmetric polynomials nonzero divisor hence form regular sequence proof iii similar seen two power sum polynomials form regular sequence would like know two power sum polynomials generate prime ideal special case answers known due theorem proposition following theorem answer certain extent two power sum polynomials generate prime ideal purely terms arithmetic conditions degree polynomials number indeterminates theorem let polynomial ring let hpa let suppose smallest prime factor factorization max prime ideal moreover form regular sequence hpa proof prime ideal follows theorem thus assume let compute jacobian scalar ignore coefficients since field characteristic zero say jacobian let denotes ideal generated minors jacobian also since generated regular sequence length determinants minors jacobian written thus claim suppose exists since may assume yni moreover also satisfies therefore equations reduce existence solution use fact roots unity say suppose smallest prime factor factorization max prime ideals regular sequences follows lemma possible solution trivial solution hence claim proved thus dim theorem product normal domain since thus write since standard graded also hence thus normal domain prime ideal remark let newton formula observe mod hence replacing xdi may also conclude hpd thus hypothesis theorem need condition hpa remark computer calculations cocoa suggest whenever hpa prime ideal form regular sequence except clear remark form regular sequence computational claim answered prove conjecture certain extent proposition let prime ideal respectively algebraically closed field let ideal generated elements prime ideal proof standard isomorphism sending see integral domains well proposition tensor product also integral domain since algebraically closed field hence claim follows remark note goal proposition generate families prime ideals given prime ideals proposition know form regular sequence also know subset regular sequence regular sequence thus also form regular sequence lemma conclude pma also form regular sequence let hpa pma let ring following theorem show prime ideal show proving normal domain using serre criterion normality theorem let polynomial ring let let hpa pma prime ideal proof follows proposition assume let compute jacobian scaler say jacobian neeraj kumar ignore coefficients since field characteristic zero since generated regular sequence length let denotes ideal generated minors jacobian determinants minors jacobian written xijmm xijaa xijbb positive integers therefore hpa pma xijmm xijaa xijbb claim suppose exists vector distinct nonzero coordinates distinct nonzero coordinates say distinct nonzero coordinates write appears times also satisfy pia thus wia system equation represented matrix form rows columns wav know neither wai choose matrix say first rows rows look solution matrix full rank since possible solution trivial solution hence claim proved similar argument used theorem conclude normal domain prime ideal know ideal prime see theorem see following theorem similar result holds complete symmetric polynomials elementary symmetric polynomials proposition let polynomial ring let one therefore ideals also prime proof follows simple observation ring symmetric polynomials algebra generated power sum polynomials algebra generated also thus one prime ideals regular sequences one may also conclude newton formula theorem prime ideal thus conclude also prime ideals remark let proposition form regular sequence provided newton formula observe mod hence see generate prime ideal form regular sequence thus need condition computer calculations cocoa suggest prime ideal follows proposition prove following example example let polynomial ring let prime ideal proof let compute jacobian say jacobian let denotes ideal generated minors jacobian also since generated regular sequence length determinants minors jacobian written lemma may write thus claim suppose exists assume none zero also assume since make let moreover either assumption thus easy simplification shows possible may argue similarly hence nontrivial solution similar argument used theorem conclude normal domain prime ideal simply note lemma proposition sequence polynomials pna hna form regular sequence respectively partially extend conclusion following proposition proposition let polynomial ring let following holds form regular sequence nak form regular sequences nak neeraj kumar proof newton formula one nekn mod mod otherwise clearly form regular sequence thus claim follows lemma newton formula one ekn mod mod otherwise clearly form regular sequence thus claim also follows lemma inal emarks following question special case question question let polynomial ring let hpa pairs integers prime ideal answer previous question extent theorem observe theorem still weaker form due arithmetic condition max fact one answer prime ideals arises using serre criterion normality vanishing sums roots unity answering following problem problem let consider roots unity field complex number natural numbers exist roots unity similar question one may also ask following question question let polynomial ring let hha pairs integers prime ideal previous question computational calculations cocoa suggest prime ideal recall following definition definition let graded artinian algebra say strong lefschetz property slp exists element multiplication map full rank call property strong lefschetz element let hpa polynomial ring field artinian ring slp see proposition following example show complete intersection ideal hha artinian ring slp example let field characteristic zero let polynomial ring standard grading deg let hha slp prime ideals regular sequences proof consider initial ideal respect lexicographic term order one stanley proved every monomial complete intersection slp strong lefschetz element using fact isomorphic cohomology ring direct product projective spaces complex number field see slp thus proposition conclude slp question let polynomial ring let true ideal hha hnak one initial ideal nak answer previous question positive one construct examples artinian ring slp joint work martino explicitly derive several examples complete intersection ideal generated complete symmetric polynomials one ask similar question complete intersection ideals conclude section one remark recent paper shapiro authors established connection regular sequences complete symmetric polynomials codimention vandermonde variety arbitrary pair positive integer sequence integers shapiro assumption discuss vandermonde variety gcd authors asked problem pairs expected codimention first case variety expected codimention equal authors conclude variety three complete symmetric polynomials form regular sequence problem three complete symmetric polynomials form regular sequence considered conjecture conjecture special case mentioned neither regular sequence clear following proposition proposition let polynomial ring form regular sequence proof newton formula one mod mod mod thus claim follows theorem neeraj kumar eferences conca krattenthaler watanabe regular sequences symmetric polynomials rend semin math univ padova cocoateam cocoa system computations commutative algebra available http dvornicich zannier newton functions generating symmetric fields irreducibility schur polynomials adv math eisenbud commutative algebra view toward algebraic geometry graduate texts mathematics new york shapiro vandermonde varities relations among schur polynomials harima watababe strong lefschetz property artinian algebras nonstandard grading algebra hochster big modules algebras embeddability rings witt vectors queen papers pure applied math kumar martino regular sequences power sums complete symmetric polynomials matematiche catania lam leung vanishing sums roots unity journal algebra macdonald symmetric functions hall polynomials second edition contributions zelevinsky oxford mathematical monographs oxford science publications clarendon press oxford university press new york mann linear relations roots unity mathematika milne algebraic geometry allied publishers poonen rubinstein number intersection points made diagonal regular polygon siam discrete math stanley weyl groups hard lefschetz theorem sperner property siam algebraic discrete methods wiebe lefschetz property componentwise linear ideals gotzmann ideals comm algebra ipartimento atematica niversit enova odecaneso enova taly address kumar
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draft optimal control linear systems limited control actions control jan burak demirel euhanna ghadimi daniel quevedo mikael johansson consider optimal control problem limited number control messages allowed sending controller actuator restrict number control actions computed transmitted controller employ mechanism decides whether control message needs calculated delivered due nature algorithms finding optimal control sequence requires minimizing quadratic cost function domain paper firstly provide exact solution problem mentioned solving exponential number quadratic programs reduce computational complexity propose two efficient heuristic algorithms based greedy search alternating direction method multipliers admm method later consider receding horizon control strategy linear systems controlled controllers also provide complete stability analysis receding horizon control uses finite horizon optimization proposed class numerical examples testify viability presented design technique index terms optimal control linear systems eventtriggered control receding horizon control ntroduction problem determining optimal control policies linear systems extensively investigated literature see standard optimal control problem assumes unlimited number control actions available actuator although assumption valid many applications hold specific scenarios communication computation resources become scarce example control systems constrained actuation resources control systems shared processor resources information exchange shared communication channel reduce communication burden controller actuator imer introduced constraint number control actions designing optimal control policies linear scalar systems later bommannavar shi extended work class systems problem proposed also formulated control problem introduction cardinality constraint changes standard control demirel quevedo faculty electrical engineering information technology university paderborn warburger str paderborn germany dquevedo ghadimi huawei technologies sweden skalhogatan box kista sweden johansson access linnaeus center school electrical engineering kth royal institute technology osquldas stockholm sweden mikaelj problem quadratic programming mixedinteger quadratic programming miqp problem proved problem therefore gao lie provided efficient branch bound algorithm compute optimal control sequence since cardinality constraint highly convex relaxation problem based optimization considered literature see although regularization cost function effective way promoting sparse solutions performance guarantees provided terms original cost function due transformed cost function control systems broadly used literature reduce amount communication controller actuator guaranteeing attainable control performance see distinct sparse control techniques mentioned earlier eventand control require design feedback controller computes control actions triggering mechanism determines control input updated vast majority works literature first designed controller without considering sparsity constraint subsequent design phase developed triggering mechanism fixed controller contrast another line research concentrates design feedback control law respecting predefined triggering condition design algorithms extremely beneficial context model predictive control mpc strategies see mpc control scheme solves optimal control problem sampling instant applies first element resulting optimal control input trajectories use algorithms therefore reduces frequency solving optimization problems transmitting control actions controller actuator consequently saves computational communication resources lehmann proposed strategy systems additive bounded disturbances controller computes new control command whenever difference actual state predictive state exceeds threshold sijs combined state estimation mpc order design estimation framework authors combined mpc sampling strategies based iss concept different aforecited works authors studied infinite horizon quadratic cost works discussed focus systems however also substantial number works considers systems literature see instance authors investigated draft stability mpc algorithms nonlinear systems yet consider disturbance later shi studied mpc problem nonlinear systems subject bounded disturbances contributions paper formulate optimal control problem thresholdbased algorithm dictates communication controller actuator propose various algorithms compute control action sequence provides optimal solutions problem general hard solve due two main reasons combinatorial nature since decisions binary variable transmit transmit introduction triggering condition leads optimizing convex cost function domain main contributions paper threefold show optimal solution control problem mentioned determined via solving set quadratic programming problems provide efficient heuristic algorithm based admm design control input sequence provides solution optimal eventtriggered control problem iii describe receding horizon implementation control algorithm including proof practical stability outline remainder paper organized follows section formulates control problem introduces assumptions section iii provides simple procedure designing optimal control laws minimize cost function section presents two heuristic methods usually achieves tolerable performance significantly reducing computational complexity section receding horizon control scheme algorithm presented section demonstrates effectiveness advantages presented approach section vii concludes paper notation write positive integers real numbers let denote set real vectors dimension set real vectors dimension vectors written bold lower case letters matrices capital letters two vectors notation corresponds inequality set real symmetric positive matrices dimension denoted let column vectors zeros vectors ones kronecker product two matrices denoted given norm defined max square matrix denotes maximum eigenvalue terms magnitude notation stands power set set written set subsets including empty set cardinality set denoted inite orizon ptimal riggered ontrol consider feedback control loop depicted fig dynamics physical plant described linear system state variable time instant control input time instant system matrices appropriate dimensions given initial condition pair assumed stabilizable necessarily schur stable paper assume sensor takes periodic samples plant state transmits samples controller node controller eventtriggered computes new control commands transmits actuator times given threshold case controller compute new control actions actuator input plant set zero worth noting denotes set time instants control computations needed turn plant runs open loop notice given set variables contrast generated initial state tentative control actions hence determine set necessary compute tentative control actions words set time instants strongly coupled tentative control inputs throughout paper aim designing admissible optimal control sequence minimize quadratic cost function matrices symmetric positive semidefinite symmetric positive definite consider constrained optimal control problem minimize subject recall set time instants computations also transmissions needed defined draft fig control system process actuator sensor controller remark note control problem described section particular class hybrid systems one may therefore convert equivalent mixed logical dynamical mld system represents system using blend linear binary constraints original variables auxiliary variables see paper instead using mld system formulation exploit geometric properties problem hand allows propose optimization problem optimal control establish stability result receding horizon implementation discussion although problem resembles optimal control problem proposed fundamental difference stems question whether scheduling exogenous input autonomously generated solving control problem one needs optimize control action scheduling sequence strongly coupled optimization process hand control systems switched systems internally forced switchings problems scheduling sequences generated implicitly based evolution state control signal major difficulty problem scheduling depends particular initial condition tentative control input explicitly determined unless specific control signal given see survey paper references therein iii omputation optimal control actions section concentrate finding control input sequence solves optimal control problem proposed therefore present framework based dividing domain convex subdomains simple yet effective procedure follow however number convex optimization problems needs solved grows exponentially length control horizon optimal control problem hard solve due restriction control space nevertheless without loss generality possible convert problem set fig decision tree diagram denote means represents set time instances new control signal need computed transmitted controller actuator figure dark colored circles represent case whereas light colored circles represent case instance set gets values optimization problems form minimize subject obtain optimal solution problem necessary solve problem subsets power set select one providing lowest cost see fig precisely written minimize number needed solved grows exponentially control horizon bear mind since independent also solved parallel addition combinatorial nature problem requires optimization convex function nonconvex domain literature exist available tools solve convex objective optimization see feasible region expressed finite union polyhedra disjunctive formulation applied particular assuming problem instance feasibility region divided disjunctive following construction write state dimension see fig example two dimensional problem disjunctive set shown different colors draft dimensional polyhedra gives sequence given rewrite optimization problem minimize subject necessary solve possible combination problem determine global optimal solution particularly problem solved sequence quadratic programming problems hence following series optimization problems minimize worth noting even though number grows exponentially problems independent therefore one parallelize problems reduce computation time corollary global optimal solution control problem computed solving set convex quadratic programming problems form parallel selecting solution provides lowest control cost among feasible solutions following examples give reader better understanding problem example let set divided four convex subsets illustrated different colors fig optimal control inputs computed solving quadratic programs different combinations convex sets order compute state variable define constant function time takes values whose value determines time state variable belong interior polyhedron divide set convex subsets time step choose one active set switching signal otherwise state variable fig set divided four convex regions illustrated different colors given scheduling sequence optimal control inputs computed solving optimization problem possible combinations five convex sets finding optimal control actions requires solving problems selecting minimum control cost among feasible solutions fig blue curve represents state trajectory red curve represents state trajectory magenta curve denotes state trajectory optimal control inputs necessary solve convex quadratic programming problems example let consider system initial condition controller transmits control messages state variable greater performance indices given diag control horizon chosen state trajectories depicted fig obtained via solving optimization problem given sets fig shows state trajectories two different sets set leads global optimal solution example demonstrates obtaining optimal solution optimal control problem mentioned earlier requires solving exponential number qps therefore next section propose heuristic algorithms significantly reduce computational complexity compromising optimality guarantees slightly omputation sub optimal control actions section propose two heuristic algorithms usually achieve low control loss significantly decreasing computation time firstly develop simple algorithm takes initial state input finds suitable sequence greedy manner increasing horizon problem secondly apply algorithm based admm compute heuristic scheduling sequence sequence used solve disjunctive problem obtain good approximate control sequence draft data compute based iteration count obtain solving compute set end return algorithm greedy heuristic techniques polynomial time complexity hence reducing computational efforts significantly numerical investigations see section indicate heuristic algorithms usually provide close optimal solutions greedy heuristic method construct intuitive greedy algorithm solve optimal control problem proposed efficiently discussed earlier given feasible switching sequence corresponding optimal control problem formulated quadratic programming problem form one solves combinatorial problem find optimal switch sequence thereby solving optimal control problem good way deal combinatorial nature type algorithms reduce computational complexity employ greedy algorithm solves global problem making series locally optimal decisions greedy algorithms necessarily provide optimal solutions however might determine local optimal solutions used estimate globally optimal solution reasonable time proposed greedy algorithm works follows since known compute corresponding switching signal set solve number qps form obtain optimal switch signal procedure repeats step solve times problem find best keeping previously found scheduling sequence unchanged algorithm provides detailed steps discussed greedy heuristic method algorithm executes number qps compared optimal control procedure result runs polynomial time heuristic method based admm section develop computationally efficient method solving approach based alternating direction method multipliers admm well developed technique solve disciplined problems idea utilizing admm heuristic solve problems recently considered literature particular employ admm approximately solve problems convex costs constraints approximate admm solutions improved multiplicity random initial starts number local search methods applied admm solutions paper using techniques introduced first apply admm original problem achieve intermediate solution algorithm general problem necessarily converge however corresponding solution gives good initial guess finding optimal solution problem second phase heuristic method perform polishing technique aim achieving feasible hopefully closer optimal solution see overview polishing techniques polishing idea based disjunctive programming problem fixed event index trajectory sequence initially developed solve convex structured problems admm recently advocated great extent effective approximation technique solve optimization problems see references therein start casting admm form essentially minimize collects state control components represents positive definite hessian quadratic cost blkdiag moreover indicator function enforcing eventtriggering control law holds otherwise detects constraint violation terms corresponding constraints twofold first constraint describes lti system dynamics rnn following blocks gij otherwise finally last constraint ensure nonconvex constraint asymptotically satisfied formulating problem iteration admm algorithm consists see complete treatment draft subject denotes iteration counter penalty parameter denotes projection onto nonconvex constraint given otherwise cost function hand convex constraints set closed convex admm algorithm converges optimal solution problem case since admm algorithm may converge optimal point moreover examples found even converge feasible point improve admm solution utilize additional convex problem polish intermediate results restrict search space convex set includes admm solution pattern let output admm algorithm sets denote trajectory restriction sequence associated index respectively heuristic polishing algoprocedure formulates rithm provides summary heuristic method input variables include threshold randomized initial state well zbest fbest store final solution comments related algorithm order argmin contrast admm algorithms convex problems converge optimum positive range parameter stability admm algorithm problems sensitive choice moreover similar convex case convergence speed also affected choice variety techniques including proper stepsize selection relaxation constraint matrix preconditioning caching employed accelerate admm procedure see reference topic one utilize adaptive iteration count procedure improve probability finding feasible solution procedure works following run admm algorithm fixed number iterations check disjunctive problem feasible solution increase number iterations double repeat procedure run admm disjunctive finding feasible solution reaching point current admm solution data zbest fbest iteration count update kgz fbest zbest end end fbest obtain zbest disjunctive problem end return zbest algorithm heuristic improve previous one terms constraint violations quadratic cost variable initialized randomly using normal distribution general repeating admm algorithm multiple initial points may improve approximate solution cost extra computational complexity suggested however application experience significant performance improvement repeating admm algorithm steps algorithm different initializations one reason could algorithm employ based admm sosolution trajectories lution disjunctive problem using caching ldl matrix factorization techniques gorithm costs order dense matrices furthermore assume max admm iteration costs order means overall cost admm algorithm number admm iterations application proper algorithm parameter selection admm algorithm converges almost always within hundreds iterations see results presented section computational complexity disjunctive step heuristic method falls generic complexity convex qps order eceding horizon control except special cases finding solution infinite horizon optimal control problem possible receding horizon control alternative scheme infinite horizon problem repeatedly requires solving constrained optimization problem time instant starting current state assuming full measurement state available current time following cost function subject system dynamics constraints involving states controls minimized finite horizon draft prediction horizon set fig shows state trajectories receding horizon implementation control system described different initial conditions seen figure number transmitted control commands depends initial condition state variable umerical examples performance control actions state variable fig evolution state trajectories system receding horizon strategy constraints set initial conditions function defines stage cost defines terminal cost denote minimizing control sequence function current state control applied plant time first element sequence time stepped forward one instant procedure described repeated another ahead optimization horizon worth noting due fundamental limitation optimization problem state never converges origin system unstable converge set including origin following result shows existence set theorem suppose neighborhood origin system constraints uniformly practically asymptotically stable lim theorem establishes practical stability system constraints shows provided conditions met plant state ultimately bounded ball radius worth noting stability results use lyapunov techniques bound tight next would like exemplify convergence state set set initial conditions example consider system given initial conditions sin cos controller transmits control message whenever condition holds performance indices chosen diag illustrate performance benefits proposed heuristic algorithms consider plant following representation assume disturbance acting plant aim minimize performance indices diag horizon length initial condition chosen sin cos sin sin cos perform simulations selected different data points equidistant half spherical surface evaluated performance greedy heuristic algorithms initial conditions data point half sphere run number qps obtain global optimal solution total run million number qps evaluation used linux machine cores intel core extreme edition solve qps parallel threads within computing resources takes approximately days perform simulations one problem setup compare true optimal control performance heuristic method performed algorithm problem set particular evaluated three sets problem generally expected control problem becomes challenging increasing run heuristic algorithm adaptive iteration count procedure described section moreover picked respectively aforementioned problem scenarios setup cases admm method produced always feasible solutions maximum number however method created infeasible solutions two initial conditions compute feasible solutions needed adjust accordingly initial conditions fig shows outcome comparison shows heuristic algorithm able find near optimal solutions relative error less draft jadmm jadmm jadmm exhaustive search admm exhaustive search admm fig relative error jadmm cardinality difference heuristic solution optimal cost problem instances third order plant model exhaustive search admm jgreedy jgreedy jgreedy fig relative error jgreedy cardinality difference greedy solution optimal cost problem instances third order plant model almost cases algorithm finds solutions relative error less problem instances average optimality gap jadmm three cases respectively finally comparing cardinality control event index optimal solutions admm method one concludes heuristic method cases finds solution optimum ones rest cases tends find sparse control actions also compared true optimal control performance greedy heuristic method shown algorithm three sets problems mentioned greedy algorithm detected optimal solutions aforementioned problems respectively also computed optimal solution relative error less problems hand three problem setups important note greedy heuristic method always provided feasible solutions problem cases average optimality gaps jgreedy three cases respectively compared heuristic method optimality gaps became larger receding horizon control implementation consider unstable system represented prediction horizon uadmm fig states input versus time model predictive controller presented section figure represent state variables whereas uadmm denote optimal heuristic control inputs computed based exhaustive search admm algorithm respectively chosen performance indices diag receding horizon implementation set fig demonstrates time responses eventtriggered receding horizonicontrol system initial conh dition two different approaches computing control input trajectories considered exhaustive search algorithm heuristic admm algorithm seen fig state trajectories associated two control approaches deviate one another except first sampling instants match together corresponding control inputs follow rather similar sparsity pattern receding horizon control via exhaustive search algorithm uses communication channel times sampling instants results reduction communication computational burden whereas receding horizon control via admm algorithm uses channel times sampling instants leads reduction communication computational burden aforementioned initial value exhaustive search achieves better control cost compared control cost heuristic however conclusion hold general particular experienced cases draft control loss programming formulation obtain optimal solution via solving exponential number quadratic programs proposed heuristic algorithms reduce computational efforts achieving acceptable performance later provided complete stability analysis receding horizon control uses optimization proposed class numerical study confirmed theory communication rate fig comparison control performance communication frequency different values shown gray scale marked curve obtained averaging monte carlo simulations horizon length samples process noise initial condition generated randomly starting different initial value receding horizon controller derived exhaustive search performs worse one based heuristic communication rate control performance consider case random disturbance modeled gaussian white process acting plant characterized injection matrix diag initial condition modeled random variable normal distribution zero mean process noise independent initial condition control performance measured quadratic function communication rate measured visualization control loss communication rate demonstrated fig different data points curve correspond different eventthreshold values ranging control loss communication rate evaluated via monte carlo simulations note communication rate decreases dramatically increased control loss threshold varies dark colors lighter color quantities become less sensitive changes threshold value also identify particularly attractive region large decrease communication rate obtained small loss control performance vii onclusion paper considered feedback control system rule dictates communication controller actuator investigated optimal control problem additional constraint communication optimization problem general used disjunctive viii ppendix begin definition practical stability control systems definition practical stability system said uniformly practically asymptotically stable positively invariant set exist nonnegative constant definition function function said function system positively invariant set exist compact set neighbourhood origin constants proof theorem prove practical stability system set control actions fulfills constraints analyze value function min seen borrow shifted sequence approach described use feasible control sequence optimality property obtain following bound investigate quantity based two possibilities suppose feasible control law following inequality holds draft schur follows implies since follows therefore iterating possible exponentially bound state evolution via using norm inequality see get used norm inequality thus lim see suppose hence follows result conclude following inequality holds inserting yields let binary variable time instant represent transmission control action follows otherwise feasible control sequence exists associated scheduling sequence denoted applying feasible control inputs upper bound becomes max denotes set possible feasible schedules generated stabilizing control sequence obtain lower bound hence establish following relation eferences anderson moore optimal control linear quadratic methods englewood cliffs prentice hall bertsekas dynamic programming optimal control athena scientific introduction stochastic control theory dover publications gallieri maciejowski asso mpc smart regulation overactuated systems american control conference hartley gallieri maciejowski terminal spacecraft rendezvous capture lasso model predictive control international journal control vol chyba grammatico huynh marriott piccoli smith reducing actuator switchings motion control autonomous underwater vehicle proceeding american control conference gupta control algorithm processor availability proceedings hybrid systems control computation conference demirel gupta quevedo johansson threshold optimization control systems proceedings international workshop discrete event systems imer optimal control limited controls proceedings american control conference bommannavar optimal control limited control actions lossy transmissions proceedings ieee conference decision control shi yuan chen finite horizon lqr control limited communication ieee transactions automatic control vol july gao cardinality constrained control chinese university hong kong master 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machine learning vol takapoui moehle boyd bemporad simple effective heuristic embedded quadratic programming appear proceedings american control conference derbinsky bento elser improved threeweight algorithm ghadimi teixeira shames johansson optimal parameter selection alternating direction method multipliers admm quadratic problems ieee transactions automatic control vol march giselsson boyd linear convergence metric selection douglas rachford splitting admmlitting admm boyd vandenberghe convex optimization cambridge university press jiang wang converse lyapunov theorem discretetime systems disturbances systems control letters vol rawlings mayne model predictive control theory design nob hill publishing bernstein matrix mathematics princeton university press
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variable screening multiple studies tianzhou zhao george department biostatistics university pittsburgh department statistics university pittsburgh oct abstract advancement technology generated abundant data allows integration multiple relevant studies due huge computational advantage variable screening methods based marginal correlation become promising alternatives popular regularization methods variable selection however screening methods limited single study far paper consider general framework variable screening multiple related studies propose novel screening procedure using estimator highdimensional regression analysis framework compared procedure sure independence screening sis procedure procedure greatly reduces false negative errors keeping low false positive rate theoretically show procedure possesses sure screening property weaker assumptions signal strengths allows number features grow exponential rate sample size addition relax commonly used normality assumption allow distributions simulations real transcriptomic application illustrate advantage method compared sis method key words phrases multiple studies partial faithfulness estimator sure screening property variable selection introduction many areas scientific disciplines nowadays omics studies including genomics transcriptomics etc biomedical imaging signal processing high dimensional data much greater number features sample size become rule rather exception example biologists may interested predicting certain clinical outcome survival using gene expression data far genes number samples advancement technologies affordable prices recent biomedical research experiments performed related hypothesis explore scientific question since data one study often small sample size limited statistical power effective information integration multiple studies improve statistical power estimation accuracy reproducibility direct merging data usually less favored due inherent discrepancy among studies tseng new statistical methodologies theories required solve issues problem integrating multiple related studies various regularization methods developed past two decades frequently used feature selection regression problems popular methods include limited lasso tibshirani scad fan elastic net zou hastie adaptive lasso zou group structure exists among variables example set gene features belonging pathway group version regularization methods applied yuan lin meier nardi one refer fan huang detailed overview variable selection group selection models number features grows significantly larger sample size regularization methods perform poorly due simultaneous challenges computation expediency statistical accuracy algorithmic stability fan variable screening methods become natural way consider first reducing lower moderate dimensional problem performing variable regularization fan first proposed sure independent screening sis method select features based marginal correlations response context linear regression models showed fast selection procedure enjoyed sure screening property since development sis many screening methods proposed generalized linear models fan chang nonparametric additive models semiparametric models fan chang quantile linear regression gaussian graphical models luo liang exploit robust measures sure screening zhu however screening methods limited single study far paper first propose general framework simultaneous variable screening multiple related studies compared single study scenario inclusion multiple studies gives evidence reduce dimension thus increases accuracy efficiency removing unimportant features screening knowledge paper first utilize multiple studies help variable screening linear regression model framework provides novel perspective screening problem opens door development methods using multiple studies perform screening different types models different marginal utilities framework natural apply selected screening procedure individual study respectively however important features weak signals studies may incorrectly screened screening performed avoid false negative errors fully take advantage multiple studies propose screening procedure one additional step combining studies potential zero correlation added procedure second check procedure potential save important features weak signals individual studies strong aggregate effect across studies screening stage compared naive multiple study extension sis method procedure greatly reduces false negative errors keeping low false positive rate merits confirmed theoretical analysis specifically show procedure possesses sure screening property weaker assumptions signals allows number features grow exponential rate sample size furthermore require data distribution via using novel statistics thus procedure applied general distribution family gaussian distribution considered fan related screening procedure single study scenarios screening apply two general applicable variable selection algorithms multiple study extension algorithm proposed well feature selection method choose final model lower dimension rest paper organized follows section present framework variable screening multiple related studies well notations propose screening procedure section section provides theoretical properties procedure demonstrates benefits multiple related studies well advantages procedure general algorithms variable selection follow screening procedure discussed section section include simulation studies real data application three breast cancer transcriptomic studies illustrate advantage method reducing false negative errors retaining important features compared sis method conclude discuss possible extensions procedure section section provides technical proofs major theorems model notation suppose data related studies observations consider random design linear model study cov var uncorrelated assume implicitly large usually assume small set covariates true predictors contribute response words assume equal zero vector addition paper assume either zero studies framework partially motivated linear random effect model considered literature jiang vector specifically first components consider random effect model true predictors study treated random effect distributed independent consequently either zero studies probability one assumption fits reality well example typical gwas study small pool snps reported associated complex trait disease among millions jiang observations model purpose identify thus define following index sets active inactive predictors target clearly setting complementary identification equivalent identification let denotes cardinality screening procedure multiple studies sure independence screening single study fan first proposed variable screening method called sure independence screening sis ranked importance variables according marginal correlation response showed great power preliminary screening dimension reduction regression problems later introduced partial faithfulness condition zero partial correlation separating set implied zero regression coefficient showed held almost surely joint normal distribution extreme case equivalent sis method purpose sure screening identify set moderate size still contain true set equivalently try identify subsets contain unimportant features need screened two potential errors may occur sure screening methods fan false negative important predictors marginally uncorrelated jointly correlated response fail selected false positive unimportant predictors highly correlated important predictors higher priority selected relatively weaker important predictors current framework variable screening multiple studies able relieve errors significantly indeed multiple studies model setting thus evidence exclude noises reduce errors single study addition sure screening used reduce dimension first stage always include second stage variable selection methods lasso dantzig selection refine set reduce errors errors occur signals falsely excluded screening suppose marginal correlation jth feature response try find set screen assumption partial faithfulness explicit definition see section variables zero coefficients sure errors guaranteed excluded however might true empirical version marginal correlation single study rule errors empirical case ratio large least order log bonferroni adjustment current setting multiple studies requirement strong signals remains naively perform screening individual study see next propose novel screening procedure allows weak signals individual studies long aggregate effect strong enough therefore procedure able reduces errors framework multiple studies closing section worthwhile mention perform screening test one usually applies fisher sample correlation however require bivariate normality assumption alternatively paper propose use estimator correlation works generally well even data shao similar ideas applied estimation large covariance matrix cai liu screening procedure multiple studies presence multiple studies evidence reduce dimension imply zero coefficient feature one hand possible features zero multiple hand studies thus aim identify following two complementary sets performing screening multiple studies min min know sure partial faithfulness assumption chance detecting zero marginal correlation least one study greatly increased increasing thus unimportant features likely screened compared single study scenario one way estimate test feature tests rejected feature exclude feature call sure independence screening procedure short viewed extension screening test multiple study scenario however reality possible important features weak signals thus small least one study features might incorrectly classified since weak signals indistinguishable null signals individual testing lead serious problem false exclusion important features final set screening significantly improved adding second step combine studies potential zero correlation fail reject null identified first step perform another aggregate test features weak signals multiple studies long aggregate test statistics large enough retained procedure conservative screening features first step alone guarantee reduce false negative errors simplicity assume observations obtained studies straightforward extend current procedure analysis scenarios different sample sizes across multiple studies thus omitted proposed aggregation sure independence screening procedure short formally described step screening study first step perform screening test study obtain estimate study set potential zero correlations xij sample covariance xij estimator covariance cdf standard normal distribution significance level study test include study step screen variables instead separates potential zero correlations preparation next step define cardinality potential zero correlation sure retain feature consider step otherwise move step remark scaling property sufficient impose assumptions stan dardized variables var var thus also treated estimator correlation thus define var cov remark analysis index set shown coincide introduced details section step aggregate screening second step wish test whether aggregate effect potential zero correlations identified step strong enough retained define statistics statistics approximately follow distribution degree freedom null thus estimate equivalently estimate cdf distribution degree freedom equal significance level second step takes sum squares studies potential zero corp relation test statistics feature test rejected conclude aggregate effect strong feature needs retained otherwise screen step performs second check addition individual testing step potentially saves important features weak signals individual studies strong aggregate effect table use toy example demonstrate idea compare two approaches example suppose five studies three features two signals one noise strong signal studies weak signal studies noise hypothesis testing small zero give small marginal correlation sometimes indistinguishable suppose used threshold corresponding table toy example demonstrate strength screening procedure signal signal noise strong signal studies large marginal correlations procedures include correctly weak signal since many studies small correlations incorrectly screened onestepsis procedure false negative however procedure saves second step noise methods tend remove screening theoretical properties assumptions conditions impose following conditions establish model selection consistency procedure condition exist constants exp tzj exp addition exist min number studies constant dimension satisfies defined next log logn logn addition require log large positive constant first condition assumes standardized variable marginally follow distribution study condition relaxes normality assumption fan second part assumes always exist positive greater minimum variance particular jointly follows multivariate normal distribution always pick second condition allows dimension grow exponential rate sample size fairly standard assumption analysis many sure screening methods like sis tpc used assumption fan though algorithm assumes polynomial growth function notice readily relaxed exponential level require product small used control errors second step screening procedure always true conditions assumes lower bound correlation features words marginal correlation zero must large enough marginal correlation detected key assumption single study many sure screening methods fan impose assumption rather condition used control type error step features condition gives assumptions features assume correlations small large studies strong weak signals well separated first step helps control type error step features studies require sum squares correlations greater threshold type error controlled step condition different methods single study scenario usually assume lower bound marginal correlation features like relax condition put restriction norm allows features weak signals study combined strong signal appreciate relaxation compare minimal requirements without step order detect feature need log large constant thus least log comparison assumption much weaker reasonable settings log consistency screening procedure state first theorem involving consistency screening step theorem consider sequence linear models satisfy assumptions conditions define event exists sequence log proof theorem found section theorem states screening first step correctly identifies set features strong weak signals well separated chance incorrect assignment low given results theorem show main theorem consistency screening procedure theorem consider sequence linear models satisfy assumptions conditions know exists sequence log log constant proof theorem found section result shows screening procedure enjoys model selection consistency identifies model specified high probability choice significance level yields consistency log partial faithfulness sure screening property first came partial faithfulness assumption theoretically justified use marginal correlation partial correlation screening follows implies set variables conditioned independence screening two conditions positive definiteness regression coefficients realization common absolutely continuous distribution showed partial faithfulness held almost surely theorem since random effect model described section also satisfies two conditions partial faithfulness holds almost surely study thus readily extend theorem scenario multiple studies corollary consider sequence linear models satisfying partial faithfulness condition study true active inactive set defined following holds every implies proof straightforward thus omitted study partial faithfulness particular since consider features zero studies case independence screening imply zero model selection consistency theorem extended partial faithfulness condition corollary sure screening property screening procedure immediately follows corollary consider sequence linear models satisfy assumptions conditions well extended partial faithfulness condition corollary exists sequence log log proof corollary simply combines results theorem extended partial faithfulness skipped algorithms variable selection multiple studies usually performing sure screening may remove enough unimportant features case since multiple studies expect screening procedure remove many unimportant features single study dimension still high applying screening procedure readily extend screening procedure iterative variable selection algorithm testing partial correlation gradually increasing size conditional set since method multiple study extension simple algorithm call algorithm section hand dimension already greatly reduced screening simply add second stage feature selection techniques select final set variables section algorithm start screening procedure build first set candidate active variables call set active set first index corresponds stage algorithm second index corresponds whether set active variables inactive variables dimensionality already decreased large amount directly apply feature selection methods group lasso remaining variables introduced section however dimension still high reduce dimension increasing size considering partial correlations given variables follow similar procedure using partial correlation order one instead marginal correlation yield smaller active set estimator partial correlation computed taking residuals regressing variables conditional set continue screening partial correlations resulting nested sequence active sets note active inactive sets stage union active inactive sets stage active set previous stage similar original algorithm perform procedure algorithm stop stage dimension already drops low moderate level common feature selection techniques used select final set alternatively continue algorithm candidate active set change anymore algorithm summarized follows algorithm algorithm variable selection step set perform screening procedure construct active set step set construct stagem active set step repeat step min feature selection alternative algorithm variable selection also introduce feature selection algorithm combining screening procedure regular feature selection methods together single study example fan performed sure independence screening first stage followed model selection techniques including adaptive lasso dantzig selector scad named procedures accordingly case since feature selection adopt model selection technique using group lasso penalty second stage min active set identified screening procedure tuning parameter chosen bic practice like regular group lasso problem call feature selection algorithm addition stages algorithm dimension already dropped moderate level group feature selection techniques always take select final set variables numerical evidence section demonstrate advantage procedure comparing multiple study extension sis named ranks features minimum absolute correlation among studies simulated data according linear model including covariates zero mean covariance matrix denotes entry first part simulation fixed sample size number studies performed replications setting assumed true active set consisted ten variables variables zero coefficients indices coefficients evenly spaced variance random error term linear model fixed randomly drew allowed different different studies considered following four settings homogeneous weak signals across studies nonzero generated unif homogeneous strong signals across studies nonzero generated unif heterogeneous weak signals across studies nonzero generated unif table sensitivity analysis choice simulation note value based average results replications heterogeneous strong signals across studies nonzero generated unif evaluated performance using receiver operating characteristic roc curves measured accuracy variable selection independently issue choosing good tuning parameters tuning parameter top number features procedure mentioned actually one special case procedure thresholding presenting procedure fixed result one point sensitivity plot also performed sensitivity analysis two cutoffs based first simulation see table found two values optimal since high sensitivity high specificity thus suggested fixing two values simulations figure showed results simulation signals homogeneously weak studies clearly outperformed procedure roc curve reached sensitivity controlled false positive errors specificity order reduce false negatives sacrifice specificity increased false positives end lost benefits performing screening end keeping many features signals became strong procedures performed equally well fit motivation theory showed strength procedure saving weak signals without much increase false positive rates signals became heterogeneous procedures performed worse procedure never outperformed tsasis procedure since examined minimum correlation among studies procedure additionally considered aggregate statistics real data application next demonstrated method three microarray datasets breast cancer tnbc sometimes aggressive subtype breast cancer usually poor prognosis previous studies shown tumor suppressor protein played important role breast cancer prognosis expression associated survival overall survival tnbc yadav purpose identify genes relevant predictive response expression level gene encodes protein three datasets publicly available authors website geo repository including metabric large cohort consisting roughly primary breast tumours curtis figure simulation results roc curve black point using homogeneous strong homogeneous weak sensitivity sensitivity heterogeneous weak heterogeneous strong sensitivity sensitivity itoh liu subset data focus tnbc cases ended tnbc samples dataset respectively routine preprocessing filtering including genes sufficiently expressed enough variation total genes remained common analysis applied algorithm compared procedure well method using log suggested paper used determined sensitivity analysis simulation algorithm ran first order stopped six features showed power screening multiple studies feature selection fit linear model study obtain coefficient estimates adjusted table showed coefficient estimates standard errors final set six genes selected procedure added three columns indicate whether also retained relative rank procedures see table six genes selected procedure missed table six genes selected procedure gene metabric est intercept triobp est rank est note indicates significant level level level methods genes typically weak signals one studies thus likely incorrectly excluded one step screening performed since metabric study larger sample size coefficients appeared significant two studies gene components ras signaling pathway responsible cell growth division ultimately lead cancer rajalingam encodes protein implicated regulation transcription found associated indirectly via heat shock factor asano encodes integrin protein essential cell adhesion downstream signaling pathways also modulated brakebusch discussion paper proposed screening procedure regression analysis multiple related studies fairly general framework weaker assumptions signal strength showed procedure possessed sure screening property exponentially growing dimensionality without requiring normality assumption shown simulations procedure consistently outperformed rankbased sis procedure independent tuning parameter far know paper first proposed procedure perform variable screening regression multiple related studies addition also introduced two applicable variable selection algorithms following screening procedure variable selection regression multiple studies studied subfield machine learning called learning mtl general procedure apply regularization methods putting group lasso penalty fused lasso penalty trace norm penalty etc argyriou zhou however dimension regularization methods usually fail due challenges computation expediency statistical accuracy algorithmic stability instead sure screening used fast algorithm preliminary feature selection long exhibits comparable statistical performance theoretically empirically computational advantages make good choice application genovese method provided alternative target learning problems current screening procedure based linear models relaxes gaussian assumption distribution one apply modified fisher ztransformation estimator rather estimator readily accommodate general elliptical distribution families biomedical applications noncontinuous outcomes categorical count survival outcomes commonly seen fan extended sis proposed general independent learning approach generalized linear models ranking maximum marginal likelihood estimates fan extended correlation learning marginal nonparametric learning screening dimensional additive models researchers exploited robust measure correlation screening zhu balasubramanian measures potential extension modifying marginal utility used screening procedure besides idea performing screening multiple studies quite general applicable relevant statistical models regression model example gaussian graphical model multiple studies leave interesting problems future study proofs start introducing three technical lemmas essential proofs main results scaling property remark without loss generality assume var var therefore proof distinguish first lemma concentration inequalities covariance lemma assumptions max logn log max log positive constant depending second third lemmas used proof theorem describe xij concentration behaviors xij lemma exists constant exp min depends lemma exists constant max max depends proofs three lemmas provided appendix proof theorem first define following error events show theorem suffices show apply lemma bound component log specifically obtain max log log second equality due lemma noting addition max max log log log inequality second line due assumption lemma assumption minj log equality third line follows lemma end obtain max log max log log inequality second line due assumptions lemma assumption distributions log particular implicitly used fact maxj upper bounded constant depending equality third line follows lemma finally complete proof combining show proof theorem first define following error events eja eja prove theorem need show log recall event defined theorem thus eja eja max max therefore given results theorem suffices show first prove equation since bound probability ready log log log log log log inequality second line due assumption min lemma inequality third line follows lemma inequality fifth line choice sufficiently large assumption last equality follows lemma lastly prove follows log log inequality third line due assumption min lemma inequality fourth line follows lemma equations min guaranteed assumption assumption upper bound term follow log first inequality inequality assumption second equality assumption next upper bound term high probability note bounded assumption addition zero mean bounded constants assumption bernstein inequality proposition vershynin constant pick exp min max large constant inequality apply reduce follows log log log log log log inequality first line obtained choice chosen assumption inequalities second line third log sufficiently large line assumption last equality lemma completes proof yields eja eja results theorem therefore complete proof theorem appendix proof lemma proof part immediately follows lemma equation cai liu prove part need bound three terms right side following inequality max max max max xij xij xij note marginal distribution assumption assumption xij mean finite orlicz see adamczak thus apply equation adamczak max exp min log large enough constant depending obtain log max used assumption log assumption make sure applying equation supplement cai liu obtain log max addition similar truncation argument proof lemma cai liu equation therein obtain picking large enough log max complete proof combining union bound argument proof lemma proof easy check var marginal distribution assumption assumption implies finite orlicz distribution finite constants therefore centered random variable finite orlicz note independent result follows equation adamczak proof lemma proof note assumption var var subgaussian bounded constants therefore first equation follows bernstein inequality definition vershynin applied centered variable noting assumption second equation follows first one bernstein inequality corollary vershynin applied sum centered variables references adamczak litvak pajor restricted isometry property matrices independent columns neighborly polytopes random sampling constructive approximation argyriou evgeniou pontil feature learning advances neural information processing systems pages asano kawase okabe tsutsumi ichikawa tatebe kitabayashi tashiro namiki kondo generates novel hypophosphorylated active form contributes 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solving resource constrained project scheduling problem using parallel tabu search designed cuda libor nov department control engineering czech technical university prague karlovo prague czech republic abstract paper parallel tabu search algorithm resource constrained project scheduling problem proposed deal combinatorial problem many optimizations performed example resource evaluation algorithm selected heuristic effective tabu list designed addition resource evaluation algorithm proposed gpu graphics processing unit version uses homogeneous model reduce required communication bandwidth according experiments gpu version outperforms optimized parallel cpu version respect computational time quality solutions comparison existing heuristics proposed solution often gives better quality solutions cite libor bukata premysl sucha zdenek hanzalek solving resource constrained project scheduling problem using parallel tabu search designed cuda platform journal parallel distributed computing volume march pages issn http source code https https keywords resource constrained project scheduling problem parallel tabu search cuda homogeneous model gpu introduction resource constrained project scheduling problem rcpsp wide range applications logistics manufacturing project management universal problem operations research domain problem briefly described using set corresponding author email addresses libor bukata manuscript version made available license http preprint submitted parallel distrib comput activities set precedence constraints describing relationships among activities activity requires defined amount resources every resource limited capacity objective find best feasible schedule according criterion rcpsp proved strong sense criterion makespan reason small instances approximately activities reliably solved exact methods like branch bound therefore heuristic required solve problem satisfactorily recent times increased interest using graphics cards solve difficult combinatorial problems since modern march graphics card usually much powerful current cpu although graphics cards restrictions global memory access new gpu architectures like kepler fermi significantly reduce bottlenecks consequence modern gpus applicable problems solvable cpus previously high computational power makes graphics cards attractive researchers practitioners also mature nvidia cuda framework enables create gpu programs effective relatively easy way since extends standard languages like adding gpu specific functions language keywords nevertheless cuda designed nvidia graphics cards implementation point view two models first one called homogeneous model required data structures stored gpu beginning algorithm results read end algorithm communication cpu gpu computations second approach heterogeneous model main logic algorithm runs cpu gpu used computationally intensive tasks disadvantage heterogeneous model frequent communication computations therefore communication bandwidth state bottleneck however heterogeneous model usually simpler implement cores every location index thread different parameters tabu tenure stopping criteria beginning search thread initializes location short operator modified version taillard robust tabu search asynchronous parallel tabu search started every thread independently reads solution parameters location possibly makes diversification runs operator solution writes back sets update flag improving solution found thread location index circularly incremented diversification takes place read solution update flag set every best global solution copied half locations circular buffer propagate elite solutions since circular buffer shared many threads access short possible locations protected critical sections relatively many authors try use gpu solving combinatorial problems example barnes implemented gpu version solve flowshop scheduling problem fsp success implementation illustrates achieved speedup cpu version gpu version times faster cpu intel xeon ghz memory nvidia tesla gpu nevertheless quality solutions investigated flowshop scheduling problem also solved authors implemented gpu version island based genetic algorithm islands used migration individuals among subset solutions individual corresponds specific solution subpopulation evaluated mutated crossed independently therefore huge parallelization achieved noted homogeneous model used maximal speedup cpu activities machines amd phenom ghz nvidia tesla proposed parallel tabu search quadratic assignment related works tabu search proposed glover hundreds publications written since time basic concept clarified gendreau author described basic terms tabu list aspiration criteria diversification intensification etc tabu search parallelization point view james proposed sophisticated solution authors use circular buffer size buffer equivalent number started threads often number cpus lem main idea start several parallel tabu search instances different parameters initial solutions tabu search instances terminate approximately time since synchronization required get promising solutions stop criterion met modified solutions read back promising solutions used initial solutions next run tabu search instance runs entirely gpu therefore communication overheads reduced minimum achieved results reveal effectiveness implementation since nvidia gtx times faster intel core ghz hofmann investigated suitability graphics cards genetic algorithms authors selected two problems solve namely weierstrass function minimization traveling salesman problem first problem effectively implemented gpu since weierstrass function comprised floatingpoint operations trigonometric functions directly supported gpu hardware contrast first problem second problem gpu therefore cpu able compete gpu authors suggest parts genetic algorithm performed fermi newer gpus homogeneous model boyer used dynamic programming solve knapsack problem gpu effective data compression proposed reduce memory occupancy achieved results show nvidia gtx graphics card faster intel xeon ghz mentioned combinatorial problems something common solution evaluation quite simple since usually simple sum hand rcpsp requires much complicated schedule evaluation methods data structures cpu version performance speedup quality solutions possible thanks effective schedule evaluation gpuoptimized simple tabu list addition required data transfers reduced minimum due homogeneous model parallel tabu search able outperform tabu search implementations quality resulting solutions paper structured follows following section introduces rcpsp mathematical formulation notation tabu search briefly described section proposed parallel tabu search algorithm cuda platform described detail sections performed experiments located section last section concludes work problem statement according standard notation classification rcpsp project described follows set activities durations number activities two dummy activities activity predecessor activities activity end activity project schedule rcpsp represented vector activities start times alternatively schedule expressed order activities activity schedule set feasible solutions rcpsp represented direct acyclic graph nodes activities edges precedence relations edge since activity scheduled activity activity requires amount renewable resources number project resources denoted set resources capacities maximal source capacity rmax equal maxm activity resource requirement means contribution paper outline proposed solution first known gpu algorithm rcpsp performed experiments revealed gpu outperforms activity activity requires resource units resource execution values positive integers resulting schedule length cmax project makespan also integer well lower bound project makespan found neglecting resources activity outgoing edges weighted duration longest path graph corresponds critical path length equal optimal project makespan condition resources unlimited capacity cmax max successors table data example instance mathematical formulation minimize cmax cmax max brief description tabu search metaheuristic move solution space transformation current solution neighborhood solution required transformation called move seen light solution modification like swap two elements order etc tabu search proposed glover improvement local search technique local search algorithm starts initial solution iteratively improves solution applying best neighborhood moves local optimum reached whereas tabu search introduces memory called tabu list reduces probability getting stuck local optimum plateau forbidding previously visited solutions consequence improving solutions permitted search process able climb objective rcpsp find feasible schedule minimal schedule length cmax schedule length latest finish time activity equations equation ensures precedence relations satisfied schedule feasible precedence relations satisfied resources overloaded activities requirements exceed capacity resource time equation instance example data example instance showed table project activities renewable resources maximal capacity corresponding graph precedences shown figure critical path highlighted bold lines length one feasible solutions instance activity order cmax resource utilization order depicted figure figure graph precedences example instance resource utilization resource utilization figure utilization resources example instance hills search space necessary due efficiency tabu list usually contains parts solutions several previously applied moves moves parts solutions unique possible forbidden move leads best solution case reasonable permit move since resulting solution visited general exceptions allowing forbidden moves called aspiration criteria quality resulting solutions improved suitable search strategy example current location solution space promising best solution found recently thorough search performed intensification accomplished concentrating computational power locality space opposite current location unpromising poor solutions found diversification performed diversification moves current search location another one better solutions could found often realized applying random moves tabu search process stopped stop criterion met stop criterion number iterations achieved quality best found solution maximal number iterations since last best solution found etc exploration solution space creating initial activity order tabu search algorithm starts initial feasible solution init created following way first longest paths graph start activity activities found weight graph edge set activities maximal distance start activity grouped levels level corresponds activities maximal distance start activity therefore activity level activity level lmax subscript max corresponds last level number final feasible schedule created levels lmax alternative feasible schedules created shuffling activities level move transformation schedule order changed tabu search algorithm swap move simple example illustrated figure two dummy activities swapped due precedence constraints therefore activity always activity always swap move defined swap swapped indices swap modifies schedule way swap swaps taken account without loss generality reduced neighborhood moves restricted swap value maximal distance swapped activities order size neighborhood parametrized two reasons feasible moves applied neighborhood size reduced without noticeable deterioration project makespan need check feasibility schedules swap filtering infeasible moves order saturate gpu feasible schedules nreduced evaluated parallel way dividing schedules equally among threads since evaluation schedule much checking whether move feasible advantageous filter infeasible moves neighborhood evaluation reduces branch divergency warps hence overall performance resources evaluation improved algorithm shown infeasible moves filtered neighborhood filter works two phases since discovered effective due lower branch divergency filter infeasible moves end part array feasible moves taken account neighborhood evaluation figure example swap move let feasible activity order feasible order means violated precedence relation feasible move move violate precedence relation therefore move applied feasible schedule modified schedule feasible well move swap feasible following equations satisfied first equation means edges activity activities indices edge activity moved position without precedence violation similar way equation states activity moved index precedence relation becomes violated neighborhood generation full neighborhood ull schedule set schedules obtained applying feasible moves since full neighborhood usually large evaluated reasonable time subset neighborhood usually taken account subset called reduced neighborhood denoted nreduced algorithm removing infeasible moves reduced neighborhood require nreduced ensure filters infeasible moves reduced neighborhood let movesarray array containing potential swaps nreduced moves satisfying equation removed set empty reorder movesarray empty moves end array remove moves satisfy equation move feasible moves beginning movesarray simple tabu list cache tabu list suitable gpu since necessary moves schedule evaluation tabu list decide whether move tabu list consequence places higher demands device memory bandwidth avoid simple efficient simple tabu list stl constant algorithmic complexity proposed access stl performed like access circular buffer size fixed equal case stl stores swap moves swap stored stl pair swapped indices special value used empty move swap iteration algorithm one move added see algorithm oldest one removed stl full evaluation precedence relations resource constraints taken account calculate activities start times cmax precedence earliest start time esprec activity calculated predecessors activity resources earliest start time esres computed using either resources evaluation algorithm algorithm completely new approach best knowledge whereas algorithm names algorithms selected respect indexed unit resource state array according heuristic probable faster resources evaluation algorithm selected schedule evaluation procedure considered precedence resource constraints final earliest start time esi max esprec esres algorithm check move stl require tabucache stl cache require swap move indices ensure returns true move stl otherwise false return tabucache tabu cache proposed effective checking move stl illustrated algorithm checking swap stl occurs much often adding new move since move added per iteration check move stl occurs every neighborhood schedule implemented boolean array synchronized stl check move stl requires one read operation thus required memory bandwidth low obvious check move stl algorithmic complexity resources evaluation required data structures difficult part project makespan evaluation computation activities start times respect resource capacities approach evaluation resources requires one array length per resource value corresponds earliest resource start time activity resource requirement resource start evaluation resources arrays set zeros activities added one one schedule according arrays updated respect activity requirements precedences resources arrays ordered descendly state resources represented set algorithm add move stl require tabulist fixed size array require tabucache stl cache require writeindex current write position require swap move indices ensure add move stl update uold vold tabulist writeindex tabucache uold vold false tabulist writeindex tabucache true writeindex writeindex earliest resources start time activresource earliest start time esres ity respect occupation resources calculated using equation guaranteed resources overloaded activity start time esres final activity start time delayed due precedence relations esres max decremented zero lines resource performed setting elements activity finish time last variable requiredeffort zero shifted right shift copy original resource array original values stored copy auxiliary variable made complexity algorithm rmax otherwise update resources arrays activity added schedule resources arrays updated algorithm resource array updated resource availability algorithm method updates state resources adding activity require require copy auxiliary array length rmax require scheduled start time activity ensure update activity added requiredeffort requiredeffort residx copyidx newtime requiredeffort residx residx newtime copyidx newtime copy copyidx end timediff newtime max residx requiredeffort timediff requiredeffort timediff copy copyidx residx residx newtime else residx max residx residx requiredeffort requiredeffort end end residx residx end end end rmax activity required effort units resource earliest start time figure example resource state update algorithm illustrated example figure one resource maximal capacity added activity requires resource units duration activity scheduled solid line corresponds original resource state dotted line corresponds updated resource activity required effort depicted square dashed border positive numbers solid dotted lines effort contributions old start time solid line changed new start time dotted line noticed sum contributions requiredeffort given activity vidually line value requiredeffort called required resource effort activity added schedule resource able provide required resource effort words variable requiredeffort resources evaluation required data structures evaluation algorithm state resource stored array element corresponds number available resource units resource able provide time ubcmax ubcmax upper boundpof makespan calculated array initialized elements value start evaluation state resources denoted update resources arrays state resources updated shown algorithm scheduled activity arrays updated interval resource values interval decreased units algorithm updating resources adding activity require require scheduled start time activity ensure updates state resources end end earliest resources start time algorithm algorithm calculates earliest start time activity require ubcmax require esprec precedence earliest start time ensure calculate earliest start time esi activity loadtime esprec ubcmax loadtime sufficientcapacity true loadtime sufficientcapacity false end end sufficientcapacity true end end return loadtime schedule evaluation procedure schedule evaluation procedure shown algorithm activities read one one activities order activity predecessors found precedence relations used update activity precedence earliest start time esprec see lines resources restrictions checked res start time adjusted swu max esprec eswu project makespan cmax finish time activalgorithm complete schedule evaluation require ensure calculate cmax activities start times cmax esprec prec esprec max eswu end prec esres getearliestresourcestime activity eswu prec res swu max eswu eswu updateresources activity swu mark current activity scheduled cmax max cmax swu dwu end return cmax earliest start time activity calculated using algorithm algorithm loadtime variable corresponds number consecutive time units resources able meet resource requirements activity loadtime time interval activity scheduled found considered variable finish time candidate interval resulting interval first interval loadtime loadtime final earliest start time lower endpoint interval ity feasible moves allowed infeasibility test resulting schedules required heuristic selection resources evaluation algorithms search started gpu probable faster resources evaluation algorithm heuristically selected decision rules required resources arrays allocated create rules jrip classifier weka datamining tool learned using attributes shown table min resource capacity avg resource capacity max resource capacity avg activity duration avg branch factor critical path length evaluation algorithm min resource capacity true time false avg resource capacity avg branch factor true time min false min resource capacity max resource capacity true time max false capacity figure example decision tree see section table attributes used learning attribute evaluation algorithm determines class evaluation algorithm classifier classify final class dependends hardware instance parameters necessary determine probable correct class measuring evaluation algorithm selected small number iterations faster one selected desired one resulting rules heuristically decide two algorithms effective given instance rules transformed decision tree shown figure rules created applied similar instances without measuring overhead well show effectivity usefulness heuristic experiments performed section parallel tabu search cuda platform parallel tabu search gpu ptsg proposed respect maximal degree parallelization since thousands cuda threads required fully loaded exploit graphics card power approach parallelization carried two ways first one parallelization performed within scope block example parallel filter see section parallel neighborhood evaluation parallel reductions second one parallelization introduced launching many blocks simultaneously basic steps ptsg described figure first instance read initial solutions created accordance section every second solution improved using improvement method method iteratively shaking schedule left right order make resource profile straight long schedule getting shorter get details method refer original article willis created solutions copied working set set shared solutions best solution working set called global best solution span denoted cmax furthermore block tabu lists tabu caches initialized auxiliary arrays allocated prepared required datastructures host ready launch kernel gpu part every block independent tabu search instance communicating others global memory see section beginning every block reads nication conditions satisfied see section search stopped specified number equal iterations achieved cmax length critical path termination kernel best global solution copied global memory host memory solution printed allocated freed create initial solutions apply improvement method every second solution cpu among cuda blocks iterations distribution assure high quality solutions cooperation among tabu search instances accomplished exchanging solutions working set fixed number solutions solution consists ork tabu der activities makespan cmax list iterations counter solutions exchange takes place last read solution improved iassigned iterations block found improvement last read solution block writes best found solution improves last read solution reads next solution since working set could accessed many blocks time necessary use locks order maintain data integrity among tabu search instances inspired james block read solution working set checked whether solution read times without improved case small number random feasible swaps applied randomize read solution diversification read solution assigned number iterations iassigned according following equation initialize fill gpu structures launch kernel read initial solution generate nreduced filter infeasible moves gpu block main loop evaluate neighborhood select best move apply add stl exchange data yes blocks stop condition yes load data cpu cpu print best gpu solution figure parallel tabu search cuda platform initial solution working set search started specified number iterations main loop main loop neighborhood generated evaluated best move selected applied added stl move leads best criterion improvement smallest criterion deterioration move stl one exception move leads global best solution end iteration solutions exchanged working set quality cmax quantity iassigned iblock itotal cmax intactness iblock iblock number iterations assigned tabu search instance itotal total number iterations calculated iblock number launched blocks part denoted quantity corresponds maximal number iterations assigned read solution ensured least solutions read working set block quality part takes account quality read solution obvious high quality solutions preferred poor ones intensification last part intactness guarantees solution given iterations prove quality cuda toolkit version microsoft visual studio sequential cpu version algorithm corresponds one tabu search instance exception solutions interchanged instead using selection heuristic see section faster evaluation algorithm selected dynamically periodic measuring every iterations parallel cpu version differs sequential version neighborhood evaluation feasible schedules neighborhood divided among cpu threads reduce evaluation time cpu gpu versions fully optimized respect memory access patterns hardware architecture cache sizes fully saturate gpu maximal number available registers per cuda thread limited due possibility launch blocks multiprocessor altogether blocks gpu block cuda threads memory model placement crucial task highly influencing effectiveness gpu program therefore decision considered thoroughly respect access pattern required bandwidth data visibility local shared data shared memory current block order wblock durations activities auxiliary arrays stored although array could located constants memory moved shared memory due higher bandwidth texture memory used storing data values predecessors activities local memory private thread located resources arrays start times activities long latency memory compensated using partial coalescing since arrays often accessed relative indices majority threads evaluate similar schedules wblock swap move finally global memory employed store working set table ptsg parameters information evaluate performance quality resulting solutions benchmark using wellknown performed number instances denoted selected ptsg parameters information stated table results shown tables cpm dev opt dev values average percentage distance critical path length average percentage distance optimal makespan respectively best sol states number optimal solutions proved optimal according experimental results experiments performed amd phenom server cores memory equipped nvidia geforce gtx cuda cores multiprocessors graphics card testing environment windows server installed cpu opt dev itotal cpm dev best sol cpm dev gpu opt dev best sol table quality solutions itotal cpm dev cpu dev best sol cpm dev gpu dev best sol table quality solutions cpu seq cpu par gpu gpu itotal comp time sched sec cpu seq cpu par gpu gpu gpu gpu speedup table performance comparison itotal cpm dev cpu dev itotal comp time sched sec speedup table performance comparison best sol cpm dev gpu dev best sol table quality solutions itotal cpm dev cpu dev best sol cpm dev gpu dev best sol table quality solutions cpu seq cpu par gpu gpu gpu gpu itotal comp time sched sec speedup cpu seq cpu par gpu gpu gpu gpu table performance comparison itotal comp time sched sec speedup table performance comparison algorithm reference genetic algorithm work nvidia geforce gtx work amd phenom cara algorithm valls ant colony optimization zhou tabu search artigues simulated annealing bouleimen lecocq cpm dev table comparison heuristics results psplib homepage http comp time total stated seconds sched sec number evaluated schedules per second obvious gpu version able achieve similar quality solutions terms cpm dev cpu version itotal doubled gpu version found optimal solutions solutions performance point view table reveals significant improvement computational time parallelization performed example parallel cpu version times faster sequential cpu version gpu almost times faster parallel cpu version itotal increased gpu still slightly faster achieves better quality solutions results shown tables dev average percentage distance best currently known upper bounds cpu version gives slightly better solutions iterations hand gpu given iterations quality solutions comparable cpu version gpu still times faster parallel cpu version lower quality gpu solutions itotal probably caused wasting work many parallel tabu search instances read solution working set one writes best improvement noted parallel cpu version times faster sequential one reason either better cache utilization amd true core scalability technology results tables dataset show gpu better utilized bigger instances gpu times faster parallel cpu version number iterations quality solutions achieved times faster gpu results shown tables noted quality gpu solutions substantially lower iterations gpu requires iterations achieve quality cpu solutions hand gpu able compete cpu since iterations performed times faster iterations parallel cpu version gpu evaluates one million schedules per second whereas cpu evaluates one hundred thousand quality solutions compared existing solutions rcpsp table proposed ptsg outperforms tabu search implementations respect quality solutions example artigues tabu search given least iterations achieves cpm dev iterations proposed ptsg reaches cpu gpu respectively addition proposed ptsg good heuristic approaches like ant colony optimization simulated annealing hand state art genetic algorithms give even better solutions ptsg performance point view difficult compare since different algorithms hardware architectures used experiments dependent characteristics instances generally determined evaluation algorithm faster evaluation algorithm usually faster long schedules low resource capacities contrast algorithm usually performs better short schedules high resource capacities example artigues tabu search requires per instance average testing configuration stated ptsg requires iterations gpu substantially higher quality solutions genetic algorithm takes per instance average intel core duo ghz processor evaluation selection heuristic heuristic see section using jrip classifier weka data mining tool decide resources evaluation algorithm faster get data learning progen generator used generate activities respectively parameters generated set exception different random seeds used generated classifier learned using command tested corresponding standard number activities achieved results table reveal accuracy decreasing number activities reason behavior smaller ratio resources evaluation time total heuristic table effect heuristic ptsg performance demonstration convergence demonstrate among blocks beneficial graph convergency figure created instance seen quality solutions getting better increasing number launched blocks therefore obvious leads better solutions ensure smoothness table accuracy selection heuristic percentage correctly classified prove proposed heuristic also improves ptsg performance measured evaluation algorithm normalized respect reference achieved using heuristic results table show heuristic accelerates ptsg times effect decreasing evaluation schedules becomes less part ptsg timeindexed algorithm seems faster algorithm standard datasets hand achieved speedup figure graph convergence gpu version graph point averaged measurements conclusion first known gpu algorithm dealing resource constrained project scheduling lem proposed performed experiments standard benchmark instances reveal merits proposed solution achieved quality solutions good outperforms tabu search implementations best knowledge addition gpu algorithm design proved effective since gpu substantially faster optimized parallel cpu version nvidia geforce gtx gpu able evaluate one million schedules per second average achieved performance boost could reached without effective structures auxiliary algorithms simple tabu list implementation adapted features gpu capacityindexed evaluation algorithm proposed many parallel reductions applied addition homogeneous model reduces required communication bandwidth cpu gpu spite fact gpus primarily designed solving combinatorial problems rising interest solutions seen reason high computational power graphics cards relatively user friendly programming api cuda offers expected gpus used operations research future references lau varakantham xiao robust local search solving durational uncertainty artif int res blazewicz lenstra kan scheduling subject resource constraints classification complexity discrete applied mathematics chaleshtarti shadrokh branch bound algorithms resource constrained project scheduling problem subject cumulative resources proceedings international conference information management innovation management industrial engineering volume iciii ieee computer society washington usa delvacq delisle gravel krajecki parallel ant colony optimization graphics processing units journal parallel distributed computing effective parallel multistart tabu search quadratic assignment problem cuda platform journal parallel distributed computing hejducki wodecki solving flexible job shop problem multigpu proceedings international conference computational science iccs glover future paths integer programming links artificial intelligence comput oper res gendreau introduction tabu search glover kochenberger eds handbook metaheuristics vol international series operations research management science springer new york james rego glover cooperative parallel tabu search algorithm quadratic assignment problem european journal operational research barnes tabu search two approaches parallel flowshop evaluation cuda platform parallel distrib comput accelerating flow shop scheduling algorithm gpu workshop models algorithms planning scheduling problems mapsp hofmann limmer fey performance investigations genetic algorithms graphics cards swarm evolutionary computation boyer elkihel solving knapsack problems gpu computers operations research brucker drexl neumann pesch project scheduling notation classification models methods european journal operational research kelley method resources planning scheduling muth thompson eds industrial scheduling englewood cliffs hall frank holmes pfahringer reutemann witten weka data mining software update sigkdd explor newsl willis iterative scheduling technique project scheduling european journal operational research resende mendes biased genetic algorithm backward improvement resource constrained project scheduling problem journal heuristics valls quintanilla resourceconstrained project scheduling critical activity reordering heuristic european journal operational research zhou wang peng aco solving rcpsp computer science computational technology iscsct international symposium vol artigues michelon reusser insertion techniques static dynamic project scheduling european journal operational research bouleimen lecocq new efficient simulated annealing algorithm project scheduling problem multiple mode version european journal operational research kolisch schwindt sprecher benchmark instances project scheduling problems handbook recent advances project scheduling kluwer brodtkorb hagen graphics processing unit gpu programming strategies trends gpu computing journal parallel distributed computing
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optimal resource allocation distributed broadband wireless communication systems yao yao mustafa shahin vakilinia department electrical computer engineering concordia university montreal canada mustafa paper concerned optimization distributed broadband wireless communication bwc systems bwc systems contain distributed antenna system das connected base station optical fiber distributed bwc systems proposed solution power constraint problem traditional cellular networks far research bwc systems advanced two separate tracks design system meet quality service requirements qos optimization location das paper consider combined optimization bwc systems consider uplink communications distributed bwc systems multiple levels priority traffic arrivals departures forming renewal processes develop analysis determines packet delay violation probability priority level function outage probability das application results renewal theory determine optimal locations antennas minimize antenna outage probability also study trade packet delay violation probability packet loss probability work helpful designing distributed bwc systems index queuing delay multiple levels priority traffic distributed antenna system das outage probability antenna placement introduction future services wireless networks requiring higher transmission rates rate expected rise next generation communication systems according objective international mobile telecommunications system set international telecommunication unionradiocommunication standardization sector however constraint transmit power limits transmission rate especially uplink communications user base station one promising solution problem distributed broadband wireless communications bwc systems distributed bwc system made distributed antenna system das radio fiber rof technology das antennas placed geographically separate locations cell contrary called centralized antenna system cas antenna set center cell rof technology reliable small delay responsible providing communication antennas central processor jointly processed way user likely antenna nearby compared cas turn reduce transmission power requirement user possible place many distributed antennas cell due relatively low cost distributed antennas adopting das bwc system solve power constraint problem naturally instead shrinking cell size conventional cas results larger overhead delay due higher frequency handovers cells number research works distributed bwc systems work dealt two main problems determining antenna set meet qos metrics delay packet loss optimal placement antennas considers downlink transmission base station user assumes central server base station maintains separate queue users work considers selection subset available distributed antennas probability packet queuing delay less threshold considers uplink transmission distributed bwc system objective selection subset distributed antennas user keep packet loss probability threshold value locations antennas fixed objective function optimized respect wrt location antennas optimal placement antennas considered works independent qos metrics objective optimize uplink capacity single cell multiple users assumed antennas placed circle centered cell center locations circle uniformly distributed independent optimal radius antenna circle determined however work appropriate cdma systems since users cell transmitting simultaneously paper considers effects interference simulation optimal location antennas cellular setting studied downlink transmission based performance metrics capacity power efficiency work takes interference account though closed form results obtained numerical results show certain cases optimal layout consists placement one antenna center remaining ones circle around circle shrinks interference coefficient increases work provide results positions antennas circle paper consider uplink communications assume user multiple levels priority traffic fig topology distributed bwc system cellular network architecture different qos requirements determine packet delay violation probability use results renewal theory type traffic function outage probability determine optimal location antennas minimizes antenna outage probability note work general compared considers multiple types traffic different qos requirements relation optimal placement antennas work closer consider uplink opposed downlink communications critical due limited transmit power users outline remainder paper follows section describes system model section presents queuing delay analysis multiple types traffic flows section gives application results system two types traffic flows section extends application system four types traffic flows next section presents work optimal placement antennas section vii presents numerical simulation results regarding analysis finally section viii gives conclusions paper system model network topology consider cellular network architecture distributed antenna system cell together neighbors referred cluster cells cluster numbered cluster size denote cell center cluster referred target cell fig shows example network cluster seven cells cell contains central processor distributed antennas placed different geographical locations linked central processor rof technology figure antennas target cell shown prevent crowding however cells also contains distributed antennas assumed antennas height users ground level target cell locations antennas wrt cell center denoted polar coordinates number antennas according increasing polar angles thus define antennas location vector frequency reuse model assume frequencies available every cell thus reuse factor one available spectrum divided number channels user assigned single channel time also assume system saturated channels always busy users using channel neighboring cells interfering corresponds worst case scenario interference major cause outage cellular system fig user target cell considered target user users neighboring cells using channel considered interferers thus single user cell using channel cluster let user refer user cell using particular channel cluster let denote location user polar coordinates relative center home cell denote location cartesian coordinates relative center cell latter may expressed terms former follows transmission model slotted transmission packet always starts beginning slot takes one slot user broadcast packet received antennas cell received signals locally decoded results forwarded central processor fiber lines respectively least one antennas decode packet successfully central processor receive transmitted packet correctly hand none antennas able decode packet successfully central processor drop packet ask retransmissions depending type packet iii queuing delay section determine queuing delay bounds user multiple types traffic different qos requirements assume user types traffic flows numbered packets flows stored separate queues traffic flow assigned priority level increases decreasing flow number packet traffic flow served queues higher priority traffic flows empty let denote independent identically distributed time service time packets traffic flow respectively system serving traffic flow let denote mean variance random variables respectively note service time packet begins reaches head queue completed either following successful transmission discarding system section determine queuing delay bounds packets different flows use delay bound violation probability theory effective bandwidth analysis metric also used violation probability defined probability packet delay exceeds certain value traffic flow represent queuing delay queuing delay threshold respectively delay bound determined utilization energy function defined asymptotic log moment generating function arrival service processes initially assume system operating single type traffic traffic flow later results extended multiple types traffic let denote number packet arrivals queue time interval defining number departures queue interval assuming saturation let denote energy functions arrival service processes violation probability packets traffic flow next extend results user multiple types traffic scenario traffic flow priority level arrival process traffic flow remain service process change services given packets higher priority flows let denote number departures flow interval denoting service time packet flow may written next letting denote energy function service process assume system stable thus none queues starving service result random variables different independent thus expressing sum two logarithms substituting infinity mean variance renewal interval application result gives energy function arrival service processes follows application central limit theorem renewal processes conditional expectation gives unique solution general may possible find energy functions defined limits applicability method case propose use asymptotic central limit theorem renewal processes evaluations states renewal process number renewal points asymptotically approaches gaussian distribution mean variance goes definition random sum mean variance given general central limit theorem approaches normal distribution large values mean variance given expectation rhs corresponds moment generating function normal random variable result becomes substituting completes derivation energy function service process flow next demonstrate accuracy asymptotic central limit theorem renewal processes example let consider binomial process whose exact energy function known exact asymptotic energy functions binomial process given respectively voice data packets stored separate queues voice given higher priority wrt data traffic assumed transmitted voice packet decoded dropped since voice delay sensitive hand failed data packets retransmitted lost unsuccessful transmissions following study delay packet losses data flow since voice flow redundant given higher priority assume voice data arrive according independent poisson processes rates respectively exact energy function poisson process known giving energy functions voice data energy functions voice data departure processes respectively success probability two results plotted fig function may seen close gives confidence use asymptotic energy function next determine mean variance service time data packets service time data packets truncated geometric distribution probability transmission packet result outage let denote pgf service time distribution pgf comparison asymptotic exact energy functions binomial process success probability finally delay violation probability data packets given unique solution equation packet loss probability data packets given fig markov fluid transition rate state state state probability system state system two types traffic flows next apply results system two types traffic flows referred voice data system four types traffic flows section extend analysis system four types traffic flows adding multimedia multimedia traffic flow formal system model new system multimedia given lower priority voice traffic higher multimedia multimedia given higher priority data traffic assume multimedia markov fluid characterized four parameters refers poisson packet arrival rate state refers transition rate state respectively similar thing multimedia traffic suffix also assume service multimedia packets service voice packets let denote pdf time packets traffic flow multimedia given probabilities markov source states respectively probabilities given next determine mean variance obtaining first second moment laplace transform expression thing traffic flow multimedia thus following equation set submitting could finally delay violation probability data packets given unique solution equation optimal distributed antenna placement section determine optimal locations antennas minimizing system outage probability seen target user first determine signal noise ratio target user derive probability system outage distance user antenna given path loss exponent independent identically distributed complex gaussian random variables rayleigh fading channels since rayleigh distribution exponential distribution mean equal one following treat interferences users noises assumed source noise consider ach channel time slotted neighbor cell users communicate synchronous manner meanwhile user main cell imposed interference users paper mainly focus scenarios user utilizes channel used set users implies channel interference level sensed user random variable dependent users transmission states model channel source alternating state active state inactive channel usage model specifies time slot neighbor user signals transmitting channel neighbor users time slot main user transmission affected less interference suppose channel changes state independently let probability neighbor user channel exists state channel state characterized markov chain let denote received signal strength antenna user transmitting distance antenna received signal strength antenna user written probability probability result instantaneous snr antenna given general outage probability antenna expressed denotes mutual information transmitter receiver antenna required transmission rate spectrum efficiency expressed substituting outage may expressed continue analysis setting let denote pdf random variable laplace transform probability outage antenna given expanding partial fraction next unconditioning result wrt location vector users marginal pdfs coordinates user respectively since assuming user locations uniformly distributed cell taking account denoting distinct value frequency set substituting gives laplace transform rvs sum independent exponentially distributed rvs assume outages antennas independent therefore system outage probability fixed set user locations given next let define taking inverse transform gives pdf substituting define antenna locations minimize expected outage probability system optimal unfortunately possible obtain minimum expected outage probability system analytically due complexity function result determined minimum value expected outage probability system numerically applying stochastic approximation method vii numerical results section present numerical results regarding analysis paper simulation results verify accuracy analysis first present numerical simulation results regarding system two types traffics consist voice traffic data traffic assumed transmitted voice packet decoded dropped since voice delay sensitive hand failed data packets retransmitted data packet dropped unsuccessful transmissions dropped packets treated packet loss voice data arrive according independent poisson processes rates respectively present delay violation probabilities data packets function delay threshold data traffic parameters fig may seen increasing increases delay violation probability fixed value similarly fig increasing increases delay violation probability fixed value fig plot numerical simulation results delay violation probability show accuracy analysis may seen good match numerical simulation results increases confidence validity analysis specially use asymptotic results renewal theory fig presents delay violation probabilities data packets function data delay threshold total number transmissions parameter may seen delay violation probability increases packet loss probability decreases increases shows tradeoff delay violation packet loss probabilities fig present similar delay violation probabilities data packets respect different parameters system four types traffics three numerical results show trend system two types traffics fig probability queuing delay greater threshold value wrt different values fig probability queuing delay greater threshold value wrt different values fig simulation numerical results probability queuing delay greater threshold value fig probability queuing delay greater threshold value wrt different values fig probability queuing delay greater threshold value slot wrt different values fig probability queuing delay greater threshold wrt different values fig probability queuing delay greater threshold value wrt different values next present numerical simulation results analysis antenna placement part paper well simulation results numerical results optimal location antennas obtained application algorithm steps given initialize antennas location vector let vector randomly generate user location vector according uniform distributions cells update optimum locations antennas applying equation let iteratively run steps converges converges optimum antenna location vector implementation algorithm chose sequence positive satisfies polyak juditsky constraints compared delay performance proposed algorithm case algorithm bandwidth slots distributed data voice traffics different ratios case algorithm considering tdma system entire slot frame potential voice data transmission entire slot data voice frames different ratios figures represent number frames data within entire slot let represent ratio data voice arrival tdma system transmit empty frames using case algorithm spite sensitivity data delay shown figure voice delay increases significantly shown figure however proposed algorithm lowest voice delay much data delay sensitivity also reduce amount computations execution algorithm assumed optimal locations antennas circle centered cell center antennas spaced evenly circle reason assumption cells symmetric every aspect number antennas assumed cluster size figures wireless channel parameters set follows path loss exponent application procedure gives optimum locations antennas results minimum expected value outage probability see table fig wireless channel parameters set follows path loss exponent case procedure gives optimum locations antennas results minimum expected value outage probability see table fig plotted simulation results expected value system outage probability function optimal radius antenna circle different values neighbor users activation probability figure may seen optimal radius antenna circle results minimum expected value outage probability comparing case antennas center optimal location provides almost improvements system average outage probability simulation numerical results show good match fig simulation result system average outage probability function first antenna location different curves represents different value table outcome optimum location first antenna show optimum antenna radius since angles changes slightly around increases fig probability queuing delay greater threshold value table outcome optimum location first antenna show optimum antenna radius since angles changes slightly around increases fig simulation result system average outage probability function first antenna location different curves represents different value fig probability voice queuing delay greater threshold value viii conclusion presently distributed broadband wireless communication bwc system promising wireless technology due low cost high performance paper considers uplink fig probability data queuing delay greater threshold communications multiple levels priority traffic value renewal arrival departure processes develop analysis determines packet delay violation probability priority level function outage probability distributed antenna system application results renewal theory determine optimal locations antennas minimize antenna outage probability also present simulation results show accuracy analysis results easy use useful design bwc systems finally packet delay violation probability result may also applicable communication systems multiple types priority traffic references mohr monserrat osseiran werner mobile networks guest editorial ieee communications magazine vol issue wang adachi gameiro gomes kitayama monteiro sawahashi xia guest editorial distributed broadband wireless communications ieee journal selected areas communications vol lee toumpakaris wei interference mitigation via joint detection ieee journal selected areas communications vol issue gao zhou wang sum rate distributed antenna system circular antenna layout proceedings ieee vehicular technology conference fall zhang selections distributed mimo links broadband wireless networks ieee journal selected areas communications vol zennaro tomasin vangelista base station selection uplink macro diversity cellular systems hybrid arq ieee journal selected areas communications vol issue feng zhou wang xia sum rate characterization distributed antenna systems circular antenna layout proceedings ieee vehicular technology conference spring firouzabadi goldsmith optimal placement distributed antennas cellular systems proceedings ieee international workshop signal processing advances wireless communications linnartz exact analysis outage probability mobile radios ieee transactions communications vol issue chang thomas effective bandwidth digital networks ieee journal selected areas communications vol rabinovitch vleeschauwer effective bandwidth available http feller introduction probability theory applications volume chapter john wiley sons zhao adve lim outage probability arbitrary snr cooperative diversity networks ieee communications letters issue cover elemenst information theory john kushner clark stochastic approximation methods constrained unconstrained systems chapter new york heidelberg berlin polyak juditsky acceleration stochastic approximation averaging siam jorunal control optimization vol july shoja taheri vakilinia new approach prevent black hole attack aodv international journal computer science information security jan kiskani khalaj vakilinia delay qos provisioning cognitive radio systems using adaptive modulation inproceedings acm workshop qos security wireless mobile networks oct acm vakilinia khalaj froushani transmission ofdma systems intelecommunications ict international conference may ieee rastegar vakilinia khalaj june rate adaptation power allocation miso rayleigh fading channel harq communications icc ieee international conference ieee hosseini lee vakilinia energy performance cool roofs impact actual weather data energy buildings jun vakilinia alvandi shoja vakilinia secure qos aware design voip wireless networks international journal business data communications networking ijbdcn oct vakilinia shahin xinyao zhang dongyu qiu analysis optimization stream processing global communications conference globecom ieee ieee vakilinia shahin iman vakilinia qos aware energy efficient resource allocation wireless cooperative ofdma relay networks wireless mobile networking conference wmnc joint ifip ieee vakilinia shahin behdad heidarpour mohamed cheriet energy efficient resource allocation cloud computing environments ieee access zemmouri samy shahin vakilinia mohamed cheriet let adapt network change towards energy saving rate adaptation sdn network service management cnsm international conference ieee vakilinia cheriet rajkumar october dynamic resource allocation smart home workloads cloud network service management cnsm international conference ieee heidarpour behdad zbigniew dziong shahin vakilinia pricing stackelberg duopoly wireless markets telecommunications network strategy planning symposium networks international ieee
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optimal locally repairable codes distance via cyclic codes jan yuan chaoping xing chen yuan abstract like classical block codes locally repairable code also obeys bound call locally repairable code optimal achieves bound breakthrough work several classes optimal locally repairable codes constructed via subcodes codes thus lengths codes given upper bounded code alphabet size recently proved extension construction length optimal locally repairable codes surprisingly presented examples optimal locally repairable codes small distance locality code length achieving roughly recently shown exist optimal locally repairable codes length bigger distance propositional thus becomes interesting challenging problem construct new families optimal locally repairable codes length bigger paper construct class optimal locally repairable codes distance unbounded length length codes independent code alphabet size technique cyclic codes particular generator polynomials carefully chosen introduction due applications distributed storage systems locally repairable codes recently attracted great attention researchers local repairable code nothing block code additional parameter called locality locally repairable code length information symbols locality see definition locally repairable codes section proved minimum distance upper bounded bound called bound locally repairable codes proved extending arguments proof classical singleton bound codes paper refer optimal locally repairable code block code achieving bound known results early constructions optimal locally repairable codes gave codes alphabet size exponential code length see also earlier construction optimal locally repairable codes given department computer sciences engineering shanghai jiaotong university shanghai china email luoyuan authors division mathematical sciences school physical mathematical sciences nanyang technological university singapore republic singapore email xingcp alphabet size comparable length however construction produced kcode specific value length thus rate code close also existence results given less restriction locality results papers require large alphabet size exponential function code length recent breakthrough construction given makes use subcodes codes construction produces optimal locally repairable codes length linear alphabet size although length codes upper bounded alphabet size construction extended via automorphism group rational function fields jin xing turns flexibility locality code length alphabet size based classical mds conjecture one wonder optimal locally repairable codes length bigger surprisingly shown exist optimal locally repairable codes length exceeding although produced optimal locally repairable codes specific parameters paves road people continue search optimal codes recently shown via elliptic curves exist optimal locally repairable codes length bigger distance proportional length result paper carefully choosing generator polynomials show exist optimal locally repairable codes distance cyclic one feature codes lengths unbounded lengths independent code alphabet sizes precisely following main result paper theorem let prime power assume positive integer gcd cyclic locally repairable code locality optimal gcd mod cyclic locally repairable code locality optimal gcd mod gcd divides remark conditions theorem easy satisfied length unbounded indeed parameters given theorem satisfy equality let verify part theorem open problems view known results result paper known optimal locally repairable codes length minimum distance satisfy either small unbounded proportional linear one natural question open problem optimal locally repairable codes length much bigger instance minimum distance proportional open problem following open problem optimal locally repairable codes unbounded length every constant organization paper paper organized follows section provide preliminaries locally repairable codes cyclic codes section present proof theorem addition also give construction optimal locally repairable codes length distance preliminaries section present preliminaries locally repairable codes cyclic codes locally repairable codes informally speaking block code said locality every coordinate given codeword recovered accessing coordinates codeword precisely speaking locally repairable code locality given follows definition let fnq block code length define subset denote projection called locally repairable code locality every exists subset cii cii disjoint apart usual parameters length rate minimum distance locality locally repairable code plays crucial role paper always consider locally repairable codes linear thus locally repairable code length dimension minimum distance locality said repairable code locality cyclic codes cyclic codes well known understood coding community briefly introduce list useful facts cyclic codes subsection cyclic code identified ideal ring every ideal principal ideal generated divisor write dimension deg furthermore codeword divisible words roots stands algebraic closure belongs let reciprocal polynomial let divisor put euclidean dual code recall facts cyclic codes without giving proof reader may refer books lemma let divisor codes equivalent let cyclic code length divisorp assume roots nonzero codeword cij xij weight determinant det zero iii let cyclic code length divisor let nth primitive root unity exist integer positive integer roots minimum distance least let denote symmetric group length automorphism block code permutation satisfying whenever automorphisms form subgroup denoted aut called automorphism group code called transitive exists automorphism aut cyclic code aut contains subgroup generated cyclic shift thus transitive constructions section first give general result locality transitive codes apply proof theorem addition also present construction locally repairable code locality general result lemma transitive linear code dual distance locality proof let codeword hamming weight let denote support let subset choose element every codeword gives implies repaired exists automorphism implies hence proof completed example let prime power let positive integers present optimal cyclic locally repairable code length minimum distance locality although code already given provide different view code observation finally leads discovery optimal locally repairable codes unbounded length given theorem let nth primitive root unity positive integer uniquely written integer integer case let obvious since roots nth roots unity distinct denote cyclic code generator polynomial due fact contains roots follows lemma minimum distance least moreover thus dimension deg thus show satisfies bound remains show locality lemma sufficient dual distance lemma sufficient observe show cyclic code minimum distance thus write mod mod codeword hamming weight implies minimum distance case case define cyclic code generated compared case degree increases due fact negative mimic proof case skip detail example let prime power let positive integers present optimal cyclic locally repairable code length locality minimum distance codes example founded already provide new view construction since implies irreducible factor either linear quadratic irreducible factor quadratic root root gives conclusion nth root unity polynomial let nth primitive root unity let first show defined root either implies roots thus product polynomials form conclude defined clear since roots nth roots unity distinct denote cyclic code generator polynomial due fact contains roots follows lemma minimum distance least moreover dimension deg thus show satisfies bound remains show locality lemma sufficient show dual distance lemma sufficient show cyclic code minimum distance observe codeword hamming weight implies minimum distance proof part theorem proof let nth primitive root unity put element let note means factor furthermore since gcd let cyclic code generated let first show locality lemma sufficient show hamming distance put lemma sufficient show hamming distance since equivalent consider codeword hamming weight hence hamming distance upper bounded finally roots minimum distance least lemma iii bound minimum distance upper bounded thus minimum distance exactly completes proof proof part theorem proof let nth primitive root unity put furthermore primitive root unity since gcd divides exist integers put must since root unity linear polynomial factor divides furthermore implies divisor let cyclic code length generated lemma show locality sufficient show hamming distance lemma sufficient show hamming put distance since equivalent consider codeword since hamming weight hence hamming distance upper bounded claim cyclic code generated minimum distance least prove claim contradiction suppose minimum distance exists nonzero polynomial divisible since roots hence roots lemma det forces since nth primitive root unity claim three divisibilities leads case since also roots lemma det using condition equation simplified observe primitive nth root unity follows equivalently case using condition equation simplified since must gives therefore well case case mimic proof case swapping three cases lead conclusion set moreover following four identities finally notice roots lemma gives det det contradiction due fact primitive root unity minimum distance bound proof completed locally reparable codes length distance finally present optimal locally repairable code length minimum distance theorem divides exists optimal locally repairable code locality proof let nth primitive root unity element put element let straightforward verify divisor let cyclic code generator polynomial similar augments proof theorem one show dual code minimum distance since roots distance least easy check code meets bound proof completed references barg tamo locally recoverable codes algebraic curves ieee trans inform theory barg haymaker howe matthews locally recoverable codes algebraic curves surfaces algebraic geometry coding theory cryptography howe lauter walker editors springer forbes yekhanin locality codeword symbols codes discrete mathematics gopalan huang simitci yekhanin locality codeword symbols ieee trans inf theory han reliable memories subline accesses proc ieee internat sympos inform theory huang chen pyramid codes flexible schemes trade space access efficiency reliable data storage systems sixth ieee international symposium network computing applications jin xing construction optimal locally repairable codes via automorphism groups rational function fields https xing optimal locally repairable codes via elliptic curves https ling xing coding first course cambridge macwilliams sloane theory codes papailiopoulos dimakis locally repairable codes ieee trans inf theory prakash kamath lalitha kumar optimal linear codes property proc ieee int symp inform theory silberstein rawat koyluoglu vichwanath optimal locally repairable codes via codes proc ieee int symp inf theory tamo barg family optimal locally recoverable codes ieee trans inform theory tamo papailiopoulos dimakis optimal locally repairable codes connections matroid theory ieee trans inform theory
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nov strong local nondeterminism spherical fractional brownian motion xiaohong lan school mathematical sciences university science technology china yimin xiao department statistics probability michigan state university november abstract let fractional brownian motion indexed unit sphere index introduced istas establish optimal upper lower bounds angular power spectrum exploit behavior establish property strong local nondeterminism key words angular power spectrum expansion spherical fractional brownian motion strong local nondeterminism mathematics subject classification introduction spherical fractional brownian motion sfbm brevity introduced istas extension spherical brownian motion well spherical analogue fractional brownian motion indexed euclidean spaces later istas established expansion studied quadratic variations spherical fractional brownian motion purpose paper investigate property strong local nondeterminism slnd spherical fractional brownian motion motivated studies sample path properties gaussian random fields indexed euclidean space currently increasing interest stochastic corresponding author research lan supported nsfc grants email xhlan research xiao partially supported nsf grants xiao modeling spherical data statistics cosmology applied areas see concept local nondeterminism lnd gaussian process first introduced berman unify extend methods studying existence joint continuity local times gaussian processes roughly speaking gaussian process said lnd property locally approximately independent increments see lemma precise description property allowed berman overcome difficulties caused complex dependence structure gaussian process studying local times pitt cuzick extended berman lnd gaussian random fields however property lnd enough establishing fine regularity properties law iterated logarithm uniform modulus continuity local times local times gaussian random fields studying many problems gaussian random fields appropriate properties strong local nondeterminism slnd proven powerful instead recalling definitions various forms strong local nondeterminism isotropic anisotropic gaussian random fields indexed applications refer xiao information recently lan marinucci xiao studied slnd property class gaussian random fields indexed unit sphere also called spherical gaussian random fields main difference aforementioned work gaussian fields indexed euclidean space takes spherical geometry full consideration method relies harmonic analysis sphere specifically lan marinucci xiao considered centered isotropic gaussian random field satisfies group rotations see systematic account random fields applying harmonic analytic tools sphere lan marinucci xiao proved slnd property isotropic gaussian field determined behavior angular power spectrum moreover applying slnd established exact uniform modulus continuity class isotropic gaussian fields since sfbm isotropic sense results slnd directly applicable approach make use expansion spherical fractional brownian motion obtained istas derive optimal upper lower bounds coefficients see definition bounds correct last part theorem useful studying dependence structures sample path properties sfbm paper provides important step towards direction specifically demonstrate coefficients play role angular power spectrum isotropic gaussian field high frequency behavior determines property strong local nondeterminism sfbm reason also call sequence angular power spectrum sfbm similarly cases gaussian random fields indexed euclidean space expect slnd property theorem useful studying regularity exact modulus continuity exact modulus nondifferentiability etc fractal properties sfbm carried subsequent paper analysis sfbm spherical gaussian random fields strongly motivated applications number scientific areas geophysics astrophysics cosmology atmospheric sciences see huge data sets satellite missions wilkinson microwave anisotropy probe wmap nasa see http planck mission european space agency see http collected made publicly available spherical random fields usually assumed gaussian proposed modeling data sets related aforementioned aspects also mention probability statistics literature various isotropic anisotropic gaussian random fields constructed studied see excursion probabilities topological properties excursion sets isotropic gaussian random fields studied many interesting questions probabilistic statistical properties anisotropic gaussian random fields sphere raised order study problems would interesting establish appropriate properties strong local nondeterminism anisotropic gaussian random fields rest paper organized follows section recall briefly background material sfbm including expansion analysis random coefficients main result section theorem provides optimal upper lower bounds angular power spectrum sfbm section combine high frequency behavior proposition establish property strong local nondeterminism sfbm sfbm asymptotic behavior angular power spectrum let north pole geodesic distance recall istas definition sfbm definition sfbm centered gaussian random field well known fractional brownian motion indexed defined every different case index set istas proved sfbm exists hurst index classical spherical brownian motion see work based following expansion proved istas theorem spherical harmonic functions eigenfunctions spherical laplacian sin sin denotes spherical coordinates precisely satisfy known form orthonormal basis space sin lebesgue measure explicit form spherical harmonics given cos denotes complex conjugate cos associated legendre functions defined terms legendre polynomials dxm negative moreover following orthonormality property holds ylm recall coefficients defined usual inner product similarly angular power spectrum isotropic gaussian random field sequence plays important role determining dependence structure probabilistic properties sfbm hence also refer angular power spectrum sfbm moreover follows theorem expansion holds sense sense every fixed hence one represent random coefficients equality holds sense notice set complexvalued gaussian random variables obviously due property moreover recall verify therefore set standard complex gaussian random variables following main result section bounds correct last part theorem behavior essential proving slnd property theorem together allow study precise analytic geometric properties sample functions sfbm see information theorem let spherical fractional brownian motion index exists uniform constant proof work spherical coordinates definition change variable cos obtain cos sin recall representation cos sin cos cos cos see becomes sin sin cos cos sin sin cos cos elementary fact sin follow sin sin hence sin sin sin sin cos cos meanwhile change variable sin inside integral sin sin sin cos cos sin sin beta function defined consequently sin sin sin readily seen make use techniques complex analysis let integral written sin sin eit unit circle first quadrant direction counterclockwise principal branch complex function obviously analytic moreover let arbitary small value two line segments defined meantime let circles radius first second quadrants defined respectively consider following integrals careful calculations show lim lim thus cauchy integral theorem complex analysis lim lim leads sin view equalities therefore combining equalities obtain sin recall formula gamma function stirling approximation error term order derive following estimates exists uniform constant let inequalities derived view strong local nondeterminism order prove strong local nondeterminism property first recall following lemma consequence proposition lemma assume sequence satisfies condition exists constant depending choices min ready state prove following theorem conclusion referred property strong local nondeterminism sfbm theorem sfbm exists constant depending hurst index integers var min var denotes conditional variance given remark note strong local nondeterminism slightly different slnd property proved isotropic gaussian random fields minimum right hand side taken also assumption definition sfbm statistics viewpoint var squared error predicting value given observations locations since already know information may reduce prediction error proof theorem known gaussian random field var inf infimum taken hence order establish sufficient prove exists positive constant holds min let follows expansion properties random coefficients left hand side equal therefore immediate consequence lemma view equalities completes proof references adler taylor random fields geometry springer new york alkhaled michalak kawa olsen wang global evaluation regional spatial variability column integrated distributions geophys res berman local nondeterminism local times gaussian processes indiana univ math cabella marinucci statistical challenges analysis cosmic microwave background radiation ann appl statist cheng xiao excursion probability gaussian random fields sphere bernoulli cuzick multiple points gaussian vector field wahrsch verw gebiete boyer madec fischer lazar iudicone mixed layer depth global ocean examination profile data climatology geophys res dodelson modern cosmology academic press estrade istas ball throwing spheres bernoulli huang zhang robeson validity commonly used covariance variogram functions sphere math geosci huang zhang robeson simplified representation covariance structure axially symmetric processes sphere stat prob lett istas spherical hyperbolic fractional brownian motion elect comm probab istas expansion spherical fractional brownian motions statist probab lett istas quadratic variations spherical fractional brownian motions stoch process appl jun stein nonstationary covariance models global data ann appl statist lan marinucci xiao strong local nondeterminism exact modulus continuity spherical gaussian fields stoch process accepted lan xiao exact moduli continuity nonwhere differentiability spherical fractional brownian motion preparation processus stochastiques mouvement brownien marcus rosen markov processes gaussian processes local times cambridge university press cambridge marinucci peccati random fields sphere representation limit theorem cosmological applications cambridge university press cambridge marinucci vadlamani asymptotics curvatures excursion sets sphere ann appl probab noda generalized radon transform brownian motion nagoya math panchenko talagrand overlap multiple spherical models ann probab pitt local times gaussian vector fields indiana univ math stein weiss introduction fourier analysis euclidean spaces princeton university press xiao strong local nondeterminism sample path properties gaussian random fields asymptotic theory probability statistics applications tze leung lai qiman shao lianfen qian editors higher education press beijing xiao sample path properties anisotropic gaussian random fields minicourse stochastic partial differential equations khoshnevisan editors lecture notes math springer new york xiao recent developments fractal properties gaussian random fields developments fractals related fields barral seuret eds springer new york
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may chromatin organization eukaryotic genome sequence extended abstract davide valeria raffaele filippo january abstract nucleosome organization eukaryotic genomes deep impact gene function although progress recently made identification various concurring factors influencing nucleosome positioning still unclear whether nucleosome positions sequence dictated determined random process postulated long time proximity tss barrier determines position nucleosome geometric constraints alter random positioning process determining nucleosomal phasing pattern fades one moves away barrier become random positioning process although statistical model widely accepted molecular nature barrier still unknown moreover far identification set sequence rules able account nucleosome organization explain nature barriers statistical mechanism hinges allow smooth transition statistical positioning back show sequence complexity quantified via various methods rule able least partially account particular conducted analyses four high resolution nucleosomal maps model eukaryotes elegans found nucleosome depleted regions well distinguished nucleosome enriched regions sequence complexity measures particular depleted regions less complex enriched ones moreover around tss complexity measures alone striking agreement vivo nucleosome occupancy particular precisely indicating positions nucleosomes findings indicate intrinsic richness subsequences within sequences plays role nucleosomal formation genomes sequence complexity constitutes molecular nature nucleosome barrier background well known chromatin organization eukaryotic genomes deep impact gene regulation function therefore comes surprise substantial amount research devoted fascinating topic particular focus identification nucleosome positions within genomes model organisms mechanisms influencing positioning although quite bit progress made identification various concurring factors influence nucleosome positioning answer question raised kornberg extend nuclesome positions sequence dictated determined random process remained elusive matter work presented sibbm seminar annual conference meeting italian biophysics molecular biology society may palermo italy dulbecco telethon institute palermo dipartimento biologia cellulare dello sviluppo palermo italy palermo dipartimento matematica informatica palermo italy palermo dipartimento matematica informatica palermo italy raffaele computational genomics group ibm watson research center yorktown heights usa futro fact object intense debate particular view result widom claiming exists genomic code nucleosome positioning mathematical terms code specific constrained object even degenarate genetic code therefore seems verbatim use term code context chromatin studies would quite misleading actually easy challenge indeed focus shifted sequence dictating sequence influencing chromatin organization finding much less amenable challenge dismissal contrast sequence debate outlined statistical positioning mechanism little challenge worth recalling based two main ingredients existence barrier statistical law governing positioning nucleosomes generalizing mentioned earlier results kornberg stryer gerland shown mechanism described quantified tonks model statistical physics particular proximity tss barrier determines position nucleosome geometric constraints alter random positioning process determining well known nucleosomal pattern dna pattern fades one moves away barrier become random positioning process analogous behaviour followed nucleosome statement results studies leave open several questions indeed given poor results finding consistent concise set explain sequence influences nucleosome positioning challenging problem show set rules exists analogously although statistical model widely accepted molecular nature barrier unknown moreover interplay statistical positioning hardly explored although expected true cellular state probably combination machanisms study mavrich performed dna regions around tss yeast addresses part latter problem indicating sequence dictates positioning nucleosomes leaving rest statistical positioning mechanism study effort also made identify compositional properties region nfr responsible formation barriers however also pointed identified sequence biases mostly dinucleotides may special cases general yet unknown properties genomic sequences downstram tss nutshell far identification concise set sequence rules able account nucleosome organization genome explaing nature barriers statistical mechanism hinges allow smooth transition statistical positioning back said code nucleosome positioning seems stringent fortunately intrinsic properties sequences particular ones measured complexity measures may play role influencing nucleosomal organization eukaryotic genome worth recalling complexity measures usually quantify intrinsic richness distinct subsequences within given sequence study whether sequence complexity quantified via various methods rule able least partially account towards end following results conducted extensive studies four high resolution nucleosomal maps three model orgamisms yeast elegans drosophila melanogaster obtain following results nucleosome depleted regions map well distinguished nuclesome enriched regions map sequence complexity measures particular depleted regions less complex enriched ones finding indicates intrinsic richness subsequences within sequences plays role influencing nucleosomal formation genomes addressing point applied methodology tss dataset mavrich also accounted additional insights gerland find nfr characterized area lower complexity respect two regions flanking tss placed proximity absolute minimum complexity curve computed two barriers towards local maxima left right tss particularly striking complexity curve obtained linguistic complexity based dictionary words length seventeen therefore much larger one considered mavrich indeed positions nucleosomes respectively proximity first maximum right left respectively tss complexity curve view point finding suggests least far tss concerned combinatorial properties nfr subsequence strongly associated creation barriers highlighting nature least part combinatorial dictated intrinsic properties sequences since complexity subsequences within sequence modulated points suggest least tss modulation smoothly accomodate sequencedictated statistical positioning continuum requires particular arrangement switching two states interest indeed low complexity area characterizing nfr indicated nucleosomes giving also indication nucleosomes must placed towards local maxima complexity curve end nfr interpretation sheds light point references felsenfeld groudine controlling double helix nature jiang pugh nucleosome positioning gene regulation advances genomics nature genetics kaplan moore gossett tillo field leproust hughes lieb widom segal nucleosome organization eukaryotic genome nature kornberg locations nucleosomes chromatin specific statistical nature kornberg stryer statistical distributions nucleosomes nonrandom locations stochastic mechanism nucleic acids research mavrich ioshikhes venters jiang tomsho schuster albert pugh barrier nucleosome model statistical positioning nucleosomes throughout yeast genome genome research mavrich jiang ioshikhes venters zanton tomsho glaser schuster gilmour albert pugh nucleosome organization drosophila genome nature gerland quantitative test barrier nucleosome model statistical positioning nucleosomes downstream transcription start sites plos computational biology segal chen thastrom field moore wang widom genomic code nucleosome positioning nature segal widom controls nucleosome positions trends genetics valouev ichikawa thaisan stuart swati peckham zeng malek costa mckernan sidow fire johnson nucleosome position map elegans reveals lack universal positioning genome research zhang moqtaderi rattner euskirchen snyder kadonaga shirley liu struhl intrinsic interactions major determinant nucleosome positions vivo nature structural molecular biology
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submitted annals statistics arxiv accuracy assessment linear sep tony cai zijian guo university pennsylvania paper considers point interval estimation loss estimator linear regression random design establish minimax rate estimating loss minimax expected length confidence intervals loss estimators regression vector including commonly used estimators lasso scaled lasso lasso dantzig selector adaptivity confidence intervals loss also studied setting known identity design covariance matrix known noise level setting unknown design covariance matrix unknown noise level studied results reveal interesting significant differences estimating loss loss well two settings new technical tools developed establish rate sharp lower bounds minimax estimation error expected length minimax adaptive confidence intervals loss significant difference loss estimation traditional parameter estimation loss estimation constraint performance estimator regression vector lower bounds difficulty estimating loss technical tools developed paper also independent interest introduction many applications goal statistical inference construct good estimator also provide measure accuracy estimator classical statistics parameter interest achieved form standard error confidence interval prototypical example inference binomial proportion often estimate proportion also margin error given accuracy measures estimation procedure also used tool empirical selection tuning parameters well known example stein unbiased risk estimate sure effective tool construction adaptive estimators normal means estimation nonparametric signal recovery covariance research supported part nsf grants nih grant msc subject classifications primary secondary keywords phrases accuracy assessment adaptivity confidence interval highdimensional linear regression loss estimation minimax lower bound minimaxity sparsity cai guo matrix estimation problems see instance commonly used methods also viewed useful tool based idea empirical assessment accuracy paper consider problem estimating loss given estimator setting linear regression one observes iid regression vector rows iid errors independent linear model well studied literature main focus estimation several minimization methods including lasso dantzig selector scaled lasso lasso proposed methods shown work well applications produce interpretable estimates assumed sparse theoretically properly chosen tuning parameter estimators achieve optimal rate convergence collections sparse parameter spaces see example loss comfor given estimator monly used metric accuracy consider present paper point interval estimation loss given note loss random quantity depending estimator parameter random quantity prediction prediction interval ususally used point interval estimation respectively however slightly abuse terminologies present paper using estimation confidence interval represent point interval estimators loss since loss depends necessary specify estimator discussion loss estimator estimation throughout paper restrict attention broad collection estimators perform well least one interior point small subset parameter space collection estimators includes estimators lasso dantzig selector scaled lasso lasso linear regression well studied two settings one setting known design covariance matrix known noise level sparse see example another commonly considered setting sparse unknown study point interval estimation loss settings specifically consider parameter space introduced accuracy assessment consists signals known design covariance matrix known noise level defined consists signals unknown contributions present paper studies minimax adaptive estimation loss given estimator minimax expected length adaptivity confidence intervals loss major step analysis establish rate sharp lower bounds minimax estimation error minimax expected length confidence intervals loss broad class estimators contains subclass estimators focus estimation loss estimators take lasso scaled lasso estimators generic examples estimators propose procedures point estimation well confidence intervals losses shown proposed procedures achieve corresponding lower bounds constant factor results together establish minimax rates estimating loss estimators analysis shows interesting significant differences estimating loss loss well two parameter spaces min log minimax rate estimating logn loss estimation much easier prior information lognp logn minimax rate estimating logn regime lognp logn practical loss estimator proposed estimating loss shown achieve optimal convergence rate adaptively say estimation loss impossible minimax rate achieved trivial estimator means estimation accuracy loss least order loss considered cases estimation loss shown impossible results indicate loss estimation difficult turn construction confidence intervals loss confidence interval loss useful even impossible estimate loss confidence interval provide upper lower bounds loss terms convergence rate minimax rate expected length confidence intervals loss estimator coincides cai guo minimax estimation rate also consider adaptivity confidence framework intervals loss estimator adaptive confidence intervals discussed detail section regarding confidence intervals loss case known procedure proposed shown achieve optimal length adaptively lognp logn furthermore shown regime adaptive confidence intervals exist even two given parameter spaces example log impossible construct confidence interval loss guaranteed coverage probability consequently also expected length automatically adjusted sparsity similarly loss adaptive confidence intervals impossible logn regarding confidence intervals loss case unknown impossibility adaptivity also holds logn establishing lower bounds requires development new technical tools one main difference loss estimation traditional parameter estimation loss estimation constraint performance estimator regression vector lower bound difficulty estimating loss introduce useful new lower bound techniques minimax estimation error expected length adaptive confidence intervals loss several important cases necessary test composite null composite alternative order establish rate sharp lower bounds technical tools developed paper also independent interest addition also study intermediate parameter space noise level known design covariance matrix unknown certain structure lower bounds expected length minimax adaptive confidence intervals parameter space established broad collection estimators shown rate sharp class estimators furthermore lower bounds developed paper wider implications particular shown lead immediately minimax lower bounds estimating expected length confidence intervals comparison works statistical inference loss specific estimators considered recent literature papers established setting limit normalized loss lasso estimator tuning parameter although provided accuracy assessment exact asymptotic expression normalized loss limit depends unknown similar setting paper established limit normalized loss lasso estimator limits normalized losses help understand properties corresponding estimators lead estimate loss results imply although normalized losses limit regularity conditions losses estimated well settings recent paper constructed confidence interval case known unknown noise level moderate dimension sparsity assumed sparsity assumption imposed method requires assumption contrast paper consider unknown known settings allowing assuming sparse honest adaptive inference studied nonparametric function estimation literature including adaptive confidence intervals linear functionals adaptive confidence bands adaptive confidence balls linear regression literature including adaptive confidence set adaptive confidence interval linear functionals paper develop new lower bound tools theorems establish possibility adaptive confidence intervals connection loss considered current paper work discussed detail section organization section establishes minimax lower bounds estimating loss shows bounds rate sharp lasso scaled lasso secb estimators respectively turn interval estimation tions present minimax adaptive minimax lower bounds expected length confidence intervals lasso scaled lasso estimators show lower bounds achieved investigate possibility adaptivity section considers estimators establishes minimax convergence rate estimating losses section presents new minimax lower bound techniques estimating loss section discusses minimaxity adaptivity another setting noise level known design covariance matrix unknown certain structure section applies newly developed lower bounds establish lower bounds related problem estimating section proves main results additional proofs given supplemental material cai guo notation matrix denote respectively row column entry matrix subset denotes cardinality denotes complement denotes submatrix consisting columns vector subvector indices vector supp denotes support norm defined kxkq max use max shorthand min shorthand matrix define spectral norm matrix norm symmetric matrix denote respectively smallest largest eigenvalue use denote generic positive constants may vary place place two positive sequences means cbn lim abnn minimax estimation loss begin presenting minimax framework estimating loss given estimator establish minimax lower bounds estimation error also show minimax lower broad collection estimators bounds achieved lasso scaled lasso estimators problem formulation recall linear model iid focus random design independent let denote observed data given estimator loss estimator denoting minimax rate convergence estimating parameter space defined largest quantity inf sup constant depending shall write confusion denote parameter consists signal design covariance matrix noise level given accuracy assessment use denote corresponding two settings considered first known design covariance matrix known noise level unknown first setting consider following parameter space consists signals second setting consider constants parameter space subset consists signals unknown minimax rate estimating also depends different estimators could lead different losses estimator general difficulty estimating loss varies first recall properties estimators specify collection estimators focus paper shown lasso dantzig selector scaled lasso lasso satisfy following property tuning parameter properly chosen log sup constant minimax lower bounds established imply logn optimal rate estimating parameter space stressed algorithms require knowledge sparsity thus adaptive sparsity provided logn consider broad collection estimators satisfying one following two assumptions estimator satisfies log constants estimator satisfies log sup constants given cai guo view minimax rate given assumption requires good estimator least one point assumption slightly stronger requires estimate well single range noise levels course estimator satisfying satisfies addition assumptions also introduce following sparsity assumptions used various theorems let constant defined sparsity levels satisfy min logn constant min lognp sparsity levels satisfy min logn constant min lognp minimax estimation loss following theorem establishes minimax lower bounds estimating loss parameter space theorem suppose sparsity levels satisfy assumption estimator satisfying assumption log inf sup min estimator satisfying assumption log inf sup constants remark assumption restricts focus estimators perform well least one point weak condition makes established lower bounds widely applicable benchmark evaluating estimators loss performs well proper subset even single point whole parameter space paper focus estimating loss similar results established loss form assumptions theorem lower bounds estimating loss hold replacing convergence rates power remains convergence accuracy assessment rate log replaced log similarly results established rest paper hold corresponding convergence rates replaced power theorem establishes minimax lower bounds estimating loss estimator satisfying assumption loss estimator satisfying assumption take lasso estimator example demonstrate implications split subsamples sample sizes respectively lasso estimator based first subsample defined arg minp log constant without loss generality assume case together imply estimation loss impossible since lower bound achieved trivial estimator log loss case regime lognp lower bound log achieved zero estimator hence estimation loss impossible however interesting case lognp proposed achieves minimax lower loss estimator log bound achieved zero estimator based second half detail construction loss estimator sample propose following estimator note first subsample used produce lasso estimator second subsample retained evaluate loss sample splitting technique similar used constructing confidence sets confidence intervals loss achieves following proposition establishes estimator minimax lower bound regime log log cai guo proposition suppose logn lasso estimator defined estimator loss proposed satisfies sequence lim sup sup minimax estimation loss turn case unknown establish minimax lower bound estimating loss parameter space theorem suppose sparsity levels satisfy assumption estimator satisfying assumption log inf sup constants theorem provides minimax lower bound estimating loss estimator satisfying assumption including scaled lasso estimator defined arg min log note scaled lasso estimator lower bound trivial achieved loss estimator log sense hence estimation loss impossible case minimaxity adaptivity confidence intervals focused last section point estimation loss showed impossibility loss estimation except one regime results naturally lead another question possible construct useful confidence intervals provide upper lower bounds loss section introducing framework minimaxity adaptivity confidence intervals consider case known establish minimaxity adaptivity lower bounds expected length confidence intervals loss broad collection estimators parameter space also show accuracy assessment minimax lower bounds achieved lasso estimator discuss possibility adaptivity using lasso estimator example case unknown focus next section framework minimaxity adaptivity confidence intervals section introduce following decision theoretical framework confidence intervals loss given parameter set space loss denote level confidence intervals inf confusion write length denoted confidence interval maximum expected length parameter space defined sup two nested parameter spaces define benchmark measuring degree adaptivity nested spaces sup inf minimax write expected length confidence intervals infimum maximum expected length mark among confidence intervals contrast considering confidence intervals words parameter lies smaller measures benchmark length conparameter space fidence intervals parameter space illustrated left figure however prior information parameter lies larger parameter space measures benchmark length confidence intervals parameter space illustrated right figure cai guo fig plot demonstrates definition rigorously define confidence interval simultaneously adapb tive condition means confidence interval coverage larger parameter space achieves minimax rate note adaptation impossible achieve otherwise possible construct confidence intervals simultaneously adaptive parameter spaces possibility adaptation parameter spaces answered vestigating benchmark quantities framework already introduced studies minimaxity adaptivity confidence intervals linear functionals highdimensional linear regression adopt minimax discussed establish minimax expected length terms minimax extation benchmark pected length adaptivity behavior exist fundamental differences case discuss separately following two sections confidence intervals loss following theorem establishes minimax lower bound expected length confidence intervals parameter space theorem suppose sparsity levels satisfy assumption estimator satisfying assumption accuracy assessment constant log min particular lasso estimator defined minimax expected length level confidence intervals log min consider adaptivity confidence intervals loss lowing theorem gives lower bound benchmark discuss theorems together theorem suppose sparsity levels satisfy assumption estimator satisfying assumption constant log min particular lasso estimator defined lower bound achieved lower bound established theorem implies theorem lower bounds hold general class estimators satisfying phase bound benchmark transition lower regime log lower bound lognp logn lower bound lasso estimator defined lower bound log log achieved confidence intervals defined respectively interval estimator also used loss minimax lower bound achieved following confidence interval log cai guo quantiles random variable degrees freedom respectively log min note confidence interval simply based observed data depending prior knowledge sparsity furthermore confidence interval tells upper bound also lower bound loss coverage property expected length established following proposition proposition suppose logn estimator defined defined satisfies lim inf inf log log log fig illustration top bottom regimes logn rightmost log leftmost log middle log log regarding lasso estimator defined discuss possibility adaptivity confidence intervals adaptivity behavior confidence intervals demonstrated figure illustrated rightmost plot figure regime log obtain log accuracy assessment implies adaptation possible regime shown proposition confidence interval defined fully adaptive regime log logn sense illustrated leftmost middle plots figure impossible construct adaptive confidence interval regimes lognp lognp logn since lognp sum adaptive confidence intervals possible regime log log comparison confidence balls note problem constructing confidence intervals related different constructingnconfidence sets confidence balls constructed form lasso estimator data dependent squared radius see details naive application confidence ball leads confidence interval loss ciinduced due reason confidence sets sought foroin theorem suffice confidence sets form achieve optimal length however since goal characterize apply unbiased risk estimation discussed theorem construct confidence interval twosided confidence interval informative confidence interval since confidence interval contain information whether loss close zero furthermore shown length confidence interval ciinduced parameter space log order confidence interval constructed expected length log log much shorter regime confidence interval provides accurate interval estimator loss illustrated figure lower bound technique developed literature adaptive confidence sets also used establish lower bound results case given present paper however new techniques needed order establish lower bounds minimax estimation error region lognp logn cai guo fig comparison confidence interval confidence interval ciinduced expected length confidence intervals region log log necessary test composite null composite alternative order establish rate sharp lower bounds confidence intervals loss consider case investigate minimax expected length adaptivity confidence intervals parameter space following theorem characterizes minimax convergence rate expected length confidence intervals theorem suppose sparsity levels satisfy assumption estimator satisfying assumption constant log particular lasso estimator defined minimax expected length level confidence intervals log construct confidence interval achieving minimax convergence rate log max log following proposition establishes coverage property expected length accuracy assessment proposition suppose logn estimator defined confidence interval defined satisfies lim inf inf log particular case also hold estimator defined result shows confidence interval achieves minimax rate given contrast loss confidence interval significantly shorter interval achieves optimal rate regime lognp logn loss confidence interval achieves optimal rate given consider adaptivity confidence intervals theorem establishes lower bound theorem suppose sparsity levels satisfy assumption estimator satisfying assumption constant log log log log log lognp logn particular lasso estimator defined lower bounds achieved lower bounds theorem imply theorem lower bounds hold general class estimators satisfying assumption however lower bound theorem significantly different meaning theorem quantifies cost adaptation without knowing sparsity level lasso estimator cai guo comparing theorem theorem obtain implies impossibility constructing adaptive confidence intervals case exists marked difference case case possible construct adaptive confidence intervals regime log logn lasso estimator defined shown proposition confidence interval defined achieves lower bound logn lower bounds logn achieved following proposed confidence interval given following result verifies claim proposition suppose logn defined defined satisfies lim inf inf log minimaxity adaptivity confidence intervals section focus case unknown establish rates convergence minimax expected length confidence intervals defined also study possibility adaptivity confidence intervals following establishes lower bounds benchmark quantities theorem theorem suppose sparsity levels satisfy assumption estimator satisfying assumption constant log accuracy assessment log particular scaled lasso estimator defined lower bounds achieved lower bounds hold satisfying assumption interior point including scaled lasso estimator special case demonstrate impossibility adaptivity confidence inbsl defined tervals loss scaled since comparison illustrated figure referring adaptivity defined impossible construct adaptive confidence intervals log log fig illustration left right theorem shows confidence interval loss estimator assumption coverage satisfying constraint expected length given must order logn contrast theorem theorem demonstrates confidence intervals must long large subset points parameter space small number unlucky points therefore lack adaptivity confidence intervals due conservativeness minimax framework following detail construction confidence intervals construction confidence intervals based following cai guo definition restricted eigenvalue introduced min min denotes subset corresponding largest absolute value coordinates outside define event log confidence interval defined log log min log max min properties established follows proposition suppose logn estimator defined defined satisfies following properties lim inf inf log proposition shows confidence interval defined achieves lower bound confidence interval defined achieves lower bound estimation loss estimators established minimax lower bounds estimation accuracy loss broad class estimators satisfying weak assumptions also demonstrated minimax lower bounds sharp lasso estimator scaled lasso estimator section show minimax lower bounds sharp class estimators satisfying following assumption accuracy assessment estimator satisfies log log sup constants say estimator satisfies assumption shown lasso dantzig selector scaled lasso squareroot lasso tuning parameter chosen properly shall stress assumption implies assumptions assumption requires estimator perform well whole parameter space assumptions require perform well single point proper subset following proposition shows minimax lower bounds established theorem theorem achieved class estimators proposition let estimator satisfying assumption exist estimators loss achieving constant factor minimax lower bounds theorem theorem estimators loss achieving constant factor minimax lower bounds theorem theorem suppose estimator constructed based subsample exist estimators loss achieving constant factor minimax lower bounds theorem theorem theorem suppose estimator constructed based subsample satisfies assumption assumption supp exist estimators loss achieving minimax lower bounds theorem reasons space discuss detailed construction confidence intervals achieving minimax lower bounds postpone construction proof proposition remark sample splitting widely used literature example condition constructed based subsample introduced constructing confidence sets constructing confidence intervals loss cai guo condition imposed purely technical reasons create independence estimator subsample used shown assumption evaluate loss estimator satisfied lasso dantzig selector general tools minimax lower bounds major step analysis establish rate sharp lower bounds estimation error expected length confidence intervals loss introduce section new technical tools needed establish lower bounds significant distinction lower bound results given previous sections traditional parameter estimation problems constraint performance estimator regression vector lower bounds difficulty estimating loss necessary develop new lower bound techniques establish lower bounds estimation error expected length confidence intervals loss technical tools may also independent interest begin notation let denote random variable whose distribution indexed parameter let denote prior parameter space use denote density given denote marginal density prior let denote distribution corresponding indicator function function write expectation specifically distance two probability distributions densities given following theorem establishes minimax lower bounds estimation error expected length confidence intervals loss constraint good estimator least one interior point theorem suppose positive definite define let denote prior parameter space estimator satisfies inf sup accuracy assessment inf min remark minimax lower bound estimation error expected length confidence intervals hold long estimator estimates well interior point besides condition another key ingredient lower bounds construct least favorable space prior marginal distributions estimation lower bound constraining well estimated due establish loss estimated well lower bound condition show much larger hence honest confidence intervals must sufficiently long theorem used establish minimax lower bounds estimation error expected length confidence intervals loss taking theorem follows properly constructed subset taking theorem follows properly constructed cases assumption implies condition several minimax lower bounds also implied theorem estimation error minimax lower bounds regime log theorem follow expected length confidence intervals minimax lower bounds theorem regions log log logn theorem follow cases assumption guarantee condition satisfied however minimax lower bound mation error region log logn expected length confidence intervals region lognp logn established using theorem following theorem requires testing composite null composite alternative establishes refined minimax lower bounds cai guo theorem let positive definite matrix let two sparsity levels assume exist parameter spaces given disti disti let denote prior space suppose exist constants inf sup min min remark long estimator performs well two points minimax lower bounds estimation error expected length confidence intervals hold note theorem belong parameter space contrast theorem theorem compares composite hypotheses lead sharper lower bound comparing simple null composite alternative simplicity construct least favorable parameter spaces points fixed distance fixed distance respectively importantly construct prior prior distinguishable introduced facilitate comparison condition construction establish loss simultaneously estimated well lower bound conditions shown loss far apart confidence interval guaranteed coverage probability must accuracy assessment sufficiently long due prior information lower bound construction involved shall stress construction comparison composite hypotheses independent interest minimax lower bound region log follows log minimax lower bound region log expected length confidence intervals follows cases taken assumption implies condition intermediate setting known unknown results given sections show significant difference terms minimaxity adaptivity confidence intervals simple setting known design covariance matrix known noise level unknown section consider minimaxity adaptivity confidence intervals intermediate setting noise level known unknown certain structure specifically consider following parameter space max constants basically assumes known noise level imposes sparsity conditions precision matrix random design parameter space similar used literature sparse linear regression random design two sparsity parameters represents sparsity represents maximum row sparsity precision matrix note special case log minimaxity adaptivity assumption lower bounds expected length confidence intervals theorems hold replaced respectively case following theorem establishes minimaxity adaptivity lower bounds expected length confidence intervals cai guo theorem suppose sparsity levels satisfy assumption constant replaced defined estimator satisfying log sup constant constant log log max min log particular constructed based subsample satisfies assumption lower bounds attained contrast theorems lower bounds case change absence prior knowledge possibility adaptivity confidence intervals similar since lasso estimator defined satisfies assumption theorem minimax lower bounds attained log log log adaptation possible regimes adaptation impossible reasons space discussion including construction tive confidence intervals regime log logn postponed supplement minimax lower bounds estimating lower bounds developed paper broader implications particular established results imply minimax lower bounds estimating expected length confidence intervals build connection sufficient note trivial estimator satisfies assumptions accuracy assessment apply lower bounds estimator establish minimax lower bounds estimating log inf sup min log inf sup log inf sup constants similarly lower bounds expected length confidence intervals established theorem theorem imply corresponding lower bounds lower bound min logn detection boundary sparse linear regression case see details estimation linear regression considered general setting unknown lower bound leads one key component lower bound logn estimating proofs section present proofs lower bound results section establish general lower bound result theorem applying theorem theorem establish theorems section reasons space proofs theorems upper bound results including propositions proofs technical lemmas postponed supplement define distance two density functions well known let denote joint probability joint density function introduce following lemma used proofs theorem theorem proof lemma found supplement cai guo lemma event write denotes loss estimator respectively recall denotes corresponding parameter proof theorem set proof assume otherwise hence follows define event obtain triangle inequality first inequality follows triangle inequality last inequality follows hence obtain inf accuracy assessment note inf since max risk lower bounded bayesian risk lower bound last term inf combined establish sup combining establish proof inf define event imply define event combined exists hence hence define event establish hence cai guo proof theorems first specify constants used proof let given define min log log min theorems follow theorem theorem suppose sparsity levels satisfy assumption suppose satisfies assumption log constant log lognp logn constant log log max particular minimax lower bound term established weaker assumption theorem establish theorem theorem regime log lower bound follow case regime lognp logn first term right hand side second term leads let min constant application leads max logn result lower bounds regions log log log log accuracy assessment follow fact lower bounds log log log logn follow following lemma shows holds defined verifying assumption holds defined verifying assumption proof found supplement regions log inf exp particular result holds assumption lemma references ery emmanuel yaniv plan global testing sparse alternatives anova multiple comparisons higher criticism annals statistics mohsen bayati andrea montanari lasso risk gaussian matrices information theory ieee transactions alexandre belloni victor chernozhukov lie wang lasso pivotal recovery sparse signals via conic programming biometrika peter bickel acov ritov alexandre tsybakov simultaneous analysis lasso dantzig selector annals statistics peter sara van geer statistics data methods theory applications springer science business media tony cai zijian guo supplement accuracy assessment highdimensional linear regression tony cai zijian guo confidence intervals linear regression minimax rates adaptivity annals statistics appear tony cai mark low adaptation theory nonparametric confidence intervals annals statistics tony cai mark low adaptive confidence balls annals statistics tony cai mark low zongming adaptive confidence bands nonparametric regression functions journal american statistical association tony cai harrison zhou block thresholding approach wavelet estimation annals statistics emmanuel terence tao dantzig selector statistical estimation much larger annals statistics victor chernozhukov christian hansen martin spindler inference linear models many controls instruments cai guo victor chernozhukov christian hansen martin spindler valid inference elementary general approach arxiv preprint david donoho iain johnstone adapting unknown smoothness via wavelet shrinkage journal american statistical association david donoho arian maleki andrea montanari phase transition compressed sensing information theory ieee transactions zijian guo wanjie wang tony cai hongzhe optimal estimation linear models arxiv preprint marc hoffmann richard nickl adaptive inference confidence bands annals statistics yuri ingster alexandre tsybakov nicolas verzelen detection boundary sparse regression electronic journal statistics lucas janson rina foygel barber emmanuel eigenprism inference ratios arxiv preprint stein unbiased risk estimates method generalized cross validation annals statistics richard nickl sara van geer confidence sets sparse regression annals statistics garvesh raskutti martin wainwright bin minimax rates estimation linear regression information theory ieee transactions james robins aad van der vaart adaptive nonparametric confidence sets annals statistics charles stein estimation mean multivariate normal distribution annals statistics tingni sun zhang scaled sparse linear regression biometrika christos thrampoulidis ashkan panahi babak hassibi asymptotically exact error analysis generalized arxiv preprint robert tibshirani regression shrinkage selection via lasso journal royal statistical society series methodological sara van geer peter yaacov ritov ruben dezeure asymptotically optimal confidence regions tests models annals statistics nicolas verzelen minimax risks sparse regressions dimensional phenomenons electronic journal statistics fei zhang rate minimaxity lasso dantzig selector loss balls journal machine learning research feng hui zou tapering estimation large covariance matrices computational statistics data analysis accuracy assessment department statistics wharton school university pennsylvania philadelphia pennsylvania usa tcai zijguo url http
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may effective separability finitely generated nilpotent groups effective separability finitely generated nilpotent groups mark june abstract give effective proofs residual finiteness conjugacy separability finitely generated nilpotent groups particular give precise asymptotic bounds function introduced measures large quotients needed separate elements bounded length identity improves work similarly give polynomial upper lower bounds analogous function introduced lawton louder mcreynolds measures large quotients needed separate pairs distinct conjugacy classes bounded word length using work blackburn mal tsev introduction say residually finite exists surjective homomorphism finite group mal tsev proved residually finite finitely presentable group exists solution word problem say conjugacy separable pair exists surjective homomorphism finite group conjugate mal tsev also proved conjugacy separable finitely presentable group exists solution conjugacy problem residual finiteness conjugacy separability subgroup separability residual properties extensively studied used great effect resolving important conjectures geometry work agol virtual haken conjecture much work literature understand groups satisfy various residual properties example free groups polycyclic groups surface groups fundamental groups compact orientable shown residually finite conjugacy separable recently several papers made effective separability properties certain classes groups main purpose article improve effective residual purdue university west lafayette mpengito effective separability finitely generated nilpotent groups finiteness results establish effective conjugacy separability results class finitely generated nilpotent groups residual finiteness finitely generated group finite generating subset see also introduced function natural numbers quantifies residual finiteness specifically value maximum order finite group needed distinguish element identity one varies elements numerous authors studied asymptotic behavior wide collection groups see state results require notation two functions write exists write finitely generated nilpotent group denote normal subgroup finite order elements see subsection dependence mild subsequently suppress dependence generating subset subsection finitely generated nilpotent groups thm proved log hirsch length first result establishes precise asymptotic behavior theorem let infinite finitely generated nilpotent group exists log additionally one compute given basis step length proof theorem done two steps establish upper bound appeal structural properties finitely generated nilpotent groups establish lower bound construct sequence order minimal finite group separates identity bounded log following consequence proof theorem corollary let finitely generated nilpotent group log introduce terminology suppose connected simply connected nilpotent lie group lie algebra say admits basis rational structure constants mal tsev completion torsion free finitely generated nilpotent group connected simply connected nilpotent lie group embeds cocompact lattice next theorem demonstrates asymptotic behavior invariant mal tsev completion theorem suppose two infinite finitely generated nilpotent groups isomorphic mal tsev completions effective separability finitely generated nilpotent groups proof theorem follows examination cyclic series comes refinement upper central series interaction topology mal tsev completion since integral heisenberg group embeds every infinite nilpotent group theorem theorem thm cor allow characterize within collection connected simply connected nilpotent lie groups asymptotic behavior residual finiteness cocompact lattice corollary let connected simply connected nilpotent lie group lie isomorphic rdim log cocompact lattice last result section need terminology say group irreducible splitting direct product function form log call polynomial logarithm degree growth theorem exists satisfying following exists irreducible torsion free finitely generated nilpotent group step length log every natural number equal realized polynomial logarithm degree growth irreducible torsion free finitely generated nilpotent group iii suppose natural numbers exist irreducible torsion free finitely generated nilpotent groups step lengths respectively natural numbers exists irreducible torsion free finitely generated nilpotent group step length log theorem consider free nilpotent groups fixed step length increasing rank make use central products filiform nilpotent groups theorem using theorem able relate polynomial logarithm degree growth residual finiteness function well known invariants class finitely generated nilpotent groups theorem implies polynomial logarithm degree growth depend hirsch length similarly theorem implies upper bound terms step length polynomial logarithm degree growth hand step size determined polynomial logarithm growth seen theorem iii conjugacy separability turn attention effective conjugacy separability introduced function natural numbers quantifies conjugacy separability effective separability finitely generated nilpotent groups precise value maximum order minimal finite quotient needed separate pair elements one varies pairs elements since dependence mild see lemma suppress generating subset throughout subsection author knowledge previous work asymptotic behavior due demonstrate surface group finite rank free group cor subsection initiate study asymptotic behavior collection finitely generated nilpotent groups first result precise asymptotic behavior integral heisenberg group theorem general nilpotent groups establish following upper bound theorem let finitely generated nilpotent group blackburn first prove conjugacy separability finitely generated nilpotent groups strategy proving theorem effectivize class groups following lower bound allows characterize virtually abelian groups within class finitely generated nilpotent groups moreoever obtain first example class groups asymptotic behavior shown dramatically different theorem let finitely generated nilpotent group contains normal abelian subgroup index log log suppose virtually abelian exists additionally one compute given basis step length proof theorem elementary prove theorem finding infinite sequence elements order minimal finite group separates conjugacy classes bounded lower finite generating subset following theorem similar nature theorem theorem let infinite finitely generated nilpotent groups step size greater equal suppose isomorphic mal tsev completions apply theorem construct nilpotent groups help demonstrate various asymptotic behaviors growth conjugacy separability may exhibit effective separability finitely generated nilpotent groups theorem natural numbers exists irreducible torsion free finitely generated nilpotent group step length theorem implies conjugacy separability function depend step length nilpotent group consider central products filiform nilpotent groups theorem acknowledgements would like thank advisor ben mcreynolds help mentor priyam patel guidance indebted karel dekimpe suggestions proposition also want thank khalid looking earlier drafts article many insightful comments finally want thank alex barrios rachel davis jonas artur jackson gijey gilliam brooke magiera nick miller conversations article background assume reader familiar finitely generated groups lie groups lie algebras notation conventions let lcm lowest common multiple convention lcm lcm let gcd greatest common multiple convention gcd denote word length respect denote identity denote order element write exists normal subgroup set natural projection write clear context subset denote subgroup generated define commutator denote commutator convention denote center centralizer define term lower central series term upper central series denote height minimal define abelianization associated projection given define denote associated projection dim dim dim given basis denote lie algebra lie ideal define natural lie projection algebra denote center term lower central series term upper central series effective separability finitely generated nilpotent groups define map ada given ada denote lie bracket convention finitely generated groups separability residually finite groups following see also define depth function finitely generated groups given def min def define max residually finite group two finite generating subsets see lem therefore suppress choice finite generating subset conjugacy separable groups following define conjugacy depth function given def min def define max conjugacy separable group lemma two finite generating subsets proof similar lem see also lem suppress choice finite generating subset lem implies order minimal finite group required separate element identity bounded order minimal finite group required separate conjugacy class identity thus conjugacy separable groups particular conjugacy separable residually finite nilpotent groups nilpotent lie groups see thorough account material subsection let def finitely generated group term lower central series defined def def upper central series defined def effective separability finitely generated nilpotent groups definition say nilpotent group step size minimal natural number equivalently step size unspecified simply say nilpotent group say finitely generated nilpotent group admissible group admissible group set finite order elements denoted finite order characteristic subgroup moreover torsion free let finite dimensional algebra term lower central series def def defined define term upper def def central series definition say nilpotent lie algebra step length minimal natural number satisfying equivalently step size unspecified simply say nilpotent lie algebra connected simply connected nilpotent lie group step length lie algebra exponential map written exp diffeomorphism thm whose inverse formally denoted log formula implies every satisfies def def log exp exp cbm cbm rational linear combination lie brackets assumption cbm may generally write log exp exp cbi cbi rational linear combination lie brackets via repeated applications formula define adjoint representation aut may write log log connected simply connected nilpotent lie group lie algebra admits codim compact lattice admits basis rational structure constants see thm details say admits cocompact lattice torsion free admissible group thm implies exists group unique isomorphism embeds cocompact lattice definition call group mal tsev completion given qdefined group tangent space identity lie bracket vector fields finite dimensional nilpotent lie algebra say connected simply connected nilpotent lie group admissible lie group effective separability finitely generated nilpotent groups polycyclic groups see material contained following subsection definition group polycyclic exists ascending chain subgroups cyclic cyclic call cyclic series say compatible generating subset respect define hirsch length denoted number indices general polycyclic group may infinitely many different cyclic series arbitrary length see however hirsch length independent choice cyclic series respect compatible generating subset lem implies may represent every uniquely second paragraph defn implies definition call collection mal tsev basis respect compatible generating subset call mal tsev coordinates admissible group may refine upper central series obtain cyclic series compatible generating subset first assume abelian may write let free basis since finite abelian group exists isomorphism generates set thus groups compatible generating subset form cyclic series given assume step length exists generating basis integers ztii basis exist ztii may choose cyclic series hzs induction implies exists cyclic series compatible generating subset set set set choose follows cyclic series compatible generating subset given moreover given construction implies rankz whenever admissible group choose cyclic series compatible generating subset way definition admissible cyclic series compatible generating subset called admissible denoted let admissible torsion free thm implies multiplication expressed polynomials terms mal tsev basis ash sociated cyclic series compatible generating subset specifih cally may write expressed polynomial mal stev coordinates respectively similarly may write effective separability finitely generated nilpotent groups expressed polynomial mal tsev coordinates integer polynomials define group product group power operation respect given cyclic series compatible generating subset unique extensions implies diffeomorphic see thm cor mal tsev completion consequently dimension step length equal hirsch length step length respectively thus may write dim furthermore may identify image set following definition use last lemma subsection definition let torsion free admissible group let subgroup define isolator subgroup given exists make observations paragraph proceeding exercise subgroup abelian may write also subgroup torsion free finish section following result given torsion free admissible group following lemma relates word length element mal tsev coordinates respect choice cyclic series compatible generating subset lemma let admissible step length mal tsev coordinates proof finite statement evident thus may assume proceed induction step length observe base case abelian groups clear suppose step length since inductive hypothesis implies exists constant let smallest integer let length cyclic series suppose assumption thus exists constant therefore may write thus increasing may assume thus cover cases letting evident thus may write hence may assume may write may write let thus need consider since implies effective separability finitely generated nilpotent groups quotients one dimensional centers following subsection define choice admissible quotient respect central element choice maximal admissible quotient define constants infinite admissible group existence torsion free quotients one dimensional center following proposition useful throughout article proposition let torsion free admissible group suppose central nontrivial element exists normal subgroup irreducible torsion free admissible group finite index subgroup primitive proof construct induction hirsch length since base case trivial may assume proposition evident letting assume exists basis letting hzi note additionally follows torsion free admissible group proposition evident defining suppose since inductive hypothesis implies exists subgroup torsion free admissible group letting natural projection induction additionally implies taking note thus third isomorphism theorem implies hence torsion free admissible construction finite index subgroup letting satisfy hypothesis proposition demonstrate irreducible suppose contradiction exists pair admissible groups since torsion free torsion free selection follows torsion free finitely generated abelian groups hence isomorphic subgroup subsequently contradiction thus either subsequently irreducible definition let central element let set subgroups satisfy proposition since set bounded exists min say choice admissible quotient respect primitive let two different choices admissible quotients respect general hand definition subsequently hirsch length choice admissible quotient effective separability finitely generated nilpotent groups respect natural invariant associated quotient corresponds torsion free quotient minimal hirsch length image generates center useful finding smallest finite quotient image definition let torsion free admissible group step length let choice admissible quotient respect let set exists observe set bounded thus exists max say corresponds choice maximal admissible quotient define infinite admissible group definition let infinite admissible group let choice maximal admissible quotient set virtually abelian define step length suppose two choices maximal admissible quotients torsion free general however definition hence well defined invariant value important corresponds polynomial logarithm degree growth similarly well defined invariant admissible groups virtually abelian moreover value correspond polynomial degree growth asymptotic lower bound natural observation additionally infinite finitely generated abelian group let torsion free admissible group primitive let choice admissible quotient respect next proposition demonstrates may choose cyclic series compatible generating subset subset compatible generating subset generates proposition let torsion free admissible group let primitive central nontrivial element let choice admissible quotient respect exists cyclic series compatible generating subset choice admissible quotient respect moreover exists subset possibly empty compatible generating subset satisfying following subgroups form cyclic series compatible generating subset given proof proceed induction note base case evident thus may assume hence may take cyclic series compatible generating subset therefore may also assume exists generating basis letting hzi note passing quotient note choice effective separability finitely generated nilpotent groups admissible quotient respect induction implies exists cyclic series exists subset compatible generating subset satisfying following subgroups form cyclic series compatible generating subset let hzs let also take take thus cyclic series compatible generating subset consider subset thus selection required subset next two propositions establish notation let torsion free admissible nilpotent group primitive element let choice admissible quotient respect demonstrate may calculate given generating basis step length proposition let torsion free admissible group let basis moreover assume exist integers basis exist ztii exists choice admissible quotient respect generally basis exists choice admissible quotient respect proof letting may write exist indices otherwise observe satisfies conditions proposition therefore min since exists thus satisfies conditions proposition thus particular min therefore min last statement follows using similar reasoning following proposition demonstrates always realized hirsch length choice admissible quotient respect central element fixed basis proposition let torsion free admissible group step length basis moreover assume exist integers ztii basis exist elements ztii exists effective separability finitely generated nilpotent groups hence max generally basis max proof let set central exists commutator bracket given set bounded exists element max proposition implies exists definition follows max last statement follows using similar reasoning properties admissible quotients demonstrate conditions choice admissible quotient respect primitive central element step length next proposition admissible group fix step length proposition let torsion free admissible group let primitive element choice admissible quotient respect particular choice maximal admissible quotient proof definition exists suppose contradiction implies ker hence since torsion free follows contradicts construction thus since every irreducible torsion free admissible group contains subgroup isomorphic integral heisenberg group following proposition relates value value quotient lower step length proposition let torsion free admissible group step length integers satisfying following proof exist elements set generates exist finally set generates also exist generate finally exist elements integers ysi let choice admissible quotient respect let choice admissible respect evident thus follows proposition implies applying proposition effective separability finitely generated nilpotent groups last proposition demonstrates definition maximum value possible hirsch lengths choice admissible quotients respect primitive central nontrivial elements proposition let torsion free admissible group step length primitive let choice admissible quotient respect max proof statement evident torsion free finitely generated abelian groups since thus may assume let let exists basis integers ztii basis moreover exist ztii definition proposition thus may assume hence evident satisfies proposition since thus choice admissible quotient respect note satisfies proposition thus definition proposition implies thus giving result definition choice torsion free admissible group choice maximal admish sible quotient cyclic series compatible generating subset satisfy proposition called admissible denoted whenever given admissible take mal tsev completion constructed defined observe vectors log span lie algebra mal tsev completion call subset induced basis definition admissible mal tsev completion lie algebra induced basis called admissible denoted always take generating subset basis given admissible natural choice admissible given whenever given admissible admissible admissible commutator geometry lower bounds residual finiteness following definitions propositions important construction lower bounds found proof theorem effective separability finitely generated nilpotent groups finite index subgroups cyclic series following proposition tells view finite index subgroups light choice cyclic series compatible generating subset proposition let admissible suppose torsion free let finite index subgroup exist natural numbers satisfying following subgroups given form cyclic series compatible generating subset given proof proceed induction hirsch length base case assume case statement proposition evident choosing compatible generating subset given thus may assume observing finite index subgroup inductive hypothesis implies exist natural numbers given form cyclic satisfying following groups series compatible generating subset given also finite index subgroup thus exists natural number groups set form cyclic series compatible generating subset given apply proposition give description subgroups form corollary let admissible torsion free let subgroups form cyclic series compatible generating subset given particular proof proposition implies exist natural numbers subgroups given form cyclic series compatible generating subset given observe also evident series cyclic series compatible generating subset given thus inductive hypothesis implies finish observe thus desired let torsion free admissible group let finite index subgroup following proposition allows understand intersects fixed choice admissible quotient respect primitive central element proposition let admissible let finite index subgroup theredexisteindices natural numbers subgroups form cyclic series compatible generating subset effective separability finitely generated nilpotent groups proof assumption cyclic series compatible generating subset satisfy conditions proposition thus exists indices subgroups form cyclic series compatible generating subset given applying proposition admissible exist natural numbers subgroups given form cyclic series compatible generating subset given since finite index subgroup groups given form desired cyclic series compatible generating subset given therefore tis desired integers reduction complexity residual finiteness first demonstrate may assume torsion free calculating proposition let infinite admissible group proof proceed induction observe base case evident thus may assume note surjective ker finite central subgroup since admissible groups linear lem implies since inductive hypothesis implies following proposition implies may pass choice maximal admissible quotient computing lower bounds torsion free admissible group proposition let admissible surjective homomorphism finite group proof nothing prove thus may assume proposition implies exists collection elements mal tsev basis moreover cyclic series ible generating subset given proposition implies exist series subgroups given forms cyclic series ker compatible generating subset given since backwards direction clear proceed forward direction specific demonstrate proceed induction observe base case clear thus may assume order apply inductive hypothesis find normal subgroup first observe may set thus may assume ker straightforward see effective separability finitely generated nilpotent groups next paragraph prove exists element compatible generating subset ker end note say ker since set non empty given finite set exists height min height claim central since assuming may assume height since height height follows thus product height height since height height definition choice implies tis thus ker subsequently hence thus since central group normal subgroup selection since may apply inductive hypothesis surjective homomorphism letting thus rank step estimates definition let admissible infinite admissible group step size let let write call simple commutator weight respect let set simple commutators weight since nilpotent group step size empty set thus set simple commutators weight denoted finite considering surjective homomorphism finite group need ensure step length equal step length assuming proposition let admissible assume torsion free let surjective homomorphism finite group step sizes respectively proof first demonstrate induction height observe height thus height hence base case follows assumption consider height assumptions proposition imply thus may assume exists element mal tsev basis induction hypothesis implies since simple commutator height thus therefore follows effective separability finitely generated nilpotent groups factors subsequently ker since contradiction subsequently surjective homomorphism finite following proposition gives conditions specific injective map restricted set central simple commutators injective map restricted central elements fixed compatible generating subset injection restricted compatible generating subset proposition let admissible torsion free let surjective homomorphism finite suppose also suppose selection exists since proof let simple commutator weight proposition implies thus demonstrate assume contradiction implies ker since ker normal finite index subgroup proposition implies exist natural numbers subgroups given form cyclic series ker compatible generating subset given ker must power may since write observe divisible since element represented uniquely terms mal tsev coordinates follows thus may write implies since divides contradiction thus thus generating subset ordq thm implies divides power hence following definition important proofs theorem theorem definition let admissible torsion free let satisfy let lcm proposition let admissible torsion free suppose surjective homomorphism finite suppose step lengths respectively proof since ordq claim suppose contradiction since power definition thus order strictly less contradiction effective separability finitely generated nilpotent groups since proposition implies hand proposition implies thus cyclic series compatible generating subset since convention hence second paragraph defn implies finish demonstrating since ascending central series refinement upper central series exists exists since simple commutator weight proposition implies given follows implies hence admissible surjective homomorphism finite group following proposition gives conditions proper quotients proposition let admissible suppose surjective homomorphism finite additionally proper quotient natural projection finally proof let first demonstrate since ker induced homomorphism since surjective homomorphism finite proposition implies therefore demonstrate since cyclic series compatible generating given follows ordq see second paragraph defn thus must ordq since ordq ordq since follows since proper subgroups given ker ker hence ker examples precise residual finiteness demonstrate techniques used proof theorem make precise calculation integral heisenberg group effective separability finitely generated nilpotent groups integral heisenberg group basics start introducing basic facts integral heisenberg group useful calculation section may write identity matrix write let standard basis choose generating subset given thus bzm lastly obtain finite presentation written commutators trivial residual finiteness since upper lower asymptotic bounds require different strategies approach separately start upper bound straight forward begin upper bound collect basic facts take presentation given equation let one see cyclic series compatible generating subset proposition log proof let seek construct surjective homomorphism finite group moreover want construct finite group log effective separability finitely generated nilpotent groups via mal tsev basis may write proceed based whether trivial image abelianization suppose since either without loss generality may assume exists prime number theorem implies exists prime log log consider map given mod first arrow abelianization map second arrow projection coordinate last arrow natural projection construction log thus log suppose implies prime number theorem implies exists prime log second paragraph defn implies since hence log thus log therefore log proposition log proof order demonstrate log construct sequence elements log proof proposition implies log thus elements looking central elements let enumeration primes let lcm claim log clear implies prime number theorem implies log subsequently log thus log given establish log demonstrating surjective homomorphisms finite groups satisfying thm implies may assume prime since may assume since follows see second paragraph defn suppose proposition implies proper quotients image vanish particular proper quotients vanish thus may assume effective separability finitely generated nilpotent groups ordq ordq implies thus second paragraph defn implies hence may disregard possibility assume since order element finite group divides order group ordq thus hence may assume exists thus may write construction since follows subsequently desired corollary let integral heisenberg group log proof theorem goal theorem demonstrate log proposition implies may assume torsion free proceed proofs upper lower asymptotic bounds separately since require different strategies start upper bound proof simpler upper bound task prove element exists surjective log homomorphism finite groupp pass thep quotient given appeal induction step length otherwise find choice admissible quotient respect primitive central element image proposition let torsion free admissible group log proof let cyclic series compatible generating subset assumption exist integers basis exist suppose using mal tsev coordinates may write lemma implies construct surjective homomorphism finite group letting suppose passing group inductive hypothesis implies exists surjective homomorphism log proposition implies thus log otherwise may assume therefore may write since exists prime number theorem implies exists prime log choice effective separability finitely generated nilpotent groups admissible quotient respect corollary implies log proposition implies thus log hence log order demonstrate log require sequence elements log independent entails finding elements high complexity respect residual finiteness elements relatively short word length comparison order minimal finite group required separate identity proposition let torsion free admissible group log proof let choice maximal admissible quotient exists admissible quotient respect moreover exists trivial exists primitive particular primitive central element let choice cyclic series compatible generating subset given satisfy proposition proposition implies choice admissible quotient respect let lcm enumeration primes greater letting claim desired sequence continuing make remarks value depends choice maximal admissible quotient specific another choice maximal admissible quotient general subsequently natural consequence sequence elements depends choice maximal admissible quotient however demonstrate given construction work choice maximal admissible quotient take claim log evident proposition implies proceed demonstrate surjective homomorphisms finite groups thm implies may assume prime ker hence may assume proposition implies thus may restrict attention surjective homomorphisms factor homomorphisms suppose nothing prove may assume proposition implies since proper quotient natural projection given two natural consequences proper quotients nontrivial image additionally surjective homomorphism finite effective separability finitely generated nilpotent groups thus may assume suppose since surjective homomorphism finite proposition implies hence may assume suppose selection follows divides since order element divides order group ordq divides implies suppose thus exists subsequently may write construction since follows given order element finite group divides order group follows ordq divides thus therefore since step length implies prime number theorem implies log hence log thus log subsequently log proof theorem let infinite admissible group proposition implies proposition implies log proposition implies log thus log cyclic series lattices nilpotent lie groups theorem let torsion free admissible group let mal tsev completion let choice maximal admissible quotient main task section demonstration value invariant mal tsev completion thus need establish properties cocompact lattices admissible lie groups start following lemma relates hirsch lengths centers cocompact lattices within admissible lie group lemma let admissible lie group two cocompact lattices dim proof proof straightforward application lem introduce notion one parameter families group elements lie group definition let connected simply connected lie group call map one parameter family group elements injective group homomorphism real line addition effective separability finitely generated nilpotent groups let admissible torsion free via exponential map lem maps given exp one parameter families group elements discussion thm implies may uniquely write mal tsev completion definition say one parameter families associated characterize discrete subgroup admissible lie group cocompact lattice based intersects collection one parameter families group elements proposition let admissible lie group suppose discrete subgroup suppose exists collection one parameter families group elements dim written dim homeomorphic cocompact lattice proof let natural projection onto space cosets suppose exists since discrete discrete subset given discrete infinite cyclic hence element sequence projects unique element thus infinite sequence convergent subsequence hence cocompact lattice suppose implies exists dim letting claim compact dim let rdim continuous map given adim since dim closed bounded subset rdim theorem implies dim compact since continuous compact dim claim coset representative let exists let write dim given construction subsequently thus since image compact set continuous map compact thm implies cocompact lattice next two propositions give structural information needed mal tsev completion torsion free admissible group structural information choices admissible quotients respect primitive central element proposition let torsion free admissible group let primitive element let choice admissible quotient respect suppose mal tsev completion let mal tsev completion isomorphic closed connected normal subgroup effective separability finitely generated nilpotent groups proof proposition exists cyclic series compatible generating subset satisfying following exists subset cyclic series compatible generating subset given let one parameter families group elements associated admissible thm implies may view connected subgroup proceed induction demonstrate closed normal subgroup follows lie isomorphic claim evident suppose may take implies thus claims evident suppose let let mal tsev completion lem follows thus closed connected normal subgroup claim mal tsev completion may write since cocompact lattice proposition implies proposition cocompact lattice prop implies cocompact lattice observe satisfies proposition thus inductive hypothesis implies closed normal subgroup since isomorphic pullback closed normal subgroup closed normal subgroup next proposition mal tsev completion choice admissible quotient respect primitive central element mal tsev completion intersects cocompact cocompact lattice proposition let torsion free admissible group let primitive element let choice admissible quotient respect mal tsev completion let mal tsev completion another cocompact lattice cocompact lattice proof proposition implies exists cyclic series compatible ing subset satisfying following exists subset groups form cyclic series compatible generating subset given let associated one parameter families group elements mal tsev construction completion may write tion implies particular proposition implies cocompact lattice desired following lemma demonstrates select cyclic series compatible generating subset cocompact lattice admissible lie group intersecting lattice collection one parameter families group elements effective separability finitely generated nilpotent groups lemma let admissible lie group cocompact lattice let coldim lection one parameter families elements homeomorphic let groups given form cyclic series compatible generating subset given proof proceed induction dimension dim statement dim evident suppose dim let proposition implies cocompact lattice inductive hypothesis implies elet ments given satisfy following groups given dim form cyclic series compatible generating subset given dim since cocompact lattice proposition implies fdim dim letting groups given series compatible generating subset given form cyclic dim let torsion free admissible group demonstrate value welldefined invariant mal tsev completion proposition let admissible lie group suppose two cocompact lattices proof proposition implies follows definition therefore may assume case demonstrate equality showing let mal tsev completion let cyclic series compatible generating subset let associated one parameter families group elements may write let let let central proposition implies cyclic series compatible generating subset given element compatible generating subset let choice admissible quotient respect let mal tsev completion since evident particular proposition implies compact lattice proposition implies closed connected normal subgroup thus prop implies compact lattice proposition implies follows satisfies conditions proposition let choice admissible quotient respect follows proposition lem implies thus inequality holds element compatible generating subset therefore proposition implies interchanging effective separability finitely generated nilpotent groups come main result section proof theorem suppose two infinite admissible groups isomorphic mal tsev completions proposition implies theorem implies log log proposition implies thus examples proof theorem free nilpotent groups theorem definition let free group rank generated define free nilpotent group step size rank set following sec construct cyclic series compatible generating subset using iterated commutators set definition call elements basic commutators weight choose arbitrary linear order weight basic commutators basic commutators weight respectively basic commutator weight basic commutators basic commutators higher weight greater respect linear order basic commutators lower weight moreover choose arbitrary linear order commutators weight say commutator contains inductively say commutator contains either contains contains note basic commutator weight greater equal must contain two distinct commutators weight necessarily additionally basic commutator weight contain distinct basic commutators weight well known number basic commutators equal hirsch length letting function may write label basic commutators respect given linear order definition one see subgroup series cyclic compatible generating subseries cor implies effective separability finitely generated nilpotent groups cyclic series basic commutators set call compatible generating subset basic commutators proposition let free nilpotent group step size rank set let basic commutator weight set contains exists normal subgroup torsion free additionally commutator contains elements cyclic series basic commutators let proof let compatible generating subset basis commutators assumption exists without loss generality may assume construct normal descending series torsion free commutator contains elements also generated basis commutators weight greater equal finally proceed induction observe commutator consider subgroup given contains elements follows construction thus base case suppose subgroup constructed let commutator bracket contains elements follows contains elements thus assumption since true let set basic commutator brackets commutator bracket central construction contains elements set hkt suppose commutator contains elements construction since normal finally evident torsion free hence induction gives construction additionally construction implies proposition thus taking proof theorem let let let free nilpotent group step size rank set theorem implies exists natural number log demonstrate log log since nilpotent group step size hirsch length greater desired result compatbe cyclic series basic commutators let ible generating subset basic commutators let choice admissible quotient respect proposition implies exists effective separability finitely generated nilpotent groups exists subset basic commutator weight contains elements proposition implies exists subgroup satisfies proposition respect moreover elements contained natural surjective homomorphism given sending elements identity therefore induced map particular thus satisfies conditions proposition proposition implies since step size additionally particular setting max proposition implies log central products applications examples contruct theorem iii arise iterated central products torsion free admissible groups whose centers hirsch length given context corollary allows compute precise residually finiteness function terms hirsch length torsion free admissible groups take central product definition let finitely generated groups let isomorphism define central product respect define central product groups respect automorphism inductively assuming already defined define central product respect induced isomorphisms write central product suppose central product two nilpotent groups since products quotients nilpotent groups nilpotent follows nilpotent group however isomorphism type dependent proposition let collection torsion free admissible groups let hzi let isomorphism given proof may assume first note torsion free admissible group normal subgroup torsion free observe since may write thus definition define torsion free admissible group generated set relations consisting commutator brackets form commutators trivial effective separability finitely generated nilpotent groups example filiform nilpotent group hirsch length step length defining hxs follows cyclic series compatible generating subset additionally proof theorem iii assume construction torsion free admissible group hirsch length corollary implies log gives theorem exist natural numbers satisfying lcm lcm respectively let identity isomorphisms respectively proposition implies thus corollary implies finite generating lastly let consider group subset scm proposition implies since corollary implies log desired review blackburn proof theorem start review blackburn proof conjugacy separability infinite admissible groups section provides motivation estimates following sections one obtains upper bound let admissible torsion free let order construct surjective homomorphism finite group separates conjugacy classes proceed induction since base case evident may assume induction implies exists surjective homomorphism finite group otherwise may assume following integer particular importance definition let torsion free admissible group cyclic series comh given patible generating subset let let define choose prime power find exists satisfying see lem consider following definition see lem definition let admissible torsion free let define smallest natural number letting set blackburn proves see however consequence choice cyclic series effective separability finitely generated nilpotent groups compatible generating subset becomes evident integer unnecessary finite order elements blackburn inducts thus suffices bound terms following blackburn method calculate asymptotic upper bound demonstrate upper bound sharp starting make following observations using cyclic series compatible generating subset given second paragraph subsection gcd thus moreover via subsection may write conjugacy class proposition proof let need construct surjective homomorphism finite group proceed based whether equal images corollary see also cor implies exists surjective homomorphism log since central elements follows thus log thus may assume particular may write let prime power divides claim contradiction suppose otherwise implies exists equation implies mod therefore exist thus contradiction hence hence prime number theorem implies exists prime log hence log thus log hence following proposition finishes proof theorem proposition proof construct sequence pairs let enumeration primes writing scalar product consider following pair elements effective separability finitely generated nilpotent groups equation implies may write conjugacy class since central elements follows moreover given claim order demonstrate claim show surjective homomorphisms finite group thm implies may assume since may assume suppose first demonstrate group exists proper quotient images since proposition implies since every choice admissible quotient respect primitive central element isomorphic trivial subgroup proposition implies exist proper quotients image thus proper quotient natural projection ker since proposition thus hence particular thus may assume proposition implies thus may assume since proposition implies equation implies mod smallest fails since unit gcd therefore hence following corollary useful proof theorem corollary let heisenberg group presentation given central let prime suppose surjective homomorphism distinct proof may write conjugacy class proposition implies hence ordq since follows given gcd exists integers may write therefore desired effective separability finitely generated nilpotent groups relating complexity groups lie algebras let torsion free admissible group finite generating subset mal tsev completion lie algebra overall goal section provide bound log klog terms log gives norm additive structure proposition let admissible suppose step length let exists log proof using mal tsev coordinates may write lemma implies exists straightforward application formula implies log writing follows kai equation implies may write log kcbi cbi rational linear combination lie brackets let via induction length iterated lie bracket one see exists maximizing possible lie brackets elements exists kcbi hence log immediate application proposition adjoint representation matrix coefficients bounded polynomial terms word length proposition let admissible let let matrix representative respect step length proof proposition implies exists log via induction length lie bracket equation preliminary estimates theorem let admissible torsion free let element let prime following section demonstrate construction integer give asymptotic bound terms independent prime first provide bound terms effective separability finitely generated nilpotent groups proposition let admissible let exists proof start make simplifying notation letting sider smooth map given suppose satisfies commutative diagram implies may write log log proposition implies strictly upper triangular matrix whose coefficients bounded since evident log backwards substitution gives result first statement following proposition originally found lem reproduce proof may provide estimates value terms proposition let admissible let prime exists moreover proof proceed induction hirsch length given statement clear setting may assume construct based value see definition induction may assume already constructed set suppose selection thus may write follows since thus let largest power set let let satisfy thus subsequently induction implies thus may write hence since definition follows hence thus proceed induction hirsch length demonstrate asymptotic upper bound since base case clear assume let suppose construction thus induction implies exist follows largest power divides proposition implies exist consequently effective separability finitely generated nilpotent groups proof theorem let infinite admissible group order demonstrate exists need show exists prime power follows first specialize torsion free admissible groups proposition let admissible exists proof let demonstrate exists induction since base case clear may assume inductive hypothesis implies exists surjective homomorphism finite group thus otherwise may assume lemma implies constant notational simplicity let since exists prime power set suppose contradiction exists implies thus proposition subsequently hence since follows implies contradiction hence proposition implies prime number theorem implies may choose log hence log proposition implies thus therefore hence max proof theorem let infinite admissible group choice cyclic series compatible generating subset let natural number proposition natural number proposition letting max claim let satisfy order show construct surjective homomorphism finite group distinguishes conjugacy classes via induction simplify following arguments let proposition implies may assume exists subgroup prime order induction implies exists surjective homomorphism finite group thus otherwise may assume effective separability finitely generated nilpotent groups suppose exists prime distinct suppose contradiction exists since basic commutator properties imply given follows contradiction hence induction implies exists surjective homomorphism finite group thus suppose exponent set suppose contradiction exists thus subm sequently proposition implies therefore may write moreover subsequently since follows contradiction sition imply thus subsequently proofs theorem theorem let infinite admissible group choice cyclic series compatible generating subset since proofs theorem theorem require different strategies approach separately start theorem since requires elementary methods assume contains infinite finitely generated abelian group index want demonstrate log log since corollary see also cor implies log thus need demonstrate log two elements want construct surjective homomorphism log proof theorem let finite generating subset let set coset representatives take generating subset may write conjugation induces map aut given thus may write conjugacy class finally exists suppose two elements taking map follows otherwise may assume corollary see also cor exists surjective homomorphism log letting ker follows log therefore log subsequently log log effective separability finitely generated nilpotent groups theorem suppose contain abelian group finite index order demonstrate desire sequence pairs particular must find elements whose conjugacy classes difficult separate elements relatively short word length comparison order minimal finite group required separate conjugacy classes first reduce calculation lower asymptotic bounds torsion free admissible groups appealing conjugacy separability two elements within finite index subgroup proposition let infinite admissible group let subgroup suppose exist proof first remark since admissible groups theorem implies suppose surjective homomorphism finite group let inclusion surjective map finite group definition thus proof theorem first assume torsion free let choice maximal admissible quotient exists choice admissible quotient respect moreover exists primitive exists particular equation implies exists let enumeration primes greater since images let central elements follows claim desired elements particular construction strate implies hence therefore need demonstrate surjective homomorphisms finite groups thm implies may assume prime since may also assume suppose consider homomorphism given natural projection since ker induced homomorphism since proposition implies hence thus may assume effective separability finitely generated nilpotent groups assume suppose subgroup since induced homomorphism proposition implies thus hence may assume suppose nothing prove thus may assume proposition implies proper quotient natural projection ker thus since particular hence may assume since corollary implies exists elements thus since exhausted possibilities follows hence suppose infinite admissible group exists finite index torsion free subgroup denote let choice maximal admissible quotient using reasoning exists choice admissible quotient respect let enumeration primes greater exist let elements construction let natural projection construction additionally finite index subgroup thus lem implies hence since central elements proposition implies construction finite generating subset hence since projection torsion free quotient injective restricted isomorphic finite index subgroup thus theorem implies since proof theorem suppose two infinite admissible step size greater isomorphic mal tsev completions proposition implies definition since theorem evident effective separability finitely generated nilpotent groups proof theorem proof let group given definition let identity morphism let consider group finite generating subset scm proposition implies since theorem implies desired references bass degree polynomial growth finitely generated nilpotent groups proc london math soc blackburn conjugacy nilpotent groups proc amer math soc approximating group solvable quotients new york math quantifying residual finiteness algebra kaletha quantifying residual finiteness arithmetic groups compos math mcreynolds asymptotic growth least common multiples groups bull lond math soc mcreynolds extremal behavior divisibility functions geom dedicata hagen patel residual finiteness growths virtually special groups appear math arxiv buskin efficient separability free groups sib math chahal solution congruence subgroup problem solvable algebraic groups nagoya math corwin greenleaf representations nilpotent lie groups applications part basic theory examples cambridge studies advanced mathematics cambridge university press cambridge dekimpe groups affine polynomial structures formanek conjugate separability polycyclic groups algebra discrete subgroups solvable lie groups type math ussr sbornik gromov asymptotic invariants infinite groups geometric group theory vol sussex london math soc lecture note cambridge univ press hall edmonton notes nilpotent groups queen mary college math notes math queen mary college london hamilton wilton zalesskii separability double cosets conjugacy classes groups lond math soc effective separability finitely generated nilpotent groups hempel residual finiteness combinatorial group theory topology alta utah ann math princeton univ press princeton holt eick brien handbook computational group theory discrete mathematics applications boca ralton chapman boca raton kassabov matucci bounding residual finiteness free groups proc amer math soc khukhro nilpotent groups automorphisms gruyter expositions mathematics walter gruyter berlin knapp lie groups beyond introduction kozma thom divisibility laws finite simple groups arxiv lawton louder mcreynolds decision problems complexity traces representations arxiv mckay structure groups prime power order london mathematical society monographs new series oxford science publications oxford university press oxford lyndon schupp combinatorial group theory mal tsev class homogeneous spaces amer math soc translation mal tsev homomorphisms onto finite groups uchen zap ivanovskogo gos ped inst mal tsev isomorphic matrix representations infinite groups amer math soc transl osin subgroup distortions nilpotent groups comm algebra patel theorem peter scott proc amer math soc raghunathan discrete subgroups lie groups math student centenary volume rivin geodesics one stories adv math remeslennikov finite approximability groups respect conjugacy siberian math segal polycyclic groups cambridge university press stebe conjugacy separability groups integer matrices proc amer math soc tenenbaum introduction analytic probabilistic number theory cambridge university press thom length laws finite groups arxiv
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pilot reuse among users underlaid massive mimo systems aug hao student member ieee wei senior member ieee zhaohui yang student member ieee jianfeng shi student member ieee ming chen member ieee underlaid massive mimo system transmitters reuse uplink spectrum cellular users cus leading cochannel interference decrease pilot overhead assume pilot reuse among pairs first derive mmse estimation channels give lower bound ergodic achievable rate cellular links mitigate pilot contamination caused propose pilot scheduling pilot power control algorithm based criterion minimizing sum mse channel estimation links show appropriate ratio well designed pilot scheduling scheme transmitter could transmit pilot maximum power addition also maximize sum rate links guaranteeing quality service qos cus develop iterative algorithm obtain suboptimal solution simulation results show effect pilot contamination greatly decreased proposed pilot scheduling algorithm scheme provides significant performance gains conventional orthogonal training scheme terms system spectral efficiency ntroduction increasing demand broadband wireless communications problem spectrum insufficiency become major factor limiting wireless system performance massive mimo transmission proposed triggered considerable research interest recently due great gains spectral efficiency energy efficiency besides communication also proven promising enhancing traditional cellular systems drawn great attention recently different conventional cellular communication traffic routed via base station communication allows two closely located users communicate directly thus distinct copyright ieee personal use material permitted however permission use material purposes must obtained ieee sending request work part supported nsfc nos nstmp six talent peaks project jiangsu province science technology project guangdong province grant scholarship china scholarship council program sponsored scientific innovation research college graduate jiangsu province grant scientific research foundation graduate school southeast university grant corresponding author hao wei yang shi chen national mobile communications research laboratory southeast university nanjing china email wxu yangzhaohui shijianfeng chenming advantages high short packet delay low energy consumption increased safety extensive research design analysis massive mimo systems uplink capacity bounds derived perfect imperfect channel state information csi tradeoff studied ref compared two prominent linear precoders respect radiated massive mimo system unlike considered simplified scenarios studied massive mimo systems underlaid communication great challenge existing cellular architecture cochannel interference due spectrum reuse lot literature working interference mitigation underlaid systems resource allocation power control algorithms proposed maximize users dus extended algorithms carried maximize dus though massive mimo communication widely studied papers investigated interplay massive mimo communication cellular links investigated perfect imperfect csi overhead acquiring csi considered massive mimo systems orthogonal pilots transmitted cellular users cus obtain csi communication introduced orthogonal pilots used transmitter channel estimation pilot overhead large significantly affect system performance furthermore transmission strategy massive mimo designed support multiple users transmitting timefrequency block though interference greatly reduced large antenna array interference still persists may worse conventional siso underlaid system order shorten pilot overhead effective strategy allowing orthogonal pilots reused among different users existing works pilot reuse mainly focus scenarios mobile users cell use orthogonal pilots users different cells reuse set pilots best authors knowledge works considered strategy within cell authors analyzed feasibility spatially correlated massive mimo channels constrained channel angular spreads authors allowed reuse pilots cus proposed mmse detector suppress interference also studied underlaid massive mimo system performance cus left consideration simplicity contrast existing works work analyzes achievable rate cellular links proposes pilot scheduling well power control algorithms optimize system performance main contributions paper summarized follows assume cus use orthogonal pilots reuse another set pilots channel estimation motivation stems pairs usually locate dispersively use low power transmission hence letting several pairs far away use pilot channel estimation would cause endurable pilot contamination first derive expression mmse estimate channels obtained imperfect csi receivers apply partial zero forcing pzf receive filters studied signal detection derive effective ratio sinr lower bound ergodic achievable rate user different estimation cellular channel vectors affected noise channel estimation links experiences effect noise pilot contamination due mitigate pilot contamination develop pilot scheduling pilot power control algorithm criterion minimizing sum mse channel estimation links first show appropriate number orthogonal pilots available dus well designed pilot scheduling scheme transmit pilot using maximum power develop heuristic pilot scheduling scheme allocate pilots dus show sum mse channel estimation links decreased significantly study sum maximization links guaranteeing quality service qos cus efficient power control algorithms often play important role reducing cochannel interference reaping potential benefits communication however algorithms usually carried based knowledge instantaneous csi links apart computational complexity algorithms require gather instantaneous csi links difficult implementation therefore paper consider performing power control algorithm periodically coarser frame level granularity based fading coefficients vary slowly simulation results show proposed algorithm converges rapidly obtains much higher sum links compared typical orthogonal training scheme note since among pairs underlaid massive mimo system less well researched consider simplified simo transmission communication mainly focus analyzing effect system performance mimo transmission similar results obtained simply modifying analysis optimization manuscript hand analysis paper focuses scenario sake clarity garding massive mimo system lot literature working mitigating pilot contamination result underlaid massive mimo system among pairs persists first use algorithms developed allocate pilots cus csi exchanged among cells straightforwardly extend proposed algorithms improve system performance paper follow common notations denote set natural numbers real space complex space respectively boldface upper lower case letters used denote matrices vectors stands dimensional identity matrix denotes vector matrix represents set subtraction operation superscript denotes operation denotes expectation operation use denote euclidean norm means element positive nonnegative rest paper organized follows section underlaid massive mimo system introduced section iii present mmse estimate channels show affects channel estimation achievable rate cellular links analyzed section section aim minimize sum mse channel estimation links maximize sum links finally numerical verifications presented section concluding remarks section vii ystem odel consider uplink underlaid massive mimo system one cus pairs set cus pairs denoted respectively equipped antennas one transmit antenna communication assume simo transmission equipped one antenna receiver equipped antennas system transmitters use resource block transmit signals leading cochannel interference dimensional received data vector data transmit power data symbol realvalued fading coefficient assumed known priori denotes fast fading vector channel similarly defined complex gaussian noise covariance misuse notation avoiding possible ambiguity complex normal distribution sign analogously dimensional received data vector given vik gik vnk gnk vik gik denote fading coefficient fast fading vector channel respectively vnk gnk similarly defined link complex gaussian noise covariance analogously derive mmse estimate follows set pairs using pilot pair denote statistically independent satisfying channel estimation similar uplink data transmission dimensional received signal matrix pilot transmission denote pilot transmit powers respectively pilot allocated pair noise matrix consists independently identically distributed gaussian elements zero mean variance mmse estimate given given channel estimate vector express true channel vector error vector represents csi uncertainty due property mmse estimation statistically independent follow orthogonal pilots usually adopted obtain csi links underlaid system reduce pilot overhead assume cus use orthogonal pilots reuse another set pilots channel estimation denote pilot matrix orthogonal column vectors length pilots also number pilots available channel estimation smallest amount pilots required without loss generality assume pilot allocated remaining pilots reused among pairs iii hannel stimation channel estimation dimensional received pilot signal matrix written vnk gnk vik gik noise matrix consisting gaussian elements zero mean variance mmse estimate gik given vik vjk denote gik mentioned statistically independent distributions given vik vjk similarly mmse estimate gnk follows vnk vnk denote channel estimation error vector statistically independent satisfy vnk vnk observed estimation gnk affected pilot noise pilot interference cus dus disappear completely due orthogonality pilots estimation gik clear apart effect pilot noise also affected pilot contamination due among pairs achievable ate nalysis section analyze achievable rate cellular links let denote unit norm receive filter used detecting signal denote unit norm receive filter adopted detecting signal since receive filter used either boost desired signal power eliminate interference signal sinr link critically depends receive filter used paper adopt pzf receivers use part degrees freedom signal enhancement remaining degrees freedom interference suppression signal detection assume uses degrees freedom cancel interference nearest cellular interferers nearest interferers using different orthogonal pilots obtain following relationship indicates estimation channels two different using pilot direction result interference interferers applying pilot eliminated simultaneously using one degree freedom since orthogonal pilots reused pairs divide pairs sets pairs set using pilot channel estimation know able cancel interference sets interferers feasible set given pzf filter obtained normalizing projection channel estimation nullspace channel estimation vectors cancelled interferers refer appendix sake convenience let denote set uncancelled cus detecting denote set uncancelled dus detecting cellular signals similarly uses degrees freedom cancel interference nearest cellular interferers nearest interferers using different orthogonal pilots following set compared mmse receivers optimally balance signal enhancement interference suppression pzf receivers suboptimal however adopt pzf receivers paper due following advantages first pzf receivers take signal boosting interference cancellation account similar mmse receivers concept shown performance pzf receivers terms system throughput approach mmse receivers much better maximum ratio combining mrc fully zero forcing receivers second simple structure pzf receivers makes analysis system tractable allows evaluate performance underlaid massive mimo system explicit way addition pzf receivers simplified mrc receivers letting also reduced fully receivers letting make analysis paper general order cancel nearest interferers obtain pzf receivers know positions transmitters relax requirement make practical obtain pzf receivers cancelling interferers largest fading coefficients lower bound achievable rate cellular links since information estimated channel vectors treated true csi using pzf receiver detecting write postprocessing received signal associated first term second equality desired information terms represent cochannel interference channel estimation error noise respectively thus effective sinr cellular link expressed pzf filter obtained first projecting chan nel estimation onto nullspace channel estimation vectors cancelled interferers normalizing projection let respectively denote sets uncancelled dus detecting signal represents desired respectively denote cochannel interference uncancelled cellular interferers given characterizes effect channel estimation error noise experienced formulated asymptotic uplink rate cellular links massive mimo underlaid massive mimo system studied system considered paper also obtain similar result asymptotic uplink rate cus shown following corollary since corollary analogously verified omit proof process brevity corollary fixed transmit powers transmitters using linear filters signal detection asymptotic uplink rate cellular link grows unboundedly goes infinity fact number antennas usually finite due multiple practical constraints hence following paper consider practical scenario equipped large finite numbers antennas consider block fading model channels remain unchanged coherence interval length based derive lower bound ergodic achievable rate cellular links shown following theorem theorem given sinr formula ergodic achievable rate cellular link lower bounded detect received signal using pzf receiver written vkk vik vnk vik vnk first term second equality desired signal terms respectively denote cochannel interference channel estimation error noise effective sinr link vkk denotes desired signal respectively denote cellular cochannel interference characterizes effect channel estimation error noise experienced given vik vnk lower bound achievable rate links affected channel estimation error cellular link channel estimation errors cancelled cellular links since cus use orthogonal pilots channel estimation value effect hence fixed pilot transmit power decreases monotonically contrast except effect also influenced estimation errors channels cancelled due increasing results smaller thereby helps increase therefore would hard determine monotonicity proof see appendix remark considering effect pilot length find vik vnk similarly cellular uplink case also derive lower bound ergodic achievable rate links theorem given sinr formula ergodic achievable rate link lower bounded vkk vik vik vik vnk vnk since orthogonal pilots reused among pairs denote pattern element pair assigned pilot otherwise denote pilot transmit power vector dus aiming minimizing arrive following problem min proof see appendix remark found feasible set implicit function due increasing results accurate channel estimations links thereby helps increase however increases number symbols available data transmission becomes smaller section show simulation results first increases decreases following focus pilot scheduling power control design based since fading coefficients vary slowly proposed pilot scheduling power control algorithms performed periodically coarser frame level granularity greatly decrease computational complexity result theorem theorem helpful following analysis ilot cheduling ower ontrol investigated cannel estimation well achievable rate cellular links underlaid massive mimo system based analysis focus two problems section first problem aims minimize sum mse channel estimation links second problem aims maximize sum rate links guaranteeing qos requirements cus pilot power control pilot scheduling mentioned section iii channel estimation cellular links affected additive noise therefore assume transmits pilot signal maximum power links apart effect additive noise channel estimation also influenced pilot contamination due location dispersion pairs shortdistance transmission preferred pilot contamination greatly reduced using effective pilot scheduling pilot power control algorithm according sum mse channel estimation links written maximum data transmit power constraints indicate pair allocated one pilot note simplicity formulated function implicit way also equivalently rewrite explicit way follows vkk vjk solve problem first give optimal condition following theorem theorem always exists optimal pilot scheduling matrix opt determined opt optimal popt satisfies proof see appendix theorem indicates proper optimal opt constraints always active explain theorem intuitive way follows number orthogonal pilots available dus appropriate relatively low ratio pilots allocated pairs using optimal pilot scheduling scheme special case pairs use different orthogonal pilots channel estimation ignorable pilot contamination would caused due dispersive positions pairs hence transmit pilot signals maximum power increase estimation accuracy contrast small relatively high ratio even using optimal pilot scheduling scheme pilot contamination may still influence estimation accuracy greatly solving may yield popt case need enlarge set pilots dus decrease ratio based theorem following assume always transmit pilots maximum power problem becomes min mixed integer programming problem optimal obtained exhaustive search recalling number scalar multiplication required compute objective function thus obtaining optimal involves complexity due exponential complexity impractical run number pairs large therefore propose low complexity pilot scheduling algorithm mitigate pilot contamination massive mimo system proposed gcpa scheme metric defined indicate interference strength among cus binary matrix used describe connections cus however obtain binary matrix suboptimal threshold needs found applying iterative grid search paper define metric evaluate potential interference strength pair vik vkk vii denote set pairs allocated pilots summarize pilot scheduling algorithm algorithm denote data transmit power vectors cus dus respectively arrive following problem max arg min end basic idea psa algorithm pair experiencing larger pilot contamination possesses higher priority pilot allocation main steps iteration explained follows first pair experiencing largest potential interference dus selected pilot causing least interference chosen finally pilot assigned pair updated algorithm carried times pairs allocated pilots data power control subsection aim maximize sum rate dus guaranteeing qos requirements ocus since exact expressions unapproachable use lower bounds replacement simulation results show gap ergodic achievable rate lower bound marginal verifying feasibility approximation respectively denote target sinr maximum data transmit power see coupled expressions moreover fractional structure sinr expressions log operation make nonconvex therefore difficult obtain optimal solution following divide optimization two consecutive parts first part optimize fixed vice versa part data power control cellular links given problem rewritten max algorithm pilot scheduling algorithm psa initialization calculate pilot allocation arg max based write sinr constraints vector form consists scaled cochannel interference interferers additive noise seen interference function remark problem feasible sinr constraint opt would hold equality optimal opt equivalently vector form qsopt readily verified reductio assume opt increase opt objective function decreasing slightly according spectral radius less opt optimal power vector form opt obtain matrix inversion spectral radius calculation required resulting high complexity result obtain qsopt using following low complexity iterative scheme represents time instant min min denote data power control algorithm cellular links dpcc proving standard readily verify dpcc algorithm converges optimal solution initial power vector detailed proof given appendix data power control links fixed arrive following problem notational simplicity constant coefficient omitted max min directly seen problem due fractional expression log operation order solve transform equivalent form admits suboptimal solutions using wellknown wmmse approach details provided appendix summarize data power control algorithm links labeled dpcd algorithm algorithm data power control algorithm links dpcd initialize accuracy repeat since propose solve original problem using jdpc algorithm operates iterative mechanism necessary characterize convergence iteration optimal first obtained using dpcc algorithm fixed determined dpcd algorithm outputs suboptimal result sum links increases iteration noting fact sum links always upper bounded conclude jdpc algorithm converges suboptimal solution problem convergence analysis jdpc algorithm obtain using bisection method min implementation complexity analysis based analysis within coherence interval transmitter first transmits pilot maximum power channel estimation transmits signal using power obtained solving successfully implement proposed psa jdpc algorithms requires information fading coefficients vnk vik directly estimated whereas vnk vik need estimated fed back collecting information runs pilot scheduling power control algorithms sends results users since fading coefficients vary slowly proposed algorithms carried coarser frame level granularity following analyze computational complexity psa jdpc algorithms since psa algorithm complexity per iteration carried times total complexity jdpc algorithm assume carried times loop iterations required dpcc dpcd algorithms converge major complexity dpcc algorithm lies computing involves complexity hence total complexity dpcc algorithm since complexity obtaining lagrange multipliers using bisection method accuracy overall complexity solving problem using dpcd algorithm therefore jdpc algorithm total complexity simulation results show small proposed algorithms involve low complexity efficient solutions based analysis alternatively optimize iteratively applying proposed dpcc dpcd algorithms suboptimal solution problem obtained let jdpc denote joint data power control algorithm iteratively carries dpcc dpcd convergence brevity omit detailed description jdpc algorithm imulation esults section present simulation results evaluate performance underlaid massive mimo system consider network square area transmitters located uniformly cell distance associated receiver uniformly distributed range dmax brevity assume equal minimum sinr requirement cus equal maximum power constraint transmitters according theorem relatively low ratio dbm dbm sum cus fully simulation fully lower bound mrc simulation mrc lower bound fig simulated sum links lower bound versus pilot length dmax pilot length pzf simulation pzf lower bound mrc simulation mrc lower bound pzf simulation pzf lower bound mrc simulation mrc lower bound sum dus table imulation parameters maximum data transmit power additive noise power path loss exponent standard deviation shadowing fading accuracy number antennas fig simulated cellular sum lower bound versus number antennas dmax transmit pilot signals maximum power increase estimation accuracy therefore following assume unless otherwise specified system parameters summarized table simulation results obtained averaging channel realizations channel realization obtained generating random user distribution well random set fading coefficients spectral efficiency versus corresponding lower bound subsection evaluate tightness simulated corresponding lower bound cellular links first compare simulated cellular sum corresponding lower bound different values pilot length different receive filters fig seen cellular sum increases number antennas considered configurations fully receivers greatly outperform mrc receivers terms cellular pilot length grows cellular sum decreases mrc case increases fully case explained remark moreover fig also shows lower bound cellular sum closely matches simulation mrc case almost coincides simulation fully case next fig compare simulated sum links corresponding lower bound different numbers antennas different receive filters note adopt pzf receivers signal detection mentioned section feasible set therefore fig assume min min seen figure pzf receivers greatly outperform mrc receivers terms sum lower bound sum dus obtained pzf receivers approximate simulated obtained mrc receivers gap simulated lower bound small considered configurations indicating feasible solve data power control problem section based lower bounds fig also shows sum increases pilot length sum decreases almost linearly pilot length orthogonal pilots reused among pairs small case channel estimation significantly influenced pilot contamination therefore links enhanced increasing however becomes large enough channel estimation accuracy hardly improved enlarging counterproductively increasing pilot length reduces number symbols available data transmission thereby decreases links following simulation assume applies fully receivers signal detection communication since number antennas may smaller number transmitters cell assume apply pzf receivers signal detection addition set performance psa algorithm subsection evaluate performance proposed psa algorithm comparison fig plot sum mse versus pilot length different pilot scheduling algorithms proposed psa scheme gcpa algorithm proposed random pilot scheduling rps benchmark also depict lower bound sum mse obtained users apply different orthogonal pilots channel estimation observed fig sum mse decreases pilot length algorithms consistent intuition moreover sum dus sum mse proposed psa gcpa rps lower bound mrc mrc number iterations fig convergence behaviors jdpc algorithm dmax mrc mrc number iterations fig convergence behaviors inner updates first outer iteration jdpc algorithm dmax proposed psa scheme approaches lower bound quickly increases outperforms two algorithms significantly terms sum mse performance jdpc algorithm subsection evaluate performance proposed jdpc algorithm denote orthogonal training scheme fig fig illustrate convergence behaviors proposed power control algorithm different configurations specifically left right panels fig respectively correspond dpcc algorithm dpcd algorithm fig corresponds jdpc algorithm seen fig sum links monotonically increases iterative procedure converges rapidly dpcc dpcd algorithms iteratively carrying two algorithms optimize data transmit power cus dus sum links effectively increased fig shows jdpc algorithm converges iterations within iterations considered configurations makes proposed data power control algorithm suitable practical applications sum system sum dus sum dus dpcd algorithm number iterations fig performances different pilot scheduling algorithms versus pilot length dmax mrc mrc pilot length dpcc algorithm proposed scheme proposed scheme proposed scheme classical massive mimo icapc maximum distance pair fig sum system versus maximum distance pair fig sum system versus maximum distance pair depicted compare proposed scheme iterative channel allocation power control icapc algorithm proposed note considered underlaid cellular system transceiver equipped one antenna therefore fair set system configurations simulating icapc algorithm also include sum curve obtained using classical massive mimo communication benchmark fig obtain benchmark instead applying direct communication pairs data signals first forwarded sent let denote uplink cellular link denote transmission working relay benchmark obtained maximizing fully receivers applied maximum obtained letting transmitters using maximum power signal transmission expected fig shows sum system decreases dmax communication adopted since icapc algorithm assumed perfect csi aimed maximize sum sum dus sum dus number pairs sum dus proposed scheme proposed scheme proposed scheme proposed scheme icapc icapc target sinr cus fig sum links versus number pairs different values coherence block length dmax coherence block length fig sum links versus coherence block length dmax dus based instantaneous csi icapc algorithm outperforms proposed scheme slightly terms system throughput adopted however among dus proposed scheme obviously increase system moreover also conclude fig due property transmission communication help improve sum massive mimo system significantly especially dmax small fig depicts sum links versus number pairs different values seen sum dus first increases approaches saturation point set grows ratio increases resulting pilot contamination channel estimations moreover since larger results cochannel interference requires longer pilot overhead fig shows sum dus decreases impact coherence block length sum dus investigated fig expected sum dus increases considered configurations proposed scheme icapc algorithm grows sum dus first lower higher case fig sum links versus target sinr cus different dmax small regime sum dus mainly influenced high regime mainly affected moreover fig also shows compared sum dus greatly increased especially small fig illustrates target sinr cus pzf parameters affect sum links several observations made figure first sum links decreases loss sum resulted increase reduces grows second pzf receivers outperform mrc receiver terms sum different choices moreover considered configuration maximum sum links obtained increasing degrees freedom interference suppression results decrease sum since less degrees freedom left signal enhancement vii onclusions paper consider underlaid massive mimo system due fact pairs usually located dispersively conduct transmission low power letting several pairs far use pilot channel estimation would feasible beneficial hence allow among dus reduce pilot overhead based setup first investigate channel estimation derive lower bound ergodic achievable rate link mitigate pilot contamination caused develop pilot scheduling algorithm criterion minimizing sum mse channel estimation links addition also maximize sum rate links based fading coefficients instead instantaneous csi iterative power control algorithm proposed obtain suboptimal solution simulation results show effect pilot contamination decreased greatly exploiting proposed pilot scheduling algorithm scheme provide significant performance gains conventional orthogonal training scheme terms system ppendix roof heorem definition respectively projections channel estimation vectors onto pzf receiver mentioned section iii channel estimation cellular links affected additive noise hence independent find desired signal dependent cochannel cellular interference using convexity applying jensen inequality denote instantaneous rate lower bound ergodic achievable rate cellular link respectively based analysis order obtain expression ano explicit need calculate convenience denote columns form orthogonal basis dimensional space specifically columns form orthogonal basis subspace spanned channel estimation vectors cancelled interferers column orthogonal unit norm pzf receiver chosen substituting yields lower bound ppendix roof heorem unlike channel estimation cellular links affected additive noise pilot contamination exists estimating channels due result estimation gkk affected noise also influenced pilot contamination caused dus obtain following relationship denote vik vik projection obviously independent column hence element follows according section central complex wishart random variable degrees freedom indicates equals norm complex gaussian random variables follows hence follows exponential distribution exp similarly exp expressing sum projections onto since norm vector independent linear combination vik vkk analogous proof process appendix easily verify independent hence relationship ergodic achievable rate link lower bound expressed meanwhile optimal values opt satisfy besides also vkk vnk popt opt opt popt vik substituting yields lower bound opt opt lemma obtain popt iteratively carrying following two steps convergence update based given update given solving last iteration parametric algorithm assume opt obtained find final popt solving reformulated opt opt opt opt opt max vki vkk ppendix roof heorem optimal pilot scheduling scheme determined allocate pilots dus problem equivalently written max vkk vik obviously problem optimization aims maximize summation fractional functions parametric algorithm often adopted solve kind problem numerator summation term concave denominator summation term convex since affine optimally solve problem using parametric algorithm according equivalently transformed following problem max since problem linear opt opt opt vki directly known popt vkk opt otherwise popt notice guarantee opt estimate channel vector link always change pilot assigned pair opt opt vki worst increasing vkk case pairs use different orthogonal pilots channel estimation case pilot contamination vanishes would transmit pilot maximum power increase channel estimation accuracy therefore theorem proven ppendix onvergence roof dpcc lgorithm ppendix order solve problem resort alternating optimization using following lemma applying lemma obtain opt lemma popt optimal solution maximization problem exists opt opt optimal solution following problem max discussed theorem iterative process converges optimal solution initial power vector function standard moreover sufficient condition standard standard therefore need prove standard since elements nonnegative positivity monotonicity scalability result standard iterative process converges optimal solution initial power vector olving via wmmse lgorithm notational convenience denote equivalently transformed following weighted minimization problem min plugging simplifying equivalent equivalence implies suboptimal solution problem obtained solving easier handle since objective function convex variable variables fixed optimal obtained based fixed given get optimal attach lagrange multiplier first constraint obtain lagrange function follows optimality condition yields min since eferences prove equivalence derive optimal checking optimality condition problem opt strictly decreases obtain using bisection method substituting get fkopt max positive weight variable meansquare estimation error wkopt according complementary slackness condition fkopt otherwise yields fcc spectrum policy task force report spectrum efficiency working group marzetta noncooperative cellular wireless unlimited numbers base station antennas ieee trans wireless vol rusek persson lau larsson marzetta edfors tufvesson scaling mimo opportunities challenges large arrays ieee signal process vol larsson edfors tufvesson marzetta massive mimo next generation wireless systems ieee commun vol xie wang gao jin strategy massive mimo cognitive radio systems ieee sel areas vol liu kato kadowaki communication networks survey ieee commun surveys vol asadi wang mancuso survey communication cellular networks ieee commun surveys vol apr lin andrews ghosh ratasuk overview proximity services ieee commun vol mar ngo larsson marzetta energy spectral efficiency large multiuser mimo systems ieee trans vol apr yang marzetta performance conjugate zeroforcing beamforming antenna systems ieee sel areas vol hoydis ten brink debbah massive mimo cellular networks many antennas need ieee sel areas vol liang zhang shen uplink interference analysis massive mimo systems mrc receivers proc ieee wireless commun netw conf wcnc bai heath asymptotic coverage probability rate massive mimo networks proc ieee globalsip zhu wang swindlehurst zhao downlink resource reuse communications underlaying cellular networks ieee sig process vol may maghsudi stanczak joint channel allocation power control underlay transmission proc ieee int conf commun icc london feng feng communications underlaying cellular networks ieee trans vol zhao wang resource allocation communication underlaying cellular networks alternating optimization method ieee commun vol jiang liu zheng gao energy efficient joint resource allocation power control communications ieee trans veh vol wang song han resource allocation underlay communication ieee trans wireless vol apr hoang resource allocation communications cellular networks proc ieee int conf london lin heath andrews spectral efficiency massive mimo systems underlay proc ieee int conf commun icc london lin heath andrews interplay massive mimo underlaid networking ieee trans wireless vol june jose ashikhmin marzetta vishwanath pilot contamination precoding tdd systems ieee trans wireless vol yin gesbert filippou liu coordinated approach channel estimation systems ieee sel areas vol ashikhmin marzetta pilot contamination precoding multicell large scale antenna systems proc ieee isit cambridge usa fernandes ashikhmin marzetta interference noncooperative tdd large scale antenna systems ieee sel areas vol ngo larsson channel estimation multicell multiuser mimo systems large antenna arrays proc ieee icassp kyoto japan gao xia peng pilot reuse massive mimo transmission spatially correlated rayleigh fading channels ieee trans wireless vol liu xiao wang pilot reuse underlay massive mimo systems ieee liu xiao wang pilot reuse interferenceaided mmse detection underlay massive mimo ieee trans veh vol apr huang yang shi chen pilot allocation power control underlay massive mimo systems ieee commun vol jindal andrews weber communication hoc networks achieving mimo gains simo transmission ieee trans vol zhu dai wang graph coloring based pilot allocation mitigate pilot contamination massive mimo systems ieee commun vol jin gao spatial pilot reuse massive mimo transmission proc ieee int conf wcsp mochaourab bengtsson adaptive pilot clustering heterogeneous massive mimo networks ieee trans wireless vol kailath sayed hassibi linear estimation prentice hall upper saddle river vol chen andrews heath ghosh uplink power control spatial multiplexing wireless systems ieee trans wireless vol july yates framework uplink power control cellular radio systems ieee sel areas vol shi razaviyayn luo iteratively weighted mmse approach distributed maximization mimo interfering broadcast channel ieee trans signal vol yang huang wang shi chen channel allocation power control uplink underlaid cellular networks proc ieee globecom workshops wkshps washington tulino random matrix theory wireless communications publishers inc vol jong efficient global optimization algorithm nonlinear problem online www optimizationonline org jiang resource allocation ofdm systems approach ieee trans wireless vol boyd vandenberghe convex optimization cambridge university press
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note penalty parameter nitsche method unfitted boundary value problems frits prentera christoph lehrenfeldb massingc department mechanical engineering eindhoven university technology netherlands numerical applied mathematics university germany department mathematics mathematical statistics university sweden sep institute abstract nitsche method popular approach implement boundary conditions situations strong imposition either inconvenient simply feasible method widely applied context unfitted finite element methods classical symmetric nitsche method stabilization parameter method chosen sufficiently large obtain unique solvability discrete systems short note discuss often used strategy set stabilization parameter describe possible problem arise show specific situations error bounds deteriorate give examples computations nitsche method yields large even diverging discretization errors keywords nitsche method finite element methods parameter accuracy stability error analysis introduction consider discretization unfitted poisson problem problem domain described separately encapsulating mesh finite element basis defined consider restriction finite element space respect problem domain approach used many methods similar virtue fictitious domain method cut finite element method cutfem finite cell method fcm immersogeometric analysis immersed boundary method unfitted discontinuous galerkin method udg extended finite element method xfem several others impose essential boundary conditions many methods apply version classical nitsche method requires stabilization penalization parameter preserve coercivity bilinear operator without additional stabilization cut elements ghost penalty terms customary methods referred cutfem penalty parameter depends shape size cut elements depends position geometry relative computational mesh typical choice sufficient provide unique solvability discrete problems choose stabilization parameter element wise constant proportional ratio surface measure intersection element boundary volume measure intersection element domain ratio become arbitrarily large stabilization parameter generally bounded mathematical literature unfitted nitsche formulations generally supplemented aforementioned additional stabilization bound stabilization parameter order prove properties corresponding author tel email addresses frits prenter lehrenfeld christoph lehrenfeld massing preprint submitted computers mathematics applications september method opinion importance sufficiently addressed literature however moreover method effectively successfully applied without additional stabilization engineering literature see note therefore detailedly treat possible problem occur additional measures taken bound stabilization parameter aim provide disclaimer directly applying classical form nitsche method immersed problems end discuss analyze method demonstrate lead poor results discretization error geometry intersects computational domain large values stabilization parameter required also investigate different situations degenerated geometry configurations yield satisfactory unsatisfactory results provide possible interpretation review alternative formulations possible modifications stabilizations method section note introduces model problem followed nitsche method consideration section section carry simple error analysis method norm natural formulation analysis reveals possible large discretization errors unfortunate cut configurations section give examples bad cut configurations show lead large even degenerating discretization errors finally list alternative approaches impose boundary conditions variants classical nitsche method circumvent possible problem section model problem model general elliptic boundary value problems consider poisson problem inhomogeneous dirichlet boundary data posed open bounded domain lipschitz boundary assume properties domain imply shift theorem form kukh kgk holds whenever smooth convex domain piecewise boundary note used denote inequality constant independent mesh problem standard weak formulation find standard sobolev spaces corresponding homogeneous formulation inhomogeneous boundary data simplicity assume dirichlet boundary conditions posed whole boundary however results presented work extend case dirichlet boundary data prescribed part boundary demonstrated section unfitted finite element method consider triangular rectilinear shape regular background discretization encapsulates domain element associate local mesh size diam define global mesh size maxt define active mesh set denoting union elements active mesh set geometric entities illustrated figure active mesh define finite dimensional function space figure example domain background mesh active mesh consisting continuous piecewise polynomials order simple prototype formulation nitschebased unfitted finite element method find denotes partial derivative normal direction boundary denotes elementwise stabilization parameter choice stabilization parameter crucial review error analysis section reveals local stabilization parameter chosen sufficiently large order guarantee discrete coercivity bilinear form common see strategy solve eigenvalue problem find element intersection see instance space functions constants let dim eigenvalue problem eigenvalues motivated forthcoming stability analysis suitable choice stabilization parameter set max max figure different cut configurations part boundary element vanishes vanishing volume sliver case measure boundary part stays bounded volume goes zero remark techniques shown scales thus implies formulation fitted mesh contrast unfitted method ratio depends cut element size also highly cut configuration see figure particular sliver cut lead arbitrarily large values relatively large parts boundary depending thickness sliver section demonstrated cut elements approximately shape result poor discretization errors remark generate solve discrete linear system associated problem several issues resolved sufficiently accurate numerical integration provided particular tuning linear algebra solver might necessary aspects important work focus choice effect numerical solution assuming numerical integration performed sufficiently accurate applying direct solver solve linear systems exactly simple error analysis section perform error analysis variational form first define meshdependent norm best approximation property holds section section discuss approximability norm consequences discretization error best approximation energy norm choice stabilization parameter stability analysis natural following norm refer energy norm note use norm error analysis essentially dictated nitsche method model problem obtain coercivity continuity following two lemmas lemma coercivity holds proof exploiting eigenvalue characterization definition obtain arbitrary positive solution lemma continuity holds proof directly obtain applying lemmas exploiting nitsche formulation consistent sense galerkin orthogonality obtain lemma lemma lemma let solution solution holds inf proof arbitrary holds claim directly follows note proof could use coercivity discrete functions end applied triangle inequality able apply coercivity estimate discrete function combined galerkin orthogonality continuity usual way version lemma suggests nitsche method provides robust method best approximation however provided norm next paragraph discuss discretization error best approximation norm approximability solution energy norm consider approximability solution finite element space energy norm inf aim paragraph show approximability discrete energy norm robust respect position domain boundary end split energy norm two parts one part robust one part robust inf inf inf assume solution smooth denoting order discretization sequel use notation inequality constant independent cut configuration first part take idea show solution optimally approximated respect parts norm independent cut position afterwards show remaining part crucial problem results missing robustness continuous extension embedding domain union let active elements bound latter part apply standard best approximation result fitted mesh estimate robust position boundary kukh due trace inequality kvkh kvkh applying gives similar results independent position boundary hence kukh consider second term crucial problem approximation energy norm let diam diam characteristic lengths cut elements parts boundary furthermore let best approximation energy norm typically zero scales denoting dimension boundary good case cut element shape regular holds would bounded constant robust bound element contribution bad case cut element shape regular however sliver cut element figure occur contribution single element becomes unbounded conclude last term unbounded thus holds left hand side therefore discretization error energy norm regardless best approximation property bounded missing robustness essentially due possible existence unfortunate cut configurations sliver cuts introduce values penalization parameter bounded terms reciprocal length boundary intersecting element energy norm natural respect symmetric nitsche formulation nevertheless one typically interested norm rather error obtain priori estimates error use inequality combine result lemma inf whilst approximability energy norm robust respect boundary position inequalities independent boundary therefore give optimal priori error estimate lemma lower bound besides showing traditional method show optimality hold nitsche method also give intuitive derivation nitche method yield bad approximations norm compare energy sobolev norms first sight norms even equivalent bounded holds even unsymmetric fitted formulations however related problem unbounded stabilization parameters targeting restrict space hui norms equivalent ckvh ckvh investigate strength equivalence examine constants first constant bounded independent cut configuration upper bound follows min min min kvh kvh kvh restriction functions intersect boundary second constant depend cut configuration max max kvh kvh due finite dimensionality contrast clearly depends therefore cut configuration inequality disregards possibility kvh due possibly large normal derivatives solution general limitation methods mentioned also case large large values however therefore depending cut configuration equivalence energy norm degenerate projections different norms deviate solution nitsche method closely resembling projection energy norm may deviate substantially optimal approximation remark using also extend kuh inf seems like satisfying result note numerical examples section demonstrate examples situations previously discussed discretization leads large values stabilization parameter aim section illustrate problem described section show problem priori analysis really yields large discretization errors practice numerical integration elements partially intersect domain performed using tessellation scheme proposed maximal refinement depth two leads exact representation domain first three examples reasonable good geometric approximation examples make sure results last examples affected geometrical errors adapt boundary conditions approximated geometry approximated geometry apply gauss quadrature functions order integrated exactly order achieve accurate integration force terms boundary conditions example overlapping square triangular quadrilateral mesh simplest example discretization sliver cuts described section squared domain structured grid aligns domain grid shown figure triangular mesh clearly visible cut elements become arbitrarily large aspect ratios triangular grid use triangular mesh isosceles right triangles legs length continuous piecewise linear basis functions apply element local stabilization parameter procedure described section set problem analytical solution equals sin sin impose dirichlet conditions previously discussed nitsche method boundaries mesh domain discretization errors figure example triangular grid resulting discretization errors energy norm norm given figure results clearly visible discretization error degenerates energy quadrilateral grid repeat example quadrilateral grid nodal positions triangular grid mesh size see figure piecewise bilinear elements used lead number degrees freedom results shown figure observe tendencies convergence observe impact small less severe quadrilateral grid compared triangular grid error several orders magnitude smaller diverging effect starts smaller value difference explained next paragraph rigorous proof gives intuitive illustration nature difference serves motivation design examples discussion difference triangular quadrilateral grids section seen numerical solution closely approximates energy norm projection analytical solution finite dimensional function space large values contribution normal derivatives energy norm diminishes consider energy norm weighted combination boundary interior increasing mesh domain discretization errors figure example quadrilateral grid balance weighted norms lost energy norm projection basically consists projection supported boundary projection interior remaining hence limit numerical solution satisfies argmin hence dim fixed independently remaining part variational formulation note triangular grid corresponding number dots minus one figure quadrilateral grid corresponding directly number dots figure besides constraining boundary values projection boundary also affect normal derivatives even complete numerical solution cut elements however let denote number basis functions support squares figure meshes note also equal number basis functions supported cut elements triangular grid implies limit one cut elements together constrained boundary condition quadrilateral mesh approximately half supported boundary constrained boundary condition along straight parts boundary normal derivatives therefore set projection interior independent boundary condition corners normal derivatives constrained boundary condition problematic area much smaller compared case triangular mesh impact large stabilization parameters although asymptotically less severe triangular grid quadrilateral grid figure sketch squares dots figures clear small values yield bad approximations actually cause diverging discretization errors explain effect note boundary data magnitudes whilst basis functions supported cut elements values magnitude boundary projection boundary condition therefore constrains coefficients basis functions value magnitude results gradients cut elements scale also observed peak stresses immersed elasticity problems large gradients introduce error sum volumes cut elements magnitude whilst error gradient squared value magnitude notice error seems unaffected sliver cuts example overlapping square mixed boundary conditions example reuse quadrilateral mesh try remove problematic corner regions end pose neumann conditions left right boundaries dirichlet conditions lower upper boundaries hypothesis yield good approximation results quadrilateral grid case projection dirichlet boundary constrain whilst number basis functions supported dirichlet boundary equals elements including corner elements exactly one constraint every two degrees freedom allows approximate boundary values without affecting normal derivatives results visible figure indeed show situation error robust respect even though error energy norm still scales also best possible approximation boundary conditions error energy norm error increases increasing furthermore notice errors energy norm reduced approximately factor accounted reduction size dirichlet boundary factor figure discretization errors example example tilted square mixed boundary conditions example demonstrate example useful academic purposes represent general case straight boundaries tensor product grids reason second example yields robust approximation errors dirichlet boundary parallel grid line number constrained projection exactly right linear bases generally case immersed methods however third example consider square boundaries consider kinked domain boundaries seen figure quadrilateral grid analytical solution previous examples configuration constraints projection dirichlet boundary basis functions supported dirichlet boundary even though kernel dimension projection boundary conditions nodes supported dirichlet boundary basis functions values magnitude boundary constrained mesh domain discretization errors figure example nodal values magnitude results figure indeed show similar behavior first example example rounded square unit ball previous examples consisted domains piecewise straight boundaries unfitted methods generally used complex domains example involves curved boundary creates cuts occur many realistic applications boundary domain defined function dirichlet conditions imposed complete boundary sketch domain visible figure discretization errors mesh domain figure example results visible figure results note example even error robust respect cut configuration furthermore peculiar effect observed errors simply increase decreasing peak certain values occurs smaller value simply imply thinner cut elements test case zones domain constitute thin sliver cut elements parts domain get thinner also get shorter decreasing intersections domain boundary grid lines form boundaries zones move towards points result cut elements become thinner decreasing intersections shift vertex removing thinnest cut element system point error decreases increases intersection shifts next vertex counting number degrees freedom system confirms errors peak values intersection boundary grid lines shift vertex values also number degrees freedom reduced indicated vertical gray dash dotted lines figure higher order basis functions domains effects observed previous examples occur higher order basis functions threedimensional domains fourth example also used investigate behavior second order discretization results visible figure note also requires larger stabilization parameter space larger higher order bases figure results example threedimensional setting visible domain defined dirichlet conditions imposed complete boundary approximated solution equals sin sin sin computation used first order trilinear basis grid size instead maximal refinement depth reduced save computation time also three dimensions effect observed figure second order figure example summary observations previous examples observed cut configurations dirichlet boundaries cause severe problems discretization discretization error measured energy norm bounded independently parameter controls volume ratios sliver cut elements error seems robust simple cases complex examples tried give interpretation origins trigger large errors small able characterize difference relative number involved projection dirichlet conditions penalty terms result example effect severe triangular discretization quadrilateral discretization specific domains instabilities due sliver cases even triggered demonstrated example similar penalty methods boundary fitting discretizations exactly right number involved penalty large penalties applied however small deviations immediately display missing robustness respect lead large errors example obtained similar results complicated domain curved boundaries quadratic basis functions three dimensions results apply piecewise linear discretizations problems simple domains conclusion resolutions problem section evident stabilization parameter needs bounded order guarantee optimal approximation properties fact several modifications nitsche method preclude previously discussed problems already exist literature divide three groups firstly discuss alternatives symmetric nitsche formulation secondly exist approaches preclude degrees freedom small supports problem domain required stabilization parameter bounded thirdly discuss stabilization mechanism called ghost penalty achieves similar effect penalizing jumps normal derivatives along boundaries cut elements alternative formulations obvious alternative symmetric nitsche formulation unsymmetric nitsche formulation however adjoint consistent methods weakly impose dirichlet conditions penalty method use lagrange multipliers related methods another approach unfitted grids manipulate function space allows strong imposition boundary conditions precluding small supports problem domain system contain functions small supports problem domain follows required stabilization parameter reduced additional advantage methods eliminate functions small supports system initially introduced order improve conditioning system matrix simplest way achieve simply excluding basis functions basis however may decrease accuracy functions small supports problem domain constrained geometrically nearby functions decreases required stabilization parameter without compromising accuracy another approach precludes basis functions small stiffness effect precluding functions small support using fictitious domain stiffness thoroughly analyzed ghost penalty ghost penalty see consistent penalty term added bilinear form customary methods referred cutfem see ghost penalty introduces penalty vicinity boundary weakly bounds polynomials neighboring elements coincide done adding terms penalize jumps normal derivatives along element interfaces local projection type penalties weakly enforces higher order continuity along element interfaces near boundary therefore relates functions small support boundary region interior functions similar strongly enforcing higher order continuity constraining basis functions small supports geometrically nearby functions effectively controls bounds stabilization parameter besides bounding stabilization parameter penalizing jumps normal derivatives gives control gradients cut elements therefore impede gradients magnitude observed cut elements remark inspired observations experiments also suggest quick fix simple implement alleviate problem unbounded stabilization parameters modify nitsche method hybrid method first manually set upper bound stabilization parameter use nitsche method elements required stabilization parameter computed smaller upper bound use penalty method flux terms penalty parameter elements nitsche method would require larger value manually set upper bound bounding penalty parameter easily motivated accuracy point view considering results analysis previously presented manuscript noted pure penalty method consistent initial problem inconsistency argued acceptable however considering generally occurs small parts domain parts boundary flux terms negligibly small compared penalty terms anyway hybrid method yields good approximations fourth test case results visible figure show quality approximation virtually independent suggests could simple straightforward fix problem consider preliminary result however adaptation method undergone rigorous testing also setting value needs thorough analysis could require different values different mesh sizes higher order discretizations problems poisson problem figure discretization errors modification nitche method example energy norm error hybrid method defined acknowledgement research prenter funded nwo graduate program fluid solid mechanics lehrenfeld gratefully acknowledges funding german science foundation dfg within project references glowinski pan periaux fictitious domain method dirichlet problem applications computer methods applied mechanics engineering burman hansbo fictitious domain finite element methods using cut elements stabilized nitsche method applied numerical mathematics burman claus hansbo larson massing cutfem discretizing geometry partial differential equations international journal numerical methods engineering massing larson logg rognes stabilized nitsche fictitious domain method stokes problem journal scientific computing burman claus massing stabilized cut finite element method three field stokes problem siam journal scientific computing parvizian rank finite cell method computational mechanics parvizian yang rank finite cell method problems solid mechanics computer methods applied mechanics engineering schillinger dede scott evans borden rank hughes isogeometric methodology based adaptive hierarchical refinement nurbs immersed boundary methods cad surfaces computer methods applied mechanics engineering rank ruess kollmannsberger schillinger geometric modeling isogeometric analysis finite cell method computer methods applied mechanics engineering ruess schillinger bazilevs varduhn rank weakly enforced essential boundary conditions nurbsembedded trimmed nurbs geometries basis finite cell methodod international journal numerical methods engineering ruess schillinger rank weak coupling isogeometric analysis trimmed geometries computer methods applied mechanics engineering schillinger ruess finite cell method review context structural analysis cad 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solution square cavity flow bingham plastic using finite volume method alexandros syrakosa georgios georgioub andreas alexandrouc oceanography centre university cyprus box nicosia cyprus department mathematics statistics university cyprus box nicosia cyprus department mechanical manufacturing engineering university cyprus box nicosia cyprus apr abstract investigate performance finite volume method solving viscoplastic flows creeping square cavity flow bingham plastic chosen test case constitutive equation regularised proposed papanastasiou rheology shown convergence rate standard simple algorithm used solve algebraic equation system produced finite volume discretisation severely deteriorates bingham number increases corresponding increase equations shown using simple algorithm multigrid context dramatically improves convergence although multigrid convergence rates much worse newtonian flows numerical results obtained bingham numbers high compare favorably reported results methods keywords bingham plastic papanastasiou regularisation cavity finite volume method simple multigrid accepted version article published journal fluid mechanics manuscript version made available license http introduction viscoplastic flows constitute important branch fluid mechanics many materials industrial geophysical biological importance known exhibit yield stress general fluids suspensions particles macromolecules pastes gels foams drilling fluids food products nanocomposites comprehensive review viscoplasticity carried barnes materials behave elastic inelastic solids certain critical shear stress level yield stress liquids otherwise flow field thus divided unyielded rigid yielded fluid regions corresponding author email addresses alexandros syrakos georgios georgios georgiou andalexa andreas alexandrou preprint submitted journal fluid mechanics april simplest constitutive equation describing viscoplasticity proposed bingham yield stress plastic viscosity stress tensor rate strain tensor velocity vector superscript denotes transpose tensor symbols denote magnitudes stress tensors respectively another viscoplastic equation casson model mostly used hemodynamics model generalisation equation involves exponent allowing shear thickening simulating viscoplastic flows poses difficulties due discontinuity constitutive equations cases necessary determine location shape yield surface flow switches one branch constitutive equation solid liquid behavior mathematically yield surface locus points common approach overcoming burden regularise constitutive equation describe two branches one smooth equation using parameter popular regularization proposed papanastasiou exp denotes stress growth parameter needs sufficiently large frigaard nouar reviewed systematically convergence papanastasiou regularized models including model another approach solving viscoplastic flows based use variational inequalities minimization stress maximization form basis augmented lagrangian method dean reviewed compared numerical methods based variational inequality approach paper adopt regularisation approach investigate performance finite method solving bingham plastic flows finite volume method fvm popular method solving fluid flows employed many cfd solvers one attractive features method applied variety physical problems relative ease however limited number published results use method solve bingham flow problems neofytou used fvm conjunction simple algebraic solver patankar spalding simulate cavity flow various fluids including bingham plastic quite low bingham numbers turan also used commercial code employing model order simulate natural convection bingham plastic square cavity fvm also used solve flows casson fluid stenosis neofytou drikakis sudden expansion neofytou drikakis rather low values also souza mendes naccache barbosa used fvm order simulate viscoplastic flow expansion followed contraction benchmark problem testing numerical methods newtonian flows cavity flow problem laminar newtonian flow reader referred works botella peyret syrakos goulas bruneau saad references therein cavity flow also used test case bingham flows sanchez mitsoulis zisis vola elias wachs olshanskii zhang dos santos used solution methods fvm mostly finite element method authors knowledge neofytou used fvm order solve driven cavity flow bingham plastic viscoplastic fluid albeit small bingham numbers although aforementioned published works contain results reynolds numbers cases results concern creeping flows objective present work apply fvm along multigrid method order improve convergence simple solver solving cavity bingham flow broad range bingham numbers multigrid methods use hierarchy grids progressive fineness applying algebraic solver grids wavelengths algebraic error reduced equal efficiency multigrid methods originally proposed fedorenko later developed brandt context fvm first used late sivaloganathan shaw hortman since employed numerous studies involving fvm many used cavity flow newtonian fluid test problem see syrakos goulas references therein newtonian studies shown using multigrid algorithms result significant performance gains knowledge finite method tested case bingham flow rest paper organized follows section integral forms governing equations presented dedimensionalized section numerical method discretisation governing equations solution resulting algebraic system discussed numerical results presented section compare well results mitsoulis zisis wachs convergence algebraic solver also studied possible ways acceleration investigated finally section contains concluding remarks governing equations consider flow square cavity side top boundary lid moves towards right uniform horizontal velocity remaining sides fixed work cartesian coordinates centered corner cavity denote unit vectors directions respectively let also denote density viscosity fluid means gauss theorem integral forms continuity momentum equations control volume follows boundary surface control volume infinitesimal element surface oriented normal vector points velocity vector pressure case papanastasiou model viscosity given exp advantage papanastasiou regularization expression used entire flow domain yielded unyielded regions high strain rates viscosity tends towards growth parameter large enough unyielded regions viscosity obtains high values result small values thus solid body motion approximated tends zero viscosity tends infinity finite value authors suggested lower values used higher yield stress values vice versa dedimensionalise governing equations scale lengths cavity side velocity components pressure stress use stars denote dimensionless variables taking account constant dimensionless equations follows exp reynolds number bingham number number sake simplicity stars denoting dimensionless variables dropped hereafter numerical method discretisation equations domain split number control volumes cvs using cartesian grid equally spaced horizontal vertical grid lines control volume continuity momentum equations approximated using algebraic expressions involving values unknowns centre centres neighbouring cvs computer code used present study ability use curvilinear grids composed quadrilateral cvs see details present section simpler form discretisation schemes acquire grid figure control volume neighbours cartesian presented although present work concerns creeping flow treatment convection terms momentum equations also described completeness surface flux integrals calculated separately face figure shows control volume neighbours letters also denote position vectors centres respective cvs first flow variables normal derivatives calculated centre face using central differences face flux integrals approximated using midpoint rule equation equations describe approximation scheme sufficiently definitions still missing denotes mass flux face defined shortly term part total viscous flux term requires viscosity already calculated somehow centres define terms first approximate gradient operator centres equation approximates normal component face centres centre control volume following approximation used boundary domain boundary coincides face horizontal component gradient calculated follows boundary value specified dirichlet boundary condition viscosity calculated centres equation calculated using velocity gradients centres thus calculated using linear interpolation since viscosity needed face centres alternative would calculate directly would require velocity gradients face centres normal component readily available tangential component would require interpolation velocity components face vertices alternative would less complex approach presently adopted reason remaining part viscous fluxes also calculated interpolating velocity gradients face centres centres example faces respectively derivatives interpolated derivatives face centres according approach convenient case curvilinear grids adopted code used present study leaves mass fluxes defined discretised using central difference scheme addition artificial pressure term whose role allow appearance spurious pressure oscillations discrete solution mass flux face form discretised without treatment artificial pressure oscillations appear velocity pressure stored centres present scheme technique known momentum interpolation first suggested rhie chow variant proposed advantage decoupled simple solution algorithm resulting equations simplify case uniform grid completeness general forms included artificial pressure term small order small impact accuracy discretisation mass flux test cases examined work involve solid wall boundaries dirichlet boundary conditions discretisation follows pressure linearly extrapolated interior product main viscous terms calculated difference boundary face approximated exact value centre face defined dirichlet boundary condition finally secondary viscous terms derivative viscosity boundary face centre taken equal values adjacent centre substituting terms continuity momentum equations discrete counterparts algebraic system obtained four equations continuity constitutive equation four unknowns per overall discretisation scheme second order accuracy meaning discretisation error decrease solution algebraic system gives values unknowns discretisation error solver used solve system described next solution algebraic system popular simple algorithm chosen solver widely used algorithm fully described interested reader referred brief simple iterative algorithm constructs solves number linear systems within iteration linear systems come breaking linearising set discretised equations systems equations momentum components converted linear systems corresponding velocity components evaluating terms including viscosity using velocity pressure values obtained previous simple iteration continuity equations system used construct approximate pressure correction linear system attempts improve current pressure estimate force velocity field simple iteration outer iteration consists successive solution linear systems velocity components pressure correction iterations inner iterations linear solver applied linear system since matrices coefficients change next outer simple iteration work used gmres linear solver velocity systems conjugate gradients pressure correction system preconditioned incomplete factorisations zero see detailed descriptions achieve convergence underrelaxation factors applied velocity pressure correction systems respectively simple iterations repeated residuals original algebraic equations drop selected threshold modification needed simple used bingham generalisednewtonian flows start every simple iteration viscosity must updated according using current estimate velocity field calculate simple algorithm converges rather slowly decided implement multigrid context accelerate convergence many algebraic solvers able quickly reduce short wavelengths error quite slow reducing longer wavelengths simple solver linear solvers used inner iterations may property also local assumptions made linearisation terms construction pressure correction equation relate direct neighbours case iterations performed short wavelengths error reduced beneficial move solution procedure coarser grid direct neighbours farther away long wavelengths error appear shorter reduced efficiently algebraic solver using cascade progressively coarser grids wavelengths error reduced equal efficiency solution procedure transferred grids follows let system algebraic equations written vector stores unknowns velocity components pressure viscosity centres grid whose spacing simple iterations approximate solution obtained satisfies equation residual algebraic system approximated coarser grid spacing grid hereafter obtained removing every second grid line grid follows operator constructed grid using discretisation schemes grid used instead restriction operator transfers variables grid grid notation comes identity matrix since variables transformed rather transferred one grid another present study restriction operator sets value variable centre control volume grid equal average values variable cvs grid overlap hereafter called children possibly different operator used transferring residuals grid present study residuals control volume grid set equal sum residuals children since residuals essentially flux imbalances means flux imbalance set equal sum imbalances children right hand side known system solved obtain correction transferred back grid obtain better estimate xnew exact solution simple iterations resume grid xnew prolongation operator transfers correction grid grid present study linear interpolation used straightforward see exact solution already obtained residuals zero therefore last term solution important used initial guess solving therefore correction zero leave solution unaltered xnew coarse grid problem solved using even coarser grid whole process going grid coarsest grid back grid constitutes multigrid cycle general number multigrid cycles required reduce residuals given amount different kinds cycles suggested used popular cycles cycles performing iterations fine grid coarse grid system solved using one multigrid cycle coarse grid correction transferred back grid according another iterations performed difference cycles cycles coarse grid problems solved using two rather one cycle present work observed sometimes useful perform extra simple iterations finest grid cycles cycles denoted respectively cycles shown schematically figure costs cycle approximately equal times cost single iteration finest grid respectively figure multigrid cycles left right cycles using grid levels numerical results creeping flow bingham plastic square cavity chosen test case finite volume method described previous section computational domain square enclosed solid boundaries equal length top boundary lid moves towards right uniform horizontal velocity rest boundaries still problem solved range bingham numbers cases exponent used except two highest bingham numbers tested used due convergence difficulties order check grid convergence domain discretised using three uniform cartesian grids cvs unless otherwise stated results presented calculated grid pressure set zero centre domain figure shows streamlines calculated selected bingham numbers bingham numbers chosen also mitsoulis zisis wachs corresponding figures unyielded regions also shown defined regions calculated course actually unyielded approximate unyielded regions ideal bingham flow results generally agree presented researchers agreement good results mitsoulis zisis whole range bingham numbers except authors predict yielded regions rounded corners present results also yield surfaces smooth may due low resolution grid employed results also agree well wachs bingham number notable differences higher bingham numbers however wachs state far capturing yield surface concerned method performs well low moderate bingham numbers less high bingham numbers direct comparison also made present results olshanskii used augmented lagrangian approach instead regularisation method agreement good zhang also used augmented lagrangian approach also provides results unyielded regions appear somewhat smaller flatter rounded compared present results although qualitatively similar main characteristic flow field vortex develops upper central region cavity similarly newtonian case two distinct unyielded regions observed larger one figure streamlines plotted intervals starting zero unyielded areas shown shaded bottom cavity smaller one vortex centre flow field symmetric respect vertical centreline number increases unyielded regions expand flow circulation becomes weaker limited upper part cavity centre vortex coming closer lid one notice figure couple secondary vortices appear lower two corners cavity completely inside lower unyielded region vortices appear slightly grow size increases artifact regularisation bingham model papanastasiou approximation fact velocity extremely low throughout lower unyielded region reality since region contact rigid walls motionless boundary condition applies unyielded material also motionless throughout situation different upper unyielded region velocity clearly seen streamlines spacing upper unyielded regions move solid bodies present case appears material upper unyielded region moves solid body particular rotates streamlines forming circular arcs inside fact streamlines cross region means fluid solidifies entry becomes fluid exit region behaviour vortex function number described detail figure vertical position strength vortex plotted functions weakening circulation increases accompanied increase stress pressure levels cavity greater stresses needed order make material flow figure also serves validate results present study researchers seen results quite close previously published data results closer mitsoulis zisis far vortex strength concerned results wachs far vortex position concerned possibly grid coarse figure shows pressure shear stress distributions left half lid increasing bingham number causes significant increase pressure stress discussed previously figure also used verify grid convergence contains results three grids used work interior lid pressure stress converge grid refinement rate convergence expected order method however grid refinement causes pressure stress tend infinity lid corners discontinuity exists corners velocity jumps zero side walls one lid due singularity observed pressure stress integrals lid converge grid refinement shear force needed drive lid calculated accurately results grid convergence discussed later increase pressure bingham number also seen pressure contours figure clear presence unyielded regions causes distortion pressure field streamwise pressure gradients greater inside upper unyielded region important issue using papanastasiou regularisation choice exponent higher value better approximation also difficult solve equations due increased degree nonlinearity therefore practical reasons kept within certain limits provide comparisons showing effect quality results issue computational effort required function investigated next section one result importance range applications location yield lines approximated present method previously stated figure shows use different values affect location yield lines significantly difference especially small figure shows lines deviate slightly yield stress value comparing figures shows lines less sensitive choice lines verifies result alexandrou vortex centre vortex strength figure coordinate centre vortex vortex strength function bingham number lid lid figure pressure shear stress left half lid results shown grids chained lines dashed lines solid lines location lines predicted almost equally well range values whereas accurately predict location yield line high value required lines nearly coincide yield stress line except near side walls lower corners upper unyielded region therefore sharp decrease stress one moves unyielded zone contrary observed lines located distance yield line meaning stress increases gradually one moves yielded zone interesting note difference actual stress yield stress less throughout fluid region separates two solid regions since contours cross region deformation rate quite small shear takes place close lid figure pressure contours various numbers thick lines outline unyielded regions note pressure scale case different figure yield lines calculated dotted dashed solid figure lines calculated dotted dashed solid unyielded regions shown shaded figure component velocity along vertical centreline closing section provide detailed results vertical centreline velocity component plotted along vertical centreline figure limits yielded unyielded zones clearly seen graph curve except two straight line segments identified one completely vertical stretches certain height segment corresponds lower unyielded zone motionless segment corresponds upper unyielded zone velocity decreases linearly height shows upper unyielded zone rotates solid body centre rotation estimated extending straight line segment intersects line detailed results shown tables velocity values obtained selected points using linear interpolation values adjacent cvs study grid convergence results various grids also included exponent given tables order grid convergence present accurate scheme calculated following formula see log log subscripts denote grid calculated tables show convergence exhibited could sign even finer grid would useful however discretisation error already relatively small seen difference solutions various grids besides use finer grid would significantly increase computational cost described next section value especially low upper parts centreline indication higher numbers flow occurs upper part domain would benefit use grids increased resolution near top adaptive grids last three columns tables defined follows calculated grid values must interpreted caution may appear large close zero measure discretisation error noticed quite small order whereas markedly larger order order times higher point shows greater variability little higher twice value upper part centreline three times value lower part general appears present choices grid density grid coarseness larger source error smallness convergence algebraic solver using simple algorithm solve bingham flows either solver multigrid context one realises severe deterioration convergence rates bingham number increases figure plot number iterations residual vector equations max residual expressed per unit volume equation control volume total number control volumes grid residual norm scaled actual numerical experiments solved dimensional version equations therefore residuals dimensional equations times larger dimensionless equations figure shows performance simple solver deteriorates either increases choices used obtain results reasonable newtonian flows figure effect varying illustrated increased convergence becomes faster especially initial stages iteration however also becomes oscillatory large spikes beyond certain value figure convergence stalls hand using small results smooth slow convergence due large number combinations grid density impossible investigate fully effect parameters observed results shown performance simple similar also values range beyond values difficult obtain convergence performance simple smoother multigrid context tested next applying multigrid coarsest grid used cycles grid according numerical experiments convergence deterioration simple algorithm increasing number also reflects multigrid performance fact figure shows solid lines multigrid method converge beyond relatively small value around remedy applied suggestion ferziger peric fluid property viscosity rans turbulence models used varies orders magnitude within computational domain may useful update property finest grid keep constant within multigrid cycle tried implementation viscosity coarse grids calculated afresh according table values velocity along vertical centreline grid table values velocity along vertical centreline grid figure norm residual plotted number simple iterations single grid solution grid restricted velocity field directly restricted interpolated immediately finer grid convergence modified multigrid method also shown figure dashed lines method converges slowly standard multigrid method robust converge wider range bingham numbers methods converge equally fast point beyond convergence modified method suddenly slows appears methods equally capable reducing certain components residual dominate residual initial stages iteration however modified method less capable reducing certain components dominate error beyond certain point modified method strictly multigrid since part solution namely updating viscosity takes place finest grid therefore method exhibits also singlegrid convergence characteristics particular figure shows convergence modified method slows grid refined true slowly converging components residual contrary standard method converges equally fast grids normal multigrid behaviour higher bingham numbers necessary use modified multigrid method even case convergence difficulties encountered however gains compared method still quite impressive figure shows convergence rates various bingham numbers using multigrid procedures case solution grid used initial guess coarsest grid used multigrid cycles grid computational effort measured terms equivalent simple iterations cases number simple iterations performed multigrid cases number cycles multiplied number simple iterations cost computationally single cycle example cycles cost approximately figure norm residual plotted number simple iterations single grid solution grid simple iterations finest grid mentioned section noted cost restriction prolongation omitted calculation since cost operations small compared cost simple iterations especially one considers numbers smoothing iterations large iterations also carried cycles therefore convergence rates directly comparable fig one may also observe number increases choice multigrid parameters becomes rather unusual compared usual multigrid practice number sweeps quite large large number simple iterations required cycles choices parameters found necessary obtain convergence may also seen small values used multigrid cases necessary otherwise procedure converge may associated fact increases simple converges faster oscillatory manner may cause problems multigrid procedure since procedure converges faster large possible results obtained large values compared cases observed multigrid extremely efficient low numbers difficulty large yet every case multigrid convergence rates much faster rates except first iterations therefore multigrid preferable case moreover note multigrid particular choices parameters converges faster indeed observed every case convergence deteriorates significantly increasing also increasing every bingham number solved problem using except used due convergence difficulties worth mentioning number inner iterations seems play important role figure figure norm residual plotted number multigrid cycles grid solid lines depict convergence standard multigrid method dashed lines depict convergence modified multigrid method viscosity updated coarse grids interpolated immediately finer grid figure norm residual plotted number multigrid cycles various grids chained lines dashed lines solid lines modified multigrid method viscosity updated coarse grids simply interpolated immediately finer grid figure norm residual grid plotted number equivalent simple iterations cases except denote respectively multigrid solution procedures initial guess solution immediately finer grid curves close vertical axis visible visible also improve readability results omitted graph seen using gmres iterations iterations velocity pressure correction systems labelled figure significantly improves convergence rate procedure compared case iterations used instead labelled seems due nonlinearity algebraic system better perform many iterations linearised velocity systems within simple outer iteration investigated detail due fact already large number parameters involved solution procedure conclusions solved creeping cavity flow bingham plastic using papanastasiou regularization finite volume method combined multigrid algorithm results obtained bingham numbers range compare favorably results methods method show proposed method provides useful tool solving viscoplastic flows wide range bingham numbers noted convergence method becomes slow high values bingham number regularization parameter use modified multigrid method convergence accelerated considerably compared simple method acknowledgements work european regional development fund republic cyprus research promotion foundation research project bie references references barnes yield stress review everything flows journal nonnewtonian fluid mechanics bingham fluidity plasticity new york papanastasiou flows materials yield journal rheology frigaard nouar usage viscosity regularisation methods fluid flow computation journal fluid mechanics donovan tanner numerical study bingham squeeze film problem journal fluid mechanics fortin glowinski augmented lagrangian methods applications numerical solution problems amsterdam glowinski numerical methods nonlinear variational problems springer new york dean glowinski guidoboni numerical simulation bingham flow old new results journal fluid mechanics neofytou order upwind finite volume method generalised newtonian fluid flows advances engineering software patankar spalding calculation procedure heat mass momentum transfer parabolic flows international journal heat mass transfer turan chakraborty poole laminar natural convection bingham fluids square enclosure differentially heated side walls journal fluid mechanics turan chakraborty poole laminar convection yield stress fluids square enclosure journal fluid mechanics neofytou drikakis effects blood models flows stenosis international journal numerical methods fluids neofytou drikakis flow instability channel sudden expansion journal fluid mechanics souza mendes naccache varges marchesini flow viscoplastic liquids axisymmetric journal fluid mechanics naccache barbosa creeping flow viscoplastic materials planar expansion followed contraction mechanics research communications botella peyret benchmark spectral results cavity flow computers fluids syrakos goulas finite volume adaptive solutions using simple smoother international journal numerical methods fluids bruneau saad cavity problem revisited computers fluids sanchez application operator splitting method bingham fluid flow simulation computers mathematics applications mitsoulis zisis flow bingham plastics square cavity journal nonnewtonian fluid mechanics vola boscardin latch laminar unsteady flows bingham fluids numerical strategy benchmark results journal computational physics elias martins coutinho parallel solution viscoplastic flows formulation computational mechanics wachs fictitious domain method dynamic simulation particle sedimentation bingham fluids journal fluid mechanics olshanskii analysis method application bingham flows computer methods applied mechanics engineering zhang augmented lagrangian approach bingham fluid flows square cavity piecewise linear finite elements computer methods applied mechanics engineering dos santos frey naccache souza mendes numerical approximations flow viscoplastic fluids cavity journal fluid mechanics fedorenko relaxation method solving elliptic difference equations ussr computational mathematics mathematical physics brandt adaptive solutions problems mathematics computation sivaloganathan shaw multigrid method recirculating flows international journal numerical methods fluids hortmann peric finite volume multigrid prediction laminar natural convection benchmark solutions international journal numerical methods fluids tsamopoulos chen borkar spin coating viscoplastic fluids rheologica acta burgos alexandrou entov determination yield surfaces herschelbulkley fluids journal rheology ferziger peric computational methods fluid dynamics springer edition syrakos goulas estimate truncation error finite volume discretization equations colocated grids international journal numerical methods fluids rhie chow numerical study turbulent flow past airfoil trailing edge separation aiaa journal patankar numerical heat transfer fluid flow hemisphere saad iterative methods sparse linear systems edition siam shaw sivaloganathan smoothing properties simple algorithm international journal numerical methods fluids burgos alexandrou flow development fluids sudden threedimensional square expansion journal rheology
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wrpn wide networks sep asit mishra eriko nurvitadhi jeffrey cook debbie marr accelerator architecture lab intel abstract computer vision applications prior works shown efficacy reducing numeric precision model parameters network weights deep neural networks activation maps however occupy large memory footprint training inference step using inputs one way reduce large memory footprint reduce precision activations however past works shown reducing precision activations hurts model accuracy study schemes train networks scratch using activations without hurting accuracy reduce precision activation maps along model parameters increase number filter maps layer find scheme matches surpasses accuracy baseline network result one significantly improve execution efficiency reduce dynamic memory footprint memory bandwidth computational energy speed training inference process appropriate hardware support call scheme wrpn wide networks report results show wrpn scheme better previously reported accuracies dataset computationally less expensive compared previously reported networks introduction promising approach lower compute memory requirements convolutional deeplearning workloads use low numeric precision algorithms operating lower precision mode reduces computation well data movement storage requirements due efficiency benefits many existing works propose deep neural networks dnns even ternary mode binary mode however majority existing works dnns sacrifice accuracy baseline networks prior works target reducing precision model parameters network weights primarily benefits inference step batch sizes small improve execution efficiency accuracy networks reduce precision activation maps model parameters increase number filter maps layer call networks using scheme wide networks wrpn find scheme compensates surpasses accuracy baseline network although number raw compute operations increases increase number filter maps layer compute bits required per operation fraction required using operations going alexnet precision doubling number filters increases number compute operations operation efficient wrpn offers better accuracies computationally less expensive compared previously reported networks report results alexnet inception dataset find sufficient training deep wide models achieving similar better accuracy baseline network activation weights find accuracy baseline making networks wider operating precision close accuracy gap previously report binary networks show art results wide alexnet wide best knowledge reported accuracies binary networks even precision highest date quantization scheme hardware friendly allowing efficient hardware implementations end evaluate efficiency benefits operations titan gpu fpga asic see fpga asic deliver significant efficiency gain operations gpu take advantage operations motivation activation maps figure memory footprint activations acts weights training inference minibatch sizes prior works proposing networks work low precision weights find activation maps occupy larger memory footprint using inputs using inputs typical training dnns cloudbased batched inference figure shows memory footprint activation maps filter maps batch size changes different networks alexnet training inference steps ofm act ofm act layer grad ofm act layer ofm grad act layer grad grad figure memory requirements feed forward convolutional deep neural network orange boxes denote weights blue boxes activations act green boxes grad increases filter reuse across batches inputs activation maps occupy significantly larger fraction memory compared filter weights aspect illustrated figure shows memory requirements canonical dnn hardware accelerator based system gpu fpga pcie connected asic device training sum activation maps act weight tensors allocated device memory forward pass along memory gradient maps backward propagation total memory requirements training phase sum memory required activation maps weights maximum input gradient maps maximum gradients inference memory allocated input ifm output feature maps ofm required single layer memory allocations reused layers total memory allocation inference maximum ifm maximum ofm required across layers plus sum batch sizes activations start occupy total memory footprint training overall reducing precision activations weights reduces memory footprint bandwidth storage also simplifying requirements hardware efficiently support operations wrpn scheme studies alexnet based observation activations occupy memory footprint compared weights reduce precision activations speed training inference steps well cut memory requirements however straightforward reduction precision activation maps leads significant reduction model accuracy conduct sensitivity study reduce precision activation maps model weights alexnet running dataset train network scratch table reports findings weights activations accuracy accuracy binary weights activations similar reported table using ttq technique data points collected using quantization scheme described later section runs training carried number epochs baseline network consistent results reported prior works quantize weights activations first last layer find general reducing precision activation maps weights hurts model accuracy reducing precision activations hurts model accuracy much reducing precision filter parameters find ttq quite effective alexnet one lower precision weights activations still lose accuracy however find scheme effective networks like resnet inception table alexnet validation set accuracy precision activations weight changes results training network scratch experiment table alexnet validation set accuracy precision activations weights changes model accuracy working operands increase number filter maps layer although number raw compute operations increase widening filter maps layer bits required per compute operation fraction required using operations result appropriate hardware support one significantly reduce dynamic memory requirements memory bandwidth computational energy speed training inference process widening filter maps inspired wide resnet work depth network reduced width layer increased operand precision still wide resnet requires network architecture work maintain depth parameter baseline network widen filter maps call approach wrpn wide networks practice find scheme simple effective starting baseline network architecture one change width filter map without changing network design parameter carefully reducing precision simultaneously widening filters keeps total compute cost network baseline table reports accuracy alexnet double number filter maps layer doubling filter maps alexnet weights activations exhibits accuracy networks operating weights activations surpasses baseline accuracy binary weights activations better accuracy xnornet compute cost product number fma operations sum width activation weight operands doubling number filter maps alexnet raw compute operations grow compared baseline network however using operands overall compute complexity fraction baseline example operands weights activations number filters alexnet total compute cost baseline compute cost comparison shown table table compute cost alexnet precision activations weights changes also experiment widening factors widening filters activation precision one low weight precision still baseline accuracy wide filters least weight activation precision required accuracy match baseline wide accuracy table shows widening filters one needs lower precision least total compute cost baseline compute cost thus widening reducing precision network parameters work higher number raw compute operations aggressively reducing precision operands involved operations activation maps filter weights sacrificing model accuracy apart benefits reduced precision activations mentioned earlier widening filter maps also improves efficiency underlying gemm calls convolution operations since compute accelerators typically efficient single kernel consisting parallel computation large opposed many small sized kernels studies deeper networks study scheme applies deeper networks study batchnormalized inception find similar trends particularly weight activations continue provide accuracy baseline use tensorflow tensorpack evaluations use train val dataset resnet filters modular layers shortcut connections filter bank width changes depth increases use variant resnet baseline accuracy implementation using data format binarizing weights activations layers except first last layer network gives accuracy binarizing resnet reorder layer done used learning rate schedule baseline network reference gap weights activations network also interesting note accuracy alexnet lower accuracy binarized experimented doubling number filters layer reduce precision activations weights table shows results analysis doubling number filters precision weights activations beats baseline accuracy activations ternary weights accuracy baseline reducing precision weights activations degrades accuracy compared baseline implementation alexnet resnet inception networks table validation accuracy compute cost precision activations weights varies width precision acc compute cost wide wide wide binarizing weights activations wide filters accuracy worse baseline network cost baseline network widening filters binarizing weights activations reduces gap wide network cost baseline network although precision seems enough wide networks advocate activation precision weight precision ternary weights one get rid multipliers use adders instead additionally configuration loss accuracy accuracy degradation tolerable one even binary circuits efficient hardware implementation saving bandwidth weights activations compared networks gains realized simpler hardware implementation lower compute cost compared baseline networks best knowledge resnet binary ternary activation accuracies results literature including unpublished technical reports similar data augmentation inception applied wrpn scheme inception network network includes batch normalization layers variant googlenet convolutional filters replaced two convolutions wide filters table shows results analysis using activations weight doubling number filter banks network produces model almost accuracy baseline network loss accuracy wide network binary weights activations within baseline network table inception validation accuracy compute cost precision activations weights varies width precision acc compute cost wide wide hardware friendly quantization scheme adopt estimator ste approach work quantizing real number ordinality set quantized numbers mathematically small finite set would zero gradients respect inputs ste method circumvents problem defining operator arbitrary forward backward operations prior works using ste approach define operators quantize weights based expectation weight tensors instance twn uses threshold scaling factor layer quantize weights ternary domain ttq scaling factors learned parameters binarizes weight tensor computing sign tensor values scaling mean absolute value output channel weights dorefa uses single scaling factor across entire layer quantizing weights dorefa uses quantizek tanh max tanh quantized version inputs quantizek quantization function quantizes number range number range transcendental tanh operation constrains weight value lie affine transformation post quantization brings range build approaches propose much simpler scheme quantizing weight tensors first hard constrain values lie within range using operation using tensorflow quantizing activation tensor values constrain values lie within range step followed quantization step real number quantized number given round round input weights activation tensor quantized versions one bit reserved case weight values hence use quantized values thus weights stored interpreted using signed activations using appropriate affine transformations convolution operations bulk compute operations network forward pass done using quantized values integer operations hardware followed scaling constants scaling operation done parallel convolution operation hardware binary weights use bwn approach binarized weight value computed based sign input value followed scaling mean absolute values binarized activations use formulation quantize gradients maintain weights reduced precision format convolution operation using wrpn forward pass training inference step involves matrix multiplication signed unsigned operands since gradient values format backward pass involves matrix multiplication operation using operand gradient weight update hard clipping tensors range maps efficiently comparator units hardware opposed using transcendental operations long latency operations ttq dorefa schemes involve division operation computing maximum value input tensor division operation expensive hardware computing maximum tensor operation additionally quantization parameters static require learning involve like ttq approach avoid costly operations propose simpler quantization scheme clipping followed rounding efficiency improvements operations gpu fpga asic practice effective performance energy efficiency one could achieve compute operation highly depends hardware runs operations study efficiency operations various hardware targets gpu fpga asic gpu evaluate wrpn nvidia titan pascal fpga use intel collect performance numbers previously reported analysis well experiments fpga implement dnn accelerator architecture shown figure prototypical accelerator design used various works fpga asic tpu core accelerator consists systolic array processing elements pes perform matrix vector operations along buffers well memory management unit pes configured support different precision operates ternary values optimized include adder since need multiplier case binary implemented using xnor bitcount rtl design targets fpga asic study synthesize design using intel process technology obtain area energy estimates jaa figure efficiency improvements operations gpu fpga asic figure summarize analysis figure shows efficiency improvements using estimates efficiency computed based number bits used operation method would expect efficient respectively however practice efficiency gains reducing precision depend whether underlying hardware take advantage figure shows performance improvement titan gpu various operations relative case gpu achieve improvements performance baseline gpu provides support operations able take advantage lower precisions contrary fpga take advantage low precisions since amenable implementations fpga reconfigurable fabric figure shows performance improvements track well estimates figure fact fpga improvements exceed estimate reducing precision simplifies design compute units lower buffering requirements fpga board reduction leads significant improvement throughput due smaller hardware designs allowing parallelism shorter circuit delay allowing higher frequency figure shows performance operations gpu fpga fpga performs quite well low precision operations terms fpga better gpu lower precisions asic allows truly customized hardware implementation asic study provides insights upper bound efficiency benefits possible operations figure show improvement performance energy efficiency various asic pes relative baseline figures show going lower precision offers orders magnitude efficiency improvements summary fpga asic well suited wrpn approach wide wrpn approach requires total operations original network however lower precision operation better efficiency fpga asic hence wrpn delivers overall efficiency win related work dnns active research area reducing precision weights efficient inference pipeline well studied works like binary connect networks twn ternary quantization inq target reducing precision network weights still using activations accuracy almost always degraded quantizing weights alexnet imagenet twn loses accuracy schemes like inq quantize network weights sacrifice accuracy much applicable training networks scratch inq shows promising results precision bnn dorefa ttq target training well ttq targets weight quantization works targeting activation quantization hurt accuracy approach reduces accuracy dorefa quantizing weights activations alexnet imagenet requires layers scheme work recent work targets activations reports accuracy within baseline precision logarithmic base quantization gap narrowed within layers quantized two operand values introduces hardware inefficiency memory accesses longer dram aligned performance aspect unclear using complicated quantization schemes target training inference using simple quantization method aim reducing precision without loss accuracy best knowledge work first study deep wide networks show accuracy baseline low precision activations weights report state art accuracy wide binarized alexnet resnet still lower compute cost conclusions present wide networks wrpn scheme dnns scheme numeric precision weights activations significantly reduced without loss network accuracy result contrast many previous works find activations detrimentally impact accuracy specifically find weights activations sufficient match baseline accuracy across many networks including alexnet batchnormalized inception achieve result new quantization scheme increasing number filter maps layer compensate loss information capacity induced reducing precision motivate work observation activations contribute significantly memory footprint weight parameters using sizes common training inference furthermore reducing precision activations weights compute complexity greatly reduced baseline weights activations wrpn quantization scheme computation low precision activations weights hardware friendly making viable system deployments well training inference servers compute fabrics compare titan gpu fpga asic implementations using wrpn show scheme increases mance across overall reducing precision allows compute units lower buffering requirements provide significant improvement throughput references https abadi agarwal barham brevdo chen citro corrado davis dean devin ghemawat goodfellow harp irving isard jia jozefowicz kaiser kudlur levenberg monga moore murray olah schuster shlens steiner sutskever talwar tucker vanhoucke vasudevan vinyals warden wattenberg wicke zheng tensorflow machine learning heterogeneous systems software available bengio courville estimating propagating gradients stochastic neurons conditional computation corr courbariaux bengio binarynet training deep neural networks weights activations constrained corr courbariaux bengio david binaryconnect training deep neural networks binary weights propagations corr graham activations corr gupta agrawal gopalakrishnan narayanan deep learning limited numerical precision corr zhang ren sun deep residual learning image recognition corr ioffe szegedy batch normalization accelerating deep network training reducing internal covariate shift corr jouppi young patil patterson agrawal bajwa bates bhatia boden borchers boyle cantin chao clark coriell daley dau dean gelb vazir ghaemmaghami gottipati gulland hagmann hogberg hundt hurt ibarz jaffey jaworski kaplan khaitan koch kumar lacy laudon law leary liu lucke lundin mackean maggiore mahony miller nagarajan narayanaswami nix norrie omernick penukonda phelps ross ross salek samadiani severn sizikov snelham souter steinberg swing tan thorson tian toma tuttle vasudevan walter wang wilcox yoon performance analysis tensor processing unit arxiv apr krizhevsky sutskever hinton imagenet classification deep convolutional neural networks pereira burges bottou weinberger editors advances neural information processing systems pages curran associates liu ternary weight networks corr lin courbariaux memisevic bengio neural networks multiplications corr mellempudi kundu mudigere das kaul dubey ternary neural networks quantization arxiv may miyashita lee murmann convolutional neural networks using logarithmic data representation corr nurvitadhi venkatesh sim marr huang ong gee hock liew srivatsan moss subhaschandra boudoukh fpgas beat gpus accelerating deep neural networks proceedings international symposium fieldprogrammable gate arrays fpga pages new york usa acm rastegari ordonez redmon farhadi imagenet classification using binary convolutional neural networks corr sung shin hwang resiliency deep neural networks quantization corr szegedy ioffe vanhoucke impact residual connections learning corr szegedy liu jia sermanet reed anguelov erhan vanhoucke rabinovich going deeper convolutions corr umuroglu fraser gambardella blott leong jahre vissers finn framework fast scalable binarized neural network inference corr vanhoucke senior mao improving speed neural networks cpus deep learning unsupervised feature learning workshop nips venkatesh nurvitadhi marr accelerating deep convolutional networks using sparsity corr zagoruyko komodakis wide residual networks corr zhou yao guo chen incremental network quantization towards lossless cnns weights corr zhou zhou wen zou training low bitwidth convolutional neural networks low bitwidth gradients corr zhu han mao dally trained ternary quantization corr
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eventual shape betti tables powers ideals jul amir bagheri marc chardin huy abstract let abelian group noetherian algebra commutative ring let ideals let finitely generated show shape nonzero betti numbers ists exhibit eventual linear behavior get large introduction celebrated result homogeneous ideal noetherian standard algebra finitely generated regularity reg asymptotically linear function asymptotic linear function stabilization index also studied case polynomial ring field precise result proved maximal degree syzygy eventually linear function interest understand eventual behavior degrees minimal generators syzygy module particular interest grading given finitely generated abelian group case result regularity powers evident analogue one particular consider cox ring toric variety graded divisor class group shall actually show setting collection nonzero betti numbers ists exhibits asymptotic linear behavior get large let also point even explicit minimal free resolution powers known instance complete intersection ideal degrees syzygies exhibit trivially linear behavior throughout paper let finitely generated abelian group let algebra commutative ring hence ggraded concentrated degree work hinges relationship multigraded betti numbers graded pieces torsi ists thus examine support torsi ists get large main result case single ideal gives following mathematics subject classification key words phrases betti numbers asymptotic linearity multigraded amir bagheri marc chardin huy theorem see theorem let let finitely generated set degg let assume noetherian exist finite collection elements finite collection integers finite collection tuples elements elements linearly independent satisfying suppg csi csi max condition linear independence stated implies csi csi csi notice important fact elements set degrees generators theorem proved last section paper proof based two following observations firstly module sometimes also referred rees modification module rees algebra ists secondly finitely generated abelian group polynomial extension degg graded complex free denotes degree corresponding complex particular free resolution free resolution follows torsi viewed complex free observations allow bring problem studying support graded modules proceed making use notion initial submodules reduce case module quotient ring obtained monomial ideal result follows considering stanley decomposition quotient ring full generality proof theorem quite technical start section considering first case fact case additional assumption noetherian ring results much stronger asymptotic linearity appears clearer proofs simpler also examine support local cohomology modules ists results case finitely generated single degree stated follows theorem theorem assume noetherian exists finite set eventual shape betti tables powers ideals torsi ists exists subset torsi ists torsi ists let ring homomorphism field function dimk tora tori polynomial function dimk torsi ists polynomial theorem theorem let ideal hbi finitely generated noetherian exists subset hbi ists let ring homomorphism field function dimk tora polynomial simplest scenario standard graded polynomial ring field homogeneous maximal homogeneous ideal generated degree theorems give following result corollary exist finite sets torsi torsi function dimk torsi polynomial function dimk hmi polynomial writing paper informed whieldon recent work similar result corollary proved later learned pooja singla also proved results second chapter thesis independently also shows graded ideal dimk torsi give results describing regularity ists general setting necessarily field necessarily standard graded problem subtle requires work different approach choose restrict noetherian base ring results seemed makes presentation clear put hypothesis statements use proof amir bagheri marc chardin huy acknowledgement work done third named author visited authors institut jussieu upmc authors would like thank institut support hospitality third named author partially supported nsa grant preliminaries section collect necessary notations terminology used paper prove auxiliary results basic facts commutative algebra refer reader denotes finitely generated abelian group ggraded algebra commutative ring identity smodule abusing notation shall use denote identity abelian groups considered paper particular group understood context use definition let collection elements say linearly independent subset forms basis free submonoid definition support defined suppg remark field let minimal free resolution numbers called multigraded betti numbers dima torsi definition maps zero maps generally shall prove following lemma relating multigraded betti numbers support lemma let free resolution summand suppg torsi assume exists homz deg finitely generated exists free resolution suppg torsj notice without restrictions one general choose suppg torsi eventual shape betti tables powers ideals proof let defined exact sequence notice onto direct summand result holds furthermore resolution graded submodule torsi implies result induction prove second statement relax finite generation following condition enable make induction deg notice module satisfies condition submodules satisfies condition set suppg torsj let augmentation denote restriction prove onto first notice surjective assume surjective let image inf inf deg set min deg choose onto exists form value follows deg deg contradicting definition prove induction end assume exists graded complex non zero homology homological degree complex exact claim proved otherwise setting homology module complex ker one suppg suppg suppg ker torsi applying argument graded onto map using induction suppg suppg one obtains graded free claimed mapping onto let shall write respectively respectively property depends parameter one says property holds exists holds following lemma use amir bagheri marc chardin huy lemma let finitely generated algebra commutative ring let finitely generated either proof let ideal generated elements strictly positive degrees one degree non zero element generator spans submodule zero degrees deg product elements among finitely many generators one finitely generated result follows shall make use notion initial modules respect monomial order natural extension familiarl notion initial ideals polynomial ring let free write sei monomial form monomial monomial order total order say monomials satisfying following condition monomial seen well ordering every subset monomials minimal element refer reader details monomial orders free modules definition let free let let monomial order initial module denoted defined generated smaller terms proposition let free let let monomial order suppg suppg proof clear definition prove proposition need show clearly monomials degree elements thus elements therefore case suppose let smallest monomial degree exists thus exists element form consists monomials smaller respect since choose degree eventual shape betti tables powers ideals monomials degree implies monomials thus terms elements particular therefore contradiction hence proposition proved one techniques take collection elements certain degree complex construction gives shall call strands complex definition let complex let often denoted obtained taking elements degrees belonging boundary maps elements since complex graded boundary maps degree particular note degree necessarily forms degree section consider case every ideal generated single degree degg keep notations section let denote finitely generated abelian group following result key ingredient proof theorem let polynomial extension let finitely generated let integer assume noetherian ring exists finite subset torsi assume torsi torsi furthermore ring homomorphism field function dimk tora tori polynomial function dimk torsi polynomial proof let graded free resolution finite rank noetherian denoted free resolution necessarily minimal term fti amir bagheri marc chardin huy let module torsi subquotient module proved prove observe first torsi zero thus equal subcomplex given graded complex finitely generated finitely generated noetherian similarly torsi finitely generated noetherian proves view lemma theorem remark context point graded spectral sequence graded second term tora tori converges follows terms finitely generated modules particular one write dimk dimk dimk provides control hilbert function polynomial terms ones tora tori ready examine asymptotic linear behavior nonzero betti numbers degrees local cohomology modules ists next two theorems let equipped structure degg element canonical basis recall degg let let ists tsts ists tsts ists tsts theorem assume noetherian exists finite set torsi ists exists subset torsi ists torsi ists eventual shape betti tables powers ideals furthermore ring homomorphism field function dimk tora tori polynomial function dimk torsi ists polynomial proof let denote shifted multirees algebra module respect natural surjective map sends makes finitely generated module observe last term thus assertion follows applying theorem theorem let ideal hbi finitely generated noetherian exists subset hbi ists furthermore ring homomorphism field function dimk tora polynomial proof since taking local cohomology respects degree hbi hbr hbr let since flat extension hbr hbr finitely generated let minimal free resolution since noetherian term finite rank implies hbr finitely generated spectral sequence hbr hbr implies hbr finitely generated multigraded proves view theorem notice hbr hbr annihilated power ideal hence holds amir bagheri marc chardin huy forms arbitrary degrees section devoted proving main result full generality ideals generated arbitrary degrees start recalling notion stanley decomposition multigraded modules definition let finitely generated abelian group let polynomial ring commutative ring let finitely generated stanley decomposition finite decomposition form direct sum elements subsets could empty variables denotes generated elements form monomial polynomial ring following lemma well known standard situations lemma let finitely generated abelian group let polynomial ring let monomial ideal stanley decomposition exists proof proof follows along lines proof corollary theorem one notices monomial homogeneous element theorem let finitely generated abelian group polynomial ring commutative ring finitely generated let denote set subsets degg whose elements linearly independent exist collection pairs suppg hep hep represents free submonoid generated elements proof since finitely generated exists homogeneous surlm jective map free write bei degg represents degree generator let ker particular suppg suppg extend monomial order monomial order bei ordering since thus proposition suppg suppg therefore suppg suppg eventual shape betti tables powers ideals observe generated monomials form let monomial ideal generated monomials clearly lemma exists stanley decomposition uij tij uij homogeneous elements variables gives tij subsets could empty uij tij thus suppg written finite union form suppg suppg uij tij let degg uij suppg suppg tij prove theorem suffices show suppg tij decomposed union free submonoids form hei linearly independent subset since tij polynomial ring whose variables variables without loss generality may assume tij tij let binomial ideal generated degg degg taking quotient identifying monomials degree thus suppg suppg suppg suppg otherwise lemma stanley decomposition exists write amir bagheri marc chardin huy elements subsets could empty variables let set degrees variables follows one monomial degree implies support disjoint set linearly independent hence letting degg suppg theorem proved hej remark would nice union theorem disjoint union however true let polynomial ring deg deg hence let suppz moreover linearly independent subsets easily seen suppz written disjoint union shifted free submonoids generated vector tuple elements shall denote empty else tuple tuples denote concatenation remark simple linear algebra arguments seen tuples elements tuple elements basis element linearly independent linearly independent equivalent conditions imply one last fact direct corollary linear independence shall prove main result recall let degg theorem let finitely generated abelian group let algebra commutative ring let ideals let finitely generated set degg let assume noetherian exist finite collection elements finite collection integers finite collection tuples linearly independent satisfying eventual shape betti tables powers ideals suppg ists maxp proof use denote let consider polynomial ring degg degg denotes canonical generator natural surjective map sends makes finitely generated module let free resolution noetherian chosen finite rank make choice degree provides free resolution thus torsi moreover taking homology respects graded structure therefore homogeneous irrelevant ideal let set degrees variables observe finitely generated module noetherian case applying theorem graded module obtain finite collection elements finite collection linearly independent subsets view fixed order tuples let one linear independence elements equivalent fact elements linearly independent taking degree get suppg ists amir bagheri marc chardin huy example let complete intersection ideal three forms degrees easily sees instance hilbert burch theorem graded free follows coker following stanley decomposition ideal generated binomials deg deg kernel map therefore generated single irreducible homogeneous binomial example relation stanley decomposition instance finally gives following decomposition setting one gets suppz suppz suppz notice one simplified expression suppz eventual shape betti tables powers ideals references berlekamp regularity defect stabilization powers ideal preprint baclawski garsia combinatorial decompositions class rings adv math bruns krattenthaler uliczka stanley decompositions hilbert depth koszul complex commutative algebra chardin powers ideals cohomology stalks fibers morphisms preprint cutkosky herzog trung asymptotic behaviour regularity composito mathematica eisenbud commutative algebra view toward algebraic geometry new york eisenbud harris powers ideals fibers morphisms math res lett eisenbud ulrich stabilization regularity powers ideal preprint asymptotic linearity regularity powers ideals math res lett herzog popescu finite filtrations modules shellable multicomplexes manuscripta kodiyalam homological invariants powers ideal proceedings american mathematical society kodiyalam asymptotic behaviour regularity proceedings american mathematical society matsumura commutative ring theory cambridge singla homological combinatorial properties graded ideals genehmitge dissertation december trung wang asymptotic behavior regularity pure appl algebra whieldon stabilization betti tables preprint institut jussieu upmc boite place jussieu paris cedex france address bagheri institut jussieu upmc boite place jussieu paris cedex france http address chardin department mathematics tulane university charles new orleans usa address tha
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mar graph divisibility integral domain jason greene boynton jim coykendall abstract factorization properties domain reflected structure group divisibility main theme paper introduce point view current understanding factorization integral domains also show connectedness properties graph topological space give rise generalization atomicity introduction let integral domain field fractions group divisibility defined partially ordered additive group principal fractional ideals multiplicative group group units order isomorphic quotient group ordering factorization properties domain reflected structure group divisibility example integral domain unique factorization domain group divisibility direct sum copies equipped usual product order also true group divisibility reflects factorization properties domain indeed hard check domain valuation domain group divisibility totally ordered refer interested reader excellent survey material regarding group divisibility cohn introduced notion atomic integral domain domains every nonzero nonunit admits finite factorization irreducible elements several years believed case atomicity integral domain equivalent ascending chain condition principal ideals accp however anne grams demonstrated atomic domain need satisfy accp grams able understand subtle difference atomicity accp using group divisibility ten years later zaks added two examples atomic domain without accp however examples atomic domains without accp still relatively scarce main theme work introduce point view current understanding factorization integral domains find graphical representation group divisibility order detect various factorization properties integral domain contents paper organized follows section recall topological structure mathematics subject classification primary secondary key words phrases atomic factorization divisibility boynton coykendall naturally associated partially ordered set addition make relevant definitions needed sequel section introduce graph divisibility integral domain show graph detects standard factorization properties studied section examine connectedness properties graph divisibility using elementary topology section see connected graph divisibility gives rise generalized atomicity also provide examples order illustrate notions definitions background section make relevant definitions graph theory topology used throughout refer reader survey known results alexandrov topology definition let topological space neighborhood base called alexandrov space arbitrary intersections open sets remain open every alexandrov space set set called minimal open set containing theorem let alexandrov space collection minimal open sets basis space space path chain connected pair points exists finite set points sense alexandrov space natural topological structure induced partially ordered set indeed partially ordered set sets form constitute basis alexandrov space conversely alexandrov space define relation given precisely following result found theorem isomorphism category alexandrov spaces continuous maps category partially ordered sets order preserving set maps let alexandrov space minimal neighborhood base one construct directed acyclic graph determined space set vertices taken underlying set define edge minimal base element properly resulting construction simple parallel edges directed acyclic graph sdag make graphical representation previous constructions precise definition let partially ordered set define intervals graph divisibility write denote alexandrov topology generated minimal neighborhood base write denote directed acyclic graph whose vertices elements edges definition let nonempty set finite directed path sequence edges finite directed path also denoted directed graph said acyclic exist path finite weak path sequence ordered pairs either finite directed path also denoted directed graph said weakly connected every pair vertices exists finite weak path graph divisibility section introduce graph divisibility integral domain see graph gives picture group divisibility used detect certain factorization properties domain although graph detect divisibility relations detect enough divisibility relation clearly differentiate atomicity accp example remainder article denote set irreducible elements atoms irr set atomic elements expressible finite product atoms denoted definition let integral domain field fractions let denote multiplicative group write denote group divisibility written additively write denote positive elements write denote group nonzero principal fractional ideals partially ordered inclusion write denote nonzero nonunit principal integral ideals recall ordering given readily follows easy check exists reverse order group isomorphism given definition hand define partial ordering set consider structure associated topological space directed acyclic graph following lemma basis remainder investigations lemma define relation given partially ordered set alexandrov space neighborhood base given collection boynton coykendall directed acyclic graph directed edges irr proof never case since unit hence product atoms similarly impossible occur finally since set multiplicatively closed follows immediately definition follows irr words condition forces therefore irr needed conversely irr certainly true suffices check forcing without loss generality remaining factors units follows needed lemma hand make definition central study definition call graph divisibility might also refer subgraph graph divisibility illustrate definition easy examples example let noetherian valuation domain pid unique nonzero prime ideal elements enumerated positive integers write chosen generator unique maximal ideal graph divisibility branchless tree looks like similarly subgraph looks like let nondiscrete valuation domain sake concreteness say corresponding value group example irreducible elements hence two elements adjacent follows collection vertices corresponding edges whatsoever fact graph divisibility antimatter domain irreducible elements consists vertices topologically speaking totally disconnected given certainly true subspace subgraph recall sink directed graph vertex arrows arrows following lemma lemma let integral domain let associated graph divisibility element irreducible node sink graph divisibility well known atomic every element written sum minimal positive elements similarly satisfies accp every descending sequence elements stabilizes group divisibility graph divisibility used characterize factorization domains close section following result theorem let integral domain let associated graph divisibility atomic every non unit element exists finite path originating terminates atom satisfies accp every every path originating terminates atom bfd every every path originating terminates atom upper bound lengths paths ffd every every path originating terminates atom finitely many paths hfd every every path originating terminates atom paths length connectedness properties section consider connectedness graph divisibility examine connectedness associated alexandrov topology conclude section examples theorem let integral domain following statements equivalent exist atoms irr points belong connected component alexandrov topology finite weak path connecting graph divisibility proof suppose exist atoms irr show belong connected component suffices show nonempty end note implies similarly follows needed belong connected component exists finite set points hence choose say irr follows directed paths boynton coykendall hence weak path weak path connecting suppose distinct points alexandrov space exists finite weak path connecting say using induction suppose result true follows existence weak path irr observe either irr follows definition similar argument handles case subgroup generated irr relate number connected components order quotient group homomorphic image group divisibility immediately get following result corollary correspondences elements connected components connected components example consider classical construction wellknown irreducible elements primes polynomials form see let write power series representations define order natural number ord min follows ord show two polynomials belong connected component ord ord indeed write ord ord allowing connected exist atoms irr equation one easily checks ord ord ord ord converse suppose using fact ord ord equivalent existence atoms irr hand implies graph divisibility follows distinct connected components given set irr words weak path whenever example let indeterminates field let determine number connected components graph divisibility domain first observe integral closure rank valuation domain value group ordered lexicographically easy check every element unit multiple hard check every element irr value easy matter check connected components given terms values example consider elements written form words expressed quotient atomic elements yxj connected alexandrov space equivalently graph divisibility weakly connected one need check integral closure discrete value group ordered lexicographically hard check every element irr value given generalizations atomicity section show connected graph divisibility gives rise generalization atomicity definition let integral domain called almost atomic every exist atoms irr called quasi atomic every exists element easy see almost atomic implies quasi atomic also quasi atomic hard show every nonzero prime ideal contains irreducible element following lemma lemma given integral domain condition implies next atomic almost atomic quasi atomic every nonzero prime ideal contains irreducible element boynton coykendall proof suffices show implies suppose quasi atomic nonzero element prime ideal irr observations give example integral domain quasi atomic example example let prime ideal contains irreducible element see recall irr ord ord follows lemma quasi atomic show connection almost atomicity connected graph divisibility theorem following statements equivalent domain almost atomic connected weakly connected proof choose two points almost atomic exist atoms irr words exist atoms follows theorem pair points belong connected component follows immediately theorem since weakly connected weak path connecting element form irr theorem implies exist atoms irr words exist atoms irr using theorem results previous section led example almost atomic domain atomic example let example since connected components correspondence certainly case almost atomic theorem however case quasi atomic indeed given write translating information back get unit note unit either follows therefore follows needed example let yxj since weakly connected must case almost atomic however atomic since graph divisibility example written combination acknowledgements authors would like thank north dakota state university department mathematics continued support references anderson anderson zafrullah factorization integral domains pure appl algebra arenas alexandroff spaces acta math univ comenianae chapman smith restricted elasticity rings polynomials determined finite subsets monatsh cohn rings subrings proc cambridge philos anne grams atomic rings ascending chain condition principal ideals proc cambridge philos joe mott group divisibility applications conference commutative algebra univ kansas lawrence pages lecture notes vol springer berlin abraham zaks atomic rings without principal ideals algebra department mathematics north dakota state university fargo address coykendall department mathematics north dakota state university fargo address coykendall
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statistical mechanics map estimation general replica ansatz oct ali bereyhi ralf hermann abstract performance estimation studied considering general distortion function observation vector received linear system additive white gaussian noise analysis considers system matrix chosen large class rotationally invariant random matrices take statistical mechanical approach introducing spin glass corresponding estimator employing replica method analysis contrast earlier replica based studies analysis evaluates general replica ansatz corresponding spin glass determines asymptotic distortion estimator structure replica correlation matrix consequently replica symmetric well replica symmetry breaking ansatz steps breaking deduced given general replica ansatz generality distortion function lets derive general form decoupling principle based general replica ansatz show structure replica correlation matrix system decouples bank equivalent decoupled linear systems followed estimators structure decoupled linear system studied replica symmetry replica symmetry breaking assumptions steps symmetry breaking decoupled system found additive system noise term given sum independent gaussian random variable correlated impairment terms general decoupling property estimator leads idea replica simulator represents replica ansatz state evolution transition system described corresponding decoupled system application study investigate large compressive sensing systems considering norm minimization recovery schemes numerical investigations show replica symmetric ansatz norm recovery fails give accurate approximation mean square error compression rate grows therefore replica symmetry breaking needed order assess performance precisely index terms estimation linear vector channel decoupling principle equivalent system compressive sensing zero norm replica method statistical physics replica symmetry breaking replica simulator results manuscript presented parts ieee information theory workshop itw ieee information theory applications workshop ita work supported german research foundation deutsche forschungsgemeinschaft dfg grant ali bereyhi ralf institute digital communications idc friedrich alexander university fau erlangen bavaria germany hermann department mathematics fau erlangen bavaria germany schuba ntroduction consider additive white gaussian noise awgn system specified independent identically distributed source vector components support set measured random system matrix corrupted gaussian noise vector variance source vector estimated observation vector using map estimator given system matrix estimator maps observation vector estimated vector via estimation function defined arg minn avk utility function estimation parameter denotes euclidean norm assumed minimum degenerate least almost order analyze performance system limit one considers general distortion function choices distortion function determines distance source estimated vector however general takes different choices asymptotic distortion lim expresses performance regarding distortion function performance analysis estimator requires explicitly computed substituted distortion function task however trivial many choices utility function source support becomes unfeasible grows large basic analytic tools fail take statistical mechanical approach investigate performance studying macroscopic parameters corresponding spin glass approach enables use replica method developed context statistical mechanics corresponding spin glass consider thermodynamic system consists particles microscopic parameter vector collecting microscopic parameters presents microscopic state system called microstate main goal statistical mechanics excavate macroscopic parameters system energy entropy analysis microstate thermodynamic limit due large dimension system statistical mechanics proposes stochastic approach microstate supposed randomly distributed support system hamiltonian due distribution assigns realization microstate energy level log denotes system entropy free energy thermodynamic system inverse temperature defined second law thermodynamics states microstate thermal equilibrium takes distribution free energy meets minimum thus microstate distribution thermal equilibrium reads normalization factor referred partition function superscript indicates distribution dependence inverse temperature distribution known distribution covers many distributions specifying correspondingly substituting distribution free energy thermal equilibrium inverse temperature reads log average energy entropy system thermal equilibrium determined taking expectation distribution log calculated terms free energy via spin glasses hamiltonian assigns energy levels randomly using randomizer resulting random interaction coefficients fact realization specifies thermodynamic system represented deterministic hamiltonian statistical mechanics known quenched randomness microstate annealed random variable analysis spin glasses takes similar steps considering given realization randomizer therefore system converges thermal equilibrium inverse temperature microstate conditional distribution given distribution specified consequently free energy reads log partition function respect hamiltonian free energy well macroscopic parameters system random however physical intuition behind analyses suggests random macroscopic parameters converge deterministic values thermodynamic limit property known self averaging property rigorously justified particular classes hamiltonians nevertheless cases mathematical proof still lacking property supposed hold analysis according self averaging property free energy spin glasses converges expectation thermodynamic limit mentioned earlier map estimator investigated using corresponding spin glass see consider spin glass whose microstate taken whose hamiltonian defined system matrix observation vector considered randomizers spin glass case given conditional distribution microstate given taking limit using laplace method integration zero temperature distribution assumption minimizer unique reduces arg minn denotes indicator function defined indicates microstate spin glass converges estimated vector map estimator zero temperature limit invoking connection study corresponding spin glass instead map estimator represent distortion system regarding general distortion function macroscopic parameter spin glass consequently replica method developed statistical mechanics employed determine defined macroscopic parameter corresponding spin glass replica method generally nonrigorous effective method developed physics literature study spin glasses although method lacks rigorous mathematical proof particular parts widely accepted analysis tool utilized investigate variety problems applied mathematics information processing coding use replica method studying multiuser estimators goes back tanaka determined asymptotic spectral efficiency mpm estimators employing replica method study demonstrated interesting properties multiuser estimators consequently statistical mechanical approach received attention context multiuser systems approach employed literature study multiple estimation problems large linear systems method also utilized analyze asymptotic properties mimo systems considering approach similar regarding multiuser estimators earlier studies mainly considered cases entries source vector binary gaussian random variables results later extended general source distribution statistical mechanical approach employed address mathematically similar problems vector precoding compressive sensing analysis superposition codes name examples despite fact replica method lacks mathematical rigor body work shown validity several results literature tanaka formula using alternative rigorous approaches later discuss rigorous results details invoking literature compressive sensing decoupling principle considering map estimator defined entries estimated vector correlated general since system matrix couples entries linearly performs several nonlinear operations performance analysis marginal joint distribution two corresponding entries interest clarify point consider case linear estimator employed instead denote matrices respectively vectors therefore written gjt gjt gjt gjt right hand side interpreted linear estimation system indexed symbol corrupted additive impairment given last two summands right hand side impairment term necessarily independent gaussian classes matrix ensembles set assumptions shown dependency derived systems index vanishes distribution impairment terms converges gaussian distribution limit result one assume system described followed linear estimator set additive scalar systems gaussian noise employed parallel words vector system considered decouple set similar scalar systems relates input entry corresponding estimated one asymptotic property estimator referred decoupling property investigated performance analysis decoupling property first studied linear estimators tse hanly noticed property determining multiuser efficiency several linear multiuser estimators limit showed system matrix effect impairment similar effect modified gaussian noise dimension tends infinity asymptotic property investigated studying asymptotics different linear receivers distributions independent work shamai also studied linear minimum mean square error mmse estimator showed conditional output distribution asymptotically gaussian authors studied asymptotics impairment term family linear estimators employed proved converges distribution gaussian random variable latter result extended larger class linear estimators regarding linear estimators main analytical tool random matrix theory fact invoking properties large random matrices central limit theorem decoupling property rigorously proved tools however fail nonlinear estimators source symbol impairment term decouple linearly due nonlinear operations estimators gerstacker employed replica method studied capacity loss due separation detection decoding authors showed additive decoupling spectral efficiency reported gaussian inputs also holds binary inputs result conjectured regardless input distribution linearity spectral efficiency always decouples additive form guo justified conjecture family nonlinear mmse estimators showed system matrix estimator decouples bank mmse estimators replica symmetry ansatz rangan studied asymptotic performance class map estimators using standard large deviation arguments authors represented map estimator limit indexed mmse estimators sequence consequently determined estimator asymptotics employing results justified decoupling property map estimators ansatz regarding decoupling property map estimators still two main issues need investigations cases system matrix analysis estimator replica symmetry breaking rsb first issue partially addressed assumption authors studied asymptotics map estimator employed recover support source vector observations received noisy sparse random measurements considered model sparse gaussian source first randomly measured square matrix measurements sparsely sampled diagonal matrix whose entries bernoulli random variables setup information rate support recovery error rate investigated considering measuring matrix belong larger set matrix ensembles results moreover could address decoupling property considered setting although class system matrices broadened considered complete generalization property presented since restricted cases sparse gaussian source loading factors less one also tried investigate first issue similar formulation compressive sensing fact authors considered linear sensing model class rotationally invariant random ansatz determined asymptotic mean square error mse recovery schemes equivalently represented formulation results however address asymptotic marginal joint distribution emphasis mse regarding second issue map estimator yet investigated rsb literature nevertheless necessity investigations mentioned various similar settings literature see example performances code division multiple access cdma detectors investigated studying rsb impact symmetry breaking onto results low noise scenarios discussed authors means entries source vector form gaussian bernoulli random variables respectively class rotationally invariant random matrices precisely defined later throughout problem formulation studied performance vector precoding rsb showed analysis yields significantly loose bound true performance replica ansatz rsb however shown lead tighter bound consistent rigorous performance bound available literature similar observation recently reported problem precoding replica analyses compressive sensing moreover discussed necessity investigating performance minimization recovery schemes rsb choices compressive sensing map estimation source vector set noisy linear observations arises several applications mimo sampling systems address one consider large compressive sensing systems employ asymptotic results analyze performance context compressive sensing represents noisy sampling system source vector sampled linearly via corrupted additive noise case source vector exactly recovered observation vector number observations large source length sampling matrix full rank number observations reduces possible answers exact reconstruction problem may increase regarding source support therefore recovered source vector observation vector necessarily unique case one needs enforce extra constraints properties source vector order recover uniquely among possible solutions compressive sensing source vector supposed sparse certain fraction entries zero property source imposes extra constraint solution allows exact recovery cases fact case one find solution denotes norm defined source sparsity latter problem unique solution even searching solution optimally however problem factor defined fraction entries depending therefore intractable main goal compressive sensing study feasible reconstruction schemes derive tight bounds sufficient compression rate exact source recovery via schemes noisy sampling systems exact recovery possible particular choices considering either cases exact recovery possible choices nevertheless source vector exactly recovered noisy observations recovery approaches sensing systems need take impact noise account classical strategy case find vector recovery distortion small consequently recovery scheme noisy sensing system based norm given arg minn map estimator defined straightforward show zero noise variance reduces optimal recovery scheme similar case scheme results optimization problem therefore computationally infeasible alternatively computationally feasible schemes introduced replacing norm cost function norm resulting recovery scheme known lasso basis pursuit denoising based formulations several iterative greedy algorithms introduced recovery taking account sparsity pattern properties sampling matrix main body work noisy compressive sensing investigates compression rate recovery distortion large compressive sensing systems common consider random sensing matrix since matrices properties restricted isometry property shown hold high probability case performance reconstruction schemes analyzed determining considered performance metric mse probability exact recovery noisy case respectively given realization sensing matrix average performance calculated taking expectation matrix distribution comparing one utilize map formulation illustrated beginning section study largesystem performance several reconstruction schemes similarity considered series papers therefore earlier replica results employed study compressive sensing systems extension analyses context multiuser estimation disadvantage assumed sampling settings limited setups consistent estimation problems literature compressive sensing systems however might require wider set assumptions thus large class settings could addressed earlier investigations result body work deviated approach applied replica method directly compressive sensing problem see example although general replica method considered mathematically several recent studies justified validity replica results context compressive sensing using alternative tools analysis widely investigated approach based asymptotic analysis approximate message passing amp algorithms context compressive sensing amp algorithms initially introduced address iteratively convex reconstruction schemes based norm minimization lasso basis pursuit low computational complexity proposed approach later employed extend algorithm large variety estimation problems including map mmse estimation see example primary numerical investigations amp demonstrated large sensing systems rate tradeoff iterative algorithms well compression tradeoff noisy cases derived state evolution recovers asymptotics convex reconstruction schemes observation rigorously justified sensing matrices using conditioning technique developed study recently extended cases rotationally invariant system matrices investigations moreover showed using amp algorithms spatially coupled measurements fundamental limits required compression rate achieved asymptotic regime methodology proposed amp algorithms state evolution also provided justification validity several earlier studies based replica method fact results given replica method recovered state evolution corresponding amp algorithms invoking approach along analytical tools recent study approved validity replica prediction asymptotic mmse mutual information linear system gaussian measurements similar results demonstrated using different approach contribution outline paper determine asymptotic distortion general distortion function cases map estimator employed estimate source vector observation given represent asymptotic distortion macroscopic parameter corresponding spin glass study spin glass via replica method general replica ansatz given arbitrary replica correlation matrix special cases studied considering rsb assumptions asymptotic distortion determined rotationally invariant random system matrices invoking results asymptotics spherical integrals using asymptotic results derive general form decoupling principle restricting distortion function special form employing moment method show system estimated decouples bank similar noisy systems followed map estimators result holds replica correlation matrix however structure decoupled system depends supposed structure correlation matrix rsb assumption steps breaking brsb noisy system given form input term added impairment term impairment term moreover expressed sum independent gaussian random variable correlated interference terms reducing assumption result reduces formerly studied decoupling principle map estimators rotationally invariant system matrix ensembles fact investigations collect previous results regarding decoupling principle addition new set setups single umbrella given general form decoupling principle precisely extend scope decoupling principle systems whose measuring matrix belongs class rotationally invariant random matrices replica ansatz general replica correlations include rsb address particular application study performance compressive sensing system minimization recovery schemes address linear reconstruction well lasso norm scheme considering sparse gaussian finite alphabet sources general setting allows investigate asymptotic performance respect different metrics multiple sensing matrices random projector numerical investigations show ansatz becomes unstable regimes system parameters predicts performance minimization recovery loosely within large range compression rates observation agrees earlier discussions necessity rsb investigations reported therefore study performance rsb discuss impact symmetry breaking throughout numerical investigations demonstrated performance enhancement obtained via random orthogonal measurements reported also holds sparse finite alphabet sources sensing via random projector matrices results phase transitions higher rates rest manuscript organized follows section problem formulated illustrate statistical mechanical approach section iii explain briefly replica method general replica ansatz well general decoupling principle given section ansatz rsb assumptions expressed sections based brsb decoupled system propose idea replica simulator vii describe given terms corresponding decoupled systems address application study consider large compressive sensing systems section viii discuss several examples numerical investigations examples given section finally conclude manuscript section notations throughout manuscript represent scalars vectors matrices lower bold lower bold upper case letters respectively set real numbers denoted indicate transposed hermitian identity matrix matrix entries equal random variable represents either probability mass function pmf probability density function pdf represents cumulative distribution function cdf denote expectation expectation random variables involved given expression real numbers superscript represent set integer indicates corresponding subset numbers sake compactness set integers abbreviated gaussian pdf denoted gaussian averaging expressed moreover many cases drop set sum minimization integral taken whenever needed consider entries discrete random variables namely support discrete results paper however full generality hold also continuous distributions announcements results manuscript presented ieee information theory workshop information theory applications workshop even though results mathematical flavor stress investigating rigor available tools replica method rather employ deriving formulas used different problems roblem ormulation consider linear system described let system satisfy following properties random vector entry distributed randomly generated rotationally invariant random ensembles random matrix said rotationally invariant gramian eigendecomposition udut orthogonal haar distributed matrix diagonal matrix given denote empirical cdf eigenvalues cumulative density states fnj define fnj denotes ith diagonal entry assume fnj converges deterministic cdf real gaussian random vector variance entry number observations deterministic function transmission length lim sake compactness drop explicit dependence independent source vector reconstructed observation vector system matrix known estimator thus given source vector recovered given scalar estimation parameter cost function referred utility function utility function supposed decouple means takes arguments different lengths positive integer order use estimator one needs guarantee uniqueness estimation output therefore impose following constraint problem given observation vector objective function unique minimizer support map estimator map estimator considered optimal estimator sense minimizes reconstruction error probability postulating source prior distribution proportional noise variance clarify argument assume finite case define reconstruction error probability entries discrete random variables estimator order minimize chosen posterior distribution input support maximized arg max arg max arg min log log comes independency prior distribution source vector let estimator postulate noise variance prior function substituting estimator reduces defined estimator models several particular reconstruction schemes compressive sensing address schemes later section viii remark case entries continuous random variables argument needs modifications fact case reconstruction error probability defined always one therefore taken measure error case map estimator considered maximize posterior pdf postulating noise variance asymptotic distortion conditional distribution many applications distortion given terms average mse others average symbol error rate considered fact former takes norm distortion function latter considers norm distortion function however form general study asymptotic performance considering general distortion function determines imperfection level estimation thus consider distortion function term average distortion usually refers case averaging weights uniform means tuple entries weighted equally distortion averaged entries tuples however possible average distortion set weights following define average distortion class binary weights includes case uniform averaging well let distortion function defined define index set subset let grow definition asymptotic distortion consider vectors defined lim average distortion index set regarding distortion function defined assuming limit exists denote lim refer asymptotic distortion limit index set definition moreover utilized investigate asymptotic conditional distribution estimator plays key role studying decoupling principle convenience define asymptotic conditional distribution map estimator follows definition asymptotic conditional distribution consider source vector passed linear system let map estimation given given take index denote conditional distribution given mass point reads pxj marginal joint distribution mass point limit define asymptotic conditional distribution given lim study asymptotic distortion limit desired index set distortion function defining macroscopic parameter corresponding spin glass employing replica method evaluate using result asymptotic distortion determine asymptotic conditional distribution investigate decoupling property estimator iii tatistical echanical pproach hamiltonian introduces spin glass corresponds map estimator spin glass zero temperature describes asymptotics map estimator convenience formally define corresponding spin glass follows definition corresponding spin glass consider integer corresponding spin glass microstate quenched randomizers given rotationally invariant random matrix constructed source vector realization hamiltonian reads corresponding spin glass inverse temperature following properties directly concluded conditional distribution microstate reads log partition function normalized free energy given expectation taken quenched randomizers entropy spin glass determined regarding map estimator one represent asymptotic distortion macroscopic parameter corresponding spin glass precisely using definition asymptotic distortion reads lim lim indicates expectation respect conditional distribution defined fact introducing macroscopic parameter temperature lim asymptotic distortion interpreted macroscopic parameter zero temperature take strategy statistical mechanics modifies partition function general function however drop arguments sake compactness case expectation taken lim lim log macroscopic parameter defined random namely depending quenched randomizer discussed section self averaging property macroscopic parameter supposed converge limit expected value quenched random variables corresponding spin glass defined definition self averaging property rigorously justified proof requires mathematical investigations however widely accepted literature assume property holds least setting specified therefore state following assumption assumption self averaging property consider corresponding spin glass defined definition almost realizations quenched randomizers using self averaging property system asymptotic distortion written lim lim lim log evaluation well normalized free energy defined confronts nontrivial problem determining logarithmic expectation task bypassed using riesz equality given random variable states log lim log using riesz equality asymptotic distortion finally written lim lim lim lim log expresses asymptotic distortion terms moments modified partition function however yet simplify problem fact one faces two main difficulties calculating right hand side moment needs evaluated real values least within right neighborhood limits need taken order stated replica method plays role replica method suggests determine moment arbitrary integer analytic function assume two following statements moment function analytically continues set integer numbers onto real axis least right neighborhood means analytic expression found integer moments directly extends real moments assumption expression determined integer moments replaced limit respect taken assumption main part replica method lacks rigor known replica continuity limits respect exchange refer assumption limit exchange order employ replica method need suppose validity two statements therefore state following assumption assumption replica continuity limit exchange spin glass defined definition replica continuity limit exchange assumptions hold means assumption calculation asymptotic distortion reduces evaluation integer moments modified partition function written refer replicas define set replicas taking expectation respect observed large limit expectation respect dropped due law large numbers inserting final expression taking limits asymptotic distortion determined proposition given esults proposition states general replica ansatz term general emphasized order indicate assumption needs considered derivation using proposition along results classical moment problem general form decoupling principle justified map estimator stating general replica ansatz let define probability distribution considering random variable corresponding stieltjes transform upper complex half plane defined denoting inverse respect composition given definition also extended matrix arguments assume matrix decomposition diagonal matrix whose nonzero entries represent eigenvalues diag matrix eigenvectors matrix defined diag general replica ansatz proposition expresses macroscopic parameters corresponding spin glass including asymptotic distortion terms parameters new spin glass finite dimension important note new spin glass referred spin glass replicas different corresponding spin glass defined definition fact spin glass replicas projection corresponding spin glass reduced support indicating number replicas macroscopic parameters spin glass replicas therefore readily determined definition spin glass replicas finite integer spin glass replicas microstate quenched randomizer defined follows entries equal random variable distributed source distribution given realization hamiltonian reads trj corresponding defined replica correlation matrix defined expectation taken thermal equilibrium thermal equilibrium microstate distributed according distribution denotes partition function system defined normalized free energy system inverse temperature given log expectation taken respect average energy entropy inverse temperature found using system thermal equilibrium replicas average distortion regarding distortion function inverse temperature expectation taken respect considering definition evaluation system parameters replicas average distortion normalized free energy needs replica correlation matrix explicitly calculated first fact describes fixed point equation terms one writes expectation using conditional distribution solution substituted distribution parameters system calculated via fixed point equation however may several solutions thus result multiple outputs system nevertheless express asymptotic distortion map estimator terms single output spin glass replicas limits exist free energy minimized proposition let linear system fulfill constraints section suppose assumptions hold consider spin glass replicas defined definition finite integer free energy corresponding spin glass defined definition given tqrj trj lim replica correlation matrix satisfying asymptotic distortion map estimator regarding distortion function determined lim lim case multiple solutions available replica correlation matrix replica average distortion evaluated via correlation matrix minimizes free energy zero temperature proof proof given appendix however explain briefly strategy following starting expectation respect noise term straightforwardly taken using results expectation respect system matrix taken discussed appendix considering following variable exchange determined terms replica correlation matrix finally employing law large numbers mth moment partition function given integral taken tends zero given moreover eni denotes probability weight vectors replicas correlation matrix explicitly determined appendix reads mrj diagonal matrix denoting trace operator respect term eni probability measure satisfies large deviations property using results large deviations integral large values written integrand saddle point multiplied bounded coefficient results denoting asymptotic equivalence exponential consequently substituting exchanging limits respect suggested assumption asymptotic distortion found proposition determines saddle point integrand function free energy determined substituting finally noting fact free energy minimized equilibrium proof concluded proposition introduces feasible way determine asymptotics map estimator validity depends assumptions restriction pursue analysis one needs solve fixed point equation replica correlation matrix calculate parameters spin glass replicas explicitly direct approach find however raises complexity analyticity issues fact finding saddle point searching set possible choices hard task moreover several solutions may use since lead analytic thus continued analytically real axis via assumption overcome issues approach restrict search parameterized set replica correlation matrices find solution within set clearly asymptotics found via approach may fail several solutions missed restriction result case becomes trustable extending restricted set replica correlation matrices several procedures restrictions considered procedures introduced literature roughly divided rsb schemes former considers replicas interact symmetrically latter recursively breaks symmetry systematic manner fact scheme first introduced due symmetric properties observed analysis spin glasses properties however force correlation matrix symmetric structure later several examples found showing leads wrong conclusions examples rsb scheme considered extension symmetric structure correlation matrix larger set consider rsb schemes manuscript however pursuing study let first investigate general decoupling property estimator concluded proposition asymptotically equivalent exponential scale set lim log general decoupling property map estimator regardless restriction general ansatz leads decoupling property map estimator fact using proposition shown almost tuple entries marginal joint distribution converges deterministic joint distribution depend entries index explicit term joint distribution however depends assumptions imposed correlation matrix proposition general decoupling principle let linear system fulfill constraints section suppose assumptions hold consider spin glass replicas defined definition replica correlation matrix asymptotic conditional distribution map estimator defined definition almost sure independent namely conditional distribution mass point consequently marginal joint distribution entries described joint distribution system specified conditional distribution input explicit form depends proof proof follows proposition moment method classical moment problem know joint probability tuple random variables uniquely specified sequence integer joint moments joint moments uniformly bounded precisely defining moment tuple sequence uniquely mapped probability distribution uniformly bounded integers consequently one infer two tuples random variables sequences joint moments identical distribution determine joint moment input output entries consider distortion function proposition evaluate asymptotic distortion limit right neighborhood zero moment determined taking limit substituting distortion function index set proposition clear asymptotic distortion independent therefore limit respect exists independent well noting evaluated moments uniformly bounded inferred asymptotic joint distribution uniquely specified depend index finally using fact source vector distribution entry independent index conclude asymptotic conditional distribution independent exact expression found determining solution fixed point equation determining joint moments proposition generalized form decoupling principle map estimators studied fact proposition indicates vector system followed map estimator always decouples bank identical systems regardless restriction replica correlation matrix consistency test one restricts replica correlation matrix special form asymptotics determined assumed structure necessarily approximate true asymptotics accurately several methods introduced literature check consistency solution primary method based calculating entropy corresponding spin glass zero temperature temperature tends zero distribution microstate tends indicator function point estimated vector consequently entropy corresponding spin glass converges one consistency check therefore zero temperature entropy given solution several works invoked consistency test showed settings ansatz fails give tight bound exact solution zero temperature entropy determined ansatz converge zero see example observation illustrates invalidity assumption hints rsb giving better bounds true solution inspired aforementioned results evaluate zero temperature entropy corresponding spin glass measure consistency order determine zero temperature entropy invoke determines entropy terms free energy inverse temperature considering free energy corresponding spin glass given proposition entropy reads tqrj trj lim denotes normalized entropy spin glass replicas determines entropy thermodynamic system lim therefore zero temperature entropy given lim lim tqrj trj obviously depends structure replica correlation matrix authors determined zero temperature entropy spin glass corresponds vector precoding problem considering assumptions observed takes form assumptions conjectured zero temperature entropy similar form general rsb structure regardless number breaking using later show conjecture true note entropy spin glass temperature denotes either conditional entropy discrete supports conditional differential entropy continuous supports random vector distributed conditioned distribution fact case dependence zero temperature entropy correlation matrix completely described via scalar corresponds diagonal entries correlation matrix see assumption illustrations nsatz ecoupling rinciple elementary structure imposed replica correlation matrix one assumes correlation matrix symmetric form means replicas spin glass defined definition invariant permutation indices using definition replica correlation matrix given consequently reads assumption structure considering spin glass replicas defined definition replica correlation matrix form real scalars considering assumption supposes substituting definition spin glass replicas specified scalars scalars moreover related via set saddle point equations obtained finally using proposition asymptotics system found proposition ansatz let linear system fulfill constraints section suppose assumptions well assumption hold let arg min defined real real variable asymptotic distortion defined reads satisfy minimize zero temperature free energy given frs gmap fig decoupled system ansatz defined proof see appendix asymptotic distortion ansatz equivalent average distortion scalar awgn channel followed estimator shown fig block diagram estimator gmap maximizes posterior probability postulated scalar awgn channel refer estimator decoupled map estimator define follows definition decoupled map estimator decoupled map estimator gmap estimation parameter utility function defined gmap arg max arg min denotes decoupled posterior distribution postulated estimator reads real constant using definition decoupled map estimator decoupled system defined next definition decoupled system define decoupled system consistent block diagram fig source symbol distributed support gaussian random variable independent estimated observation gmap gmap decoupled map estimator estimation parameter utility function defined definition defined proposition using proposition equivalency asymptotic distortion extended asymptotic conditional distribution well fact considering decoupling principle definition describes structure decoupled system assumption proposition decoupling principle let linear system fulfill constraints section estimated via map estimator consider decoupled system defined definition suppose assumptions hold tuple converges distribution almost sure proof using proposition two different indices mass point therefore index lim consequently asymptotic moment assumption determined letting distortion function ansatz determining asymptotic distortion substituting proposition asymptotic joint moment reads mjk defined considering definition assuming describes moment well noting mjk uniformly bounded pair integers concluded asymptotic joint distribution joint distribution equivalent proposition gives general form decoupling principles investigated fact restricting system matrix source distribution one recover formerly studied decoupling principles zero temperature basic measure ansatz consistency evaluate zero temperature entropy assumption following discussion section substituting taking steps appendix zero temperature entropy determined lim function defined taking derivative first limit finally reads later see zero temperature entropy takes form rsb assumptions rsb rsb ecoupling rinciple parisi introduced breaking scheme broaden restricted set correlation matrices scheme recursively extends set matrices larger sets breaking scheme employed broaden structure correlation matrices therefore obtained structure identified broken rsb structure key feature parisi breaking scheme starting structure new structure breaking reduced structure breaking thus set fixed point solutions found assuming broken structure includes solutions previous structure well definition parisi breaking scheme let multiple integer represent correlation matrix parisi breaking scheme constructs new correlation matrix real scalar choosing correlation matrix definition matrix finds rsb structure one step breaking steps breaking increased recursively inserting next breaking scheme determining new correlation matrix repeating procedure start correlation matrix extend result higher rsb steps breaking assumption structure considering spin glass replicas defined definition replica correlation matrix form real scalars regarding parisi breaking scheme assumption considers letting structure parameters structure reduces setting therefore set correlation matrices contains one considered assumption proposition ansatz let linear system fulfill constraints section suppose assumptions well assumption hold let arg min defined real real variables asymptotic distortion reads defined scalars satisfy solution fixed point equation log moreover minimize zero temperature free energy frsb reads log frsb defined proof see appendix gmap fig decoupled scalar system ansatz similar approach ansatz employ proposition introduce decoupled system describes statistical behavior map estimator entries assumption decoupled system differs within new impairment term correlated source noise symbols joint distribution impairment term intuitively plays role correction factor compensates possible inaccuracy ansatz decoupled map estimator however follows structure definition decoupled system fig defines decoupled system source symbol distributed support gaussian random variable random variable correlated independent defined proposition estimated observation gmap gmap decoupled map estimator estimation parameter utility function defined definition defined proposition proposition decoupling principle let linear system fulfill constraints section estimated via map estimator consider decoupled system defined definition suppose assumptions hold tuple converges distribution proof proof takes exactly steps decoupling principle using proposition decoupled system general provides accurate approximation estimator asymptotics searching larger set solutions include ansatz investigate latter statement consider case case structure reduces setting proposition becomes zero consequently moreover fixed point equations hold choice scalars couple set equations ansatz zero temperature free energy system furthermore reduces form assumption denoting parameters ansatz qrs concluded qrs solution fixed point equations stable solution fixed point exists solution however give necessarily ansatz stable solution fixed point equations minimum free energy may occur point investigate impact replica breaking examples later numerical results parisi breaking scheme employed extend structure correlation matrix rsb structure steps breaking recursively repeating scheme fact one start structured employ breaking scheme steps determine correlation matrix case replica correlation matrix referred brsb correlation matrix assumption brsb structure considering spin glass replicas defined definition replica correlation matrix form scalars sequences reals satisfies following constraint considering correlation matrix proposition form indicated assumption previous extended general ansatz reduce well ansatz proposition expresses replica ansatz brsb assumption proposition brsb ansatz let linear system fulfill constraints section suppose assumptions well assumption hold define sequence let arg min defined real scalar sequences sequence real variables asymptotic distortion defined reads recursively determined scalar sequences satisfy sequence given arg min log dzb function defined defined pzb gmap fig decoupled scalar system brsb ansatz denoting case multiple solutions ansatz chosen free energy zero temperature frsb log dzb minimized proof see appendix one simply observe proposition reduces propositions letting respectively ansatz moreover extends corresponding decoupled system estimator considering general decoupling principle investigated proposition taking steps proposition decoupled brsb system found represents extended version system additive impairment taps fact considering impairment terms intuitively play role correction factors brsb ansatz takes steps correction account decoupled map estimator moreover remains unchanged definition brsb decoupled system define brsb decoupled system system illustrated fig source symbol distributed support gaussian random variable random variable correlated independent defined proposition estimated observation gmap gmap decoupled map estimator estimation parameter utility function defined definition proposition proposition brsb decoupling principle let linear system fulfill constraints section estimated via map estimator consider brsb decoupled system defined definition suppose assumptions hold tuple converges distribution proof using proposition takes steps proposition rsb zero temperature appendix shown brsb assumption replica correlation matrix free energy corresponding spin glass inverse temperature reads denotes normalized free energy spin glass replicas defined limit function defined following discussion section entropy zero temperature reads hrsb lim reduces hrsb justifies conjecture states zero temperature entropy number breaking steps including case similar form depends scalar fact hamiltonian reduces one considered vector precoding considering deterministic vector zeros substituting zero temperature entropy reduces one determined within factor factor comes difference prior assumption support vii eplica imulator haracterization via ingle ser epresentation general brsb decoupling principle determines equivalent system describes inputoutput statistics map estimator brsb ansatz order specify exact parameters decoupled system set fixed point equations needs solved explicitly section propose alternative approach describes ansatz terms corresponding decoupled system inputoutput statistics define exact form decoupled system steady state transition system named replica simulator proposed approach enables investigate properties rsb studying replica simulator order clarify idea replica simulator let define set statistics regarding brsb decoupled system definition consider system consistent block diagram fig denote joint distribution source impairment terms system correlation defined mse denoted mse invoking definition brsb ansatz completely represented terms statistics decoupled system fact means definition fixed point equations expressed mse moreover factor given authors considered complex vector reduces indicating decoupled posterior distribution defined definition second term right hand side extended form likelihood ratio defining reads determined recursively defined fixed point therefore rewritten accordingly alternative representation brsb ansatz leads new interpretation fact one define transition system vector replica parameters denotes state decoupled system defines transition rule refer transition system replica simulator define formally following definition replica simulator let integer consider vector entries satisfying corresponding constraints proposition denote support transition rule trb maps prior state posterior state following way trb realizes brsb decoupled system considering entries replica parameters constructs entries determining new set replica parameters statistics decoupled system using equivalent representation fixed point equations given replica simulator breaking order defined transition system simr trb structure replica simulator illustrated fig replica simulator breaking order sequence states considered process state called steady state setting results regarding proposition trb brsb decoupling mse fig replica simulator breaking order brsb ansatz fact steady state replica simulator minimizes free energy function conclusion also extends case set considering definition well discussions brsb ansatz numerically investigated using methods developed literature transition systems approach may reduce complexity numerical analysis however impact computational fact assuming one realizes brsb decoupled system desired state vector denoted via methods realization monte carlo simulation brsb ansatz found means iterative algorithm designed find steady state transition system latter statement clarified scheme scheme analysis via replica simulator begin set replica simulator breaking order simr trb corresponds decoupled system else trb corresponds brsb decoupled system end initiate initial state evaluate break else consider mapping rule return end output end scheme trb step realized via different methods one may determine distribution system analytically simulate system generating impairment source samples numerically via monte carlo technique another degree freedom step different mapping rules conjecture cases brsb decoupled system represents asymptotic system validity conjecture reduce computation complexity different convergence speeds employed algorithms designed based scheme computational complexity depends realization method convergence speed mainly restricted given mapping rule replica simulator introduces systematic approach investigating replica based decoupling principle moreover gives intuition impact symmetry breaking clarify latter statement let consider example example ansatz let thus fixed point equations read mse equations assumption moreover given mse determined fixed point equation dkl dkl denote mutual information divergence respectively assuming system matrix setting independent right hand side tends zero therefore solutions concluded consequently becomes ineffective fixed point equations reduce latter observation interpreted terms state evolution replica simulator precisely assume initial state replica simulator breaking order one chosen corresponding decoupled system sufficiently correlated source noise symbols case assuming mapping rule converging correlation iteration scheme reduces thus steady state becomes independent discussion extended replica simulators larger breaking orders moreover properties rsb could studied using methods developed literature transition leave investigations possible future work concept replica simulator may clarify connection rsb message passing based algorithms investigations however skipped considered possible future work viii arge ompressive ensing ystems considering setting represented section large compressive sensing system studied results restricting source cdf form large limit source vector distributed entries equal zero remaining entries distributed case vector thus considered represent large compressive sensing system sensing matrix considering prior different recovery schemes investigated restricting prior setup system correspondingly section study asymptotics several recovery schemes using brsb decoupling principle cases continuous finite alphabet sources continuous sources assuming describes continuous random variable multiplied random variable case varying utility function different reconstruction schemes considered address linear lasso norm recovery schemes results however employed investigate general norm recovery scheme example linear recovery scheme map estimation reduced linear recovery scheme utility function set fact case map estimator postulates prior distribution gaussian performs considerably inefficient source sparse using brsb decoupling principle conclude limit source entry estimated entry converge probability sparse random variable distributed estimated symbol gmap decoupled system reduces gmap given scalars defined proposition letting result reduces decoupling principle reported literature see however result holds wider set sensing matrices source distributions example lasso recovery scheme study lasso recovery scheme set regarding brsb decoupling principle prior distribution decoupled system input postulated laplacian double exponential unit variance postulation results better performance recovery scheme many cases since laplacian pdf could realistic approximation sparse source distribution consequently decoupled system output found gmap gmap sgn denoted null window function window width defined sgn sign indicator decoupled estimator often referred operator recovers earlier results setting sensing matrix example norm recovery scheme norm recovery scheme considers perform significantly better latter schemes sparsity increases case prior distribution brsb decoupled system input found limit finite values considered sparse distribution symbols occur uniformly prior explains better performance norm recovery scheme compared linear lasso schemes case output decoupled system reads gmap gmap denoted null window function gmap operator recovers analysis wider class settings examples also studied earlier replica based studies given results analytically derived expressions considering ansatz properly substituting corresponding address results two important cases reported literature following special case authors addressed case sparse source sampled sensing matrix matrix entries random variables variance vanishing proportional asymptotic performance estimator addressed linear lasso norm recovery schemes employed using map decoupling principle results reported derived directly setting proposition special case results extended larger set sensing matrices prediction asymptotic mse determined sparse gaussian sources given results recovered considering distortion function source distribution representing gaussian cdf finite alphabet sources result employed study sampling problem finite alphabet sources considering symbol occurs probability outcomes distributed due consequently source distribution reads interpreted multiplication discrete random variable distributed random variable sake brevity denote sorted version symbols support notational compactness define respectively similar continuous case different choices utility function address different types reconstruction schemes investigate sequel example linear recovery scheme consider case finite alphabet source reconstructed via linear recovery scheme introduced example using brsb decoupling principle source estimated symbols converge random variables distributed defined found gmap gmap scalar indicates observation symbol equivalent decoupled system boundary point defined example lasso recovery scheme replacing reconstruction scheme example lasso estimator brsb decoupled system form gmap boundary point reads example norm recovery scheme finite alphabet sources norm recovery scheme optimal terms symbol error rate since realizes sparse uniform distribution fact case symbols source uniformly distributed norm utility function exactly models source true prior therefore considered optimal scheme norm recovery scheme brsb decoupled system reduces gmap boundary point defined umerical esults arge ompressive ensing ystems section numerically investigate examples large compressive sensing systems known setups purpose simulate decoupled systems setting source distribution sensing matrix specific form determine expected distortion equivalent scalar system discuss validity rsb assumptions examples simulation setups settings distortion functions considered numerical investigations follows sensing matrices throughout numerical investigations consider two important cases random projector matrices random matrix case entries supposed generated arbitrary distribution without loss generality assume entries random variables variance structure primary also discussed case random matrix theory matrix regardless well known asymptotic empirical eigenvalue cdf gramian follows law states returns zero otherwise using definition straightforward show reads projector matrix constraint sensing matrix row vectors orthogonal matrices also referred row orthogonal matrices sensing matrix assume case row vectors normalized number rows thus outer product aat reads aat consequently gram matrix takes two different eigenvalues multiplicity multiplicity considering definition factor asymptotic empirical cdf eigenvalues read results form source model consider continuous finite alphabet sources distributed sparse gaussian sparse uniform distributions respectively precisely assume entries continuous finite alphabet sources generated gaussian uniform distribution respectively multiplied bernoulli random variable probability take take moreover assume nonzero outcomes finite alphabet source symmetric form positive real integer distortion function regarding continuous sources determine performance different estimators considering mse distortion function fig predicted normalized mse versus estimation parameter linear lasso norm recovery schemes considering compression rate sparsity factor set noise variance set source power noise power ratio becomes dashed solid lines respectively indicate cases random projector measurements curves match numerical results reported finite alphabet sources moreover determine probability error measure well mse considering either case clear mse obtained norm recovery scheme bounded mmse bound reported literature numerical results continuous sources considering examples consider case sparse gaussian source sparsity factor sampled via random sensing matrix fig shows prediction normalized mse defined mse mse function estimation parameter compression rate set random projector sensing matrices considered curves match results reported seen recovery scheme optimal choice estimation parameter outperforms lasso scheme projector fig predicted normalized mse versus estimation parameter different compression rates considering lasso recovery sparsity factor set true noise variance set dashed solid lines respectively indicate random projector matrices compression rate increases converges however choice estimation parameter make norm performance even worse lasso moreover contrast noiseless case projector matrix always outperforming matrix noisy case fact also reported authors showed noiseless sampling case gaussian matrix ansatz linear lasso recovery schemes locally stable perturbations break symmetry replica correlation matrix result fact agrees general belief convex optimization problems exhibit rsb result however indicated norm reconstruction ansatz becomes unstable therefore rsb needed accurately assessing performance order investigate observation plotted normalized mse lasso recovery scheme predicted ansatz terms estimation parameter fig considering different compression rates observed given estimation parameter normalized mse increases compression rate grows large compression rates normalized mse converges agrees fact asymptotically large compression rates source observation vectors independent thus mse converges source power investigate norm recovery scheme plot corresponding curves norm scheme considering fig reference norm recovery fig shows normalized mse predicted ansatz function estimation invalid projector fig normalized mse function estimation parameter determined via ansatz different compression rates considering norm recovery scheme sparsity factor noise variance considered respectively dashed solid lines respectively indicate random projector matrices compression rate increases ansatz starts give invalid solutions low estimation parameters fact grows fixed point equations valid solutions reduces interval solution invalid becomes larger compression rate increases parameter two different compression rates system setup set similar one considered fig curves plotted projector measurements contrast lasso recovery scheme ansatz starts give invalid predictions norm scheme compression rate increases observed normalized mse drops unexpectedly interval estimation parameters compression rate grows fact interval fixed point equations either unstable solution solution illustrate result let consider examples ansatz sensing matrix employed case equivalent noise power estimation parameter read increasing compression rate interference increases thus mse takes larger values therefore small one consider takes large values write considering example substituting ansatz fixed point equations grows large written following form tending zero bounded real scalar defined taking limit valid bounded real value reduces large compression rates noting setup mse one concludes agrees results given fig similar approach norm recovery scheme example however results following contradicting equations scalar defined clearly set equations solution approximated fixed point equations explain invalidity predicted performance norm recovery scheme large compression rates order investigate ansatz plot optimal normalized mse function compression rate fig consider case sensing matrix sparsity factor set source noise power ratio normalized mse minimized numerically estimation parameter figure illustrates mse ansatz starts drop moderate compression rates observation confirms discussion stability saddle point given fact unexpected drop ansatz caused limited stability region fixed point solutions precisely given source noise power ratio estimation parameter fixed point equations stable solutions within certain interval compression rates interval widens grows fig compares norm recovery figure optimal normalized mse plotted case sensing matrix sparsity factor considered noise linear lasso norm invalid compression rate fig optimal normalized mse versus compression rate assumption considering linear lasso recovery schemes sparsity factor source noise power ratio considered respectively minimized numerically estimation parameter solid dashed lines show results random orthogonal measurements respectively grows ansatz norm recovery starts give invalid solutions variance set ansatz considered two cases namely minimized possible estimation parameters interval ansatz valid latter case referred restricted curve figure depicts difference ansatz quite small low compression rates prediction however deviates larger compression rates observation indicates performance analysis norm recovery exhibits rsb sake comparison also plot curve lasso recovery scheme well mmse bound observed normalized mse lasso bounds curve ansatz moreover consistent mmse bound rigorously justified literature restricted norm curve moreover crosses lasso curve deviation indicates large compression rates optimal estimation parameter minimizes mse lies somewhere interval ansatz returns valid approximation mse check consistency solutions also sketch zero temperature entropy corresponding spin glass function compression rate fig considering setup fact consider point fig normalized mse curve differentiable restricted mmse bound compression rate fig versus prediction optimal normalized mse considering norm recovery scheme random measurements sparsity factor source noise power ratio considered respectively green dashed line shows prediction optimal numerically minimized however green solid line indicates restricted predicted minimized numerically intervals ansatz stable black solid line denotes ansatz deviates restricted curves sake comparison normalized mse lasso recovery well mmse bound plotted fig fig zero temperature entropy also confirms better accuracy ansatz based investigations exact performance norm recovery scheme needs rsb accurately studied however useful see validity region system parameters prediction given ansatz lies approximately curve purpose define norm recovery break compression rate rbr function system parameters compression rate prediction via ansatz starts deviate precisely given setting parameters set region given defined vbrrs indicate respectively normalized mse compression rate setup specified set calculated ansatz respectively consequently break zero temperature entropy hrsb compression rate fig hrsb functions compression rate considering spin glass corresponds norm recovery scheme setting considered setup investigated fig fig better approximation performance via ansatz demonstrated figure compression rate rbr maximum deviation given rbr max vbrrs general set includes several setup parameters sensing matrix source distribution sparsity factor noise variance estimation parameter considering case sensing matrix sparse gaussian source given sparsity factor break compression rate function snr estimation parameter rbr snr define source noise power ratio snr snr case region vbrrs snr area enclosed snr axes well rbr snr surface fig illustrates intersection region planes snr snr break compression rate increases respect snr agrees intuition moreover snr snr rbr snr vbrrs fig rbr snr terms estimation parameter snr snr considering matrix employed sensing break compression rate starts saturate estimation parameter grows break compression rate takes values close one observation agrees instability reported rbr snr starts saturate grows another extreme case estimation parameter tends zero map estimator reduces arg min case break compression rate converges minimum compression rate observation large snr limit agrees instability ansatz reported error probability bound matrix projector matrix snr fig error probability versus snr norm recovery scheme employed compression rate set measurements error probability deviates bound within interval snr random orthogonal measurements meets bound almost every snr alphabet size sparsity factor estimation parameter considered respectively numerical results finite alphabet sources considering finite alphabet source given boundary points linear recovery read case lasso reconstruction boundary points given finally employing norm recovery source noise power ratio snr defined snr projector projector matrix matrix obability mmse bound norm compression rate fig normalized mse versus compression rate snr sparsity factor considered norm recovery scheme employed estimation mse numerically minimized considering sensing matrix red curve indicates mmse bound normalized mse figure demonstrates phase transition normalized mse fig shows predicted error probability finite alphabet system function snr unit compression rate norm recovery scheme employed cases considered random orthogonal measurements sparsity factor set figure illustrates either sensing matrix error probability meets bound relatively large snr regime deviation bound case moreover occurs larger interval snr grows contrast measurements coincidence occurs almost every snr projector sensing matrix employed observation intuitively justified due orthogonality rows latter case mse similar behavior fig observed fig compression rate increases considering either mse error probability deviation bound observed random orthogonal measurements considering fixed snr observed rate discontinuity point mse suddenly jumps lower value upper value certain compression rate phenomenon known phase transition macroscopic parameters suddenly change phase within minor variation setting compression rate phase transition occurs referred transition rate increases snr grows phase transitions reported literature communications information theory several problems turbo coding cdma systems fig illustrates phase transition assumption case sparse binary source source vector sensed via random matrix snr estimation norm recovery scheme employed equivalent lasso particular example estimation parameter moreover optimized numerically mse minimized determine point curve fixed point equations solved specific compression rate possible solutions found within set solutions physically stable considered stable solution minimum free energy taken ansatz normalized mse compression rate plotted using ansatz figure shows normalized mse suddenly jumps upper bound converges sake comparison plotted mmse bound well shows samilar behavior higher rates fig plotted normalized mse terms compression rate setting sensing matrix replaced random projector matrix figure shows phase transition orthogonal measurements higher transition rate compared case random sensing moreover improvement terms mse observed fig employing projector matrix extends conclusion given sparse gaussian sources sparse finite alphabet sources well order obtain accurate approximation performance transition point one may investigate free energy entropy corresponding spin glass also considering rsb ansatz however gives accurate approximation mse specific setup considered onclusion manuscript considered performance map estimator limit respect general distortion function taking statistical mechanical approach replica method employed determine asymptotic distortion system deviated earlier approaches evaluating general replica ansatz corresponding spin glass general ansatz let derive well brsb ansatz considering class rotationally invariant random matrices results recover previous studies special cases justifies uniqueness zero temperature entropy expression brsb assumption conjectured replica ansatz evaluated led general form decoupling principle fact invoking general replica ansatz shown tuple entries marginal joint distribution converges empirical distribution asymptotically means limit marginal joint distribution entries determined replica ansatz decouples set identical joint distributions form asymptotic decoupled distribution however depends structure imposed ansatz brsb ansatz awgn system estimated map estimator decouples awgn channels followed correlated impairment terms intuitively model interference system vanish reducing brsb assumption general decoupling principle justified confirms conjecture decoupling generic property map estimators since validity relies replica continuity assumption recent results statistical mechanics shown failures finding exact solution via replica method mainly caused structure projector projector matrix matrix obability projector compression rate fig terms compression rate snr considering projector sensing matrices mse numerically minimized respect estimation parameter figure shows projector matrix enhances reconstruction performance respect mse transition rate imposed ansatz replica continuity decoupling property enabled represent equivalent replica simulator interpretation replica ansatz fact brsb fixed point equations completely described statistics corresponding decoupled system brsb ansatz therefore represented state evolution transition system takes initial set ansatz parameters input determines new set parameters via simulating brsb decoupled system output example systems considered noisy compressive sensing system studied different sparse recovery schemes including linear lasso norm recovery schemes numerical investigations showed sparse gaussian sources performance linear lasso recovery schemes accurately approximated ansatz within large interval compression rates prediction norm scheme performance however observed lack stability within moderate high compression rates result ansatz scheme considered deviates ansatz significantly rate grows sparse finite alphabet sources prediction error probability mse reported phase transitions respect compression rate numerical results moreover showed better performance random orthogonal measurements compared random sensing cases previously reported sparse gaussian sources current work pursued several directions example replica simulator introduced section vii studied methods developed context transition systems analysis may result proposing new framework simplifies evaluation fixed point equations another direction analysis conditional distribution correlated impairment terms brsb decoupled system study lead understand necessary sufficient conditions ansatz gives accurate approximation system performance inspired model spin glasses full rsb ansatz proved give stable solution system parameters conjecture exact solution large compression rates cases exhibiting rsb given large number breaking steps nevertheless numerical investigations depicted even cases ansatz brsb ansatz small give good approximations solution moderate compression rates latter study furthermore gives insights accuracy approximation given finite number rsb steps investigating connection replica simulator message passing based algorithms another interesting topic future work acknowledgment authors would like acknowledge german research foundation deutsche forschungsgemeinschaft dfg support work grant ppendix roof roposition starting define moment function therefore lim lim lim lim log taking expectation respect noise term first moment function extended replica vector gramian system matrix defined satisfies properties stated section comes taking expectation factor defined true noise variance specified section remark considering one could drop term kzk since plays rule optimization problem precisely one could redefine hamiltonian enew without loss generality case coefficient right hand side reduces reads new however clear tends zero approaches result result considering expression define random variable consequently moment function given taking expectation respect multiplying however later show almost realizations source vector scalar converges deterministic value therefore expectation respect dropped call unbiased since expresses deviation replicas source vector defined considering eigendecomposition gramian udut expectation expressed terms spherical integral integral measure probability measure regarding system setup specified section matrix distributed orthogonal group haar probability distribution therefore corresponding spherical integral itzykson zuber integral integral extensively studied physics mathematics literature see example brief discussion spherical integral closed form solution given appendix invoking theorem long rank expectation written defined order employ result substitute need check rank condition lemma considering defined following argument holds rank proof first rewrite matrix columns unbiased replicas considering obvious could rank assumption indicates analytically continues real axis limit respect taken right neighborhood therefore values indicates growth order respect composition lim exists constant consequently one write lim rank lim lim concludes rank lemma ensures always holds therefore noting fact eigenvalues expectation right hand side reduces function defined mrj square matrix deterministic matrix given correlation matrix defined remark note although symmetric tqv symmetric matrix general however due symmetry eigenvalues tqv real therefore sequence integrals real axis exists indices substituting given order determine sum follow technique employed split space replicas subshells defined correlation matrices vectors replicas subshell correlation matrix precisely given source vector subshell matrix defined nqab qab denoting entry sum determined first subshell subshells following dqab integral taken nqab term eni determines probability weight subshell defined eni remark one may define subshells transferred correlation matrix instead correlation matrix case subshells defined rotate space respect rotation however impact analysis last step determine eni represent dirac impulse function using inverse laplace transform defining sab complex frequency corresponding nqab dsab esab nqab integral taken imaginary axis consequently defining frequency domain correlation matrix defined matrix sab reads sab sab dsab integral taken substituting eni reduces eni enm sab defined sab log thus finally reads dsdq consequently one needs evaluate expectation respect order determine moment function however almost realizations converges deterministic asymptotic thus expectation respect dropped show latter statement note term depends therefore sufficient study convergence lemma justifies property using law large numbers decoupling property functions lemma consider system specified section let assumption hold defined given log log vector elements random variable distributed source distribution expectation taken defined proof consider decoupling property functions define vector support coefficients reads sab vai vai log log sab functions defined log log vector elements define compactness assumption suggests limits respect exchanged thus one consider asymptotics given tends large limit regarding fact collected distribution term right hand side converges expectation distribution due law large numbers precisely defined substituting lemma concluded remark considering lemma eventually says probability weight eni given correlation matrix converges deterministic weight tends infinity statement equivalently states almost given realization source vector correlation matrix converges expectation fact considering correlation matrix defined entries functions therefore variate randomly due source distribution given lemma however indicates entries converge deterministic asymptotics almost realization alternative approach one could study convergence property correlation matrix means law large numbers first conclude lemma rewriting proper way replacing expectation using fact probability weight eni needs converge deterministically nevertheless approach taken seems straightforward using lemma drop expectation respect replacing asymptotic distortion found taking limits assumption suggests exchange order limits take limit respect first denoting probability measure defined eni satisfy large deviation properties use saddle point approximation evaluate integral says dsdq drop given regarding fact vanishes large limit saddle point integrand function exponent bounded coefficient indicates asymptotic equivalency exponential scale defined following definition functions defined set said asymptotically equivalent exponential scale lim log unbounded sequence mth moment replaced asymptotic equivalent consequently substituting equivalent term exchanging limits order log lim lim lim lim lim lim lim lim lim comes fact bounded functions point found letting derivatives exponent zero using standard definition saddle point given following fixed point equations reduces results replacing expression asymptotic distortion saddle point correlation matrix considered expectations conditional distribution defined trj trj simplifies expressions given proposition general fixed point equation satisfied several saddle points therefore multiple asymptotic distortions might found case one note valid solution one minimizes free energy spin glasses zero temperature using mth moment free energy system reads lim lim lim log lim lim lim trj log trj comes facts bounded limits respect supposed exchange deduced lemma finally considering definition proposition concluded ppendix roof roposition starting assumption replica correlation matrix real considering definition hamiltonian spin glass replicas given trj defined denoting trj shown appendix structure correlation matrix thus one write eim real functions denoting eigendecomposition vdq vdt vdr diagonal matrices eigenvalues therefore equivalently states ath column matrices eigenvalue corresponding two different corresponding eigenvalues namely occur multiplicity substituting occur multiplicity taking limit found note symmetric pursue analysis rewrite hamiltonian using ekx therefore partition function given using gaussian integral thus partition function reduces parameters spin glass replicas determined using partition function starting normalized free energy reads log noting takes expectation gaussian distribution one use riesz equality show varies vicinity log tends expectation taken consequently normalized free energy reads lim log next parameters specified determining conditional distribution substituting following fixed point equations deduced found taking trace sum entries sides respectively one directly evaluate right hand sides however considering straightforward show substituting taking limit fixed point equations finally read defined order determine replicas average distortion defined regarding distortion function replace hamiltonian ehr given take steps find modified form normalized free energy replicas average distortion evaluated depend thus taking limit needed last step take zero temperature limit using laplace method summation fixed point equations reduce defined arg min taking approach replicas average distortion zero temperature reads order avoid multiple solutions need find normalized free energy corresponding spin glass given proposition fact fixed point equations may different solutions therefore several asymptotics distortion obtained case fixed point solution minimizes zero temperature free energy system corresponding asymptotic distortion taken substituting proposition free energy corresponding spin glass inverse temperature found function defined taking limit zero temperature free energy reads defined defining proposition concluded ppendix roof roposition take approach appendix considering replica correlation matrix form real need evaluate parameters spin glass replicas defined definition starting hamiltonian trj given discussed appendix given matrix trj following form eim found terms using eigendecomposition straightforward show denote ath eigenvalues respectively regarding structure three different sets corresponding eigenvalues namely multiplicity multiplicity multiplicity thus substituting taking limit given next step evaluate partition function substituting hamiltonian reads ekx partition function determined substituting using equalities partition function found denoted argument partition function indicate expression determined given normalized free energy spin glass replicas reads log using riesz equality taking limit normalized free energy reduces lim log order find fixed point equations use therefore concluded taking trace sum diagonal blocks sum entries sides respectively evaluate right hand sides take alternative approach express expectations taking derivatives limit fixed point equations finally reduce defined replicas average distortion regarding distortion function determined modifying hamiltonian ehr given taking steps find modified form normalized free energy replicas average distortion reads lim analysis concluded taking zero temperature limit read defined arg min denoting lim moreover asymptotic distortion given reads well determined terms moreover given multiple solution fixed point equations found proposition suggests choose solution minimizes free energy therefore one needs find optimal corresponding free energy meets minimum value second law thermodynamics satisfied inverse temperature initially search optimal considering given find corresponding minimize zero temperature free energy using proposition free energy inverse temperature given written function defined find thermal equilibrium let using equalities concludes satisfies prj qrj qrj log reduces prj qrj qrj log denoting solution free energy corresponding spin glass given zero temperature reads log lim proposition concluded finally defining ppendix roof roposition strategy extend approach appendix general number breaking steps following appendix considering frequency domain correlation matrix trj written eim considering given vector integers found terms letting denote ath corresponding eigenvalues long constraint holds different sets corresponding eigenvalues specified multiplicity multiplicity multiplicity multiplicity substituting determined terms define hamiltonian spin glass replicas determined substituting hamiltonian using equalities partition function finally reads sequences define dzb consequently one evaluates free energy using riesz equality reduces log defined sake compactness fixed point equations moreover found via found taking trace sum entries sides respectively moreover concluded adding entries diagonal blocks size right hand sides evaluated using equalities thus fixed point equations finally concluded denote recursively defined replicas average distortion regarding distortion function determined using hamiltonian modification technique employed appendix modifying hamiltonian taking derivatives average distortion inverse temperature given finally taking limit find asymptotic distortion defined arg min denotes limiting factor considering definition reads moreover fixed point equations reduce ansatz set extreme point free energy given inverse temperature order satisfy second law thermodynamics solution needs found set real vectors satisfy constraint parameters ansatz however finally taken zero temperature free energy minimized using proposition free energy corresponding spin glass given vector written function defined therefore vector given set arg min set real vectors satisfying constraint substituting reduces arg min reads vector determined minimizes free energy finally taking limit zero temperature free energy evaluated log dzb define lim defining sequence denoting proposition concluded ppendix eneral rsb requency omain orrelation atrix consider spin glass replicas defined definition hamiltonian reads trj referred frequency domain correlation matrix appendix show general rsb assumption including case frequency domain correlation matrix structure different scalar coefficients show let correlation matrix form integer represents brsb well structures setting coefficients correspondingly considering defined written defining vector vector entries equal clear eigenvector therefore denoting eigendecomposition vdq reads uut diagonal matrix diagonal entries expect entry corresponding eigenvector zero consequently reduces states span eigenspace eigenvalues also distributed frequencies fact eigenvalue corresponding occurs multiplicity second term right hand side change distribution eigenvalues modifies eigenvalue corresponding therefore also represented different scalar coefficient extend scope analysis note function strictly increasing different single mass point consequently eigenvalues distribution remains unchanged thus real case single mass point cdf becomes constant function results kim constant therefore represented setting concludes structure single mass point cdf real constant ppendix symptotics pherical ntegral consider haar measure orthogonal group unitary group let matrices integral form entr ugn known spherical integral integral extensively studied mathematics literature well physics often called itzykson zuber integral variety problems evaluation spherical integrals asymptotic regime interesting therefore several investigations done asymptotics asymptotics integral investigated matrices distinct eigenvalues converging spectrums assumptions closed form formula given however final formula complicated hard employ authors showed asymptotics integral written directly terms corresponding asymptotic eigenvalue distribution long replica analysis considered utilize result since number replicas considered small enough theorem shown matrix assumption spectrum asymptotically converges deterministic cdf compact finite length support asymptotics integral written terms log lim denotes single nonzero eigenvalue authors showed theorem case rank assumption theorem spherical integral asymptotically factorizes product integrals therefore log lim denoting nonzero eigenvalues rank appendix one employ order evaluate asymptotics system matrix consistent system setup illustrated section moreover using discussion investigations appendix extended case complex variables spherical integral asymptotics found references therein eferences edwards anderson theory spin glasses journal physics metal physics vol pastur shcherbina absence order parameter model journal statistical physics vol guerra toninelli thermodynamic limit mean field spin glass models communications mathematical physics vol infinite volume limit generalized mean field disordered models arxiv preprint korada macris tight bounds capacity binary input random cdma systems information theory ieee transactions vol merhav statistical physics information theory publishers inc parisi replica analysis travelling salesman problem journal physique vol anderson application statistical mechanics problems 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asking difficult questions visual question generation via intermediate rewards junjie chunhua jian jianfeng anton van den nov australian centre robotic vision university adelaide australia faculty engineering information technology university technology sydney australia school computer science technology nanjing university science technology china abstract robby get cup cupboard despite significant progress variety problems developing method capable asking intelligent questions images proven inscrutable challenge towards end propose deep reinforcement learning framework based three new intermediate rewards namely progressive informativeness encourage generation succinct questions turn uncover valuable information towards overall goal directly optimizing questions work quickly towards fulfilling overall goal avoid tendency existing methods generate long series insane queries add little value evaluate model guesswhat dataset show resulting questions help standard guesser identify specific object image much higher success rate tall one ikea elsa pink yes handle get brilliant figure two illustrative examples potential conversations human robot left conversation clearly makes people frustrated right one makes people happy robot achieves goal quicker way via less informative questions question extracts informative answer towards achieving particular goal thus reflects knowledge asker estimate capabilities answerer although information would beneficial identifying particular object image little value agent asking human exact values particular pixels statistics gradients aspect ratio corresponding bounding box fact answerer incapable providing requested information makes questions pointless selecting question significant probability generating answer helps achieve particular goal complex problem asking questions essential part way humans communicate learn intelligent agent seeks interact flexibly effectively humans thus needs able ask questions ability ask intelligent questions even important receiving intelligent judge man questions rather answers although visual question answering vqa attracted attention visual question generation vqg much difficult task obviously generating facile repetitive questions represents challenge generating series questions draw useful information towards overarching goal however demands consideration image content goal conversation thus far could generally also seen requiring consideration abilities motivation participant conversation yes introduction pink short one yes work done visiting university adelaide first two authors contributed work equally tionable answers robot example fig given task realized missing critical information required carry needs ask question limited number attempts human gets frustrated carries task scenario applies equally intelligent agent seeks interact humans surprisingly little tolerance agents unable learn asking questions ask many result visual question generation vqg started receive research attention primarily problem methods approach problem manner tend generate arbitrary sequences questions somewhat related image bare relationship goal reflects fact methods means measuring whether answers generated assist making progress towards goal instead paper ground vqg problem version game guesswhat introduced method presented play guesswhat game made three components questioner asks questions oracle guesser tries identify object oracle referring based answers quality generated questions thus directly related success rate final task training uses game setting used visual dialog generation previously however work focus generating dialogs helping agent achieve goal better question generation moreover previous work uses final goal reward train dialog generator might suitable dialog generation rather weak undirected signal control quality effectiveness informativeness generated question task words cases want talk robot want finish specific task hold meaningless boring chat therefore paper use intermediate rewards encourage agent ask short informative questions achieve goal moreover contrast previous works consider overall goal reward assign different intermediate rewards posed question control quality achieved fitting vqg reinforcement learning paradigm devising three different intermediate rewards main contributions paper explicitly optimize question generation first reward designed encourage agent achieve final goal pick object oracle thinking via asking multiple questions however different considering whether goal achieved additional rewards awarded agent use fewer questions achieve reasonable setting need robot finish task ask hundreds questions second reward proposed progressive reward established encourage questions generated agent progressively increase probability right answer intermediate reward individual question reward decided change answer probability negative reward given probability decreases last reward informativeness reward used restrict agent ask useless questions example question leads identical answer candidate objects question eliminate ambiguous show whole framework fig evaluate model guesswhat dataset standard oracle guesser show novel questioner model outperforms baseline model large margin also evaluate reward respectively measure individual contribution qualitative results show produce informative questions related works visual question generation recently visual question generation problem brought computer vision community aims generating questions works treat vqg standalone problem follow image captioning style framework translate image sentence case question example mora use model generate questions answers directly image visual content zhang focus generating questions grounded images use densecap region captioning generator guide question generation mostafazadeh propose dataset generate natural questions images beyond literal description image content view vqa vqg dual learning process jointly training framework although works generate meaningful questions related image motivation asking questions rather weak related goals moreover hard conduct quality measurement type questions instead work aim develop agent learn ask realistic questions contribute achieving specific goal visual dialogue generation attracted many attentions recently das introduce reinforcement learning mechanism visual dialogue generation establish two agents corresponding question answer generation respectively finally locate unseen image set images question rounds dialogue oracle person cnn image feature guesser yes oracle funiture guesser oracle drink image feature vqg question generator guesser vqg yes intermediate rewards success vqg figure framework proposed vqg agent plays whole game environment target object assigned oracle unknown vqg guesser vqg generates series questions answered oracle training let oracle answer question based objects round measure informativeness reward also let guesser generate probability distribution measure progressive reward finally consider number rounds set reward based status success intermediate rewards adopted optimizing vqg agent reinforce agent predicts feature representation image reward function given measuring close representation compared true feature however focus encouraging agent generate questions directed towards final goal adopt different kinds intermediate rewards achieve question generation process moreover question generation agent model asks questions based dialogue history involve visual information florian propose employ reinforcement learning solve question generation guesswhat game introducing final status success sole reward share similar backbone idea several technical differences one significant differences previous work considers using whether achieving final goal reward assign different intermediate rewards posed question push vqg agent ask short informative questions achieve goal experimental results analysis section show model outperforms also achieves higher intelligence using questions possible finish task question generation nlp long history works grammar question generation text domain natural language processing nlp authors focus automatically generating gapfill questions crowdsourcing templates manually built templates used question generation respectively works focus constructing formatted questions text corpus reinforcement learning reinforcement learning adopted several vision language problems including image captioning vqa aforementioned visual dialogue system etc ren use policy network value network collaboratively generate image captions different optimization methods image captioning explored called spider sequence training zhu introduce knowledge source iterative vqa employ learn query policy authors use learn parameters model images structured knowledge bases works solve related problems employing optimization method focus using carefully designed intermediate rewards train vqg agent tasks vqg ground vqg problem guess game specifically guesswhat dataset guesswhat interactive game roles observe image rich visual scene contains multiple objects view game three parts oracle questioner guesser game random object scene assigned oracle process hidden questioner questioner ask series questions locate object list objects also hidden questioner rounds questioner gathered enough information guesser start guess game considered successful guesser selects right object questioner part game vqg problem question generated based visual information image previous rounds pairs goal vqg successfully finish game case locate right object paper fit vqg reinforcement learning paradigm propose three different intermediate rewards namely reward progressive reward informativeness reward explicitly optimize question generation reward established lead dialogue achieve final goal progressive reward used push intermediate generation process towards optimal direction informativeness reward used ensure quality generated questions better express generation process first introduce notations guesswhat game used throughout rest sections game defined tuple observed image dialogue rounds pairs list objects image target obj ject question sequence tokens sampled vocabulary composed word tokens question stop token dialogue stop token end answer yes set yes applicable object object category segment mask learning environment build learning environment generate visual dialogues based guesswhat dataset since focus vqg fair comparison oracle guesser produced referring original baseline models guesswhat also introduce vqg supervised learning model referred baseline rest paper oracle oracle requires generating answers kinds questions objects within image scene build neural network architecture oracle referring bounding box obtained segment mask object encoded eight dimensional vector represent spatial feature ospa xmin ymin xmax ymax xcenter ycenter wbox hbox indicates box coordinates width height category embedded using learned table current question encoded lstm three features concatenated single vector fed one hidden layer mlp followed softmax layer produce answer probability guesser given image series questionanswer pairs guesser requires predicting right object list objects referring consider generated dialogue one flat sequence tokens encode lstm last hidden state extracted feature represent dialogue also embed objects spatial features categories mlp perform dialogue object features softmax operation produce final prediction vqg baseline given image history pairs vqg requires generating new question build vqg baseline based rnn generator rnn recurrently produces series state vectors transitioning previous state current input token use lstm transition function sjm case state vector conditioned whole image previous questionanswer tokens add softmax operation produce probability distribution vocabulary baseline conducted employing supervised training train vqg minimizing following negative log loss function log log test stage question sampled model starting state new token sampled probability distribution embedded fed back lstm repeat operation end question token encountered reinforcement learning vqg use established oracle guesser vqg baseline model simulate complete guesswhat game given image initial question generated sampling vqg baseline stop question token encountered oracle receives question along assigned object category spatial information output answer questionanswer pair appended dialogue history repeat loop end dialogue token sampled number questions reaches maximum finally guesser takes whole dialogue object list inputs predict object consider goal reached selected otherwise failed efficiently optimize vqg towards final goal generate informative questions adopt three intermediate rewards introduced following sections framework state action policy view vqg markov decision process mdp vqg noted agent dialogue generated based image time step state agent defined image visual content history pairs tokens current question generated far action agent select next output token vocabulary depends actions agent takes transition two states falls one following cases current question finished oracle environment answer appended dialogue history next state end dialogue finished guesser environment select object list keeps otherwise new generated token pending current question next state maximum length question mmax maximum rounds dialogue jmax therefore number time steps dialogue mmax jmax model vqg stochastic policy represents parameters deep neural network used vqg baseline produces probability distributions state goal policy learning estimate parameter set components mdp significant aspect define appropriate reward function pair emphasized vqg aims generate questions lead achieving final goal therefore build three kinds intermediate rewards push vqg agent optimized towards optimal direction whole framework shown fig reward one basic rule appropriate reward function conflict final optimal policy primary purpose vqg agent gather enough information soon possible help guesser locate object therefore define first reward reflect whether final goal achieved moreover take number rounds consideration accelerate questioning part let reward nonzero game successful given state end token sampled maximum number rounds jmax reached reward pair defined jmax guesser otherwise set reward one plus weighted maximum number rounds jmax actual rounds current dialogue dialogue successful zero wise based want final goal motivate vqg generate useful questions moreover intermediate process considered reward function rounds pairs guarantees efficiency generation process fewer questions generated reward vqg agent get end game game succeed quite useful setting realistic want use fewer orders guide robot finish tasks weight balance contribution successful reward dialogue round reward progressive reward based intuition observation human interactive dialogues find questions successful game ones progressively achieve final goal long questions asked answered confidence referring target object becomes higher higher therefore round define intermediate reward pair improvement target probability guesser outputs specific interact guesser round obtain probability predicting target object probability increases means generated question positive question leads dialogue towards right direction set intermediate reward called progressive reward encourage vqg agent progressively generate positive questions round record probability returned guesser compare last round difference two probabilities used intermediate reward way question considered positive reward leads higher probability guess right object otherwise reward negative informativeness reward human ask questions especially guess game expect answer help eliminate confusion distinguish candidate objects hence imagine posed question leads answer candidate object question useless example candidate objects red posed question red get answer however pair help identify target want avoid kind questions case need evaluate question based answer oracle given generated question interact oracle answer question since oracle takes algorithm training procedure vqg agent image current question target object inputs outputs answer let oracle answer question objects image answers different consider useful locating right object otherwise contribute final goal therefore set reward positive called informativeness reward useful questions formally round oracle receives image current question list objects outputs answer set ajo ajon element corresponds object informativeness reward defined ajon identical otherwise input oracle ora guesser gus batch size update generate episodes select image one target object generate pairs jmax qjh number total objects ahjohn ora qjh ohn giving positive reward pair improve quality dialogue encouraging agent generate informative questions training policy gradient three different kinds rewards take intermediate process consideration stateaction pair add three rewards together final reward function considering large action space game setting adopt policy gradient method train vqg agent proposed intermediate rewards goal policy gradient update policy parameters respect expected return gradient descent since episodic environment given policy generative network vqg agent case policy objective function takes form ahjohn identical else gus end qjh break gus argmaxoh jmax else define evaluate update vqg agent evaluate update baseline substituting notations vqg agent following policy gradient parameters optimized following gradient update rule reinforce algorithm gradient estimated batch episodes sampled policy log value function returns expectation cumulative reward baseline function help reduce gradient variance chosen arbitrarily use mlp takes state input vqg agent outputs expected reward baseline trained mean squared error min log whole training procedure shown experiment section present vqg results conduct comprehensive ablation analysis intermediate reward mentioned proposed method evaluated guesswhat game dataset standard oracle guesser comparing baseline model show proposed model efficiently generate informative questions serve final goal dataset evaluation metric guesswhat dataset composed dialogues grounded images unique objects pairs dialogues vocabulary size use standard split training validation test following report accuracies games evaluation metric given dialogue target object located guesser game noted successful indicates vqg agent generated qualified questions serve final goal two kinds test runs training set test set respectively named newobject newimage newobject randomly sampling target objects training images restrict use new objects seen newimage sampling objects test images unseen report three inference methods namely sampling greedy beam size two test runs implementation details standard oracle guesser vqg baseline reproduced referring error trained oracle guesser test set respectively vqg baseline referred baseline initialize training environment standard oracle guesser vqg baseline start train vqg agent proposed reward functions train models epochs stochastic gradient descent sgd learning rate batch size respectively baseline function trained sgd time epoch training image sampled one objects inside randomly assigned target set maximum round jmax maximum length question mmax weight dialog round reward set progressive reward set results ablation analysis section give overall analysis proposed intermediate reward functions better show effectiveness reward conduct comprehensive ablation studies moreover also carry human interpretability study evaluate whether human subjects results reported https original authors use grid search select find produces best results table results training images newobject newobject baseline sampling greedy table results test images newimage newimage baseline sampling greedy understand generated questions well human use pairs achieve final goal note vqg agent trained reward trained progressive rewards trained informativeness rewards final agent trained three rewards noted overall analysis tab show comparisons vqg agent optimized proposed intermediate rewards model proposed noted uses indicator whether reaching final goal sole reward function see proposed intermediate rewards combinations vqg agents outperform compared models evaluation metrics specifically final agent surpasses accuracy newobject sampling greedy respectively obtains higher accuracy newimage sampling greedy respectively moreover agents outperform supervised baseline significant margin fully show effectiveness proposed intermediate rewards train three vqg agents using rewards respectively conduct ablation analysis see already outperforms baseline model means controlling dialogue round push agent ask wise questions combination reward respectively performance vqg agent improved find improvement gained reward higher reward suggests intermediate progressive reward contributes experiment final agent combines rewards achieves best results fig shows qualitative results results found supplementary material including baseline vqg baseline vqg person food plate table person food left front yes person food drink yes front yes donut yes left person food yes left middle food yes right yes front one yes failure table failure table success cup failure donut failure donut success donut phone book remote left middle remote laptop yes right yes front yes food plate spoon food bow yes left yes front yes food drink yes right front yes failure table failure keyboard success laptop failure knife failure fork success glass figure qualitative results vqg agent green comparisons baseline blue model brown elements middle array indicate probabilities successfully locating target object round better viewed color baseline baseline question informativeness also investigate informativeness questions generated different models let oracle answer questions objects round count percentage questions successful game define question one lead answer candidate objects experimental results show vqg agent questions higher baseline confirms contribution reward human study figure comparisons success ratio agent soler well baseline model different dialogue round left indicates number successful dialogues corresponds bar chart right indicates success ratio corresponds line chart better viewed color fail cases dialogue round conduct experiment investigate relationship dialogue round game success ratio specifically let guesser select object round calculate success ratio given round comparisons different models shown fig see agent achieve goal fewer rounds compared models especially round three progressive trend prove vqg agent learn progressive trend generated questions count percentage successful game progressive ascending trend target object observing probability distributions generated guesser round agent achieves baseline respectively indicates agent better generating questions progressive trend considering introduce progressive reward qualitative results progressive trend shown fig probability right answer progressively increasing conduct human study see well human guess target object based questions generated models show human subjects images generated pairs baseline final vqg agent let guess objects replacing guesser real human ask three human subjects play split game recognized successful least two give right answer based experiments averagely subjects achieve highest accuracy based agent achieves accuracies baseline questions respectively results indicate agent generate higher qualitative questions benefit human achieve final goal conclusion ability devise concise questions lead two parties dialog satisfying shared goal effectively possible important practical applications theoretical implications introducing suitably crafted intermediate rewards deep reinforcement learning framework shown possible achieve result least particular class goal method devised achieves guess goal reliably succinctly also outperforms however since oracle guesser fixed inaccurate certain extent consider main objective paper show effectiveness proposed intermediate rewards vqg problem leave work train three components jointly reinforcement learning framework references agarwal mannem automatic question generation text books proc workshop inno use nlp buil educ pages association computational linguistics andreas rohrbach darrell klein learning compose neural networks question answering arxiv preprint antol agrawal mitchell batra lawrence zitnick parikh vqa visual question answering proceedings ieee international conference computer vision pages becker basu vanderwende mind gap learning choose gaps question generation proc conf north amer chap asso comp ling human lang pages association 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reinforcement learning survey arti intell research labutov basu vanderwende deep questions without deep understanding acl pages duan zhou chu ouyang wang visual question generation dual task visual question answering arxiv preprint liu zhu guadarrama murphy optimization image description metrics using policy gradient methods arxiv preprint kannan yang parikh batra best worlds transferring knowledge discriminative learning generative visual dialog model arxiv preprint mazidi nielsen linguistic considerations automatic question generation acl pages mora puente nieto towards automatic generation question answer pairs images mostafazadeh misra devlin mitchell vanderwende generating natural questions image arxiv preprint harada russell policy invariance reward transformations theory application reward shaping proc int conf mach volume pages ren wang zhang deep reinforcement image captioning embedding reward proc ieee conf comp vis patt rennie marcheret mroueh ross goel sequence training 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mar autonomous norm ham bounded michael brandenbursky jarek abstract prove autonomous norm group compactly supported hamiltonian standard bounded let symplectic manifold let ham group compactly supported hamiltonian diffeomorphisms recall hamiltonian diffeomorphism map flow generated vector field xft defined xft dft smooth compactly supported function see section details function called hamiltonian depend second variable time independent called autonomous known every hamiltonian diffeomorphism product autonomous ones autonomous norm ham defined min autonomous conjugation invariant norm known unbounded group compactly supported hamiltonian diffeomorphisms oriented surface finite area paper concerned group ham compactly supported hamiltonian diffeomorphism euclidean space equipped standard symplectic form prove following result theorem diameter autonomous norm ham bounded proof let ham let ham autonomous diffeomorphisms compactly supported hamiltonian functions let euclidean ball radius centered origin containing union supports functions lemma exists autonomous diffeomorphism ham displacing ball times michael brandenbursky jarek statement lemma means follows lemma exists ham following equality holds ahm ahi diffeomorphism lemma observe since supports pairwise disjoint obtain composition ahm autonomous hamiltonian function equal since commutator product two autonomous diffeomorphisms obtain product three autonomous diffeomorphisms proof lemma let smooth function satisfying following conditions let induced hamiltonian vector field equal thus induced hamiltonian diffeomorphism displaces ball many times like taking appropriate cut function obtain required compactly supported diffeomorphism remarks theorem contained kernel calabi homomorphism see section definition argument shows product three autonomous diffeomorphisms trivial calabi invariant known hofer norm ham unbounded stably bounded kernel calabi homomorphism admit nontrivial quasimorphisms however stably unbounded difficult see diameter autonomous norm ham least best knowledge open question whether exists hamiltonian diffeomorphism autonomous norm equal autonomous norm ham bounded acknowledgements thank center advanced studies mathematics ben gurion university supporting visit second author bgu references vladimir arnold boris khesin topological methods hydrodynamics volume applied mathematical sciences new york michael brandenbursky jarek kedra autonomous metric group algebr geom michael brandenbursky jarek kedra egor shelukhin autonomous norm group hamiltonian torus available http michael brandenbursky egor shelukhin autonomous hamiltonian surfaces math res lett appear available http dmitri burago sergei ivanov leonid polterovich norms groups geometric origin groups diffeomorphisms volume adv stud pure pages math soc japan tokyo gambaudo ghys commutators surfaces ergodic theory dynam systems helmut hofer eduard zehnder symplectic invariants hamiltonian dynamics advanced texts basler advanced texts basel textbooks verlag basel morimichi kawasaki relative quasimorphisms stably unbounded norms group symplectomorphisms euclidean spaces journal symplectic geometry leonid polterovich geometry group symplectic diffeomorphisms lectures mathematics eth verlag basel ben gurion university israel address brandens university aberdeen university szczecin address kedra
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macroscopic modeling simulation managed networks dec matthew wright roberto horowitz alex kurzhanskiy march abstract help mitigate road congestion caused unrelenting growth traffic demand many transit authorities implemented managed lane policies managed lanes typically run parallel freeway standard lanes restricted certain types vehicles originally thought managed lanes would improve use existing infrastructure incentivization behaviors like carpooling implementations often characterized unpredicted phenomena often detrimental system performance development traffic models capture sorts behaviors key step helping managed lanes deliver promised gains towards goal paper presents several macroscopic traffic modeling tools used study freeways equipped managed lanes managed proposed framework based kinematic wave theory model managed lanes modeled parallel links connected nodes certain type traffic may switch managed lane links two types managed lane configuration considered vehicles switch managed lanes anywhere separated switching allowed certain locations called gates incorporate two phenomena model particular managed networks inertia effect friction effect inertia effect reflects drivers inclination stay lane long possible switch would obviously improve travel condition friction effect reflects driver fear moving fast managed lane traffic adjacent links moves slowly due congestion calibration models large road networks difficult dynamics depend many parameters whose numbers grow network size present iterative approach calibrating model physical parameters finally validate model calibration methodology case studies simulations two managed california freeways keywords macroscopic first order traffic model first order node model traffic managed lanes hov lanes dynamic traffic assignment dynamic network loading inertia effect friction effect introduction traffic demand developed developing worlds shows sign decreasing resulting congestion remains costly source inefficiency built environment one study lomax estimated delays due congestion cost drivers billion hours united states alone leading burning billion extra gallons fuel historical strategy accommodating demand construction additional infrastructure recent years planners also developed strategies improve performance existing infrastructure improved road operations demand management seeks lower number vehicles road kurzhanskiy varaiya one strategy widely adopted united states developed countries creation managed lanes obenberger managed lanes implemented freeways restricting use one lanes certain vehicles example hov lanes intended incentivize carpooling reduces total number cars road demand management outcome chang addition demand management managed lanes provide opportunity improved road operations responsive traffic control example tolled express lanes give drivers opportunity pay toll drive parallel lanes presumably express lane traffic management authorities opportunity adjust toll amount response state traffic network potential managed lanes instruments reactive realtime traffic addition made popular among transportation authorities kurzhanskiy varaiya however effects managed lanes always straightforward rehabilitative expected presence create complex traffic dynamics jang cassidy even freeway simple geometry dynamics traffic flow complex fully understood adding managed lanes alongside lanes exacerbates effect adding managed lane creates two parallel distinct coupled traffic flows physical structure used intended managed lanes carry flows different characteristics strictly hovs compositions freeway vehicles move two lane flows complex phenomena unobserved freeways emerge see daganzo cassidy liu jang cassidy cassidy others making better use managed lanes requires understanding macroscopic behavior induce one tool understanding macroscopic traffic flow behavior macroscopic traffic flow model rich literature exists macroscopic models flows long roads junctions roads meet extension situation created placing managed lane parallel freeway managed network straightforward model describe vehicles enter exit managed lane capture interactions caused two flows proximity friction effect liu jang cassidy paper presents macroscopic flow modeling tools used simulation managed networks begin section discussion relevant modeling tools literature make use section describes network structures two common managed lane configurations ungated lanes section also describes models friction inertia effects arise managed lanes section outlines one may take managed lanespecific constructions add general macroscopic traffic simulation process section describes calibration methodologies network configurations section presents case studies two networks one typical macroscopic simulation results managed lane modeling modeling techniques presented paper based kinematic wave macroscopic traffic flow model models describe aggregate traffic flows fluids following conservation law briefly introduce notation discuss basics class models detailed reviews available many references modeling basics simulation framework road divided discrete cells refer links links drawn nodes link begins one node ends another many links may begin end node link characterized density number cars link model traffic flows fully prescribed density timestep link density updates according equation fil flj length link fil flow number vehicles leaving link entering link time flj flow leaving link entering link time number links end link beginning node number links begin link ending node computing flows requires use two intermediate quantities link link demand number vehicles wish exit link timestep link supply number vehicles link accept time functions density model computes often called fundamental diagram link model model computes flows links supplies demands often called node brief outline macroscopic simulation road network sometimes called dynamic network loading simulation performed could time use link model link compute link demand supply function density use node model node compute flows fij incoming links outgoing links functions information vehicles desired movements update state link using increment repeat desired simulation end time reached tools described paper compatible link model make use particular node model studied wright aspect node model particular relevance managed lane modeling relaxed fifo rule construction wright necessary modeling flows managed lanes without fifo relaxation congestion one two lane groups could block traffic may unrealistic see wright section detailed discussion far presented ingredients model able express many simple network topologies joining links nodes capture several important behaviors managed networks next three sections briefly overview additions standard model explained greater detail remainder paper multiple classes vehicles drivers describe number vehicles link single number formulation vehicles treated however simulation managed network makes sense break different classes vehicles drivers example freeway hov lane facility might consider two classes hovs end rewritten ncl ncl fljc indexes vehicle classes often called commodities traffic literature extending density update equation multiple classes means link node models must also extended produce flows fij paper specify particular link model assume use one produces demands sic overall supplies node model computing fij responsible splitting available supply among different demanding vehicle classes examples type link model include considered wong wong daganzo van lint examples link models include hoogendoorn bovy types models analogs supply demand toplogical expression managed networks describe link terms total density breakdown percommodity portions ncl discretizing road links lose information differences vehicle proportions across lanes well behavior becomes problem unmodeled behavior interest setting means modeling freeway managed lane done single link following would impossible study managed network behavior interest modeling differences vehicle density across lanes natural microscopic mesoscopic see example treiber hoogendoorn bovy ngoduy others models macroscopic models simplicity less avenues including differences one straightforward method model lane separate link bliemer shiomi however method drawbacks first requires addition method prescribe proportions vehicle class lane logit model used farhi shiomi requires data calibration second drastically increasing number links macroscopic model necessarily increase size state space model complexity sense incompatible overall goal selecting macroscopic model mesoscopic model macro macroscopic model lost instead paper choose model lanes lane group one link parallel managed lanes managed lane group another link applied entire length road creates network topology two parallel chains links one one managed two chains share nodes chains permitted locations physical access physical barriers policy access double solid lines traffic markings possible use logit model instead driver behavior model first introduced wright model similar one described liu though reference lane crossflows considered observed phenomena managed networks friction effect one several emergent behaviors thought important understanding traffic phenomena present managed networks blamed underperformance managed lanes jang cassidy expand earlier description friction effect describes tendency vehicles hov lane reduce speed vehicles adjacent lane congest point slow hypothesized occurs due hov drivers uncomfortable traveling drastically higher speed adjacent vehicles slowing reduce speed differential jang cassidy liu authors created macroscopic model managed network friction effect model proposed modeling friction effect two separate link models managed lanes one lane threshold density value section propose expressive model friction effect replacing piecewise implementation linear implementation reduction demands slc managed lane proportional speed differential managed lane lane linear relationship like hypothesized jang cassidy fig magnitude friction effect degree managed lane drivers slow match lane drivers speed thought dependent type separation managed lane lane painted lines concrete barrier jang encoded selecting linear coefficient speed differential managed network topologies consider two types managed network configurations full access separated gated access configuration managed lane physically separated lane eligible vehicles may switch two lane groups location often managed lane certain periods day times serve lane hov lanes often accessible outside rush hour hand configuration traffic may switch managed lane lane certain locations called gates locations two lane groups separated road markings double solid line physical barrier usually managed lanes times implemented managed lane access scheme depends jurisdiction example lanes common northern california separated lanes common southern california differences physical geometry access points two access types requires two different types topology constructing network macroscopic model note link node models used section discussed previous section attempt agnostic regards particular link model fundamental diagram used implementations however model friction effect section parameterize friction effect particular link particular vehicle class time adjusting link demand discussion specify particular link model link model reviewed appendix link model also one used simulations presented section node model used remainder paper particular form necessary one discussed wright wright use node model handles traffic optimizes utilization downstream supply makes use input link priorities relaxation conservation turning ratios fifo constraint node models last feature allows describe set lanes upstream offramp one link handle condition congested offramp congestion spill back onto lanes link opposed entirety link see wright section wright discussion figure freeway managed lane managed lane managed lanes managed lane configuration presented figure managed links recall discussed lanes managed lanes collapsed one link parallel geometry share beginning ending node pairs traffic flow exchange managed lanes happen every node note figure use slightly irregular numbering scheme clear whether link link managed lane link ramp link parallel links graph numbers made digit terminating node links one digit link managed lane links two link ramp links three link note also use convention managed lane left lanes ramps far right links long modeling purposes create model may broken smaller ones creating nodes nodes figure fundamental diagrams parallel managed lane links may different liu introduce two vehicle classes corresponds traffic corresponds special traffic managed lane active confined link whereas use managed lane links denote portion vehicles class link attempt enter link quantity called split ratio split ratios managed lanes make assumption vehicle classes take offramps rate example node figure might say strictly speaking necessary assume equal however practice offramp split ratios typically estimated flow count data taken detectors offramp freeway generally speaking detectors identify vehicle type quantity estimable average quantities estimate split ratios one needs extra knowledge tendency class take offramp example vehicles half likely special traffic take certain offramp assuming class exits freeway rate simple assumption problem unidentifiability typical data appears several split ratios first may possible tell many vehicles taking offramp link come upstream managed lane present onramp link case assumptions must made example three different assumptions may reasonable vehicles link take offramp rate vehicles managed lane able cross lanes take offramp node vehicles entering via onramp one present exit via offramp node vehicles managed links take offramp rate vehicles coming onramp directed offramp looking back node figure assumption would say equal assumption would say assumption would say best assumption node depend road geometry particular part road near offramp onramps many lanes vehicle managed lane would cross second crossflows managed lane links observable even exist detectors immediately upstream downstream node traffic switch managed lanes impossible uniquely identify crossflows simulation crossflows must governed driver choice model putting together assumptions policy managed lane summarize necessary split ratios needed computing flows node model example node figure means split ratios applicable vehicles class managed lane split ratios marked dash flows unobservable come driver choice model course whatever method chosen compute unknown split ratios must similarly node onramp offramp dash meaning previously mentioned managed lanes often certain time periods nonspecial traffic allowed managed lane model change policy simply changing split ratios nodes node example nonrestrictive policy encoded node words managed lane link treated additional lane split ratios governing crossflows two links found driver choice model vehicle classes node model managed networks mentioned section make use node model discussed wright wright describe freeway network managed lanes node model differentiates others deals traffic flow optimally utilizes available supply makes use input link priorities relaxation common fifo constraint default link priorities taken proportional link capacities explain relaxed fifo say link vehicles wish enter links link jammed accept vehicles strict fifo constraint would say vehicles wish enter queue exit block vehicles wish enter road however certain lanes may queue traffic may still pass lanes relaxation encoded mutual restriction intervals interval partly describes overlapping regions link exit serve links portion lanes serve also serve blocked cars queueing enter congested example served three lanes three lanes leftmost also serves would example consider node figure say links four lanes managed lane links two lanes onramp merges offramp diverges rightmost lane say managed lane link congested vehicles lanes wish enter managed lanes queue leftmost lane hand link congested vehicles managed lanes wish enter lanes queue rightmost managed lane finally suppose jammed offramp cause vehicles queue rightmost lane taking together statements mutual restriction intervals example read tables recall written congested restricting link restricted link chosen restriction intervals allow expected behavior network congested link causing possible queueing right managed lane drivers trying enter link spillback left managed detailed discussion mutual restriction intervals included node model flow calculations solution algorithms see wright wright friction effect friction effect phenomenon situations managed lanes relatively uncongested traffic still slow adjacent lanes congest example assume managed lane two sublanes left right slow see daganzo cassidy liu cassidy hypothesized jang cassidy phenomenon arises drivers fear vehicles suddenly dangerously change managed lane ahead suggest modeling friction effect based feedback mechanism uses difference speeds parallel managed lane links scale flow therefore speed managed lane link necessary explain concept refer figure consider parallel links managed lane recall model speed traffic link time otherwise vlf theoretical free flow speed link time say friction effect present managed lane link following notation figure time min means link congestion speed current free flow speed speed link less speed managed lane link denote speed differential observed jang magnitude friction effect degree drivers slow towards lane traffic speed depends physical configuration road example less friction effect present managed lanes separated lanes buffer zone contiguous lanes jang presence concrete barrier would practically eliminate friction effect factors may affect magnitude include example whether one managed lane whether shoulder lane left managed lane drivers could swerve necessary encode variability magnitude friction effect managed lane link introduce friction coefficient link friction coefficient reflects strength friction value depends particular managed configuration chosen modeler value means friction may appropriate perhaps managed lane separated lanes concrete barrier means managed lane link speed tracks link speed exactly friction effect active true adjust fundamental diagram managed lane link scaling theoretical free flow speed vlf propagate change rest fundamental diagram parameters exact mathematical changes course different every different form fundamental diagram particular fundamental diagram discussed appendix means adjusting free flow speed capacity follows high critical density see appendix definition using adjusted values calculation sending function min fundamental diagram appendix must also check whether applying friction lead managed lane link speed falling link speed unadjusted sending function used possibility happening due use two critical densities create appendix fundamental diagram backwards lambda shape evidence friction effect reducing managed lane speed form linear speed differential found jang exact form sending function friction depend link original fundamental diagram model separated managed lanes gated access separated managed lane configuration presented figure unlike configuration managed lane link chains necessarily meet every node instead need meet locations vehicles move managed lane note unlike configuration need managed lane links aligned labeled figure nodes two link chains meet called gates aside one way describe managed lane configuration would every node gate similar construction excluding traffic managed lane case disable flow exchange given gate fixing split ratios keep traffic lanes example disable gate flow exchange two lanes node figure set means thus managed lane easily converted separated managed lane setting split ratios everywhere designated practice gate stretch freeway may hundreds meters long cassidy potentially designate two three sequential nodes gates paper however model gate single node figure freeway separated managed lane gates managed lane configuration suggest setting mutual restriction coefficients manner managed lanes section compared managed lane configuration configuration smaller friction effect jang drivers separated managed lane feel somewhat protected buffer whether virtual double solid line real concrete vehicles changing abruptly slow moving lane therefore drop speed dramatically degree friction effect mitigated disputed see footnote cassidy overall bottlenecks created gates much greater instigators congestion cassidy inclusion friction effect modeling separated managed lane configurations thus essential modeling case modeling flow vehicles managed lanes offramps recall section managed lane model model vehicles moving managed lane link offramps straightforward manner setting corresponding split ratios example node configuration figure configuration however modeling traffic moving managed lanes offramps complicated generally gates coincide offramp locations fact typically two five offramps two gate locations denote offramps road segment connecting two gates exits see figure offramps accessed directly managed lane instead vehicles traveling managed lane link intend take one exits must switch managed lane link link directed correct offramp creating simulation vehicles behave like requires involved modeling construction resolve challenge model introduces new vehicle classes addition special traffic used model section additional classes used distinguish subsets special traffic population destination offramp largest number offramps two adjacent gates altogether vehicle classes indicates class vehicles exit offramp leaving managed lane gate definition traffic type may exist lane segment gate offramp traffic type either link segment offramp gate managed lane link movement pattern ensured setting constant split ratios direct traffic link gate direct traffic offramp send traffic offramps denotes input link node output link see figure vehicles class enter network via onramps upstream boundary instead converted special traffic leaves managed lane gate perform part link model computation total demand switching link conversion remains switch exact link switching takes place managed lane link immediately upstream gate link figure say amount traffic change vehicle class vehicle class time using link example units vehicles per simulation timestep factor included switching done proportional link outflow rather density split ratio link immediately upstream exit exit time statement based assumption vehicles managed lane link exit freeway exit rate special vehicles happened stay lanes portion vehicles intend leave lanes exit portion vehicles managed lane leave managed lane link closest upstream gate leave network exit reach note offramps two particular gates vehicles switch type upstream gate would ramp exit using figure reference formally describe procedure destination assignment traffic managed lane link given vehicle counts per commodity free flow speed offramp split ratios given segment connecting two adjacent gates offramps assume initialize assign traffic set return step update state switches class traffic done may unresolved split ratios classes gates similar dashed split ratios tables section split ratios filled tools section sort driver lane choice behavior inertia effect end previous section mentioned split ratios likely undefined switching class vehicles classes particular class vehicles upstream link link figure remaining vehicles upstream managed lane link link figure need decide whether pass gate remain current lane sample tools modeling driver lane choices include class logit logistic regression models mcfadden dynamic split ratio solvers one presented wright reviewed appendix applied lane choice farhi logit models produce set portions one lane sum one lane value equilibrium portion vehicles travel lane dynamic simulation context considered article differences logit model equilibrium portions current distribution vehicles across lanes time used select split ratios time actual distribution approaches logit equilibrium argue though use sort split ratio solver unmodified form might inappropriate computing gate split ratios managed lane configuration unmodified value function either managed lane link might include terms link speed traffic density etc however often managed lane separated lanes buffer zone barrier makes switching two links hazardous switching two contiguous lanes therefore one uses model would appropriate modify logit model value function staying current link movements figure positive value drivers gains travel time movements figure must value value refer model inertia effect may also incorporate inertia effect dynamic split ratio solvers one introduced wright reviewed appendix time particular algorithm selects split ratios attempt balance density ratio next timestep ncl njl jam figure node input links form travel facilities output links managed lane density maximum number vehicles link hold example applied node figure algorithm would attempt make equal possible details see appendix wright modify several steps solver goal balanced goal towards enforcing inertia effect illustrate changes particular example node figure ensuring split ratio assignment algorithm gives preferences movements done step original algorithm setting oriented priorities specifically modify particular example original formula gives since get according yields input link forms one lane output link class vic modified follows split ratio defined priori form one lane different lanes parameter called inertia coefficient indicates strong inertia effect reduces original formula priori unassigned traffic link must stay lane directed output link choice lies modeler case example figure modified formula yields picking would give preference movement way choosing multiple input links obvious arbitrary choice may result unbalanced flow distribution among output links therefore suggest picking one pair input link setting input links setting input link must lane expected positive net inflow vehicles result split ratio assignment flows computed node model arg min input links vic pair links belongs lane denotes output link lane input link particular example node figure need determine whether flow link proceed link flow link link priori undefined split ratios computed according split ratio assignment algorithm priori unassigned traffic managed lane stay managed lane priori unassigned traffic coming links distributed links according dynamic split ratio solver hand priori unassigned traffic lane stay lane priori unassigned traffic coming links distributed links according dynamic split ratio solver putting together pieces managed network simulation model section present unified simulation algorithm managed networks considered simplified macroscopic simulation method briefly outlined section extensions made incorporating additional items described sections one definitions network consisting set links set nodes node always least one incoming link one outgoing link link may upstream node downstream node different vehicle classes traveling network classes indexed let denote simulation timestep addition covered modeler may optionally choose define control inputs simulated system modify system parameters system state control inputs may represent operational traffic control schemes ramp metering changeable message signs variable policy etc context paper suggest including parameterization friction effect construction managed lane model control actions link model definitions link let state vector denotes density link vehicle classes time element vector ncl updates timestep timestep according also define initial condition system define three types links ordinary links links beginning ending nodes origin links links ending node links represent roads vehicles use enter network destination links links beginning node links represent roads vehicles use exit network link define link model computes demands slc link supply function appendix describes particular example link model used example simulations section origin link define vector element dcl denotes exogenous demand class network link node model definitions node let denote incoming links denote outgoing links node define split ratios triplet split ratio may fully defined partially defined fully undefined split ratios node undefined also define node split ratio solver logit dynamic fill undefined split ratios time node define node model time takes incoming links demands sic split ratios outgoing links supplies nodal parameters computes flows fij paper refer specific node model relaxed fifo construction additional parameters mutual restriction intervals incoming links priorities state update equation definitions links update states according equation ncl ncl slightly generalized form ordinary destination links filc beginning node link origin links dcl ordinary origin links fljc ending node link destination links slc simulation algorithm initialization ncl perform control inputs optionally specified modeler purposes includes managed lane link modify sending function link model accordance friction effect model recall particular link model appendix use managed lane link whose downstream node gate node managed lane configuration perform class switching detailed section link commodity compute demand slc using link link model ordinary destination link compute supply using link model origin links supply used node one undefined split ratios use node split ratio solver complete set split ratios note inertia effect model used modified split ratio solver one described section used appropriate node use node model compute throughflows fij every link compute updated state ordinary link use origin link use destination link use stop otherwise increment return step calibrating managed network model typically traffic modeler set data collected traffic detectors velocity flow readings create network topology parameter values allow model reproduce values simulation parameters tweaked perform prediction analysis managed networks parameters interest fundamental diagram parameters link calibration fundamental diagram typically agnostic node model network topology exists abundant literature topic purposes simulations following section used method dervisoglu method appropriate percentage special able access managed lane vehicles traffic flow entering system parameter depends time day location well type managed lane could roughly estimated ratio managed lane vehicle count total freeway vehicle count periods congestion given location inertia coefficients parameters affect traffic different classes mixes different links effect total vehicle counts produced simulation friction coefficients tune parameters open question jang cassidy dependency managed lane speed lane speed investigated different densities managed lane presented data suggests although correlation two speeds exists overwhelmingly strong therefore suggest setting friction coefficients values exceeding mutual restriction intervals also open question estimate mutual restriction intervals measurement data see discussion section guidelines offramp split ratios calibrating traffic model identifying best values parameters match data typically involved process simplest network topologies particular consider single unbroken stretch freeway nonlinear nature network effects systems means estimating parameter isolation might lead unpredictable behavior instead nonlinear optimization techniques genetic algorithms poole kotsialos particle swarm methods poole kotsialos others ngoduy maher fransson sandin etc employed managed networks discussed key unknown parameters introduced offramp split ratios item may particularly hard estimate typically timevarying explicitly represent driver behavior rather physical parameters road items difficult identify using methods literature remainder section describes iterative methods identification offramp split ratios gatedaccess configurations split ratios managed lane consider node one whose output links offramp depicted figure shall make following assumptions total flow entering offramp given time known measurements restricted offramp supply portions traffic sent offramp managed lane lane given time equal none flow coming onramp link flow exists directed toward offramp words distribution flow portions directed offramp managed lane output links known written well known demand sic supply given given time unknown found known node model would compute flows particular define goal find equation sic obviously solution exist best case match set directing traffic links offramp suppose given assume monotonically increasing function assumption true particular node model wright moreover thus solution within given interval exists obtained using bisection method algorithm finding follows initialize enough vehicles satisfy offramp demand set stop use node model evaluate set stop use node model evaluate set stop update else update set return step represents lower bound search interval iteration upper bound split ratios separated managed lane gated access configuration node offramp one output links simpler case separated hov lane shown figure traffic directly hov lane link thus deal node caveat however recall section separated managed lane case traffic classes split ratios destinationbased traffic fixed figure node link onramp inputs link offramp outputs shall make following assumptions total flow entering offramp given time known measurements restricted offramp supply flow coming onramp link flow exists directed toward link words demand sic supply given denote set classes split ratios known let split ratios determined assume classes exit rate first three assumptions reproduce assumptions made managed lane case assumption reminder portion traffic flow direct away offramp account similarly managed lane case define function determined node model first term side depends assume monotonically increasing function look solution equation interval solution exists iff algorithm finding one presented previous section except initialized assumed iterative full calibration process purposes simulations presented following section placed iterative split ratio identification methods sections within larger iterative loop remaining parameters model calibration follows flowchart shown figure initial input data link fundamental diagrams input demand special traffic portion input demand initially guessed split ratios demand run simulation normal mode compute split ratios distribute traffic managed lane links start run simulation split ratio solver mode compute split ratios flows match demand run simulation normal mode produce density flow speed vmt vht yes tune input data demand input demand special traffic portion input demand simulation results match benchmark yes stop figure calibration workflow start assembling available measurement data fundamental diagrams assumed given mainline onramp demand specified per periods together special vehicle portion parameter indicating fraction input demand able access managed lane initially know offramp split ratios measured directly instead use arbitrary values represent call values initially guessed offramp split ratios instead offramp split ratios flows directed offramps refer offramp demand run network simulation outlined section entire simulation period point step simulation priori undefined split ratios traffic managed lanes assigned using split ratio solver using split ratios run network simulation time instead using initially guessed offramp split ratios compute given offramp demand described sections result step obtain new offramp split ratios run network simulation originally step time new offramp split ratios record simulation results density flow speed well performance measures vehicle miles traveled vmt vehicle hours traveled vht check resulting offramp flows match offramp demand yes proceed step otherwise repeat steps experience case studies following section takes process described steps two iterations converge evaluate simulation results correctness bottleneck locations activation times correctness congestion extension bottleneck correctness vmt vht simulation results satisfactory stop otherwise proceed step input data order shown block figure simulation results managed lane case study interstate north consider stretch north freeway contra costa county california postmile postmile shown figure test case managed lane configuration freeway managed lane hov lane allows entry vehicles two passengers stretch contains two hov lane segments whose beginning ending points marked map first hov segment miles long converted hot spring metropolitan transportation commission second hov segment miles long onramps offramps hov lane active rest time hov lane open traffic behaves lane figure map north contra costa county build model used data collected corridor system management plan csmp study system metrics group bottleneck locations well activation times congestion extension identified study using video monitoring tachometer vehicle runs onand offramp flows given increments assume hov portion input demand model calibrated typical weekday suggested csmp study simulation used fundamental diagram described appendix parameters follows capacity ordinary lane vehicles per hour per lane vphl capacity auxiliary lane vphl capacity hov lane vphl active vphl behaves lane free flow speed varies mph measurements came partially california performance measurement system pems california department tranportation partially tachometer vehicle runs congestion wave speed link taken free flow speed modeling results presented figures showing density flow speed contours respectively hov lanes plot top contour corresponds hov lanes bottom lanes plots traffic moves left right along absolute postmile axis vertical axis represents time bottleneck locations congestion areas identified csmp study marked blue boxes lane contours hov lane get congested speed drop due friction effect friction effect vehicles hov lane slow slow moving lane traffic seen hov lane speed contour figure figure shows example well offramp flow computed simulation matches target referred offramp demand recorded detector offramp crow canyon road see beginning end day computed flow falls target corresponding areas marked red circles due shortage mainline traffic simulation offramp demand satisfied finally table summarizes performance measurements vehicle miles traveled vmt vehicle hours traveled vht delay computed simulation versus collected course csmp study delay computed vehicles speed mph lane vmt hov lane vmt total vmt lane vht hov lane vht total vht lane delay hov lane delay total delay simulation result collected data table performance measures north figure north density contours hov lanes produced simulation density values given vehicles per mile per lane blue boxes lane speed contour indicate congested areas identified csmp study figure north flow contours hov lanes produced simulation flow values give vehicles per hour per lane blue boxes lane speed contour indicate congested areas identified csmp study figure north speed contours hov lanes produced simulation speed values given miles per hour blue boxes lane speed contour indicate congested areas identified csmp study figure flow crow canyon road offramp hours collected offramp demand computed simulation offramp flow managed lane case study interstate east consider stretch east los angeles county california shown figure test case separated managed lane configuration freeway managed lane also hov lane freeway stretch consists miles east postmile postmile merges miles east postmile postmile gate locations traffic switch hov lanes marked map site hov lane always active onramps offramps largest number offramps two gates thus freeway model vehicle classes lov hov figure map east freeway los angeles county build model used pems data corresponding segments east east monday october california department tranportation fundamental diagrams calibrated using pems data following methodology dervisoglu north example assume hov portion input demand modeling results presented figures showing density flow speed contours respectively hov lanes plot top contour corresponds hov lanes bottom lanes plots traffic moves left right along absolute postmile axis vertical axis represents time hov lane get congested dashed blue lines contour plots indicate hov gate locations figure shows pems speed contours east hov lanes used target simulation model plots traffic also travels left right horizontal axis representing postmiles vertical axis represents time figure shows example well offramp flow computed simulation matches target referred offramp demand recorded detector offramp north hill avenue simulated offramp flow matches offramp demand fairly closely similar results found offramps finally table summarizes performance measurements vmt vht delay computed simulation versus values obtained pems pems data come east east vmt vht delay values computed sums corresponding values two freeway sections delay values computed vehicles traveling slower mph lane vmt hov lane vmt total vmt lane vht hov lane vht total vht lane delay hov lane delay total delay simulation result pems data table performance measures east figure east density contours hov lanes produced simulation density values given vehicles per mile per lane figure east flow contours hov lanes produced simulation flow values give vehicles per hour per lane figure east speed contours hov lanes produced simulation speed values given miles per hour figure east speed contours hov lanes obtained california department tranportation monday october horizontal axis represents absolute postmile vertical axis represents time hours four contours share color scale figure flow north hill avenue offramp hours pems data offramp demand computed simulation offramp flow conclusion paper discussed modeling procedures two managed lane configurations full access special traffic switch managed lanes node separated special traffic switch two lanes specific nodes called gates introduced friction effect section inertia effect section friction effect reflects empiricallyobserved drivers fear moving fast managed lane traffic adjacent links moves slowly due congestion inertia effect reflects drivers inclination stay lane long possible switch would obviously improve travel condition presence interchanges freeways managed lanes produces modeling complications beyond one would see generic freeway model example separated managed lane requires special modeling trick sending vehicles traveling managed lane offramps locations gates offramps generally coincide paper proposed mechanism employs destinationbased traffic classes populated modeling links approaching gates using offramp split ratios section approach natural use feature many traffic models freeways managed lanes feature many parameters calibrating difficult presented iterative learning approach estimating parameters simulation results comparing model calibration results showed good agreement two case studies validating modeling techniques sequel paper extend results include traffic control simulation reactive tolling controller managed lane acknowledgement would like express great appreciation several colleagues elena dorogush ajith muralidharan sharing ideas gabriel gomes pravin varaiya critical reading help clarifying theoretical issues research funded california department transportation link model figure backwards lambda fundamental diagram majority paper remain agnostic particular functional relationship density demand per commodity ncl slc supply flow also called fundamental diagram used macroscopic model particular fundamental diagram required simulation results presented section example implementation friction effect use fundamental diagram horowitz shown figure fundamental diagram captures traffic hysteresis behavior backwards lambda shape often observed detector data koshi slc min link capacity vlf free flow speed congestion wave speed njl jam density vlf vlf vlf called low high critical densities respectively written used paper units per simulation timestep variable congestion metastate encodes hysteresis ncl examining see link density goes becomes congested ability receive flow reduced density falls image overlaid giving schematic image fundamental diagram shown figure unless shape fundamental diagram assumes triangular function density alone without admits two possible flow values dynamic split ratio solver throughout article made reference method solving partiallyor split ratios wright split ratio solver designed implicitly solve split ratio problem sij exp sij exp solved explicitly sij also functions problem chosen problem rely information link model beyond supplies demands thus independent choice link model wright solution algorithm follows reproduced wright discussion available reference define set commodity movements split ratios known set commodity movements split ratios computed unknown given input link commodity sic assume split ratios known define set output links exist unknown split ratios assuming given input link split ratios must sum define commodity unassigned portion flow given input link commodity exists least one commodity movement assume otherwise undefined split ratios trivially set every output link define set input links unassigned demand portion directed toward output link given input link commodity define set output links split ratios computed vic assume nonempty set contains least two elements otherwise single split ratio trivially set equal assume input link priorities nonnegative define set input links zero priority uzp enable split ratio assignment inputs zero priorities perform regularization denotes number elements set uzp expression implies regularized input priority consists two parts original input priority normalized portion input links positive priorities uniform distribution among input links normalized portion input links zero priorities note regularized priorities algorithm distributing among commodity movements assigning values priori unknown split ratios aims maintaining output links uniform ratios possible iteration two quantities identified largest oriented demandsupply ratio produced split ratios assigned far smallest oriented ratio whose input link denoted still unclaimed split ratio two quantities found commodity smallest unallocated demand demand directed corresponding bring possible due insufficient demand demand directed summarize iteration algorithm attempts bring smallest oriented thatpall oriented ratio largest oriented ratio turns ratios become perfectly balanced ratios sij well algorithm initialize otherwise split ratios undefined case could assigned arbitrarily remaining set input links unassigned demand may directed output link remaining set output links demand may directed stop split ratios calculate remaining unallocated demand sic link pairs calculate oriented demand sic link pairs calculate oriented priorities sic split ratio defined priori otherwise denotes number elements set vic examining expression one see split ratios fully defined yet complemented fraction inversely proportional number output links among flow commodity input link distributed note step using regularized priorities opposed original done ensure inputs ignored split ratio assignment find largest oriented ratio max max define set output links minimum oriented ratio achieved arg min min set pick output link smallest output ratio multiple minimizing output links minimizing output links may chosen arg min define set input links minimum oriented ratio output link achieved arg min set pick input link commodity smallest remaining unallocated demand arg min define smallest oriented ratio oriented demands created split ratios assigned iteration perfectly balanced among output links maintain remaining unassigned split ratios distributed proportionally allocated supply algorithm ends point emptied done else assign min take minimum remaining unassigned split ratio portion split ratio portion needed equalize better understand latter second term min rewritten right hand side last equality interpreted flow must assigned input output commodity equalize minus flow already assigned divided remaining unassigned portion demand commodity coming input link assigned split ratio portion incremented unassigned split ratio portion decremented computed set return step references bliemer dynamic queueing spillback analytical multiclass dynamic network loading model transportation research record california department 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verified compiler probability density functions manuel eberl johannes tobias nipkow jul technische bhat developed inductive compiler computes density functions probability spaces described programs simple probabilistic functional language work implement compiler modified version language within theorem prover isabelle give formal proof soundness semantics source target language together isabelle code generation inductive predicates yields fully verified executable density compiler proof done two steps using standard refinement approach first abstract compiler working abstract functions modelled directly theorem prover logic defined proven sound compiler refined concrete version returns expression introduction motivation random distributions practical significance often expressed probabilistic functional programs studying random distribution often desirable determine probability density function pdf used determine expectation sample distribution sampling method monte carlo mcmc bhat presented compiler computes probability distribution function program probabilistic functional language fun evaluated compiler number practical problems concluded reduces amount time effort required model mcmc system significantly compared models bhat also stated eventual goal formal verification compiler theorem prover advantage providing guaranteed correctness result compilation provably pdf source expression according formal semantics greatly increases confidence one compiler final publication available springer via https master thesis implemented compiler similar probabilistic functional language interactive theorem prover formally proved correctness related work work based work bhat presents density compiler probability spaces described expressions language fun small functional language basic arithmetic boolean logic product sum types conditionals number discrete continuous distributions however support lists recursion correctness proof purely pen paper authors stated formalisation proof assistant coq ultimate goal park developed probabilistic extension objective caml called work focuses sample generation using infinite stream random numbers compute samples distribution described probabilistic program bhat generate density functions functional programs park generate sampling functions functions map sample space sampling function effectively uses finite number random numbers interval returns sample desired random variable approach allows handle much general distributions even ones distributions density function allow precise reasoning distributions determining exact expectation attempt formal verification made pgcl another probabilistic language formalised proof assistant first hol hurd david cock pgcl contains rich set language features recursion probabilistic nondeterministic choice david cock formally proved large number results semantics language developed mechanisms refinement program verification however focus work verification probabilistic programs compiling density functions bhat mention reconciling recursion probability density functions difficult language pgcl probably suited developing density compiler audebaud also implemented probabilistic functional language recursion theorem prover namely coq focus also verification probabilistic programs cock uses shallow embedding type operation surrounding logic theorem prover hol used audebaud use deep embedding restricted type system expression structure predefined primitive functions etc like bhat also explicitly reference giry monad basis work utilised tools stated work interactive theorem prover isabelle generic proof assistant supports number logical frameworks object logics logic hol set theory widely used instance use heavily rely johannes isabelle formalisation measure theory already part library also use sudeep kanav formalisation gaussian distribution number libraries proof central limit theorem avigad namely notion interval set integrals fundamental theorem calculus integrals moved library respective maintainers well outline sect explain notation use give brief overview mathematical basics require particular giry monad sect section contains definition semantics source target language section defines abstract compiler gives outline soundness proof section explains refinement abstract compiler concrete compiler final correctness result evaluates compiler simple example section gives overview much work went different parts project difficulties encountered also lists possible improvements future work summarising results obtained finally appendix contains table notation auxiliary functions used work reference preliminaries notation following explain conventions notation use typographical notes use following typographical conventions mathematical constants functions datatype constructors types typeset regular roman font max return free bound variables including type variables italics dst etc embedded regular text often set types constants surrounding text italics distinguish isabelle keywords bold print lemma datatype primrec etc typeset calligraphic font etc categories typeset fraktur set grp meas etc file names isabelle theories set monospaced font deviations standard mathematical notation order maintain coherence notation used isabelle deviate standard mathematical notation following ways convention functional programming therefore isabelle function application often written instead operation binds strongest associates left integral integrand variable measure written instead special notion used integral ignores negative part integrand effectively max lambda abstractions used specify functions corresponding standard mathematical notation would table lists number additional mathematical notation semantics notation always use denote type environment function variable names types denote state function variable names values note variable names always natural numbers use bruijn indices letter upsilon used denote density contexts used compiler also employ notation resp denote insertion new variable type resp value typing environment resp state used entering scope bound variable newly inserted variable index variables incremented precise definition central concept measure theory library isabelle mostly use functions anyway distinction purely formal note analogy notation prepending element list contexts bruijn indices generally represented lists element entry variable inserting value newly bound variable simply prepending operation table general notation otation escription efinition undefined image set arbitrary value merge merging disjoint states density distr indicator function measure density measure undefined otherwise true otherwise result applying density otherwise use notation inserting new variable set variables shifting variables general notation variations thereof always mean expression type type environment whereas variations thereof mean expression compiles context mathematical basics category theory part section based mainly presentation ernsterich doberkat detailed introduction see textbook original paper giry basic measure spaces two basic measure spaces use counting space counting space countable set carrier set measurable sets subsets measure measure subset simply number elements set borel space borel space measure space real numbers carrier set borel measurable sets smallest containing open subsets borel measure measure uniquely defined measure maps real intervals lengths spaces space measurable space measure every set measure equivalently technical reasons also assume required later order define bind operation giry monad convenient way within isabelle nonemptiness condition always trivially satisfied measure spaces used work category meas note measurable space identity function measurable spaces function function function therefore measurable spaces form category meas objects category measurable spaces morphisms category measurable functions identity morphism identity function morphism composition function composition kernel space kernel space measurable space natural measurable space measures certain property purposes property measures defined sect additionally natural property kernel space satisfy measuring fixed set varying measure within function formally simply define smallest measurable space carrier set measure fulfils property let borel additionally measurable function define denotes measure image measure maps objects meas objects meas morphisms meas morphisms meas thus see endofunctor category meas giry monad giry monad naturally captures notion choosing value according probability distribution using parameter another distribution observing result consequently return yields dirac measure probability measure probability lies single element bind integrates input values compute one single output measure formally measurable spaces measure value measurable function return otherwise unfortunately restrictions due isabelle type system require determine resulting measurable space bind since information provided type done additional parameter convenient define bind way chooses arbitrary value takes count space empty choice somewhat difference practical significance use bind empty measure spaces simplify proofs bind instead define join operation also known category theory first use define bind join operation flattens objects maps element one operation naturally defined join note isabelle join additional explicit parameter measurable space result avoid problem bind makes expressions function thus obtained independent value chosen definition kernel space note containing join complicated however justified easier proofs unproblematic later since never use join directly bind bind defined using join following way modulo handling empty measure join join syntax better readability employ notation operations giry monad syntax notation defined recursively stands monadic expression patterni stands arbitrary raw text patterni pattern language syntax semantics source language used formalisation modelled language fun described bhat similarly target language almost identical target language used bhat however made following changes languages variables represented bruijn indices avoid handling freshness captureavoiding substitution related problems sum types supported consequently match command replaced else command furthermore booleans primitive type rather represented unit unit type double called real represents real number absolute precision opposed ieee floating point number beta gamma distributions included following sections give precise syntax typing rules semantics source language target language types values operators source language target language share type system operators figure shows types values exist additionally standard arithmetical logical operators exist lifting function values function measure spaces done distr function isabelle function already defined library simplifies proofs bind seeing proof work already done proofs distr note bool int real stand respective isabelle types whereas stand types datatype unit datatype val unitval boolval bool intval int realval real val val datatype fst snd add mult minus less equals pow fact sqrt exp inverse cast figure types values source target language operators total meaning every input value parameter type return single value result type requires definitions operations division logarithm square root define outside domain could also handled implementing operators giry monad letting return either dirac distribution single result evaluated parameter defined null measure however would probably complicate many proofs significantly increase readability use following abbreviations true false stand boolval true boolval false respectively realval intval etc omitted expressions presence implicitly clear context stands auxiliary definitions number auxiliary definitions used definition semantics full list auxiliary functions see table following two notions require detailed explanation measure embeddings measure embedding measure space obtained tagging values measure space injective function fact always datatype constructor instance set values type naturally measured stripping away realval constructor using measure real numbers measure resulting set reals formally stock measures stock measure type natural measure values type defined follows countable types unit count measure corresponding type universes type embedding measure realval embedding product measure note order save space increase readability often write instead integrals state measure using stock measure also construct measure states context typing environment state variables function maps variable value state maps every variable value type every variable undefined fix finite consider set states another representation states tuples component value variable natural measure given tuples finite product measure stock measures types variables source language figures show syntax resp typing rules source language figure defines source language semantics primitively recursive function similarly abbreviations mentioned sect omit val presence implicitly obvious context context expression constant real number write val realval distributions language contains distributions bernoulli uniformint uniformreal gaussian poisson uniform distributions parametrised pair numbers return uniform distribution interval gaussian distribution parametrised pair real numbers mean standard deviation table auxiliary functions unction escription dst dst dst dst det semantics operator applied result type operator input type parameter type distribution dst result type distribution dst distribution dst parameter density distribution dst parameter value unique type value intval integ set values type true iff countable set free variables expression true iff contain random fail returns realval analogous int pair etc return measure measurable space sets datatype expr var nat val val let expr expr expr expr expr random expr expr else expr fail figure source language syntax invalid parameters bernoulli uniformreal distributions return null measure deterministic expressions call expression deterministic written det contains occurrence random fail expressions particular interest free variables fixed value return precisely one value define function given state deterministic expression returns single value isabelle expression randomfree used instead deterministic hence suffix order emphasise syntactical nature property additionally worth noting syntactically deterministic expression truly deterministic variables contains randomised case sometimes val var fail val var fail pair let else rand let dst random dst dst figure typing rules expressions definition obvious leads following equality assuming deterministic valid state return property later enable also convert deterministic expressions equivalent expressions target language target language modelled closely one used bhat type system operators source language key difference expressions source language return measure space result type expressions target language always return single value since source language lacks sum types target language additionally target language differs used bhat following respects language function types since functions occur integrands final results compilation result density function simply define integration introduce integration variable bound variable let final result contain single free variable bruijn index implicit abstraction around compilation result evaluation expressions target language never fail language bhat failure used handle undefined integrals hand use convention isabelle measure theory library returns integrals functions advantage keeping semantics simple makes proofs considerably easier target language let bindings since contrast source language would semantically superfluous however still useful practice since yield shorter expressions avoid multiple evaluation term could added little effort figures show syntax typing rules semantics target language primrec state expr val measure val var let else true else random dst dst fail figure semantics expressions datatype cexpr cvar nat cval val cexpr cexpr cexpr ifc cexpr cexpr else cexpr cexpr figure expressions target language cet val cet var cval cvar cet cet pair cet cet int ifc else figure typing rules target language primrec state cexpr val cval cvar ifc else true else realval figure semantics target language converting deterministic expressions auxiliary function used rules compiler handle deterministic expressions particular interest mentioned earlier deterministic expressions converted equivalent function precisely definition obvious satisfies following equality deterministic expression abstract compiler density contexts first define notion density context holds acquired context data compiler require compute density expression density context tuple contains following information set random variables current context variables randomised set parameter variables current context variables may occur expression randomised treated constants type environment density function returns common density variables parameters function space extended real numbers density context describes parametrised measure states let space set parameters write measure obtain taking transforming merging given state parameter state finally applying density resulting image measure isabelle definition definition state density distr merge bhat say deterministic expression also expression target language syntax silently treat informally describes measure obtained integrating variables treating variables parameters evaluation expression variables context effectively function density context implemented isabelle locale finite disjoint space space measure definition first step implemented abstract density compiler inductive predicate density context expression function type val state val ereal first parameter state assigns values free variables second parameter value density computed compiler therefore computes density function parametrised values free variables source expression compilation rules virtually identical bhat except following adaptations bhat handle else case match rule sum types support sum types dedicated rule else use bruijn indices requires shifting variable sets states whenever scope new bound variable entered unfortunately makes rules somewhat technical provide compiler support deterministic let bindings semantically redundant always expanded without changing semantics expression fact unfolded compilation regarded feature adds convenience expressivity following list shows standard compilation rules adapted bhat plus rule multiplication note functions compute marginal density one resp two variables integrating away variables common density function computes probability current branch execution integrating variables common density val val var var pair var var fail fail let let undefined rand random dst dst dst rand det det random dst dst true false else det additionally three congruence rules required technical reasons rules required abstract concrete result may differ null set outside domain det truei falsei else fst fst snd snd snd fst discr neg addc multc det val realval add inv exp exp else figure abstract compilation rules soundness proof show following soundness result abstract compiler lemma assumes shows undefined abbreviation following four facts applying density yields measure domain prove following generalised auxiliary lemma lemma assumes shows predicate simply means borelmeasurable parameter state predicate holds proof straightforward induction following inductive definition abstract compiler many cases monad laws giry monad allow restructuring induction goal way induction hypothesis applied directly cases definitions monadic operations need unfolded goal essentially show two integrals equal output produced proof given bhat analogous much concise due fact side conditions measurability integrability proven explicitly many important uninteresting steps skipped hinted note since abstract compiler returns parametrised density functions need parametrise result state undefined even expression contains free variables concrete compiler approach concrete compiler another inductive predicate modelled directly abstract compiler returning expression compilation result instead hol function use standard refinement approach relate concrete compiler abstract compiler thus lift soundness result abstract compiler analogous one concrete one effectively shows concrete compiler always returns expression represents density space described source language concrete compilation predicate written lists variables typing environment targetlanguage expression describing common density random variables context may parametrised variables definition concrete compilation rules course direct copy abstract ones abstract hol operations replaced operations expressions due bruijn indices lack functions explicit objects target language rules somewhat complicated inserting expression scope one bound variables integral requires shifting variable indices inserted expression correctly reason print rules found isabelle theory file refinement refinement relates concrete compilation abstract compilation set set words take abstract compilation predicate variable sets refined variable lists typing context expression remain unchanged common density context compilation result refined hol functions expressions applying target language semantics main refinement lemma states concrete compiler yields result equivalent abstract compiler modulo refinement informally statement following ground concrete density context abstract versions proof conceptually simple induction definition concrete compiler practice however quite involved every single induction step intermediary expressions needs shown congruence lemmas abstract compiler need applied integration involved integrability shown order convert integrals lebesgue integrals integrals product spaces iterated integrals combining main refinement lemma abstract soundness lemma easily show concrete soundness lemma lemma assumes shows real informally lemma states ground expression compiling empty context yield targetlanguage expression representing density function measure space described final result summarise soundness lemma proven concise manner convenience define symbol read type compiles includes groundness requirements well compilation unit unit final soundness theorem compiler stated isabelle syntax definition choice typing environment course completely arbitrary since expression contains free variables precise lemma statement isabelle slightly different better readability unfolded one auxiliary definition omitted type cast real ereal lemma assumes fixes shows density real words result means following theorem let expression compiler determines type returns expression measure obtained taking stock measure using evaluation density measure obtained evaluating input values type free variables indeed type type context free variable except parameter variable function isabelle code generator allows execute verified compiler using values generate code one target languages standard haskell evaluation example test compiler choose expression chosen bhat let random uniformreal let random bernoulli else compiler inherently since may return zero one many density functions seeing expression may matching compilation rules one therefore must use values command instead value command receive set compilation results val realval omitted better readability symbolic variable names used instead bruijn indices abbreviate expression display result compilation using isabelle command values result singleton set contains pair real long complicated expression simplifying constant subexpressions expressions form fst replacing bruijn indices symbolic identifiers obtain else else else hbi else else else simplification yields following result result bhat reached compiler generates much larger expression one printed reason printed version compiler output particular constant subexpressions evaluated simplification course useful using compiler practice implemented since conceptually interesting outside scope work conclusion breakdown formalisation compiler took three months contains total roughly lines isabelle code definitions lemma statements proofs examples table shows detailed breakdown table breakdown isabelle code type system semantics abstract compiler concrete compiler general measure theory auxiliary lemmas lines lines lines lines seen table sizeable portion work formalisation results general measure theory substitution lemmas lebesgue integrals auxiliary notions lemmas measure embeddings since utility formalisations restricted particular project moved isabelle measure theory library difficulties main problems encountered formalisation missing background theory mentioned previous section sizeable amount measure theory auxiliary notions formalised notably existing measure theory library contain integration substitution side product formalisation central limit theorem avigad proved fundamental theorem calculus building thereupon integration substitution however lemma supported continuous functions whereas required theorem general functions using proof fundamental theorem calculus proved lemma initially comprised almost lines proof shortened significantly thereafter proving side conditions many lemmas measure theory library require measurability integrability etc proofs often implicitly dismissed trivial formal proof proving blow proofs render complicated technical measurability proofs particular ubiquitous formalisation measure theory library provides tools proving measurability automatically quite helpful many cases still work progress require tuning lambda calculus bhat use language symbolic identifiers target language paper proof obvious choice since leads concise familiar definitions formal setting always comes typical problems deal variable capture related issues reason chose use bruijn indices instead however makes handling terms less intuitive since variable indices need shifted whenever several terms combined another issue lack function type target language allowing firstorder functions bhat would made many definitions easier natural would complicated others significantly due measurability issues furthermore effectively needed formalise number properties lambda calculus used implicitly paper proof future work following improvements isabelle formalisation could probably realised little effort sum types match statement compiler rule deterministic let bindings let bindings target language preprocessing stage allow normal variables instead bruijn indices postprocessing stage automatically simplify density expression far possible first interesting substantial weakness formalisation opposed bhat remaining four merely make using compiler convenient efficient additionally long term monte carlo mcmc sampling method algorithm could also formalised isabelle integrated density compiler however would involved project require formalisation additional probability theory isabelle summary using formalised semantics simple probabilistic functional programming language predefined probability distributions compiler returns probability distribution program language describes modelled closely given bhat used existing formalisations measure theory formally prove correctness compiler semantics source target languages shows compiler given bhat correct also formal correctness proof compiler done reasonable effort general measure theory library particular suitable references audebaud proofs randomized algorithms coq mathematics program construction lecture notes computer science vol springer berlin heidelberg http avigad serafin formally verified proof central limit theorem corr bhat agarwal vuduc gray type theory probability density functions proceedings annual acm symposium principles programming languages popl acm new york usa http bhat gordon russo deriving probability density functions probabilistic functional programs tools algorithms construction analysis systems lecture notes computer science vol springer berlin heidelberg http best paper award bhat gordon russo deriving probability density functions probabilistic functional programs todo springer berlin heidelberg cock verifying probabilistic correctness isabelle pgcl proceedings systems software verification november cock pgcl isabelle archive formal proofs jul http formal proof development doberkat stochastic relations foundations markov transition systems studies informatics chapman doberkat basing markov transition systems giry monad http giry categorical approach probability theory categorical aspects topology analysis lecture notes mathematics vol springer berlin heidelberg http construction stochastic applications measure spaces logic phd thesis technische institut informatik hurd mciver morgan probabilistic guarded commands mechanized hol electron notes theor comput sci jan http park pfenning thrun probabilistic language based upon sampling functions proceedings acm symposium principles programming languages popl acm new york usa http appendix operator semantics perator add minus mult inverse nput type utput emantics type sqrt exp unit otherwise otherwise otherwise fact equals less pow otherwise fst snd truei real number truei cast real cast int distributions istribution param omain ype ensity function true false bernoulli uniformint uniformreal gaussian poisson exp exp otherwise figure distributions source language density functions given terms parameter type given column parameter type product type stand two components function variable point density function evaluated target language auxiliary functions definitions following functions see corresponding isabelle theory file print since rather technical section always variable always value always target language expressions always target language expressions interpreted functions implicit abstraction around functions table take bruijn indices account shift variables enter scope integral unction escription else transforms variables function suc prepares expression insertion scope new variable prepares function insertion scope new variable shifts variables deleting variable substitutes substitutes cvar effectively applying interpreted function argument function composition alias converts deterministic source language expression target language expression see sect integrates free variable integrates free variables list returns expression computes returns expression computes returns expression computes dst returns expression computes density distribution dst parametrised evaluated concrete compiler rules edc val val hcvar cval edc var set var edc pair set set edc fail fail cval realval edc let map suc map suc let edc rand random dst dst cvar cvar dst edc rand det set det random dst dst cvar edc truei falsei else edc det det else edc discr oper oper hoper cvar cvar edc fst fst cvar cvar edc snd snd cvar cvar edc neg edc addc det set cvar edc multc val realval cvar edc add cvar cvar cvar edc inv cvar edc exp exp cvar cvar cvar else
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flare native compilation heterogeneous workloads apache spark ruby james kevin kunle tiark purdue university stanford university mar gesserte rtahboub tiark kjbrown kunle abstract sql still performs least order magnitude worse relational query engines like hyper part due fact relational query engines optimize query plans aggressively relational operator level also compile queries native code thus operating closer metal spark current techniques one might argue fair comparison due added expressiveness spark different nature systems show actually possible bring performance spark much closer relational engines without sacrificing added expressiveness present flare new spark yields significant speedups compiling entire query plans obtained spark query optimizer native code bypassing inefficient abstraction layers spark runtime system need modern data analytics combine relational procedural functional processing widely recognized systems like spark added sql relational query optimization promise increase expressiveness performance good extensions extracting high performance modern hardware platforms spark made impressive progress show relational workloads still significant gap compared query engines stepping outside relational world query optimization techniques ineffective large parts computation treated functions udfs present flare new spark brings performance closer best sql engines without giving added expressiveness spark demonstrate order magnitude speedups relational workloads well range machine learning kernels combine relational iterative functional processing flare achieves results compilation native code replacing parts spark runtime system extending scope optimization code generation large classes udfs heterogeneous workloads flare native code translation alone already provides excellent performance sql queries dataframe operations performance still suffers heterogeneous workloads executing many small queries interleaved user code combining relational iterative functional processing common machine learning pipelines extract transfer load part etl pipelines often implemented dataframes compute kernels often supplied functions udfs appear black boxes query optimizer thus remain unoptimized library code workloads flare takes advantage delite existing compiler framework languages dsls alternative generating target code single step flare map spark query plans delite intermediate language dmll enables spark interface udfs written existing delite dsls cover domains machine learning optiml graph processing optigraph pde solvers optimesh dsls embedded scala provide apis comparable built top spark unlike scala code uses normal spark apis delite dsl code amenable optimization native code generation including gpu code computational kernels show combining spark sql queries dataframes machine learning kernels written optiml particular results order magnitude speedups compared standard spark implementation introduction modern data analytics applications require combination different programming paradigms spanning relational procedural functional processing shortcomings expressiveness performance key reasons excitement around early mapreduce tools heralded solution big data problems tapered instead systems like spark added sql apis enable relational query optimization good extensions demonstrate standard relational benchmarks spark preprint march copyright authors user programs sql optiql java scala python jdbc console optiml optigraph spark sql dataframe api export query plan delite catalyst optimizer dmll code generation jni flare code generation flare code generation lms code generation optimized scala flare runtime spark native code delite runtime resilient distributed dataset spark sql flare level flare level flare level figure flare system overview architecture spark spark sql adapted flare level replaces spark jvm code generation native code generation granularity spark query stages flare level generates code entire queries eliminating rdd layer orchestrating parallel execution optimized shared memory architectures flare level translates spark query plans delite established framework languages enables optimization relational queries together udfs implemented one dsls implementation flare revisit design decisions spark sql rooted legacy spark earlier systems show alternative implementations start different assumptions provide better performance expressive api one key assumption mapreduce hadoop spark distributed architecture focusing primarily makes easy scale computation adding machines also means individual machine may used efficiently immediate consequence increase datacenter bills global scale effects approaches nobody ever got fired running hadoop cluster inefficient use energy may accelerating global warming today machines dozens cores memory range readily available rent purchase time writing amazon instances offer main memory cores hardware threads machines dell configured cores hardware threads nvidia advertises latest system supercomputer box compute power equal hundreds conventional servers powerful machines becoming increasingly commonplace large clusters less less frequently needed many times big data big often computation bottleneck small cluster even single large machine sufficient used full potential scenario primary target heterogeneous workloads small clusters powerful machines potentially accelerators gpus flare prioritizes performance levels scaling option bypassing mechanisms execution thus flare strengthens spark role unified efficient big data platform opposed mere cluster computing fabric overall architecture flare three possible levels integration shown figure paper first reviews design spark spark sql section continues main contributions present flare new accelerator spark identify key impediments performance spark sql particularly related spark java execution environment immediate remedy flare level generates native code instead java improved performance section identify performance issues spark sql specifically related joins environments flare level drastically reduces overheads making different starting assumptions spark focusing instead reducing constant factors throughout particular flare level compiles whole queries opposed individual query stages results optimized data path section extend flare accelerate heterogeneous workloads consisting relational queries combined iterative machine learning kernels written userdefined functions flare level uses delite compiler framework optimize relational queries together udfs means intermediate language section evaluate flare comparison spark reducing gap relational query engines benchmarks involving heterogeneous machine learning workloads settings flare exhibits speedups evaluation spans numa cluster gpu targets section background spark apache spark today popular big data framework core programming abstraction immutable implicitly distributed collection data type called rdd resilient distributed dataset rdds serve programming interfaces also transparently manage quick example counts number errors potentially distributed log file physical plan filter startswith value error scan text value format textfileformat inputpaths readschema struct value string val lines val errors error println total errors spark rdds provide deferred api example calls textfile filter merely construct computation graph actual computation takes place point invoked sometimes rdds described lazily evaluated somewhat misleading second call entire however rdds support memoization via explicit calls mark data set kept memory future operations dataframe operations spark sql internally computes query plan much like relational dbms spark sql optimizes query plans using relational query optimizer called catalyst may even generate java code runtime accelerate parts query plan using component named tungsten see section hard overstate benefits kind api generates complete program query representation runtime first enables various kinds optimizations including classic relational query optimizations second one use api multiple exposes spark languages python api also serve translation target literal sql spark sql dataframe api directed acyclic computation graph represented rdd describes distributed operations coarsegrained way granularity map filter level detail enough enable computation scheduling via selective recomputation along lineage result provide full view computation applied element data set example code snippet argument normal scala closure makes easy integrate rdd code arbitrary external libraries also means given closure needs invoked every element data set thus performance rdds suffers two limitations first limited visibility analysis optimizations especially standard optimizations like join relational workloads expressed rdds second interpretive overhead function calls processed tuple recent spark versions aim ameliorate issues introduction spark sql subsystem lines val errors select lines value like error println total errors third one use full host language structure code use small functions pass dataframes build logical plan optimized whole course true long one stays relational world use udfs part contribution show section dataframe model extends udfs flare flare uses existing generative programming frameworks make larger classes expressions available programming patterns general piece program code builds intermediate representation specific computation runtime like dataframes query plans argue apis including spark dataframes real key success unified efficient big data platforms completely independent rdds show section underlying rdd layer actually impediment performance spark flare demonstrates spark dataframe api successfully supported generic dsl compiler framework removes confinement relational query optimizations enables optimization entire data processing pipelines combine relational processing iterative machine learning kernels would require unoptimized udfs plain spark power apis dsls core addition spark sql alternative api based dataframe conceptually equivalent table relational database collection rows named columns however like rdds dataframe api records operations compute result right away write example val lines val errors value error println total errors like rdd api call trigger execution unlike rdds however dataframes capture full executed obtain internal representation using catalyst tungsten spark sql queries optimized using catalyst optimizer supports optimizations starting logical query plan tree catalyst performs optimizations tree transformations order realize optimized plan important note time writing catalyst yet perform kind join reordering instead simply joins tables order appear clause hence users need manually encode good join order avoid big intermediate tables bad surprises tungsten execution aims improve spark performance reducing allocation objects jvm java virtual machine heap controlling memory management employing data produces following output original definition term lazy evaluation means term evaluated needed internally spark distinguishes type dataset provides typesafe collection api elements type type dataframe dataset row provides untyped api rows arbitrary arity column names purpose paper difference immaterial hence use terms dataset dataframe interchangeably val select sum revenue lineitem data loading elided size double double double long revenue figure query benchmark spark figure code tures generating java code compiled jvm bytecode runtime optimizations simultaneously improve performance spark sql libraries dataframe operations spark sql preload direct csv preload csv flare preload csv figure running times spark without compared code flare flare adding fuel fire present flare new spark sql achieves better performance without changing spark userfacing flare efficiently compiles spark sql queries native code reduces internal overhead brings performance closer sql engines flare integrate spark three levels illustrated figure level enables native compilation within tungsten level granularity level goes compiling whole queries instead query stages effectively bypassing spark rdd layer runtime operations like hash joins environments finally level goes beyond purely relational workloads adding another intermediate layer query plans generated code flare level employs delite compiler framework efficiently compile query pipelines heterogeneous workloads consisting code multiple dsls sql motivate describe three levels following sections query benchmark pick simplest query benchmark query shown figure define schema table lineitem provide source file finally register temporary table spark sql steps shown experiments use scale factor data set means table lineitem stored csv file following setup mcsherry run tests fairly standard run query straight csv file input record running time scala val revenue double scala time completed clearly result seconds best see figure could convert data columnar apache parquet format increased performance preload data subsequent runs purely since mainly interested computational part opt preload note passing preloading quite slow almost min may due variety factors execute query get much better result running query times yields speedups timings stagnate around key question good result fast spark spark performance studies primarily focus scaleout performance running big data benchmarks clusters performing terabyte sorting etc however mcsherry isard murray eloquently argued hotos paper accompanying blog post big data systems spark tend scale well often lot internal overhead particular mcsherry demonstrate straightforward native implementation pagerank algorithm running single laptop outperform spark cluster cores using version since query simple perfectly feasible write program performs exactly computation map input file memory using mmap system call load data columnar representation execute main query loop shown figure compile program gcc run take total including data loading actual query computation compared spark program runs faster good reason code needs faster spark believe fact running laptop cluster inspired setup interested gauging inherent overheads spark spark sql absolute terms second computer shown know use first paul barham via focus simple relational query profit query optimization code generation thus scenario spark launch spark shell configured use single worker thread macbook pro retina ghz intel core mhz ssd spark java hotspot master local double samples jobs stages storage details query submitted time duration succeeded jobs reset zoom bytebuffer dec nativecolumnaccessor mutablerow mutablerow int mutablerow int apache spark sql execution columnar nullablecolumnaccessor super nullablecolumnaccessor mutablerow int mutablerow int generatedclass generatediterator anonfun anon inmemorytablescan spark join join exchange flare hash join number output rows wholestagecodegen inmemorytablescan filter number output rows number output rows wholestagecodegen broadcastexchange filter number output rows data size bytes time collect time build time broadcast broadcasthashjoin number output rows project figure join lineitem orders cost different operators left spark hash join plan right shows three separate code generation regions communicate spark runtime system figure cpu profile spark sql lineitem table time large hump middle right spent accessing decoding inmemory data representation hashaggregate number output rows aggregate time total min med max exchange data size total min med max wholestagecodegen hashaggregate number output rows aggregate time total min med max collectlimit details emitted generated code lms compared systems like llvm operates higher level abstraction context query compilation lms used part legobase engine query accelerated flare yields exactly performance code holds rdbmss like hyper takes comparable machine identifying bottlenecks understand mance gains need investigate spark sql spends time two reasons performance difference particularly visible case low computational content uses trivial query operators first observe spark tungsten layer generates java code query fact two pieces code generated one piece main query loop iterator traverse data structure looking cpu profile figure see execution time spent accessing decoding data representation going back forth two pieces generated code code paths part spark runtime second even remove indirection replace unified piece java code performance remains lower difference gets pronounced queries tight control data structures memory management required flare level already provide significant speedups number queries also constrained architecture spark runtime flare level takes liberties revisits key design assumptions spark sql spark sql operates legacy spark workloads assumed big reasonable way processing humongous amounts data large number distributed unreliable machines furthermore spark sql inherits spark rdd abstraction encodes lineage support fault tolerance necessity distributed environments machines however many realistic workloads fact modern hardware faults statistically highly improbable fault tolerance often supported hardware raid arrays redundant power supplies similar facilities accelerating spark sql performance way level modern query engines hyper requires fully compiled removes remaining inefficiencies legacy spark file page flare level architecture flare level figure adds native compilation within tungsten startup flare installs number rules catalyst optimization logic query optimized catalyst rules trigger tungsten invoke flare generate code supported operators combinations operators flare invokes compiler loads generated native code running jvm java native interface jni spark runtime execute generated code part query evaluation flare level modify spark internals memory management data format rdd layer etc hence flare level serve lightweight accelerator increases performance certain queries necessarily interfere sparks execution model transparently supports standard cluster execution faul tolerance behavior like tungsten flare query compiler implements neumann model fuses pipelines operators need materialize intermediate results code generation flare realized using lightweight modular staging lms generative programming compiler framework uses scala type system distinguish normal code expressions generate code lms special type constructor rep used denote staged expression cause expression type flare level bottlenecks complex queries concerns granularity code generation interfacing runtime system become pronounced previous example query fact queries joins exhibit unfortunate consequences execution due spark design primarily cluster computing framework figure shows timings simple join query joins lineitem orders tables tpch benchmark spark query optimizer picks expensive join default may right choice distributed execution suboptimal right flags possible tune spark catalyst query planner prefer hash join instead efficient even hash join operator follows broadcast model high overhead internal exchange operator present physical plan even running single core looking spark query plan hash join query reveals tungsten split query three code generation regions dark blue areas right side figure need communicate spark runtime system therefore surprise flare achieve much faster query execution generating tight code entire query environment executors sql val tol val def findnearestcluster rep densevector double rep densematrix double rep int maprowstovector row dist row square relational etl val data data input val val mat flare dsl user code training loop val densetrainingset mat densevector double val val val randomint val newmu tol iter val findnearestcluster val allwp dist square relational dsls val select data class findnearestcluster group class flare memory columnar data figure thread pinning specific cores numa flare level architecture architecture flare level illustrated figure spark sql dataframe api catalyst optimizer remain optimized query plan exported wholesale catalyst flare flare compiler iterates recursively operator node query plan tree maps flare internal optimized data structures query execution logic represented lms lms performs light optimizations like common subexpression dead code elimination generates invokes compiler flare launches resulting binary either inside jvm like level separate process cuts rest spark relies solely flare runtime trigger execution generated code contrast flare level operates transparently user flare level exposes dedicated api let users pick dataframes evaluate flare figure clustering flare combining sql spark dataframes optiml val create dataframe sql direct val flare turn flaredataframe execute query plan flare flare level currently support cluster execution would possible extend spark runtime set hooks would delegate flare within single machine coordinate multiple cluster nodes necessary data exchanges existing spark fabric setup would similar hadoopdb combines mapreduce processing cluster nodes fast dbms instances node able handle computation efficiently hadoop parallel numa execution query engines realize parallelism either explicitly implementing special split merge operators internally modifying operator internal logic orchestrate parallel execution flare level uses latter realizes parallelism using openmp architectural level flare operators implementations take care splitting work internally across multiple threads accumulating final results etc instance parallel scan operator starts parallel section produce method sets number threads invokes consume downstream operators parallel join aggregate operators turn implement materialization points implement consume method way parallel invocations possible without conflict either data structures merged parallel section data structures flare also contains specific optimizations environments memory access numa including pinning threads specific cores optimizing memory layout various data structures reduce need accessing memory instance workloads perform small amounts computation given large number threads single cpu socket flare code generation supports workloads various data partitioning strategies order maximize local processing reduce need threads access memory illustrated figure section optimizing data loading data loading often overlooked factor data processing seldom reported benchmarks however recognize data loading csv often dominant performance factor spark sql queries apache parquet format attractive alternative modeled dremel binary columnar format offers opportunities compression queries load required columns instead data keeping running theme better parquet allows irrelevant data ignored almost entirely spark code read parquet files generic resulting undue overhead generality primarily due supporting multiple compression encoding techniques also exists overhead determining column iterators needed sources overhead seem somewhat unavoidable reality resolved generating specialized code flare implement compiled csv parquet readers generate native code specialized given schema result flare level compile data paths flare level many data analytics applications require combination different programming paradigms relational procedural functional processing example machine learning application might use relational apis etl dedicated libraries computations spark provides specialized libraries pipelines supports functions support applications unfortunately spark performance falls proverbial cliff dataframe operations interleaved user code currently spark sql optimization code generation treats user code black box hence flare level focuses generating efficient code heterogeneous workloads data point assigned partition nearest mean application mixes multiple dsls sql optiml user code lines spark sql reads data file preprocesses input lines show processing code using optiml delite provides optimized data types vectors expressive libraries assist computations instance maprowstovector dist optimized methods final result using sql illustrated lines user defined functions udf spark sql uses scala functions appear black box optimizer mentioned section flare internal code generation logic based technique called lightweight modular staging lms uses special type constructor rep denote staged expressions type become part generated code extending udf support flare achieved injecting rep rep functions dataframes way normal functions plain spark example udf sqr squares given number define register udf def sqr flareudfcontext import rep int sqr sqr use udf query val select partsupp sqr flare relational query execution first set experiments focuses standard relational workload demonstrates inherent overheads spark sql cause slowdown least compared best available query engines execution single core experiments show flare level able bridge gap accelerating spark sql level performance query compiler systems retaining flexibility spark dataframe api also compare parallel speedups effect optimization evaluate performance benefits optimized data loading environment conducted experiments single numa machine sockets xeon cores per socket ram per socket total operating system ubuntu lts use spark scala postgres hyper gcc optimization flags dataset use standard benchmark scale factor sequential execution parallel execution notice definition sqr uses additional argument type flareudfcontext import overloaded operators work rep int rep types staged function become part code well optimized along relational operations provides benefits udfs general purpose code embedded queries enables queries optimized respect surrounding code queries run within loop experimental evaluation assess performance acceleration potential flare comparison spark present two sets experiments first set focuses standard relational benchmark second set evaluates heterogeneous workloads consisting relational processing combined machine learning kernels udfs experiments span numa cluster gpu targets heterogeneous workloads dsls flare level figure goes even leverages existing compiler framework called delite built top lms efficiently compile applications mix multiple dsls query delite particular focus parallel distributed applications well heterogeneous hardware gpus delite frontend provides several scala dsls covering various domains databases machine learning linear algebra etc since dsls embedded scala provide apis similar libraries built top spark flare level exports spark sql optimized query plan maps delite optiql similar process explained section core delite built around functional intermediate language called distributed language dmll models parallel patterns name suggests dmll represents parallel patterns highly flexible loop abstraction fuses multiple loops maximize pipeline horizontal fusion furthermore dmll provides implicitly parallel nested collection operations collect group well optimizations like loop fusion loop nesting optimizations finally dmll performs analysis optimizations multiple hardware targets numa clusters gpu figure shows code highlights popular application partitions data points clusters running time experiment compare absolute running time flare level postgres hyper spark using benchmark scale factor case spark use single executor thread though jvm may spawn auxiliary threads handle compilation postgres hyper implement optimizers avoid inefficient query plans particular reordering joins spark catalyst optimizer also perform kind join hence match join ordering query plan spark sql flare hyper small number exceptions spark sql original join ordering given reference outperformed hyper plans spark sql flare queries kept original join ordering spark sql difference mainly due catalyst picking sortmerge joins hash joins worth pointing hyper postgres plans use indexes primary keys may give additional advantage figure gives absolute execution time postgres hyper spark sql flare queries postgresql running time spark hyper flare postgres spark sql hyper flare figure performance comparison postgres hyper spark sql flare level systems data loading time excluded execution time reported spark flare use persist ensure data loaded memory first glance performance flare hyper lie within range notably outperform postgres spark queries similarly spark performance comparable postgres queries unlike systems postgres compile queries runtime relies volcano model query evaluation incurs significant overhead hence see spark query compilation provide significant advantage standard interpreted query engines queries closer look flare outperforms spark sql aggregate queries respectively observe spark order magnitude slower flare nested queries like found examining execution plans found catalyst plan detect patterns help avoiding table previously scanned sorted join queries flare faster spark sql likewise join variants outer join flare faster respectively performance gap spark sql flare attributed bottlenecks identified sections first overhead associated data access jvm second spark sql strategy employs costly distributed operators sortmerge join broadcast hash join even running single core third internal bottlenecks processing overhead rdd operations communication spark runtime system compiling entire queries instead isolated query stages flare effectively avoids bottlenecks hyper compiled relational query engine precursory look shows flare faster hyper moreover flare faster hand hyper faster flare moreover hyper faster performance gap part attributed hyper use specialized operators like groupjoin employing indexes primary keys seen whereas flare spark sql currently support indexes order understand performance gained using indexes disabled indexing hyper benchmarks detailed sults omitted observed flare outperformed hyper matched performance hyper outperformed flare performance gap shrunk finally flare outperformed hyper performance gap increased summary flare hyper generate native query code runtime subtle implementation differences query evaluation code generation result faster code instance hyper uses proper decimal precision numbers whereas flare follows spark using double precision floating point values native architecture furthermore hyper generates llvm code whereas flare generates code compiled gcc compilation time compared compilation time query spark flare detailed results shown spark measured time generate physical plan includes java code generation compilation quantify jit compilation hard measure code may recompiled multiple times flare measured code generation compilation gcc systems spend similar amount time code generation compilation average flare compilation time depends complexity query less queries well line interactive exploratory usage parallel scaling experiment compare scalability spark sql flare level experiment focuses absolute performance configuration outperforms single thread cost metric proposed mcsherry pick four queries represent aggregate join variants figure presents speedup numbers scaled cores first glance spark appears good speedups whereas flare speedup drops high core counts however examining absolute running times flare faster spark sql furthermore takes spark sql estimated cores match performance flare single core flare consistently faster cores similarly flare continues outperform spark steady number cores reaches notice cost metric last two queries infinity spark configuration matches flare performance spark csv speedup spark parquet flare csv flare parquet figure speedup streaming data ssd single thread tuples postgres hyper csv csv spark spark flare flare csv parquet csv parquet customer lineitem nation orders part region supplier spark aggregate join partsupp flare speedup table spark sql flare level running time table loading time postgres hyper sparksql seemingly good scaling spark reveals runtime incurs significant overhead particular would expect become increase level parallelism flare directly observe effect sharp drop cores since machine cores per socket cores start accessing memory numa reason spark scales better internal overhead contribute anything query evaluation trivially parallelizable hides memory bandwidth effects summary flare scales expected memory workloads reflects hardware characteristics workload means query execution takes good advantage available resources exception multiple cpu sockets problem address next next step evaluate numa optimizations flare show enable scale queries like higher core numbers particular pin threads individual cores lay memory accesses local memory region attached socket figure performs better threads dispatched different sockets due computation bounded memory bandwidth dividing threads multiple sockets multiply available bandwidth proportionally however computation bound dispatching threads different sockets little effect see scaling capacity machine tests cores seen maximum speedup respectively running time cores cores figure flare spark sql without numa optimizations spark good nominal speedups top flare better absolute running time configurations bottom systems numa effects cores clearly visible running time one socket two sockets four sockets cores cores figure flare numa optimizations different configurations threads pinned one two four sockets speedups relative single thread shown top bars faster one case reading nation table rows cases spark parquet reader faster however flare highly optimized csv reader operates closer level performance parquet reader tables except supplier benefit less speedup using parquet performing queries figure shows speedups gained executing queries without preloading data systems whereas reading entire table gives spark flare marginal speedups reading required data gives speedups range spark excluding performed slower optimized data loading often overlooked part data processing data loading flare contains optimized implementation csv files columnar apache parquet show loading times tables table full table read data table see spark flare parquet file readers outperform csv file readers scenarios despite scenario parquet spark csv reader parquet files tested uncompressed encoded using parquet plain encoding logreg speedup speedup gene gda cores cores figure level machine learning kernels scaling shared memory numa thread pinning data partitioning heterogeneous workloads udfs logreg speedup spark turn attention heterogeneous workloads combine relational processing iterative functional computation study range machine learning kernels input data loaded via dataframes kernel computation expressed udf written optiml dsl used part sql query gpu cluster numa recent study brown spark logreg figure level machine learning kernels running node amazon cluster left center node gpu cluster connected within single rack compared spark delite regards numa scalability single machine cpu sockets total cores specifically gaussian discriminant analysis gda logistic regression logreg kmeans clustering gene barcoding application gene chosen machine learning benchmarks shown figure delite gains significant speedups spark every application studied thread pinning numaaware data partitioning contributing different ways application stated previously flare level generates code delite intermediate language dmll verified generated code matches perfectly code thus guaranteeing performance data iterations likewise spark tackles issue data reuse among mapreduce jobs applications explicitly persisting intermediate results memory along lines need expressive programming model perform analytics structured semistructured data motivated hive dremel impala shark spark sql many others snappydata integrates spark transactional mainmemory database realize unified engine supports streaming analytics transactions moreover asterix stratosphere tupleware provide data programming models support user defined functions udfs flare distinguishes maintainng expressiveness achieving performance close highly optimized systems like hyper flare keeps spark sql intact integrates code generation runtime level replacing spark rdds runtime system furthermore flare level integrates delite execution order support applications design flare main target clusters faults improbable query compilation recently code generation sql queries regained momentum historic efforts back way system query compilation realized using code templates daytona hique general purpose compilers hyper hekaton dsl compiler frameworks legobase dryadlinq dblab embedded dsl frameworks intermediate languages address compromise productivity performance writing programs run diverse programming models voodoo addresses compiling portable query plans run cpus gpus voodoo intermediate algebra expressive captures hardware optimizations multicores simd etc clusters gpu brown also showcase speedups spark running small cluster machines employing gpu accelerators kmeans logreg applications despite running spark intended architecture run amazon using nodes code generated delite demonstrated speedup spark approximately speedup logreg see figure adapted applications moved cluster higherend machines cpu cores well gpus performance gap widened even study shows running gpu cluster nodes cores performance jumped speedup application completeness flare level generates dmll verified generated code identical used spark nopin pin numa spectively flare across systems however flare parquet reader demonstrated speedup spark spark csv reader speedup spark lessens slightly higher scale factors found flare parquet reader consistently performed average least one order magnitude faster across query regardless scale factor nearly every case reading parquet file flare approximately slower processing expected however reading parquet file spark rarely significantly slower processing results show reading parquet certainly provides performance gains spark compared reading csv overall performance bottleneck spark surely lie cost reading ssd compared processing related work numerous cluster computing frameworks implement combination parallel distributed relational procedural mapreduce computations mapreduce model realized hadoop performs big data analysis unreliable machines twister haloop support iterative mapreduce workloads avoiding reading unnecessary data keeping invariant thermore voodoo used alternative monetdb delite general purpose compiler framework implements dsls sql machine learning graphs matrices provides parallel patterns generates code heterogeneous targets distributed multiloop language dmll provide rich collections parallel patterns supports numa machines weld another recent system aims provide common runtime diverse libraries sql machine learning steno performs optimizations similar dmll compile linq queries furthermore steno uses dryadlinq runtime distributed execution nagel generates efficient code linq queries weld similar dmll supporting nested parallel structures however weld focuses larger set data science frameworks spark tensorflow numpy pandas support large enough set joins operators run queries query compilation inside flare level based delite optiql dsl dmll intermediate language performance evaluation data analytics frameworks aims identify performance bottlenecks study parameters impact performance workload resources probability faults etc recent study single spark cluster revealed cpu source bottlenecks mcsherry proposed cost configuration outperforms single thread metric showed many cases programs outperform big data processing frameworks running large clusters decision support benchmark consists analytical queries address several choke points aggregates large joins arithmetic computations etc flare level performance evaluation done using conclusion modern data analytics need combine multiple programming models make efficient use modern hardware large memory many cores accelerators gpus introduce flare new spark brings relational performance par best sql engines also enables highly optimized heterogeneous workloads importantly comes without giving expressiveness spark apis believe apis spirit dataframes compiler systems like flare delite play increasingly important role future satisfy increasing demand flexible unified analytics high efficiency references openmp http accelerating queries join groupjoin pvldb abouzeid abadi silberschatz rasin hadoopdb architectural hybrid mapreduce dbms technologies analytical workloads pvldb alexandrov bergmann ewen freytag hueske heise kao leich leser markl stratosphere platform big data analytics vldb journal apache hadoop http apache parquet https armbrust xin lian huai liu bradley meng kaftan franklin ghodsi zaharia spark sql relational data processing spark sigmod pages acm astrahan blasgen chamberlin eswaran gray griffiths king lorie mcjones mehl system relational approach database management tods behm borkar carey grover onose vernica deutsch papakonstantinou tsotras asterix towards scalable semistructured data platform models distributed parallel databases berkhin survey clustering data mining techniques grouping multidimensional data pages springer boncz neumann erling analyzed hidden messages lessons learned influential benchmark technology conference performance evaluation benchmarking pages springer boncz zukowski nes query execution cidr brown lee rompf sujeeth aberger olukotun abstraction eat performance optimized heterogeneous computing parallel patterns cgo pages acm brown sujeeth lee rompf chafi odersky olukotun heterogeneous parallel framework languages pact howe balazinska ernst haloop efficient iterative data processing large clusters pvldb crotty galakatos dursun kraska binnig zdonik architecture compiling workflows pvldb dean ghemawat mapreduce simplified data processing large clusters osdi pages diaconu freedman ismert larson mittal stonecipher verma zwilling hekaton sql server oltp engine sigmod pages acm ekanayake zhang gunarathne bae qiu fox twister runtime iterative mapreduce acm hpcd pages acm graefe extensible parallel query evaluation system tkde greer daytona language cymbal acm sigmod record volume pages acm henderson lazy evaluator popl pages acm press klonatos koch rompf chafi building efficient query engines language pvldb kornacker behm bittorf bobrovytsky ching choi erickson grund hecht jacobs joshi kuff kumar leblang pandis robinson rorke rus russell tsirogiannis yoder impala modern sql engine hadoop cidr krikellas viglas cintra generating code holistic query evaluation icde pages ieee laskowski mastering apache spark technical report stanford infolab november palkar thomas shanbhag narayanan pirk schwarzkopf amarasinghe zaharia infolab weld common runtime high performance data analytics cidr pirk moll zaharia madden voodoo vector algebra portable database performance modern hardware vldb ramnarayan mozafari wale menon kumar bhanawat chakraborty mahajan mishra bachhav snappydata hybrid transactional analytical store built spark sigmod pages rompf odersky lightweight modular staging pragmatic approach runtime code generation compiled dsls commun acm shaikhha klonatos parreaux brown dashti koch architect query compiler sigmod pages acm sujeeth brown lee rompf chafi odersky olukotun delite compiler architecture embedded languages tecs sujeeth lee brown rompf atreya odersky olukotun optiml implicitly parallel language machine learning icml sujeeth rompf brown lee chafi popic prokopec jovanovic odersky olukotun composition reuse compiled languages ecoop transaction processing council version thusoo sarma jain shao chakka anthony liu wyckoff murthy hive warehousing solution framework pvldb xin rosen zaharia franklin shenker stoica shark sql rich analytics scale acm sigmod pages isard fetterly budiu erlingsson gunda currey dryadlinq system distributed computing using language osdi zaharia chowdhury franklin shenker stoica spark cluster computing working sets usenix hotcloud zaharia xin wendell das armbrust dave meng rosen venkataraman franklin ghodsi gonzalez shenker stoica apache spark unified engine big data processing commun acm https lattner adve llvm compilation framework lifelong program analysis transformation cgo lee brown sujeeth chafi rompf odersky olukotun implementing languages heterogeneous parallel computing ieee micro lee brown sujeeth rompf olukotun mapping nested parallel patterns gpus micro mcsherry scalability cost http mcsherry isard murray scalability cost hotos melnik gubarev long romer shivakumar tolton vassilakis dremel interactive analysis datasets proceedings vldb endowment murray isard steno automatic optimization declarative queries acm sigplan notices volume pages nagel bierman viglas code generation efficient query processing managed runtimes vldb neumann efficiently compiling efficient query plans modern hardware pvldb nvidia world first supercomputer box http ousterhout rasti ratnasamy shenker chun making sense performance data analytics frameworks nsdi pages page brin motwani winograd pagerank citation ranking bringing order web
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language software defined network programming based openflow dec mehdi mohammadi ala zijiang james yang computer science department western michigan university paper proposes network policies xdnp new language based xml notation describe network control rules software defined network environments rely existing openflow controllers specifically floodlight novelty project separate complicated apis policy descriptions separation makes possible extend current work northbound higher level abstraction support wide range controllers based different programming languages approach believe network administrators develop deploy network control policies easier faster index defined networks openflow floodlight sdn compiler sdn programming languages sdn abstraction ntroduction network sdn technology network engineers administrators control manage network services abstraction lower level functionality end achieved splitting system makes decisions send packets control plane underlying systems known data plane charge forwarding packets selected destinations order practical sdn needs mechanism control plane interact data plane openflow protocol used network research community recent years couples openflow controller frameworks emerge openflow simplified network management providing abstractions control set switches remotely openflow framework requires network engineers administrators write programs control manage data traffic network control network devices however one potential problem network engineers face programming languages support openflow complex administrators required http http https http http http http know much irrelevant information develop deploy control policy problem prohibitive beginner network engineer good background programming language controller challenges programmers including interaction concurrent modules interface switch hardware programming model simple unified language higher abstraction level depend specific language fill gap developed format based xml semantic operations developed policy description xml used widely network management configuration protocols like netconf work annexed current openflow controller top layer service two contributions paper proposing script language describing network control policies implementing translator converts xml file java source code containing controller program floodlight emulation experiments affirms applicability xml notation policy describer sdns elated ork xml documents widely used describe systems configure control one attractive usage code generation several works try use xml notation representation source codes javaml work represents java source code xml notation source code representation javaml way constructs like superclasses methods message sends literal numbers directly represented elements attributes document content xml notation also used another work named srcml structural information added unstructured source code files srcml aims enhance source code representation adding syntactic information obtained parse tree liang proposed proof concept use xml description scheme openflow networking experiments defined networking experiments hierarchical model experiment highest level experiment contains information topology deployment control output components however paper presented poster section geni engineering conference gec washington march provided format description mentioned way generated xml parser furthermore provided full evaluation system real environments work differs work way design platform network engineers describe network control rules xml notations regardless underlying topology network elements words liang work focuses description network environment control network traffic behavior several works projects aim propose higher level abstraction openflow apis development frameworks frenetic implemented python emerged simple rules including pairs actions support filtering forwarding duplicating modifying packets later included complicated operators like packet processing functionalities pyretic modern sdn programming language based frenetic beyond current parallel composition operator presents two complex abstractions sequential composition operator applying control policies abstract topologies abstractions development modular control programs becomes simpler netkat another language sdn solid mathematical foundation supports primitives like filtering modifying transmitting packets well union sequential composition operators works concentration provide language policy description procera fml designed based aim declarative policy language procera based functional reactive programming language relies haskell able use constructs data types defined haskell fml built datalog language support declarative programming interface part resilient network management system authors proposed policy description language based management patterns management patterns describe handling controlling individual openflow resilience services iii oftware efined etworking networking defines two separate layers new architecture network environments data plane supposed operations like buffering packets forwarding dropping tagging collecting packet statistics control plane hand may algorithms track dynamic topology network route processing capability control plane usually consists separate powerful machine called controller control plane uses computed data data plane statistics manage govern set dependent switches installing removing packet forwarding rules openflow realization sdn follows architecture typical openflow network switch find rule match received packet flow table proceeds rule otherwise packet sent controller processing controller examines packet fig overall architecture system header establishes rule based packet next similar packets arrived switches required controller since switches appropriate rule process although forwarding packets controller increases latency occurs much roposed ystem xml translator consists lexical analyzer lexer syntax analyzer parser lexer tokenizes input file matches tags attributes identifiers constant values based regular grammars syntax analyzer hand performs syntactic analysis input file find problem step generates appropriate java source code overall architecture system depicted figure language specification overall design xml file used control program follow format figure rule description defined hierarchical structure top level define class name sdn element list rules contacting zero rule elements declared inside rule one conditions expressed compositional conditions condition elements support logical operators stated attribute connector example condition element connector joined previous condition logical operator java source code conditions comply simple pattern variable value variables chosen source destination source port destination port current supported operator equal sign value port number address example xml file shown figure construct lexer example define pattern address define regular expressions lexer file octet tag names considered keywords script required comply exactly xml specification syntax analyzer fig format policy description file grammar section translator defined syntax analyzer used yacc tool generate translator input file define tokens introduced lexer grammar production rules examine syntax xml file output associated production rule leading code generation xml file syntax analyzer description follows form bnf notation grammars example following production rules show nesting iteration nature xml script control policies ruleslist rule rule nothing rule conditioncounter fig sample xml policy description xml sample contains two rules first one says packets going address coming address forwarded port switches assigned address second rule indicates packet originated telnet service port dropped specified port example shows possible implement network policy management schemes services like firewalls load balancing xml notation conditionlist conditionlist condition condition condition condtext number string input xml file matched designed tokens rules cause error program termination program error message lexical analyzer designed lexer lex format defined matching patterns xml elements attributes literals needed typical network controller source lexer file fed flex generate source code language used regular expression notations xperiments lex yacc input files ready generate final translator calling following commands used equivalent versions lex yacc called flex bison respectively following command lexer parser grammar respectively flex bison gcc translator file named order generate java source code desired xml file use following command console command line output command file named containing java class control policies network file pushed controller machine compiled used network management used mininet network emulator examine behavior proposed translator mininet supports openflow register generated java source code floodlight controller attach new controller mininet topology explain setup experiments generating java source code form xml file added floodlight controller use floodlight source code add generated class main package needed set floodlight load class startup first tell loader module exists adding fully qualified module name configuration file named path tell module loaded need modify floodlight module configuration file append demo file full name class added file compile floodlight project openflow controller designed services running next step running mininet setup simple network topology containing one switch three hosts connected switch also set controller remote configuration obtained command sudo remote test module calling ping command since first rule forwarding packets going port ping install rule switches receive reply hosts ping like simple rule block traffic running command pingall get packet loss means responding none ping requests table shows comparison xml file versus generated java file terms number code lines size code kilo byte using system reduce coding efforts original java source code size rule description generated source code onclusion paper describes new approach networks presents higher level abstraction compared current network programming languages table omparison xml java source codes xml file java file line code size kilo byte defined scripting language based xml notation network administrators define control policies without concerning complexities underlying controller framework indeed make networking easier attractive network administrators work opens opportunity using service oriented architecture web services model collaboration sdn controllers controller offers load balancing firewalling extending work support apis complicated scenarios intended done future works technically examining manipulating openflow headers possible approach supporting controller frameworks different languages also another direction future works eferences mckeown anderson balakrishnan parulkar peterson rexford shenker turner openflow enabling innovation campus networks acm sigcomm computer communication review vol foster harrison freedman monsanto rexford story walker frenetic network programming language acm sigplan notices vol acm badros javaml markup language java source code computer networks vol maletic collard marcus source code files structured documents program comprehension proceedings international workshop ieee liang lin openflow networking experiment description language based xml web information systems mining monsanto foster harrison walker compiler system network programming languages acm sigplan notices vol acm monsanto reich foster rexford walker composing software defined nsdi vol anderson foster guha jeannin kozen schlesinger walker netkat semantic foundations networks acm sigplan notices vol acm voellmy kim feamster procera language highlevel reactive network control proceedings first workshop hot topics software defined networks acm hinrichs gude casado mitchell shenker practical declarative network management proceedings acm workshop research enterprise networking acm smith hutchison mauthe management patterns network resilience management network operations management symposium noms ieee ieee
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approximate generalized matching covers jul dawei huang university michigan seth university michigan abstract paper present linear time approximation schemes several generalized matching problems nonbipartite graphs results include algorithms maximum weight minimum weight cover byproduct also obtain direct algorithms exact cardinality versions problems running time technical contributions work include efficient method maintaining relaxed complementary slackness generalized matching problems reductions cover problems supported nsf grants introduction many combinatorial optimization problems known reducible computing optimal matchings graphs problems include computing edge covers undirected shortest paths negative cycles bidirected flows see problems investigated heavily since tutte work however existing reductions graph matching often inadequate blow size input use auxiliary space piggyback specific matching algorithms like algorithm moreover existing reductions destroy dual structure optimal solutions therefore approximation preserving paper design algorithms computing covers direct fashion efficient reductions algorithms based formulations problems contrast approaches using shortest augmenting walks easily adapt weighted approximate variants problems let define problems formally assume graphs may multiple parallel edges loops subset degf perfect degree constraints hold cover cover subset degf perfect degree constraints hold equality unweighted graphs objective maximize cover minimize weighted graphs possibly subject additional constraint perfect classic reductions classical reduction standard graph matching uses problem stepping stone function indicates many copies matching number matches edges incident counting multiplicity problem reduced subdividing edge path new vertices original vertices reduction blows number vertices approximation preserving problem easily reduced standard matching replicating vertex times replacing edge bipartite clique endpoints replicas step reduction approximation preserving blows number vertices edges reductions together reduce factor graph matching problempon vertices fmax edges gabow gave method solving time using calls single iterations algorithm observe cover complementary problems cover complementary edge set necessarily deg complementarity implies algorithm one problem solves polynomial time says nothing precise complexity solving exactly approximately indeed phenomenon well known realm problems literature term often refers definition perfect example maximum independent set minimum vertex coverpare complementary problems completely different approximation profiles gabow cardinality algorithm implies cover computed time says nothing approximability cover far aware fastest approximation algorithms cover see treat general weighted set cover problem sets analysis shows greedy algorithm harmonic number interest approximate cover problem inspired new application anonymizing data environments users different privacy demands see data records correspond edges privacy demand measured goal anonymize records satisfy everyone privacy demands new results give new algorithms computing covers approximately exactly show folklore reduction minimum weight cover maximum weight matching sense matching gives edge cover implies cover time one apply number simple practical algorithms approximate cover simple reduction extend covers give algorithm weighted cover algorithm follows two results somewhat surprising first approximate weighted algorithm reports dual solution transformed weighted cover algorithm second algorithm exists running time first claim clearly false drop approximate dual solution requirement reason approximate vertex cover translate maximum independent set second surprising running time independent demand function magnitude edge weights corollaries reductions obtain new exact algorithm minimum cardinality cover running time ratherpthan time direct algorithm cardinality runs time without reduction algorithm blossom structure characterization considerably simpler corresponding cover interest simplicity one might want efficient code solves approximate directly without viewing problem multigraph implicitly infinite supply edge know direct algorithm indeed structure blossoms seems rely strict complementary slackness incompatible main technical tool relaxed complementary using relaxed complementary slackness matched unmatched edges different eligibility criteria included augmenting paths blossoms whereas blossoms require copies unmatched eligible ineligible structure paper section give introduction generalized matching problems structure blossoms augmenting walks section show folklore reduction cover section reduce approximate cover approximate section give algorithm graphs weights speed log independent weight section gives linear time algorithm compute maximal set augmenting walks basis cover section reviews basic algorithmic concepts matching theory generalizations cover problems lps blossoms augmenting walks tools let generalize algorithm approximate maximum weight matching approximate maximum weight approximate minimum weight cover notation input multigraph let sets edges exactly one endpoint endpoints respectively denotes intersection definition degt formulation maximum weight problem expressed maximizing following constraints subject blossom constraint generalization blossom conj straint ordinary matching reason subset incident edges sum subset allow distinguish matched edges endpoints inside exactly one endpoints basic feasible solution integral therefore interpreted membership vector ensure optimality solution algorithm works dual minimize subject yzf aggregated dual yzf defined yzf unlike matching associated combination vertex set subset incident edges minimum weight cover problem expressed minimizing subject dual program maximize subject yzc yzc cover algorithms maintain dynamic feasible solution satisfies primal constraints call edges matched edges unmatched referred type edge vertex saturated degf degf given deficiency def vertex defined def degf similarly cover surplus vertex defined surp degf blossoms follow gabow definitions terminology blossoms augmenting walks etc blossom tuple vertex set edge set base vertex base edge set may empty often refer blossom referring vertex set blossoms defined inductively follows definition single vertex forms trivial blossom singleton use yzf yzc denote aggregated dual cover respectively omit subscript clear context inductively let sequence disjoint singletons nontrivial blossoms suppose exists closed walk starting ending mod vertex set identified blossom following satisfied base requirement singleton two edges incident must matched unmatched alternation requirement fix singleton exactly one matched nontrivial blossom must either edge set blososm ebi base singleton otherwise may either empty contain one edge opposite type blossoms classified singleton otherwise note blossoms usual matching problem always light purpose blossoms identify part graph behaves unit searching augmenting walk property formally stated follows lemma let arbitrary vertex exists even length alternating walk odd length alternating walk using edges moreover terminal edge incident must different type exists proof prove induction base case blossom consisting singletons one two walks must odd must even consider cycle suppose claim holds inductively nontrivial blossoms let arbitrary vertex use pbi denote walk guaranteed blossom two cases case contained singleton examine two walks notice walks graph obtained contracting subblossoms inductive hypothesis extend replacing walk pbi suitable connecting endpoints particular different types replace even length walk pbi otherwise replace pbi alternation requirement one must odd must even case contained blossom without loss generality consider contracted walk extend alternating walk terminating similar case obtained concatenating alternating walk pbk pbk right parity notice cases base requirement definition guarantees starting edge alternating walk alternates base edge main difference blossoms generalized matching problems blossoms ordinary matching problem meaningful finding augmenting walks blossoms ordinary matching since vertex matched edge incident alternating walk enters blossom base vertex via matched edge must leave unmatched edge result subwalk inside blossom always even generalized matching problems subwalk either even odd may contain cycle general alternating walk enters blossom base edge leave blossom nonbase edge also define notion maturity blossoms cover simplicity let focus first complementary slackness assign positive pair satisfies constraint equality requirement captured following definition definition mature blossom blossom mature satisfies following every vertex saturated degf def def must light blossom def motivated two observations complementary slackness dual variables positive primal constraint satisfied kequality blossom positive topology augmenting walks unsaturated heavy blossoms start end augmenting walk since extend augmenting walk starts unmatched edge several remarks made blossom mature may contain augmenting walk specifically suppose light unsaturated nonbase vertex also unsaturated odd length alternating walk satisfies definition augmenting walk moreover deficiency odd length alternating walk augmenting reasons never contract immature blossoms maturity blossom imply set similarly mature cover blossom satisfies augmentation never destroys maturity particular never creates unsaturated heavy blossom maturity cover blossoms defined similarly similar properties definition mature blossom cover blossom mature cover satisfies following every vertex saturated degf surp surp must heavy blossom surp figure two example contractable blossoms bold edges matched thin ones unmatched blossoms circled border base edges represented arrow pointing away blossom algorithm keeps track laminar set mature blossoms maintains nonnegative value one subset neighborhood defined follows symmetric difference operator xor subsets mature blossom cover walk augmenting walks analogy augmenting paths ordinary matching complications arise fact blossom treated identically single vertex contracted example figure two edges incident blossom type augmenting along never happen ordinary matching moreover augmenting walks begin end vertex visit vertex multiple times hence naive contraction blossom single vertex loses key information internal structure blossoms definition characterizes walk contracted graph extended augmenting walk graph obtained contracting laminar set blossoms let definition let edges blossoms walk singletons say augmenting walk respect following requirements satisfied figure example blossom changes augmentation augmenting walk notice rematching base edge blossom changes blossom turns heavy blossom light one terminal vertices requirement terminals must unsaturated singletons unsaturated light blossoms closed walk must singleton def otherwise either singletons blossoms deficiency must positive terminal edges requirement terminal vertex singleton incident terminal edges must unmatched otherwise either matched unmatched alternation requirement let internal singleton blossom singleton exactly one matched nontrivial blossom must one extended natural consequence definition augmenting walk augmenting walk proved exactly lemma rematching along augmenting walk means updating symmetric difference decrease total deficiency moreover rematching along every blossom still satisfies definition blossoms different base vertices edges cover corresponding notion called reducing walk definition reducing walk naturally obtained definition replacing unsaturated deficiency light oversaturated surplus heavy also worth pointing cover complement deg blossom set augmenting walk also reducing walk complementary slackness characterize approximately optimal solution maintain dual functions short explicitly maintain edge dual since maximizing value explicitly given max following property characterizes approximate maximum weight property approximate complementary slackness let nonnegative parameters say duals set blossoms satisfies complementary slackness following hold approximate domination unmatched edge approximate tightness matched edge blossom maturity blossom unsaturated vertices duals unsaturated vertex following lemma states property characterizes approximately optimal solution lemma let along duals let maximum weight satisfy property parameters proof first define otherwise approximate tightness moreover gives following degf property unsaturated vertices duals blossom maturity definition equal degf connection covers classical approach solving cover problem reducing specifically looking minimum weight cover seen choosing edges maximum weight deg main drawback reduction yields inefficient algorithms example gabow algorithms solving maximum weight scales linearly makes undesirable small even gabow algorithm runs log time moreover reduction words complement arbitrary maximum weight guaranteed cover section establish two results first prove folklore reduction cover matching approximation preserving allows use efficient approximate matching algorithm solve weighted cover problem establish connection approximate approximate cover using approximate complementary slackness previous section give minimum weight cover algorithm approximate maximum weight algorithm approximate preserving reduction cover edge cover problem special case cover everywhere minimum weight edge cover problem reducible maximum weight matching simply reweighting edges let edge minimum weight let define new weight function follows schrijver showed following theorem theorem let maximum weight matching respect weight function minimum weight edge cover respect weight function show reduction also approximation preserving theorem let weight matching respect weight function weight edge cover respect weight function proof let optimal edge cover matching defined previously construction similarly last inequality holds reduction naturally extend cover next section show obtain cover algorithm within framework cover show algorithm computing used compute cover start giving approximate complementary slackness cover similar property characterizes good approximate cover property approximate complementary slackness cover let positive parameters say cover duals blossom family satisfies approximate complementary slackness following requirements holds approximate domination unmatched edge yzc approximate tightness matched edge yzc blossom maturity blossom oversaturated vertices duals oversaturated vertex recall using aggregated duals yzc cover yzc proof following lemma identical lemma lemma let cover duals satisfying property parameters let minimum weight cover suppose blossoms duals satisfying property let cover complement key observation blossom set duals used show property specifically following lemma lemma duals satisfy property parameters duals complementary cover satisfies property parameter proof trivial see property oversaturated vertices duals property unsaturated vertices duals equivalent show property equivalent property suffices show function yzf agrees function yzc cover complement recall yzc yzf refer blossom respect cover suffices show figure illustration relation complementary cover figures left demonstrate complementary right circled dashed therefore yzf yzc summed set blossoms words yzf yzc claim follows blossom maturity property property argue one equality implies suppose therefore edge set contains edges endpoints inside plus edge degc lower bound set edges inside many endpoints equality implies degc every implies also saturates every vertex therefore equality property holds edge set contains edges endpoints inside similarly degc lower bound total number endpoints equality implies every vertex saturated property satisfied notice arugment also applies case completes proof property implies property direction symmetric approximation algorithms cover section prove main result giving approximation algorithm computing maximum weight crux result implementation edmonds search relaxed complementary slackness eligibility criterion notion approximate complementary slackness introduced gabow tarjan bipartite matching general matching gabow gave implementation edmonds search exact complementary slackness problem finds augmenting walks one time main contribution section adapt approximate complementary slackness facilitate finding augmenting walks batches illustrate works first give approximation algorithm graph small edge weights let positive weight function algorithm computes maximum weight time independent also show use scaling techniques transform algorithm run log time independent approximation small weights main procedure time algorithm called edmonds search one iteration edmonds search finds set augmenting walks using eligible edges creates dissolve blossoms performs dual adjustment maintaining following invariant invariant approximate complementary slackness let parameter multiple granularity multiples multiples approximate domination unmatched edge blossom edge tightness matched blossom edge blossom maturity blossom unsaturated vertices unsaturated vertices strictly less vertices notice relax property allow unsaturated vertices positive purpose edmonds search decrease unsaturated vertices maintaining invariant following define following eligibility criterion criterion edge eligible satisfies one following key property definition asymmetric matched unmatched edges result augment along eligible augmenting walk edges except contracted blossoms become ineligible let gelig graph obtained discarding ineligible edges let elig gelig obtained gelig contracting blossoms initialization set edmonds search repeatedly executes following augmentation blossom formation dual adjustment blossom dissolution steps unsaturated vertices see figure let define mean reachable vertices steps algorithm well labelling blossoms singletons start defining notion alternation follows definition augmenting walk say two edges incident alternate either singleton different types augmenting path augmentation blossom formation find maximal set gelig maximal set reachable mature blossom gelig preimage update step new elig contains augmenting walk well reachable blossoms dual adjustment let set vertices elig reachable unsaturated vertex via eligible alternating walk classify vertices set inner vertices set outer vertices let vin vout set original vertices represented vout adjust values follows vout vin root blossom vbout root blossom vbin blossom dissolution dual adjustment root blossoms might zvalue remove none exists update gelig notice root blossoms generated root blossoms decremented dual adjustment step root blossoms formed augmentation step become unreachable augmentation thus get incremented figure algorithm small integer weights nontrivial blossom alternating path contracted graph one every two consecutive edges alternates search forest consists blossoms singletons reachable unsaturated via eligible alternating path elig furthermore require preimage paths gelig start unsaturated vertex unmatched edge follows roots unsaturated singletons unsaturated light blossoms label root vertices outer nonroot vertex search forest let edge pointing parent status vertices defined follows definition vertex outer one following satisfied root search tree observe since augmenting along eligible augmenting walk make ineligible edges eligible blossoms reachable augmentation must also reachable hence contract blossom augmentation actual implementation reuse label previous augmentation blossom formation step dual adjustment reason augmenting along eligible augmenting path create eligible edges make unreachable vertex reachable moreover depth first nature search tree guarantees vertices augmenting path remains reachable still inherit label augmentation singleton nontrivial blossom otherwise one following holds classified inner singleton nontrivial blossom call grown repeatedly attaching child individual search tree let denote root blossom parent using edge eligible containing say edge eligible eligible one following satisfied outer singleton outer blossom inner singleton inner blossom hence consists singletons blossoms reachable unsaturated singleton light blossom via eligible alternating path whose preimage starts unsaturated vertex unmatched edge simplicity call blossoms singletons reachable singletons blossoms unreachable vertex original graph gelig reachable unreachable reachable unreachable elig version edmonds search primal dual variables initialized way property approximate domination always satisfied property approximate tightness vacuous initially empty property unsaturated vertices reason large gap primal dual objective beginning algorithm evaluated following def goal algorithm seen bridging gap primal objective dual objective preserving complementary slackness properties achieve two ways augmentations enlarge augmenting along augmenting walk reduce total deficiency vertex set dual adjustments change dual variables way decreases unsaturated vertices maintaining complementary slackness condition algorithm progress edmonds search measured latter overall reduction unsaturated vertices correctness algorithm reduces showing augmentation blossom formation blossom dissolution dual adjustment preserve invariant lemma augmentation blossom formation preserves invariant figure example eligible alternating search tree outer blossoms singletons labeled using solid boundaries inner blossoms singletons dashed boundaries proof first show identity invariant augmentation particular augmenting walks intersect affect result function invariant augmentation use denote base edge augmentation definition augmenting walks intersects let augmentation combining equations hence invariant blossom edge satisfies approximate domination well tightness continues satisfy invariants augmentation eligible edge criterion thus augmentation duals satisfy approximate domination unmatched duals satisfy tightness augmentation augmentation preserves maturity blossoms vertex nonterminal blossom degf maturity naturally preserved terminal blossom degf degf moreover augmentation always base edge therefore also mature augmentation newly formed blossom step must mature value function unchanged invariants preserved lemma blossom dissolution preserve invariant proof discarding blossoms preserves value function hence preserves invariants crux proof show dual adjustment also preserves invariant particular approximate domination tightness proving correctness dual adjustment first prove following parity lemma first used generalize lemma parity let search forest defined let preimage every vertex parity multiple proof claim clearly holds initialization vertices every eligible edge straddles two singletons blossoms must multiple always multiples always share parity multiple therefore suffices show every vertex blossom parity prove need show blossom formation step groups vertices parity together new blossoms formed edges eligible criterion means endpoints share parity hence induction vertices also share parity dual adjustment step also preserves property vertices blossom classification thus incremented decremented lemma dual adjustment preserves invariant proof focus approximate domination tightness invariants affected dual adjustment much cases consider compared ordinary matching different cases generated edge considering classification endpoints whether matched whether base edge respective endpoints blossoms whether eligible following analysis narrow number meaningful cases consider edge unreachable unsaturated vertex root blossom clearly remains unchanged dual adjustment therefore assume least one say reachable every reachable endpoint contribute change adjustment ofpy plus adjustment let net change quantity omit use edge considering clear perform dual adjustment following scenarios occurs inner singleton outer blossom inner blossom occurs outer singleton inner blossom outer blossom consider effect dual adjustment edge first consider case case introduce change exactly one say case inner singleton case approximate domination preserved need worry approximate tightness hence assume since eligible would included child hence granularity therefore dual adjustment case outer singleton case tightness preserved need worry approximate dominination similar case must ineligible dual adjustment case inner blossom divide cases according whether matched subcase case tightness preserved eligible since otherwise must search tree dual adjustment subcase basically symmetric subcase eligible therefore dual adjustment approximate domination preserved case outer blossom subcase must parent search tree thus since contradicting fact reachable eligible dual adjustment afterward subcase similarly similarly reachable eligible therefore dual adjustment completes case exactly one endpoints reachable following part complete argument endpoints reachable argue three scenarios happen either opposite signs cancel sign sign aligns property wish keep cases hold use lemma parity argue enough room dual adjustment violate approximate domination tightness case assume parent first examine tree edges parent edge hence must eligible argue sign case three cases inner singleton outer blossom inner blossom first observe three cases straightforward inner singleton outer blossom know since outer therefore inner singleton since inner combined fact notice since parent outer singleton outer blossom inner blossom second case third case three cases remains unchanged case case symmetric case either outer singleton inner blossom outer blossom cases fact must eligible implies inner singleton outer blossom inner blossom hence still remains constant suppose tree edge still break cases according sign need consider since otherwise cancel remains constant case case incremented therefore need worry tightness notice inner singleton outer blossom inner blossom outer blossom inner blossom since holds endpoint easy verify cases eligible eligible notice augmentation blossom formation steps augmenting walk reachable blossom elig edge eligible endpoints since otherwise find augmenting walk new reachable blossom thus ineligible invariant granularity lemma parity must multiples therefore implies dual adjustment case decremented similar case assume focus approximate domination outer singleton inner blossom outer blossom since therefore eligible must eligible similar case eligible endpoints lead discovery additional blossom augmenting walk gelig impossible augmentation blossom formation therefore conclude case ineligible lemma parity dual adjustment approximate domination still holds dual adjustment theorem computed time proof setting unsaturated vertices takes iterations reach thus satisfying property invoking lemma optimum running time iteration augmentation blossom formation dual adjustment blossom dissolution implemented linear time defer detail implementation section total iterations running time result lemma lemma also obtain following result corollary cover computed time scaling algorithm general weight section modify weighted algorithm work graphs general weights modification based scaling framework idea divide algorithm log scales execute edmonds search exponentially diminishing goal scale use edmonds searches halve unsaturated vertices maintaining relaxed version approximate complementary slackness manipulating weight function approximate domination weak beginning strengthened scales approximate tightness weakened exchange assume without loss generality powers two define error parameter scale scale works new weight function old weight function rounded nearest multiple last scale maintain scaled version invariant scale invariant scaled approximate complementary slackness positive unsaturated vertices scale log maintain blossoms duals satisfy following invariant granularity multiple multiple approximate domination approximate tightness let index earliest scale became matched unmatched rematched afterwards mature blossoms blossom unsaturated vertices duals free vertices always smaller equal vertices based invariant edmonds search use following eligibility criterion criterion scale edge eligible one following holds nonnegative multiple start scale algorithm initializes similar algorithm small edge weights scale duals unsaturated vertices start execute iterations edmonds search parameter criterion scale terminates unsaturated vertices decreased last iteration terminates reach notice although invariant eligibility criterion changed fact edmonds search preserves complementary slackness invariant particular dual adjustment preserves approximate domination approximate tightness still holds proof lemma long definition eligibility guarantees following parity property lemma point scale let set vertices gelig reachable unsaturated vertices using eligible edges vertex parity multiple dual adjustment never changes edges inside blossom following effect edge endpoints lying different blossoms ineligible might decrease never exceed threshold eligibility drop eligible never decrease ineligible might increase never exceed threshold eligibility raise eligible never increase order words dual adjustment destroy approximate domination approximate tightness therefore edmonds search within scale preserve invariant also need manipulate duals different scales ensure invariant formally completion scale increment ensure approximate domination approximate tightness holds scale increasing ensures approximate domination approximate tightness since algorithm terminates unsaturated vertices reach terminates corresponding duals satisfying following property property final complementary slackness approximate domination approximate tightness let index earliest scale becomes matched unmatched rematched afterwards blossoms maturity blossoms unsaturated vertices duals free vertices implies domination tightness satisfied within factor approximate domination easy see since thus approximate tightness lower bound weight first enters scale throughout scale least since similar proof lemma show characterizes minimum weight running time algorithm log total log scales scale consists iterations edmonds search implemented linear time theorem maximum weight computed log time corollary minimum weight cover computed log time linear time algorithm also point applying techniques algorithm modified run time independent main idea force algorithm ignore edge log scales let maximum possible weight edge matched unmatched eligible scale let scale notice ignore scale scale moreover also forcibly ignore scale scale log ignoring otherwise eligible edge might cause violation approximate tightness approximate domination however since free vertices point violation amount see notice also upper bound amount change caused dual adjustment scale hence scale scale total amount violation approximate tightness approximate domination bounded guarantees still get approximate solution therefore every edge takes part log scales cost per scale total running time log theorem maximum weight minimum weight cover computed log time independent weight function linear time augmenting walk algorithm section presents linear time augmenting walk algorithm used augmentation step edmonds search goal find maximal set edge disjoint augmenting walks graph gelig main idea apply modified depth first search finds maximal set vertex disjoint augmenting walks ordinary matching linear time several difficulties adapting algorithm instead vertex disjoint looking maximal set edge disjoint augmenting walks might lead intersecting search trees algorithm needs able identify augmenting cycles present standard matching problem depth first search tree branches outer inner singletons well outer blossoms moment maintain input contracted graph minor search forest ordered set depth first search trees vertices tbi minor defined dynamic laminar set blossoms singletons maximal blossoms vertex use denote largest blossom contains use refers set vertices contained blossoms tbi since search trees rooted unsaturated vertices use definition define vertices search forest also gives set edges eligible vertex search forest similar depth first search search trees previously terminated searches either successful search tree contains augmenting walk exhausted search tree maximal search tree lead augmenting walk case final output algorithm along newly discovered blossoms maximal set augmenting walks invariant vertex edge eligible later search tree words vertex exhausted every edge eligible scanned depth first search procedure notice vertex already exhausted unexhausted classification changes change introduces new set eligible edges therefore say vertex outer inner currently outer inner edges eligible scanned edmonds algorithm set blossoms discovered search augmenting walk standard matching corresponds property exist unmatched edge connecting two outer vertices different search trees walks must edge disjoint necessarily vertex disjoint suppose scan neighborhood order given adjacency list edeg algorithm edge one following three states explored indicating edge explored building search tree tbi either included tbi included tbi used enlarge tbi augmented indicating edge explored tbi part unexplored indicating explored tbi yet algorithm terminates unexplored edges declared unreachable removing edges edge reached via alternating path starting unsaturated vertex stated lemma main complication algorithm alternating structure search tree necessitates maintenance maximal set blossoms maintaining maximal set blossoms allows completely decide vertex whether even odd length alternating path unsaturated vertex reduce maintaining following invariant invariant edge scanned must blossom referred active search tree search last search tree search forest tree algorithm currently growing algorithm grows search tree depth first consisting vertices manner extending active walk pactive walk finish exploring yet rest invariant states topology search tree well depth first property invariant consists edge disjoint alternating trees graph sequence rooted unsaturated blossoms edges tbi explored edge explored augmented edges eligible must also explored augmented parent edge must also augmented vertices tbi singletons blossoms tree edge eligible descendant active walk pactive consists walk root search tree initially starts single unsaturated vertex root vertex active path suppose blossom last blossom active path iteration scan next unexplored edge eligible depending endpoint located one following augmentations form augmenting walk two situations unmatched saturated yet active search tree unmatched singleton root active search tree def cases extend active walk terminate search make state every pactive edge augmenting dfs extensions exploration lead augmentation step either nontrivial blossom search tree singleton extend active walk child corresponds grow step edmonds search put blossom formation descendant edge also eligible form path contract blossom consisting blossoms singletons blossom call place blossom active path label vertices outer ready scanned note previously inner vertices exhausted exhausted outer vertex dfs retractions every edge eligible already scanned mark exhausted remove active walk vertex active walk terminate search search terminates returns search tree update deficiency function case discover augmenting unsaturated vertices become saturated start new search unsaturated vertex exhausted search tree root long exists notice root successful search tree reused long unsaturated augmentations far algorithm terminates unsaturated vertices exhausted vertex either saturated unsaturated discover augmenting path starting first show search terminates unexplored edges reachable alternating path starting unsaturated vertex guarantees ignore edges might lead new augmenting path lemma search terminates unexplored edges unreachable alternating path contain edge proof suppose unexplored edge reachable alternating path consisting explored augmented edges without loss generality let first reachable unexplored edge let edge precedes alternating path since explored augmented unexplored edges must explored leading contradiction lemma operations preserve invariants proof invariant natural consequence depth first search invariant immediate maintain blossoms active walk invariant holds explored eligible search along finds augmenting walk every edge eligible must also explored search retracts show invariant notice depth first nature search tree must ancestor one another otherwise first explored search tree dfs extension put search tree child suppose ancestor scan edge eligible blossom step performed edge must made eligible later blossom step ancestor descendant puts blossom invariants show following lemma lemma augmenting walk edge must intersect augmenting walk proof suppose augmenting walk intersect edge lemma assume edges must explored ineligible termination algorithm must exhausted unsaturated vertex must search tree root definition augmenting walk must eligible must explored show induction must explored must search forest definition search tree root explored eligible means must also search tree search tree explored consider vertex since augmenting walk alternation requirement augmenting walk one must eligible eligible must explored done otherwise eligible endpoints thus must blossom case blossom formation step makes eligible since search tree preceding edge eligible predecessor particular unsaturated inner singleton unsaturated light inner blossom otherwise augmentation step performed contradicts fact augmenting walk theorem maximal set augmenting walks found linear time proof algorithm runs linear time edge explored twice direction moreover blossom structures also maintained linear time using disjoint set data structure say set nested blossoms maximal contracting blossoms contractible blossom reachable unsaturated vertex found since augmenting walk eliminated augmentation equivalent saying edge eligible thus implement blossom step edmonds search suffices run linear time find augmenting walk return blossom set discovered blossom formation step invariant maximal corollary maximal set nested blossom found linear time corollary one iteration edmonds search criterion approximate complementary slackness implemented linear time algorithms unweighted cover section give algorithm maximum cardinality minimum cardinality cover direct consequence algorithm weighted problem algorithm matches running time rely reduction algorithm moreover much simpler state analyze illustration purposes focus maximum cardinality algorithm consists two phases first phase referred batch augmentation finds close optimal using instance algorithm close optimum discard dual variables dissolve blossoms uses linear time augmenting walk algorithm increase cardinality becomes optimum stated formally following theorem theorem maximum cardinality computed time proof view maximum cardinality problem maximum weight problem weight function choose theorem compute maximum cardinality matching time maximum cardinality means augmentations away optimal hence discard blossom structure duals approximate run linear time augmenting pwalk algorithm lemma respect optimal discussion iterations suffice total running time algorithm theorem minimum cardinality cover computed time proof similar theorem first use algorithm cover given section find minimum cardinality cover viewing graph weighted graph weight everywhere choosing give cover notice always taking arbitrary incident edges taking union always give trivial cover cardinality hence means reductions needed make optimal therefore run augmenting path algorithm lemma find reducing paths reducing path augmenting path reducing path found iterations phase total running time algorithm references choromanski jebara tang adaptive anonymity via proceedings international conference neural information processing systems pages choromanski khan pothen jebara large scale robust adaptive anonymity via cover parallel computation chvatal greedy heuristic problem mathematics operations research drake hougardy simple approximation algorithm weighted matching problem information processing letters duan pettie linear time approximation maximum weight matching acm duan pettie scaling algorithms weighted matching general graphs proceedings annual symposium discrete algorithms soda pages edmonds maximum matching polyhedron vertices res nat bureau standards edmonds paths trees flowers canadian journal mathematics pages edmonds johnson matching class integer linear programs combinatorial optimization eureka shrink pages gabow efficient reduction technique subgraph bidirected network flow problems proceedings fifteenth annual acm symposium theory computing pages gabow data structures weighted matching extensions factors corr gabow sankowski algebraic algorithms shortest undirected paths ieee annual symposium foundations computer science pages gabow tarjan algorithm special case disjoint set union proceedings fifteenth annual acm symposium theory computing stoc pages gabow tarjan faster scaling algorithms network problems siam journal computing gabow tarjan faster scaling algorithms general graph matching problems acm khan pothen new algorithm cover problem proceedings seventh siam workshop combinatorial scientific computing pages lawler combinatorial optimization networks matroids dover books mathematics series micali vazirani algoithm finding maximum matching general graphs annual symposium foundations computer science pages pettie sanders simpler linear time approximation maximum weight matching inf process pulleyblank faces matching polyhedra phd thesis university waterloo ontario canada schrijver combinatorial optimization polyhedra efficiency algorithms combinatorics springer tutte problem decomposing graph connected factors journal london mathematical society
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robust toric ideals jun adam boocher elina robeva abstract call ideal polynomial ring robust minimally generated universal basis paper show robust toric ideals generated quadrics essentially determinantal discuss two possible generalizations higher degree providing tight classification determinantal ideals counterexample natural extension lawrence ideals close discussion robustness higher betti numbers introduction let polynomial ring field call ideal robust minimally generated universal basis collection polynomials form basis respect possible monomial term orders robustness strong condition instance robust number minimal generators initial ideal term orders general expect inequality trivial reasons monomial principal ideals robust simple considerations show robustness preserved upon taking coordinate projections joins see section however nontrivial examples robust ideals rare difficult result recently extended shows ideal maximal minors generic matrix indeterminates robust toric case class lawrence ideals naturally includes determinantal ideals largest known class robust ideals paper discusses robustness toric ideals main result following theorem let set irreducible binomials minimally generate ideal assume partitioned disjoint sets polynomials distinct variables consists polynomials degree following equivalent ideal generated robust consists minors generic matrix rescaling variables higher degree true robustness implies determinantal even lawrence adam boocher elina robeva motivation studying robust toric ideals threefold first study ideals minimally generated basis term order ubiquitous literature conca studied certain classical ideals determined minimally generated basis study koszul algebras one fruitful approaches via ideals generated quadratic basis aware however systematic study ideals minimally generated universal basis robust ideals second focus toric ideals natural testing ground algebraic questions toric ideals possess binomial universal bases hence rich theory basis theory see finally passing ideal initial ideal particular type flat deformation phrasing robustness almost equivalent property minimal number generators preserved interpretation suggests might geometric interpretation robustness terms hilbert scheme paper organized follows section prove main result theorem characterizing robust toric ideals generated degree two methods mainly combinatorial sections pose two questions concerning extensions theorem using lawrence ideals provide negative positive answers respectively section closes discussion robustness higher betti numbers original motivation project quadratic robust toric ideals determinantal sequel toric ideal always mean prime ideal generated minimally homogeneous binomials nonzero coefficients support polynomial mean set variables appearing terms definition set polynomials called robust universal basis elements minimally generate ideal set polynomials written union polynomials disjoint sets variables say robust component admits decomposition say set irreducibly robust notice robustness preserved disjoint unions classify robust ideals suffices study irreducible ones remark ideal corresponds join varieties corresponding goal section prove following theorem theorem let irreducibly robust set consisting irreducible quadratic binomials robust consists minors generic matrix rescaling variables equality general weaker robustness example consider ideal robust toric ideals remark statement theorem assume generators irreducible turns sufficient show ideal generate prime notice one direction follows immediately results show minors universal basis prove converse technique essentially eliminate certain combinations monomials appearing simplify notation omit writing coefficients proofs clear affect argument particular treat issue coefficients tackling proof theorem earlier lemmas lemma let robust set prime quadratic binomials monomial appears term two different polynomials proof suppose monomial appears polynomials let lex term order taking support first since prime select lead term applying buchberger algorithm would obtain degree zero syzygy elements contradicting minimality proposition robust proof write clear minimally generates ideal let term order extend term order taking last since basis respect basic properties bases know basis respect proposition extremely useful analysis helpful assume working ring variables use reduction extensively following main technical lemma use letters convenient ease reading lemma let robust set prime quadratic binomials contain two polynomials form monomial contain two polynomials form assume distinct contain two polynomials form adam boocher elina robeva monomial two polynomials whose supports share variable terms squarefree contains two polynomials whose supports share variable coefficients must contain minors generic matrix proof prove first statement proofs others similar suppose notice primality contain factor let lex term order variables whose lead term since basis respect must polynomial whose lead term divides since lead term must two distinct polynomials appearing contradicts lemma suppose restricting subring proposition tells assume involves variables notice primality part know distinct let lex term order variables whose lead term part must polynomial whose lead term divides possible monomials since appears lead term must polynomial contains applying part primality know part know must squarefree since already appears say renaming necessary choosing lex term order variables see whose lead term divisible monomials neither lead term polynomial lemma prove first statement second proof similar suppose first restrict using proposition assume working variables factors let lex term order variables taking obtain wyz whose lead term wyz since must divisible lead term polynomial without loss generality assume consider possibilities primeness contain factor part contain factor hence possible options left factors contained factors means new distinct variables conclude impossible follows immediately parts assume contains two polynomials whose supports intersect assume polynomials squarefree write represents variable notice primality part neither robust toric ideals since would repetition hence may well assume different variables call two cases either also new variable case see without loss generality rewriting two polynomials whose supports intersect must contain either distinct variables case variables restrict subring computing lex order variables obtain lead term must divisible lead term polynomial lead term presence hence must either case contains polynomial form primality part lemma allows say along squarefreeness implies required case term similar considerations show parts allow case variables restrict subring computing lex term order variables obtain ebg lead term must divisible lead term polynomial lead term presence hence must either symmetry assume contains polynomial form primality part restricts since already appears conclude symmetry may assume notice two polynomials whose supports intersect involve variables hence previous part proof conclude contains polynomial contradiction lemma proof theorem suppose since irreducible must contain two polynomials whose supports intersect lemma conclude contains polynomials form coefficients however computing lex order obtain bdf cde nonzero coefficients reducing obtain either zero constant multiple cde latter case order continue algorithm would another polynomial whose lead term divided cde presence terms prohibited lemma also prohibited hence must reduce zero two subtractions means fact polynomials precisely determinants matrix adam boocher elina robeva nonzero constants complete proof suppose since irreducible one polynomial must share variable say renaming variables necessary say variable ignoring constants moment applying proof lemma polynomials conclude contains minors matrix applying technique well shows fact get minors full matrix inductively continue process obtain notice proof every term order used lex term order ended prime ideal hence following corollary set prime quadratic binomials minimally generate ideal following equivalent basis respect every lex term order basis respect every term order iii generates prime ideal irreducible robust components generic determinantal ideals hypersurfaces corollary generic matrix set minors universal basis remark almost case every irreducibly robust component determinantal indeed every prime binomial rescaling either former determinantal thus possible robust component determinants lawrence ideals encouraged result previous section natural ask extent robustness classifies generic determinantal ideals indeed easy see ideal minors matrix whose entries relatively prime monomials robust two questions consider question robust toric ideal generated minors matrix monomials precisely matrices monomials provide robust ideals minors answer first question negative examples provided lawrence ideals studied toric ideal corresponding variety ideal corresponding called lawrence lifting ideal generated polynomials following form robust toric ideals sublattice field xann binomial ideals form called lawrence ideals following result theorem proposition following sets binomials lawrence ideal coincide minimal set binomial generators reduced basis universal basis graver basis hence lawrence ideals provide large source robust toric ideals naturally include class generic determinantal ideals given natural rephrase first part question question robustness characterize lawrence ideals answer negative example ideal adf bde polynomial ring robust lawrence example found using software gfan gra jen toric ideal corresponding lattice defined kernel given counterexample ask question nice combinatorial description robust toric ideals remark heuristically easy find robust ideals lawrence starting lawrence ideal given lattice gives rise lattice modifying slightly often case resulting toric ideal robust though often ubiquity examples computationally suggests nice combinatorial description robustness may require imposing hypotheses matrices monomials section answer question theorem suppose monomials degree least given set variables let adam boocher elina robeva suppose set consists irreducible binomials robust monomials relatively prime proof technical begin fixing notation since assume prime gcd gcd gcd therefore define zij gcd write zij zij thus gcd gcd gcd zij zkl whenever iii gcd zkl gcd zkl goal show zij lemma exist permutations term orders xim yjm xjm yim xim yjm xkm ylm xkm ylm xlm ykm moreover xim xim zim xim ylm ylm zmlm ylm proof build several steps begin take lex term order variables first consider lcm lcm since variables different variables exist term orders refining leading term robust toric ideals leading term consider first since form basis respect exist xim yjm xim yjm xjm yim ordering true since yim xim chosen variables first xjm yim yim xim xim yjm true thus similarly deduce thus since gcd xim gcd xim assuming xim xim since gcd either thus particular ifqn already contradiction assume since xim xim zim zkm properties iii allow conclude xim xim zim xim zim going back chose ordering consider ordering symmetric argument find exist xkm ylm thus ylm ylm zmlm ylm zmlm hence exist xim xim zim xim zim ylm ylm zmlm ylm zmlm lemma even rearrange numbers proof property lemma permutations fixed points thus exists zim zim exists zml fix since assumed repeating whole argument instead recall would get similar permutations property permutations agree permutations set thus transpositions permutations particular since run argument permutations agree composed transpositions particular adam boocher elina robeva even rearranging numbers thus matrix looks follows rest statement lemma follows lemma proof theorem suffices show zij without loss generality assume lemma xkm ylm xim yjm proof lemma know hold substitute xim yjm xkm ylm rewriting expressions form xim xim zim ylm ylm zmlm canceling repeating factors get zim xim yjm zljm xkm zkm lkm ylm zmlm therefore every zim zljm zkm lkm zmlm noting similarly switching using shows zim zljm zkm lkm zmlm property way happen case xim zlm zim zmlm ylm zmlm iii switching xim zim zim zmlm ylm zmlm cancelations xim ylm thus xim ylm since permutations thus matrix looks like example prime contradiction thus zij thus symmetry zij gcd combined assumptions theorem statement get gcd robust toric ideals gcd gcd entries pairwise relatively prime assume entries pairwise relatively prime let monomial term order show minors basis respect need show reduce zero result reduction guaranteed exist generic matrix xij since entries relatively prime clear reduction extend simply ring map xij xij robustness higher betti numbers interest robust ideals originated following classical inequality natural ask ideals term orders equality holds setting determinantal ideals conca proved ideal maximal minors generic matrix initial ideal property also gave examples determinantal ideals initial ideal property different vein conca herzog hibi showed generic initial ideal gin gin gin also agree interest instead approach inequality universal setting consider equality holds term orders case say robust betti numbers following result due first author theorem ideal maximal minors generic matrix term order particular every initial ideal squarefree monomial ideal linear free resolution resolution obtained complex taking appropriate lead terms syzygy combination theorems yields corollary let toric ideal generated degree two robust robust betti numbers original hope project robust toric ideals robust betti numbers unfortunately situation seems much delicate example using gfan jen able check lawrence ideal corresponding lattice given matrix initial ideals different betti numbers adam boocher elina robeva acknowledgments many results paper discovered via computations using macaulay gra gfan jen thank david eisenbud bernd sturmfels many useful discussions finally thank seth sullivant introducing lawrence ideals starting path toward example references boocher free resolutions sparse determinantal ideals math res letters bernstein zelevinsky combinatorics maximal minors algebraic combin conca negri gorla universal groebner bases maximal minors arxiv conca herzog hibi rigid resolutions big betti numbers comment math helv conca thomas nice initial complexes classical ideals algebraic geometric combinatorics gra grayson macaulay software system research algebraic geometry available http jen jensen gfan software system fans tropical varieties sturmfels bases convex polytopes university lecture series vol american mathematical society providence sturmfels zelevinsky maximal minors leading terms adv math
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segmentation nearly isotropic overlapped tracks photomicrographs using successive erosions watershed markers jan alexandre fioravante siqueiraa wagner massayuki nakasugaa sandro guedesa lothar ratschbacherb departamento raios cronologia ifgw university campinas institut geologie bergakademie freiberg abstract major challenges automatic track counting distinguishing tracks material defects identifying small tracks defects similar size detecting overlapping tracks address latter issue using wusem algorithm combines watershed transform morphological erosions labeling separate regions photomicrographs wusem shows reliable results used photomicrographs presenting almost isotropic objects tested method two datasets diallyl phthalate dap photomicrographs compared results counting manually using classic watershed mean efficiency ratio using wusem test datasets keywords automatic counting diallyl phthalate digital image processing fission track dating corresponding author phone email addresses siqueiraaf alexandre fioravante siqueira wamassa wagner massayuki nakasuga sguedes sandro guedes lothar ratschbacher authorship statement designed algorithms wrote code analyzed data wrote paper counted tracks images analyzed data wrote paper obtained images samples analyzed data wrote paper supervised research project supervised research project preprint submitted computers geosciences january introduction solid state nuclear track detectors ssntd materials inorganic crystals plastics glasses known record path charged particles several applications ssntd nuclear science instance measurements radon gas boron neutron capture therapy age determination fission track dating charged particle collide ssntd ionization collision atoms modify path particle damage ssntd structure called latent track track becomes visible optical microscope convenient etching process tracks counted photomicrographs also used could improve counting accuracy procedures measuring counting tracks involve practical problems variation observer efficiency automatic method based image processing techniques could increase track counting rate improve counting reproducibility however separating elements nontrivial images one hardest tasks image processing automatic systems separating counting measuring tracks studied several solutions presented still precision automatic methods satisfactory yet automatic analysis could need manually adjusted operator time consuming usual measure major challenges automatic track counting detecting overlapping tracks distinguishing tracks material defects surface scratches due polishing identifying small tracks defects comparable size background photomicrographs address problem identifying overlapping ion tracks photomicrographs propose algorithm based watershed transform using morphological erosions markers similar method used separate packings ellipsoidal particles represented using tomography tested method two datasets diallyl phthalate dap photomicrographs used results relate incident track energy mean gray levels mean diameter products sample first dataset material methods employed isodata threshold obtain binary images used tests wusem algorithm based following techniques morphological erosion watershed transform labeling describe algorithms section isodata threshold iterative data analysis technique isodata threshold method returns threshold separates image two pixel classes threshold intensity halfway mean intensities applied two classes isodata always convergent case classes tracks regions interest roi background used algorithm isodata implemented structuring elements morphological erosion morphological image processing structuring element matrix representing mask shape used retrieve information shapes input image erosion basic operation image processing uses chosen structuring element shrink border roi binary image shrinkage factor correspondent structuring element size tests section used disks structuring elements since tracks dap mostly round shape algorithms used contained watershed transform watershed algorithm method defines contours watershed gradient modulus gray levels input image considered relief surface detection method algorithm input image seen perspective two dimensions correspond spatial coordinates third represents gray levels interpretation consider three kinds points points regional minima points drop water would flow common minimum points drop water would flow different minima set points named watershed line aim watershed algorithms find watershed lines method used paper implemented function labeling algorithm image processing labeling algorithm labels connected regions binary input image according sense eight pixels surrounding reference pixel pixels receive label connected value used algorithm implemented described watershed using successive erosions markers wusem algorithm present wusem watershed using successive erosions markers algorithm combines morphological erosions watershed transform labeling algorithms separate regions interest roi binary images wusem algorithm steps follow figure user define initial radius iterative radius structuring element figure figure representing wusem functions employed implementation input binary image eroded using structuring element radius equal erosion used marker watershed algorithm resulting image labeled first segmentation new structuring element defined radius input binary image eroded new structuring element erosion used marker watershed result labeled second segmentation process continues eroded image objects segmentations summed result labeled reorder roi labels area smaller pixels excluded ensure noise affect results tracks counted according lower right corner method objects touch bottom right edges counted leads accurate unbiased measure wusem implemented function segmentation wusem available supplementary material receives arguments str initial radius delta radius representing structuring elements respectively exemplify wusem capabilities used separate overlapping tracks within test photomicrographs figure dap photomicrographs used two sets diallyl phtalate dap photomicrographs test wusem algorithm one obtained detectors irradiated tracks induced fission tracks figure representation disks structuring elements defined segmentation employing wusem interest region radius first second third structuring elements respectively tracks first dataset contains photomicrographs tracks nine different dap plaques irradiated ions nominal fluence beam perpendicular detector surfaces gsi darmstadt irradiation detectors covered aluminum foils forming moderation layers thicknesses varying zero cover detectors named cover thicknesses ions arrived setup initial energy mev slowed aluminum cover hitting detector surface incidence energies photomicrographs contained folder orig available supplementary material figure wusem algorithm application input image first image binarized using isodata threshold image eroded using initial radius iterative radius equal respectively ease visualization process continues eroded image regions erosions summed result labeled regions area smaller excluded labeled track receives color nipy spectral colormap finally function enumerate objects used number found tracks final results shown equal respectively animation also available https calculated using software srim varied mev cover mev cover detectors etched pew solution koh ethanol water solution min images captured ccd camera coupled zeiss microscope reflected light mode nominal magnification detectors etched minutes total min new images captured way obtained subsets images tracks within photomicrographs almost isotropic orientation resembling circles figure figure photomicrograph first dataset presenting tracks dap sample induced fission tracks second dataset contains photomicrographs two different magnifications dap plaque used external detector coupled apatite sample irradiated thermal neutrons induce fission atoms inside fission process two fragments released eventually detected dap plaque fragments arrive detector surface different energies incidence angles resulting variety track formats hence counting tracks photomicrographs complex previous case figure license reusability wusem algorithm several functions implementation lie within packages numpy scipy matplotlib among others code published paper written python photomicrographs contained folder orig available supplementary material figure input test photomicrograph second dataset presenting tracks dap sample available gnu gpl license photomicrographs figures distributed paper available license experimental processing photomicrographs tracks exemplifying methodology use photomicrograph first exemplify wusem figure binarized photomicrograph using isodata threshold figure different gray levels tracks may properly separated complicating extraction track features address issue filled regions binary image using function fill holes scipy image folder orig incid available supplementary material figure input photomicrograph figure binarized using isodata threshold threshold region filling colormap gray binarizing input image wusem algorithm separates overlapping tracks figure example chose initial radius delta radius parameters tracks counted using lower right corner method border tracks counted lie right bottom edges image used function clear border given supplementary material remove tracks edges wusem returns labeled region used parameter function enumerate objects figure comparison manual automatic counting experienced observer easily distinguish tracks photomicrograph even several tracks superimposed reason manual counting considered control comparison automatic ing following wusem algorithm applied photomicrograph set processing parameters studied established arbitrary value two tracks less mean manual counting tolerance wusem parameters become candidates automatic counting lies within tolerance interval track number obtained approach figure wusem best parameters study ones within tolerance interval samples according stated comparison best parameters initial radius delta radius min samples initial radius delta radius min samples using best parameters defined compared manual classic flooding watershed wusem counting sample figure table classic watershed algorithm used implemented imagej full method consisted binarizing input image applying watershed removing small objects source code operations instructions use given supplementary material since tracks dataset shapes attribute efficiency manual counting wusem counting initially set obtain smaller number tracks compared manual counting however wusem counting returns false positives incorrectly labels background regions roi points line figure avoid false positives one could use restrictive criterion eccentricity section counting reproducibility important considering reliability track counting despite variations track characteristics efficiencies wusem automatic counting remained constant within uncertainties compared manual counting manual counting wusem counting sample min min min min min efficiency min table mean standard deviation efficiency ratio sample first dataset relating ion energies diameter product mean gray levels application use wusem relate track energies product major minor diameters respectively mean gray level track considered approximately circular tracks analysis based eccentricity criterion equal less figure additional criterion used also counting tracks ensure false positives spurious objects counted tracks avoided separating track obtain features gray levels diameters mean gray level diameters track photomicrograph obtained relate incident energy sample mean gray level sample obtained getting mean gray levels tracks images sample adopted similar process obtain mean diameters results related incident energy figure diameter products roughly reflect electronic energy loss curve dap figure calculated software image processing eccentricity object number interval lower value region becomes closer circle srim bragg peak appears around mev scatter points attributed poor control etching conditions uncertainty temperature may cause large variation etching results variations gray level means impaired probably set acquired reflected illumination mode privileges surface details depth effects processing photomicrographs fission tracks dap photomicrographs first dataset main concern track superposition however tracks similar created collimated beam tracks one step ahead applying wusem images tracks present variety shapes image dataset obtained dap plaques irradiated thermal neutrons coupled apatite mounts fission fragments born interior mineral emitted different directions towards detector instance round tracks created perpendicular incident fragments elliptical ones created particles hitting dap surface shallower angles figure previous analysis manual counting performed experienced observer taken reference expect observer able recognize tracks efficiently however case expect observer efficiency fragments hitting detector lower energies originate tracks difficult distinguish detector surfaces experienced observer would avoid counting tracks keep counting efficiency constant repeating previous processes photomicrographs second dataset first binarize test figure using isodata threshold binarized image generated two scenarios considering ignoring border tracks regions binary image also filled using function fill holes scipy apply image folder orig able supplementary material wusem algorithm chose initial radius delta radius parameters wusem result parameter function enumerate objects figure dataset established arbitrary tolerance five tracks less mean manual counting case wusem parameters become candidates track number obtained approach according stated comparison best parameters initial radius delta radius first magnification initial radius delta radius second one using best parameters defined compared manual classic watershed wusem counting sample figure table magnification manual counting wusem counting efficiency table mean standard deviation efficiency ratio magnification second dataset wusem succeeded avoiding false positives application compared classic watershed black line line figure still user could apply restrictive criteria also worth noting efficiency variation two image sets bigger objects easier treated thus automatic track counting greater magnification images resulted higher counting efficiency table discussion structuring elements study used disks structuring elements processing images datasets since tracks photomicrographs dataset almost isotropic seen figure disks suitable structuring elements used segmentation however tracks within images dataset defined format figure employing different structuring elements rotated cones ellipses could improve segmentation automatic counting result false positives false negatives counting efficiency track counting reproducibility counting every track image counting types tracks every time primary concern ftd instance efficiencies counting tracks standard sample vary using zeta age calibrations absolute methods determining neutron fluence efficiency constant counting tracks unknown age samples areas ftd using laser ablation inductively couple plasma mass spectrometer electron microprobe efficiency issues could also benefit automatic counting major challenge reproducibility avoiding false positives spurious objects scratches detector mineral surface etching figures automatic counting algorithms could misrepresent tracks situations preferable restrict criteria thus increasing number false negatives counted tracks even implying efficiency reduction even experienced observers expect decreasing efficiency due superposition counting tracks high track density samples loss tends severe automatic counting applying algorithms wusem separation tracks objects formed several tracks always possible counting less tracks actual number cluster acceptable however processing large number clusters higher track density samples expect lower efficiency compared lower track density samples effect assessed calibrating efficiency function track density therefore efficiencies presented wusem tables hold track densities used samples perspective future development wusem algorithm represents advance automated track counting opening several possibilities differently direct application classic watershed wusem allows adaptation figure user determine optimal structuring elements efficiencies using training image subset apply parameters sample photomicrographs convenient fission track dating tracks present shapes regardless track source efficiency may vary mainly due superposition tracks higher track density samples another possibility applying wusem separate parts input image allowing determination specific parameters track configuration could bring even better results dealing superposition two three tracks conclusion paper present watershed algorithm using successive erosions markers call wusem employ wusem separate overlapping tracks photomicrographs wusem performs well images containing overlapping circular tracks collimated beam photomicrographs fission tracks various orientations dap results encouraging mean counting efficiency ratio using wusem test datasets show also diameter eccentricity criteria may used increase reliability method since wusem using circles structuring elements aimed isotropically shaped regions technique suitable separating etched tracks dap etching velocity mineral surfaces depends crystal orientation yielding complex etching figures also natural minerals richer scratches etching figures mistaken fission tracks especially using image processing techniques wusem could studied separate tracks mineral surfaces would need use different structuring elements consider orientation shape track acknowledgements authors would like thank raymond jonckheere second photomicrograph dataset matthias insights also christina trautmann sample irradiation gsi darmstadt work supported paulo research foundation fapesp grants supplementary material supplementary material contains complementary results code published study available https references references brown solomon tinker development energy discriminate nuclear track etch dosimeter gas measurements journal environmental radioactivity smilgys guedes morales alvarez hadler coelho siqueira alencar soares curvo boron thin films detectors bnct method measure reaction rate radiation measurements url http wagner van den haute dating springer netherlands dordrecht url http goto danhara zircon dating hiyoriyama cryptodome kuttara volcano southwestern hokkaido japan bulletin volcanological society japan doi imayama takeshita cho kitajima tsutsumi kayama nishido okumura yagi itaya sano partial melting contrasting cooling history within higher himalayan crystalline sequence nepal himalaya lithos url http takagi tsutsui arai iwano danhara geological framework fission track dating pseudotachylyte atotsugawa fault magawa area central japan island arc doi url http yuguchi sueoka iwano danhara ishibashi sasao nishiyama spatial distribution apatite ages toki granite central japan exhumation rate cretaceous pluton emplaced east asian continental margin island arc url http ninomiya taniguchi shimoyama watanabe danhara iwano dunkley shiraishi gouzu dating submarine pyroclastic rock southwest japan island arc url http fleischer price walker hubbard track registration various nuclear track detectors physical review url https moriwaki shimpuku makimura improvement track counting accuracy efficiency fission track radioisotopes url https gleadow gleadow belton kohn krochmal brown coincidence mapping key strategy automatic counting fission tracks natural minerals geological society london special publications url http gonzalez woods digital image processing edition pearson upper saddle river usa williams automatic fission track counting using quantimet nuclear instruments methods beer pipper heinrich automatic measurements nuclear tracks plastic foils journal physics scientific instruments url http price krischer measurement etched tracks nuclear track detectors nuclear instruments methods physics research section accelerators spectrometers detectors associated equipment trakowski dreute sonntag brechtmann beer drechsel heinrich automatic measuring system particle tracks plastic detectors nuclear instruments methods physics research masumoto wadatsumi computerized system dating geoinformatics wadatsumi masumoto measurement fissiontracks principles example zircon fish canyon tuff international journal radiation applications instrumentation part nuclear tracks radiation measurements url http fews fully automated image analysis etched tracks nuclear instruments methods physics research section beam interactions materials atoms url http petford miller imaging fission tracks using confocal scanning laser microscopy american mineralogist petford miller briggs automated counting fission tracks external detector image analysis computers geosciences url http espinosa gammage meyer dudney nuclear track analysis digital imaging radiation protection dosimetry boukhair haessler adlo nourreddine new code digital imaging system track measurements nuclear instruments methods physics research section beam interactions materials atoms dolleiser fully automated optical microscope analysis particle tracks solids nuclear instruments methods physics research section beam interactions materials atoms bedogni development reader routine analysis fast neutron dosemeters improvement dosimetric performance using automatic vision motion tools radiation measurements url http yasuda namiki honma umeshima marumo ishii benton development high speed imaging microscope new software nuclear track detector analysis radiation measurements tawara eda takahashi doke hasebe kodaira ota kurano yasuda development automated multisample scanning system nuclear track etched detectors nuclear instruments methods physics research section accelerators spectrometers detectors associated equipment siqueira nakasuga pagamisse tello saenz job automatic method segmentation fission tracks epidote crystal photomicrographs computers geosciences url http ruan zhang zhang chen song liu liu image mosaic method recognition proton track nuclear emulsions nuclear instruments methods physics research section accelerators spectrometers detectors associated equipment url http enkelmann ehlers buck schatz advantages challenges automated apatite fission track counting chemical geology url http beucher lantuejoul use watersheds contour detection doi url http serra image analysis mathematical morphology vol academic press schaller neudecker saadatfar delaney mecke tomographic analysis jammed ellipsoid packings aip conference proceedings vol url http ball hall clustering technique summarizing multivariate data behavioral science ridler calvard picture thresholding using iterative selection method ieee transactions systems man cybernetics dias velasco thresholding using isodata clustering algorithm ieee transactions systems man cybernetics url http van der walt boulogne warner yager gouillart image processing python peerj url https fiorio gustedt two linear time strategies image processing theoretical computer science doi russ image processing handbook edition crc press url https ziegler ziegler biersack srim stopping range ions matter nuclear instruments methods physics research section beam interactions materials atoms url http van der walt colbert varoquaux numpy array structure efficient numerical computation computing science engineering url http jones oliphant peterson scipy open source scientific tools python url http hunter matplotlib graphics environment computing science engineering url http van rossum python reference manual tech centrum voor wiskunde informatica cwi amsterdam url https rueden schindelin hiner dezonia walter arena eliceiri imagej next generation scientific image data bmc bioinformatics url https guedes hadler iunes paulo tello reproducibility ssntd track counting efficiency nuclear instruments methods physics research section accelerators spectrometers detectors associated equipment url http hurford green zeta age calibration dating chemical geology url http hurford standardization fission track dating calibration recommendation fission track working group mission geochronology chemical geology isotope geoscience section soares alencar guedes takizawa smilgys hadler alpha spectrometry study makrofol measurements track diameter radiation measurements hadler iunes tello chemale kawashita curvo santos gasparini moreira guedes experimental study methodology dating without neutron irradiation radiation measurements url http hasebe barbarand jarvis carter hurford apatite chronometry using laser ablation chemical geology soares guedes hadler zack iunes novel calibration thermochronology physics chemistry minerals doi soares guedes stockli zack characterisation apatites potential uranium reference materials fissiontrack dating geostandards geoanalytical research url http gombosi garver baldwin development electron microprobe zircon geochronology chemical geology url http dias chemale soares guedes new approach electron microprobe zircon fission track thermochronology chemical geology january url http figure processing input image using wusem line presents segmentation step left column original binary image red erosion obtained according variation delta radius white center column erosion results labeled used markers right column watershed results using generated markers parameters wusem algorithm initial radius delta radius figure labels generated tracks figure using wusem algorithm tracks enumerated using enumerate objects tracks lower right corners counted according lower right corner method parameters wusem algorithm initial radius delta radius colormaps nipy spectral gray figure manual counting mean top blue bar values right sample automatic counting results mean within orange points outside gray points tolerance interval blue bar first dataset figure comparison manual automatic counting min etching samples min etching samples white manual counting gray flooding watershed counting red line distribution median white signal distribution mean dashed line red line regression wusem counting data black line regression flooding watershed counting data figure regions figure complying labeled regions tracks correspondent original gray levels colormaps nipy spectral magma figure relation incident energy versus mean diameter product min min samples left axis incident energy versus mean gray levels min min samples left axis cyan dashed line electronic energy loss calculated srim right axis figure input photomicrograph figure binarized using isodata threshold threshold region filling tracks separated figure using wusem algorithm enumerated using function enumerate objects parameters wusem algorithm initial radius delta radius figure comparison manual automatic counting photomicrographs dataset white manual counting gray flooding watershed counting red line distribution median white signal distribution mean blue dots outliers dashed line red line regression wusem counting data black line regression flooding watershed counting data figure using suitable input parameters wusem may perform better certain regions classic watershed return reliable results instance highlighted region presents three tracks wusem separates correctly region oversegmented classic watershed highlighted region turn undersegmented classic watershed returns two tracks wusem returns three tracks closer real number four tracks left input photomicrographs highlighted regions center tracks separated using wusem right tracks separated using classic watershed parameters wusem algorithm initial radius delta radius
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sparse activity detection massive connectivity jan zhilin chen student member ieee foad sohrabi student member ieee wei fellow ieee paper considers massive connectivity application large number potential devices communicate sporadic fashion detection device activity pattern together estimation channel central problems scenario due large number potential devices network devices need assigned signature sequences main objective paper show using random signature sequences exploiting sparsity user activity pattern joint user detection channel estimation problem formulated compressed sensing single measurement vector smv problem multiple measurement vector mmv problem depending whether single antenna multiple antennas efficiently solved using approximate message passing amp algorithm paper proposes amp algorithm design exploits statistics wireless channel provides analytical characterization probabilities false alarm missed detection using state evolution consider two cases depending whether component channel fading known design minimum mean squared error mmse denoiser amp according channel statistics simulation results demonstrate substantial advantage exploiting statistical channel information amp design however knowing fading component offer tangible benefits case employ two different amp algorithms namely amp vector denoiser parallel quantify benefit deploying multiple antennas index activity detection channel estimation approximate message passing compressed sensing internet things iot communications mtc ntroduction one key requirements wireless cellular networks provide massive connectivity communications mtc envisioned support diverse applications environment sensing event detection surveillance control communications two distinctive features compared conventional communications overall system needs support massive number devices connected cellular may order traffic pattern device may given time small fraction potential devices active network accurate user activity detection channel estimation manuscript accepted appear ieee transactions signal processing work presented part ieee international conference acoustics speech signal processing icassp march work supported natural sciences engineering research council nserc canada discovery grant steacie memorial fellowship authors edward rogers department electrical computer engineering university toronto toronto canada zchen fsohrabi weiyu crucial establishing successful communications devices identify active users estimate channels user must assigned unique signature sequence however due large number potential devices limited coherence time frequency dimensions wireless fading channel signature sequences users mutually orthogonal signature sequences superimposed pilot stage causes significant interference simple matched filtering correlation operation applied user activity detection channel estimation key observation paper sporadic nature traffic leads sparse user transmission patterns exploiting sparsity formulating detection estimation problem independent identically distributed random nonorthogonal pilots compressed sensing problem multiuser interference problem overcome highly reliable activity detection accurate channel estimation made possible compressed sensing terminology equipped single antenna activity detection channel estimation formulated single measurement vector smv problem multiple antennas problem formulated multiple measurement vector mmv problem paper proposes use compressed sensing techniques joint user activity detection channel estimation problem due nature massive device communications paper adopts computationally efficient approximate message passing amp algorithm main technique amp iterative thresholding method key feature allows analytic performance characterization via state evolution main contributions paper novel amp algorithm design user activity detection exploits statistical information wireless channel characterization probabilities false alarm missed detection smv mmv scenarios related work user activity detection problem massive connectivity studied information theoretical perspectives algorithmic point view problem closely related sparse recovery compressed sensing studied variety wireless communication settings example assuming prior knowledge channel state information csi joint user activity detection channel estimation considered specifically proposes efficient greedy algorithm based orthogonal matching pursuit sporadic communication exploiting statistics channel joint sparsity structures proposes modified bayesian compressed sensing algorithm cloud network context orthogonal frequency division multiplexing ofdm systems introduces random access protocol employs basis pursuit denoising detection method detection error bound based restricted isometry property performance schemes practical setting illustrated perfect csi assumed joint user activity data detection code division multiple access systems cdma investigated designs maximum posteriori detector accounting sparsity finitealphabet constraints signal proposes greedy orthogonal least square algorithm exploiting block sparsity among several symbol durations differing works consider cellular systems study user activity detection wireless hoc networks node system identifies neighbor nodes simultaneously authors propose scalable compressed neighbor discovery scheme employs random binary signatures group testing based detection algorithms authors propose dedicated scheme uses signatures based code chirp decoding algorithm achieve better performance contrast aforementioned works paper adopts computationally efficient amp algorithm user activity detection channel estimation suitable networks large number devices amp algorithm first proposed iterative algorithm conventional compressed sensing signals measurements framework state evolution tracks performance amp iteration introduced amp algorithm extended along different directions example generalizes amp algorithm broad family iterative thresholding algorithms provides rigorous proof framework state evolution deal signals measurements proposes complex amp algorithm camp exploiting input output distributions generalized amp gamp algorithm designed similarly bayesian approach used design amp algorithm accounting input distribution compressed sensing problem multiple signals sharing joint sparsity mmv problem designs amp algorithm via vector form message passing designs algorithm directly using message passing factor graph although use amp algorithm user activity detection previously proposed statistical information channel exploited prior work also performance analysis yet available paper makes progress showing exploiting channel statistics significantly enhance bayesian amp algorithm moreover analytical performance characterization obtained using state evolution finally amp algorithm extended case main contributions paper considers user activity detection channel estimation problem uplink network large number potential users given time slot small fraction active exploit sparsity user activity pattern paper formulates problem compressed sensing problem proposes use random signature sequences computationally efficient amp algorithm device activity detection paper provides design analysis amp cases equipped single antenna multiple antennas paper considers two different scenarios fading coefficients user known detector designed based statistics fast fading component fading coefficients known detector designed based statistics fast fading fading components function distribution device locations cell proposed detector exploits statistics wireless channel specifically designing minimum mean squared error mmse denoiser paper provides analytical characterization probabilities false alarm missed detection via state evolution scenarios case equipped single antenna numerical results indicate analytic performance characterization via state evolution close simulation exploiting statistical information channel user activity significantly improve detector performance iii knowing fading coefficient actually bring substantial performance improvement compared case statistical information fading available case equipped multiple antennas paper considers amp vector denoiser parallel amp vector denoiser paper exploits wireless channel statistics denoiser design analytically characterizes probabilities false alarm missed detection based state evolution parallel algorithm suitable distributed computation performance characterization difficult obtain simulation results show multiple antennas significantly improve detector performance predicted performance amp vector denoiser close simulated performance iii amp vector denoiser parallel achieve approximately performance paper organization notations remainder paper organized follows section introduces system model section iii introduces amp algorithms smv mmv problems section considers user activity detection channel estimation section considers case simulation results provided section conclusions drawn section vii throughout paper letters denote random variables realizations respectively boldface letters denote vectors boldface uppercase letters denote matrices random vectors context make distinction clear superscripts denote transpose conjugate transpose inverse operators respectively denotes identity matrix appropriate dimensions denotes expectation operation denotes definition denotes either magnitude complex variable determinant matrix depending context denotes norm ystem odel consider uplink wireless cellular system one located center devices located uniformly circular area radius coherence block subset users active let indicate whether user active purpose channel probing user identification user assigned unique signature sequence sln length sequence paper assumes signature sequence generated according complex gaussian distribution zero mean variance sequence normalized unit power normalization factor incorporated transmit power consider channel model channel static block paper consider two cases equipped either single antenna multiple antennas one antenna received signal modeled channel coefficient user effective complex depends background sian noise whose variance noise power normalized user transmit power aim recover entries based received signals interested regime number potential users much larger pilot sequence length user pilot sequences mutually orthogonal due sporadic traffic small number devices transmit block resulting sparse recovering single antenna case form smv problem compressed sensing paper also considers case equipped antennas case received signal expressed matrix form channel vector user effective complex gaussian noise denotes point mass measure denotes probability density function pdf channel vector given prior information pdf denotes prior information user note use denote random channel vector denote realization based pdf entries rtn nth row vector also use represent mth column vector note indicates whether entire row vector zero words columns share sparsity pattern interested detecting user activity well estimating channel gains active users correspond rows matrix based observation regime problem recovering form mmv problem compressed sensing key observation paper design recovery algorithm significantly enhanced taking advantage knowledge statistical information toward end provide model distribution entries distribution rows since special case focus model assume user accesses channel small probability fashion correlation different users channels row vectors follow mixture distribution model distribution assume users randomly uniformly located circular coverage area radius center channels users follow independent distribution depends distance specifically includes pathloss shadowing rayleigh fading user modeled distance measured meter fading coefficient exponent shadowing follows gaussian distribution zero mean variance rayleigh fading assumed complex gaussian zero mean unit variance across antennas fading includes shadowing denoted whose pdf modeled distribution distance shadowing parameter paper considers case statistics fading known well case exact fading coefficient known latter case motivated scenario devices stationary shadowing estimated stored prior information known captures distribution rayleigh fading component known drop write captures distribution fading rayleigh fading iii amp lgorithm amp iterative algorithm recovers sparse signal compressed sensing introduce amp framework smv mmv problems section amp smv problem amp first proposed smv problem starting amp proceeds iteration index iteration estimate iteration residual appropriately designed function known denoiser operates nth entry input vector first order derivative respect first argument averaging operation entries vector note third term right hand side correction term known onsager term statistical physics amp algorithm matched filtered output modeled signal plus noise including multiuser interference gaussian due correction term denoiser typically designed reduce estimation error iteration compressed sensing literature prior distribution usually assumed unknown case minimax framework worst case leads soft thresholding denoiser prior distribution known bayesian framework used account prior information paper adopt bayesian approach design mmse denoiser massive connectivity setup shown next section amp algorithm analyzed asymptotic regime fixed via state evolution predicts performance amp algorithm iteration follows referred state random variables following following complex gaussian distribution zero mean unite variance following expectation taken denote random variables capture distributions entries entries factor prior entries respectively characterizing mse estimate iteration amp mmv problem amp vector denoiser one extension amp algorithm solve mmv problem proposed employs vector denoiser operates row vector matched filtered output vector denoiser operates nth row vector notations similar used state evolution amp algorithm mmv problem also similar form random vector following random vector following expectation taken minimize estimation error iteration also design vector denoiser via bayesian approach parallel different extension amp algorithm dealing mmv problem parallel algorithm proposed basic idea solve mmv problem iteratively using multiple parallel exchanging soft information parallelization allows distributed implementation algorithm computationally advantageous especially number antennas large outline parallel algorithm illustrated algorithm operates basis columns denoted respectively denoiser used mth antenna iteration note add index notation denoiser indicate index outer iteration index smv stage respectively first phase called phase messages calculated passed mth stage messages convey current belief probability active user first iteration since information available next phase called within conventional amp algorithm applied received signal antenna note denoiser amp algorithm function current belief activity users obtained based information sharing stages finally phase estimate channel gains used refine belief activity users ser activity etection ingle ntenna ase main point paper exploiting statistics wireless channel significantly enhance detector performance section proposes mmse denoiser design amp algorithm wireless massive connectivity proof see appendix algorithm parallel method initialize execute execute within initialize end end execute calculate probability user active based decision mth stage end problem specifically takes wireless channel characteristics consideration case two scenarios considered fading user either directly available statistics available section studies optimal detection strategy analyzes probabilities false alarm missed detection using state evolution amp algorithm mmse denoiser amp algorithm scenario statistics largescale fading known distributions channel coefficients independent identical devices scenario devices stationary shadowing coefficients estimated thus exact fading known distributions channel coefficients form complex gaussian variance parameterized independent identical across devices derive mmse denoisers via baysian approach cases first characterize distributions follows proposition consider circular wireless cellular coverage area radius center uniformly distributed devices channels devices modeled fading parameters shadowing fading parameter defined system model follows distribution exp constants depending parameters exp proposition denote channel coefficient contains fading rayleigh fading known pdf given dgn exp known pdf given exp proof see appendix note first scenario channel distribution depends parameters exponent model standard deviation shadowing model assumed known estimated practice second scenario channel distribution rayleigh fading model parameterized fading fading information obtained tracking estimated channel reasonable period second scenario applicable case users mostly stationary fading changes slowly time worth noting although paper restricts attention rayleigh fading model approach developed equally applicable rician statistical channel model following design mmse denoisers amp algorithm exploiting statistical knowledge fading since unknown distribution available denoiser reduces indicates denoiser entry matched filtered output using pdf entries expressed exp dgn mmse denoiser given conditional expectation random variable realization note denoiser depends expression conditional expectation given following proposition proposition based pdf model iteration amp conditional expectation given given functions defined exp dgn exp exp dgn hypothesis testing problem inactive user active user optimal decision rule given llr log mmse denoiser soft thresholding soft thresholding mmse denoiser fig mmse denoiser soft thresholding denoiser indicator function proof see appendix note implement iteration value needed practice empirical estimate kzt denotes norm used although complicated form note precomputed stored table lookup add runtime complexity gain intuition illustrate shape mmse denoiser compared widely used soft thresholding denoiser fig observe mmse denoiser plays role similar soft thresholding denoiser shrinking input towards origin especially input small thereby promoting sparsity exact knowledge fading available substitute pdf entries simplified exp mmse denoiser given conditional expectation exp compared mmse denoiser add left hand side emphasize dependency prior information user activity detection amp algorithm converged employ likelihood ratio test perform user activity detection llr denotes ratio denotes decision threshold typically determined cost function performance metrics interest probability missed detection defined probability device active detector declare null hypothesis probability false alarm defined probability device inactive detector declare active consider threshold two cases depending whether fading coefficient available statistical knowledge fading first derive likelihood probabilities following proposition suppose follows follows complex gaussian distribution zero mean unit variance likelihood given given exp agn exp dgn proof see appendix based ratio given llr log exp dgn defined observing llr monotonic simplify decision rule indicating user activity detection performed based magnitude based likelihood probabilities threshold probabilities false alarm missed detection characterized follows exp simplified using note since statistical information fading known averaged false alarm missed detection probabilities depend exact knowledge fading known distribution simplified likelihood probabilities become exp exp ratio given llr log exp notation llr emphasizes dependency prior information similar case statistics known llr also monotonic means user activity detection performed based also use denote threshold detection based probabilities false alarm missed detection probability given follows exp exp use notation indicate prior known note false alarm probability form even value may different due different denoisers natural question arises design threshold function known fading theory treat user separately set thresholding value user separately according cost function example specific target false alarm probability needed user design thresholding parameter using expression order bring fairness paper considers common target false alarm probability users condition users share thresholding parameter since expression depend case different users may different probabilities missed detection depending fading measure performance detector entire system employ average probability missed detection exp exp distribution given large given averaged performance depends statistics fading state evolution analysis characterized probabilities false alarm missed detection user activity detection parameter represents standard deviation residual noise still needs determined amp proceeds converges compute use state evolution interpreted mse denoiser note mse also expressed var var conditional variance given expectation taken note drop fading coefficient unknown using conditional variance characterize mse designed denoisers following propositions proposition mse denoiser case statistics known given dgn mse functions defined respectively function defined exp dgn proof see appendix worth noting first term right hand side corresponds mse estimate fading coefficient well user activity assumed priori known integral corresponds averaging possible second term represents cost unknown unknown user activity reality similarly case exactly known mse characterized follows proposition mse denoiser case known exactly mse dgn dgn function defined exp exp proof see appendix also observe first term right hand side corresponds averaged mse user activity assumed known second term corresponds extra error brought unknown user activity based expressions mse state evolution expressed mse based evaluated according functions iteration number amp algorithm converges converges fixed point equation compare resulting mses two cases according decomposition variance var var var var indicates knowing fading help improve estimation given however simulation results section show surprisingly model fading considered paper performance improvement actually minor indicating knowing fading help get much better estimation knowing exact value crucial user activity detection statistical information sufficient device detection ser activity etection ultiple ntenna ase section designs amp algorithms account wireless channel propagation massive connectivity problem case mentioned earlier two different amp algorithms used mmv problem amp vector denoiser operating row input matrix parallel divides mmv problem parallel smv problems iteratively solves smv problem antenna separately soft information exchange antennas amp vector denoiser admits state evolution allows easier characterization performance whereas implemented distributed way helpful reducing running time algorithm especially equipped large antenna arrays user activity detection amp vector denoiser scenario single antenna consider cases statistical knowledge exactly knowledge fading known design denoisers first characterize pdfs row vectors following proposition denote row vector known pdf given exp dgn known pdf exp proof results extensions considering multivariate random variables given utn following complex gaussian distribution zero mean covariance mmse denoisers cases given conditional expectation following proposition known conditional expectation given dgn dgn defined follows exp exp exp known conditional expectation defined follows exp proof see appendix covariance matrix tracked via state evolution simplified following proposition proposition based pdfs state evolution initial covariance matrix diagonal matrix identical diagonal entries stays diagonal matrix identical diagonal entries determined mse known mse given mse dgn functions defined respectively gamma function exact fading known mse given mse dgn dgn function defined proof see appendix note noise covariance matrix first matched filtering indeed diagonal matrix identical diagonal entries based proposition mmse denoiser simplified defined respectively mmse denoiser simplified exp fading unknown probabilities false alarm missed detection given respectively exp defined note let reduce denoisers case mentioned also store functions table lookup amp algorithm converged use likelihood ratio test perform user activity detection recall follows complex gaussian distribution case fading unknown based derived appendix likelihood probabilities given user inactive active respectively exp exp dgn case fading coefficient known noting follows distribution follows mixed gaussian distribution likelihood probabilities computed respectively exp exp cases immediately obtain llrs respectively exp dgn llr log llr log exp observing llrs monotonic set threshold perform detection exp dgn agn dgn lower incomplete gamma function simply obtained noticing integral interpreted cumulative distribution function cdf distribution degrees freedom since regarded sum squares identical real gaussian random variables using approach large scale fading known probabilities false alarm missed detection evaluated exp exp easy verify reduce respectively based design threshold achieve probability false alarm probability missed detection proposed thresholding strategy case still used scenario user activity detection parallel outline parallel algorithm presented algorithm section adopt parallel algorithm problem setup present expression denoiser probability device active based decision mth stage discuss case largescale fading known extension scenario fading unknown similar since parallel algorithm employs parallel expression scalar denoiser form mmse denoiser case however instead using prior probability active user algorithm access better estimate probability activities exp algorithm proceeds therefore expression mmse denoiser written exp elements nth row mth column respectively end ith outer iteration likelihood probabilities given user inactive active written exp using equations probability user active based decision mth calculated exp parallel terminated use likelihood ratio test perform user activity detection shown llr user calculated llr log exp seen llr expression algorithm similar form therefore discussion show probabilities false alarm missed detection simplified form similar respectively complete performance prediction analysis also need determine parallel algorithm however due soft information exchange antennas deriving analytic state evolution challenging numerical experiments section show performance parallel similar amp vector denoiser observation suggests parameter algorithm similar final value amp vector denoiser briefly discuss complexities amp vector denoiser algorithms computational complexities mainly lie matched filtering residual calculation depend problem size iteration advantage parallel computation allowed due division mmv problem several smv problems imulation esults evaluate performance proposed method cell radius potential users among active channel fading amp simulated amp predicted lower bound amp simulated amp predicted lower bound amp simulated amp predicted lower bound fig performance amp based user activity detection statistical knowledge fading parameters background noise first consider case statistical knowledge fading fig shows tradeoff probabilities missed detection false alarm amp mmse denoiser pilot sequence length set transmit power set see predicted match analysis well also plot lower bound using lower bounds close actual performance indicating convergence amp able almost completely eliminate multiuser interference remaining error dominated background noise fig shows performance amp algorithm mmse denoiser exact fading coefficients known comparison performance statistical knowledge fading also demonstrated simulated performance included since predicted performance almost depicted fig fig shows predicted curves match simulated curves well interestingly indicates performance improvement knowing exact fading coefficients negligible suggests knowing distribution fading rather exact value already enough good user activity detection performance next simulation compares amp algorithm mmse denoiser two algorithms widely used compressed sensing cosamp amp soft thresholding denoiser compare amp cosamp based matching pursuit technique compare amp mmse denoiser amp soft thresholding exploit statistical knowledge fig shows amp mmse denoiser outperforms cosamp amp soft thresholding denoiser partly due fact cosamp amp soft thresholding denoiser amp simulated amp simulated amp predicted amp simulated amp simulated mmse denoiser mmse denoiser mmse denoiser soft thresholding denoiser soft thresholding denoiser amp predicted amp simulated amp simulated amp predicted soft thresholding denoiser cosamp mmse denoiser soft thresholding denoiser cosamp mmse denoiser soft thresholding denoiser cosamp mmse denoiser power dbm fig performance amp based user activity detection knowledge fading fig performance comparison amp mmse denoiser amp soft threshoding denoiser cosamp exploit statistical knowledge note amp soft thresholding implicitly solves lasso problem sparse signal recovery problem least squares optimization therefore results fig indicate amp mmse denoiser also outperforms lasso fig compares performance amp algorithm mmse denoiser amp algorithm soft thresholding denoiser function transmit power pilot length convenience set properly choosing threshold observe first mmse denoiser outperforms soft thresholding denoiser significantly importantly observe minimum needed drive zero transmit power increases mmse denoiser whereas fig impact transmit power length pilot user activity detection performance mmse denoiser soft threholding denoiser minimum soft threshloding denoiser indicating clear advantage accounting channel statistics user activity detector design finally consider case assuming knowledge fading coefficients fig illustrates probabilities false alarm missed detection different numbers antennas amp vector denoiser parallel algorithms comparison case also included fig shows amp vector denoiser simulated results match predicted results well shows performances amp vector denoiser parallel ampmmv approximately indicating although two algorithms employ different strategies exploit statistical knowledge channel way resulting similar performances impact pilot length number antennas probabilities false alarm missed detection transmit power increases shown fig set make convenient comparison properly choosing threshold note scenario exact fading coefficients known different users probabilities false alarm different probabilities missed detection thus use average probability missed detection users fig shows increasing brings significant improvement specifically tend remain unchanged transmit power increases however either increasing increasing driven zero transmit power increases words minimum required drive zeros reduced increasing vii onclusion work shows compressed sensing viable strategy sporadic device activity detection massive case case simulation results validate analysis show exploiting statistics channel amp denoiser design significantly improve detection threshold deploying multiple antennas also bring significant performance improvement power ppendix amp simulated amp predicted bound vamp simulated simulated vamp predicted bound vamp simulated simulated vamp predicted bound proof proposition power fig performance amp vector denoiser vamp user activity detection case denote distance user shadowing shadowing linear scale respectively note appendices slightly abuse notations appeared paper due limited alphabet however cause confusion since used derivations appendices denote fading derive pdfs follows assuming users uniformly distributed cell radius pdf since follows gaussian distribution pdf derived zero mean variance exp using get pdf vamp simulated vamp simulated pdf vamp simulated vamp simulated vamp simulated vamp simulated power dbm exp constants given proposition fig impact length pilot number antennas transmit power proof proposition connectivity applications random signature sequences specifically propose user activity detection algorithm exploiting statistics wireless channel uplink cellular system large number potential users small fraction active time slot show using state evolution performance characterization terms probabilities false alarm missed detection accurately predicted particular consider cases equipped single antenna multiple antennas present designs mmse denoisers scenarios fading either available exactly statistics available case adopt two amp algorithms amp vector denoiser parallel tackle detection problem derive performance analysis denote fading following rayleigh fading real imaginary parts respectively denote channel coefficient accounting fading rayleigh fading pdf phr pxr exp phr pxr joint pdfs jacobian determinant known follows complex gaussian distribution zero mean variance proof proposition omit superscript subscript notation simplicity conditional expectation given expressed dgdx exp exp dgdx exp obtained using gaussian integral substituting expression derived appendix get proof proposition omit superscript subscript following note random variable defined random variable defined following distribution since follows complex gaussian distribution zero mean unit variance get exp compute derive follows exp defined appendix obtained using gaussian integral using mse var algebraic manipulations obtain proof proposition similar first derive conditional expectation exp defined appendix based var mse var obtained using gaussian integral plugging finally get obtained obtained substituting combine results get since need derive expressed exp dgdx dgdx exp exp exp exp proof proposition omit superscript subscript notation simplicity conditional variance given var proof proposition omit superscript subscript case large scale fading unknown proof similar appendix except need deal random vectors rather random scalars using derived exp exp plugging using multivariate gaussian integral algebraic manipulations obtain case fading known conditional expectation found proof proposition state evolution simplified since proofs cases similar following focus case fading known omit subscript use subscript instead superscript convenience use induction assuming holds evaluate right hand side first derive distribution based exp completes induction exp defined compute conditional covariance matrix given cov rtt rtt rtt rtt taking expectation obtain cov diagonal matrix due fact element integral odd function symmetric interval zero furthermore easy observe integral diagonal elements cov identical note mmse denoiser hand side rewritten cov leads result also diagonal matrix identical diagonal elements following derive explicit expression end first compute cov denote ith diagonal entry cov ith entry based exp exp fist term last step obtained using gaussian integral second term obtained integrating spherical coordinates instead cartesian coordinates function defined expected depend indicating diagonal elements indeed identical replacing get random variable depends eferences andrews buzzi choi hanly lozano soong zhang ieee sel areas vol june fundamental limits massive connectivity information theory application ita workshop san diego usa donoho maleki montanari algorithms compressed sensing proc nat acad vol chen chen guo capacity gaussian channels ieee trans inf theory vol june schepker bockelmann dekorsy exploiting sparsity channel data estimation sporadic communication inter symp wireless commun sys iswcs ilmenau germany rao lau active user detection channel estimation uplink cran systems ieee inter conf commun icc london june wunder jung ramadan compressive random access using common overloaded control channel ieee globecom workshops san diego usa wunder boche strohmer jung sparse signal processing concepts efficient system design ieee access vol zhu giannakis exploiting sparse user activity multiuser detection ieee trans vol schepker dekorsy compressive sensing detection orthogonal least squares ieee veh technol conf vtc spring yokohama japan may luo guo neighbor discovery wireless hoc networks based group testing allerton conf control computing usa zhang luo guo neighbor discovery wireless networks via compressed sensing performance evaluation vol july bayati montanari dynamics message passing dense graphs applications compressed sensing ieee trans inf theory vol maleki anitori yang baraniuk asymptotic analysis complex lasso via complex approximate message passing camp ieee trans inf theory vol july rangan generalized approximate message passing estimation random linear mixing ieee inter symp inf theory isit petersburg russia july donoho maleki montanari message passing algorithms compressed sensing motivation construction ieee inf theory workshop itw cairo egypt schniter turbo reconstruction structured sparse signals annual conf information sciences systems ciss princeton usa mar kim chang jung baron belief propagation joint sparse recovery online available http ziniel schniter efficient inference multiple measurement vector problem ieee trans signal vol hannak mayer jung matz goertz joint channel estimation activity detection multiuser communication systems ieee inter conf commun icc workshop london june donoho johnstone montanari accurate prediction phase transitions compressed sensing via connection minimax denoising ieee trans inf theory vol june montanari graphical models concepts compressed sensing compressed sensing theory applications eldar kutyniok eds new york cambridge university press needell tropp cosamp iterative signal recovery incomplete inaccurate samples appl comp harmonic vol may donoho maleki montanari phase transition compressed sensing ieee trans inf theory vol zhilin chen received degree electrical information engineering degree signal information processing beihang university buaa beijing china respectively currently pursuing degree university toronto toronto canada main research interests include wireless communication signal processing foad sohrabi received degree university tehran tehran iran degree mcmaster university hamilton canada electrical computer engineering since september student university toronto toronto canada form july december research intern alcatellucent stuttgart germany main research interests include mimo communications optimization theory wireless communications signal processing received ieee signal processing society best paper award wei received degree computer engineering mathematics university waterloo waterloo ontario canada degrees electrical engineering stanford university stanford respectively since electrical computer engineering department university toronto toronto ontario canada professor holds canada research chair tier information theory wireless communications main research interests include information theory optimization wireless communications broadband access networks wei currently serves ieee information theory society board governors serves chair signal processing communications networking technical committee ieee signal processing society ieee communications society distinguished lecturer currently serves area editor ieee ransactions ireless ommunications served associate editor ieee ransactions nformation heory editor ieee ransactions ommuni cations editor ieee ransactions ireless ommunications guest editor number special issues ieee ournal elected reas ommuni cations eurasip ournal pplied ignal rocessing technical program ieee communication theory workshop technical program committee communication theory symposium ieee international conference communications icc wei received ieee signal processing society best paper award ournal ommunications etworks best paper award steacie memorial fellowship ieee communications society best tutorial paper award ieee icc best paper award mccharles prize early career research distinction early career teaching award faculty applied science engineering university toronto early researcher award ontario wei recognized highly cited researcher fellow canadian academy engineering
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distributed colour reduction revisited jukka kohonen university helsinki janne korhonen aalto university christopher purcell aalto university jukka suomela aalto university sep przemyslaw eth abstract give new simple distributed algorithm graph colouring paths cycles algorithm fast need globally consistent orientation reduces number colours three iterations introduction present fast simple distributed algorithm colouring paths cycles algorithms colouring paths cycles key primitives used subroutines many distributed parallel algorithms also material typically covered introductory courses distributed algorithms yet best currently known algorithms tend inefficient restricted inelegant problem setting focus iterative colour reduction algorithms paths cycles path properly coloured colours algorithm relabel nodes path properly coloured colours naturally iterate algorithm find colouring colours colours reach fixed point use notion colour reduction interested algorithms nodes based current colours immediate neighbours algorithm interpreted function maps three old colours new colour new colour node coloured two neighbours colours want keep algorithm symmetric apply even global orientation naturally require algorithm produces proper colouring output assuming proper colouring input simple example colour reduction algorithm reduces number colours min example properly path input colours apply function local neighbourhoods get properly path endpoints apply standard trick pretend nodes near endpoints neighbour different colour contribution present simple approach enables iterative colour reduction follows steps reduce number colours astronomical numbers naturally getting colours possible local rule paths inherently global problem aware prior algorithm equally fast need assume global orientation see next section algorithm also much simpler prior algorithms algorithm also completely one safely skip next section prior work typical theoretical presentation topic gives number algorithms complexity iterations colour reduction however highlight differences algorithms use concrete numbers goal reduce number colours least note chosen use addresses colours see seemingly innocent part actually dominates prior algorithms proceed two steps first develop colour reduction algorithm assuming global orientation node one predecessor one successor put otherwise develop algorithm satisfies necessarily using asymmetric algorithm black box design symmetric algorithm also satisfies asymmetric algorithms classical example asymmetric algorithm algorithm cole vishkin modern form colour reduction algorithm iterating rule get point algorithm stuck colours switch naive algorithm eliminates one colour per round overall obtain steps reduce number colours typical introductory lecture topic stops however better particularly elegant approach algorithm naor stockmeyer gives colour reduction rule fixed point colours one round naive algorithm overall get steps way beyond colours also simple trick applicable algorithms algorithms use colour predecessor ignore colour successor simulate two steps algorithm one iteration obtain obtain iterations colour reduction beyond rounds optimal values parameters also last step optimal simple computer search shows algorithm however step leaves room improvement algorithm asymmetric symmetric algorithms equipped efficient asymmetric algorithm example apply undirected paths somewhat fashion concrete example use local minima local maxima colours split path fragments consist strictly increasing colours orient fragment direction increasing colours apply efficient asymmetric algorithm fragment iteratively find everything except local minima local maxima properly coloured label local minima colour local maxima colour obtain proper run two rounds naive colour reduction algorithm get back colours care needed make sure lose many rounds step example want useful colour reduction already first round addition identifying local minima maxima little bit thought combined obtain something along lines refers local minima local maxima use two additional colours rounds new algorithm need rounds much streamlined symmetric algorithms design efficient algorithms symmetric design available algorithms derived asymmetric algorithms following scheme similar one sketched however colour reduction algorithms designed general undirected graphs naturally apply also case graphs maximum degree classical example linial algorithm algorithm based socalled set families obtain slightly different algorithms based specific choice set family unfortunately constructions give best exponential colour reduction log per round example one constructions used linial gives log new algorithm much faster provides colour reduction log log per round asymptotically optimal symmetric asymmetric algorithms algorithm fix target number colours consider subsets families consist subsets collections consist families turns certain collections directly interpreted colour reduction algorithms call collections colourful collections section gives definition colourful collection section shows colourful collection gives colour reduction algorithm section shows converse also true colour reduction algorithm gives colourful collection size finally section shows construct large colourful collections colourful collections say collection colourful following holds family subsets families example colourful collection simple example keep examples easier read leave two innermost levels brackets commas write simply colourful collections algorithms show use colourful collection colour reduction algorithm proceeds follows convenient imagine undirected edge pair directed edges plan label node family using labels nodes label directed edge subset using labels incoming edges label node colour detailed description apply bijection label node family consider edge assign labels consider node incoming edges note set correctness already argued algorithm assuming colourful let show indeed produces proper colouring colours would edge construction contradiction algorithms colourful collections show colourful collections sufficient design oneround colour reduction algorithms also necessary given colour reduction algorithm construct colourful collection size proceed follows two colours define subset colour define family define collection correctness first consider family let definition also hence common element holds second let consider families common element would also exist contradicts path colours order algorithm would fail colour middle nodes properly therefore shows holds together argument also shows hence size collection indeed explicit constructions colourful collections already seen section example colourful collection present general scheme even use case running example simplify notation let take subsets size subsets split pairs subset complement collection contains families form picking one half pair collection formed augmenting family subsets size form collection contains families contain one subset size finally set note running example hence colour reduction algorithm assuming collection indeed colourful correctness collection satisfies consider let let complement set size satisfies satisfies augmented collection still satisfies added subsets intersect everything satisfying purposes ignore extra subsets trivial collection clearly also satisfies finally need argue also satisfies property follows directly fact satisfy remains checked holds even pick vice versa trivial contains one subset one element complement contained every analysis construction every even colour reduction scheme concrete example obtain following colour reduction scheme asymptotically scheme log log remarks colour reduction assign input colours singleton families way colour reduction algorithm additional nice property nodes already colour change colours natural property prior efficient algorithms guarantee references leonid barenboim michael elkin distributed graph coloring fundamentals recent developments synthesis lectures distributed computing theory morgan claypool publishers richard cole uzi vishkin deterministic coin tossing applications optimal parallel list ranking information control frankl families finite sets set covered union others israel journal mathematics dec fabian kuhn roger wattenhofer complexity distributed graph coloring proc podc pages nathan linial locality distributed graph algorithms siam journal computing moni naor larry stockmeyer computed locally siam journal computing david peleg distributed computing approach society industrial applied mathematics philadelphia usa joel rybicki exact bounds distributed graph colouring master thesis department computer science university helsinki may joel rybicki jukka suomela exact bounds distributed graph colouring proc sirocco pages jukka suomela distributed algorithms online textbook roger wattenhofer lecture notes principles distributed computing online lecture notes
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logical methods computer science vol submitted published jun generic fibrational induction neil ghani patricia johann fumex university strathclyde glasgow address abstract paper provides induction rule used prove properties data structures whose types inductive carriers initial algebras functors results semantic nature inspired hermida jacobs elegant algebraic formulation induction polynomial data types contribution derive slightly different assumptions sound induction rule generic inductive types polynomial induction rule generic kinds properties proved well like hermida jacobs work general fibrational setting accommodate general notions properties inductive types rather particular syntactic form establish soundness generic induction rule reducing induction iteration show generic induction rule instantiated give induction rules data types rose trees finite hereditary sets hyperfunctions first lies outside scope hermida jacobs work polynomial far aware induction rules known exist second third general fibrational framework instantiation hyperfunctions underscores value working general fibrational setting since data type interpreted set introduction iteration operators provide uniform way express common naturally occurring patterns recursion inductive data types expressing recursion via iteration operators makes code easier read write understand facilitates code reuse guarantees properties programs totality termination supports optimising program transformations fold fusion short cut fusion categorically iteration operators arise initial algebra semantics data types data types regarded carriers initial algebras functors lambek lemma ensures carrier initial algebra least fixed point initiality ensures given unique homomorphism denoted old initial algebra algebra functor map acm subject classification key words phrases induction algebraic semantics data types fibrations category theory paper revised significantly expanded version research partially supported epsrc grant logical methods computer science ghani johann fumex creative commons ghani johann fumex fold iteration operator data type initial algebra semantics thus provides theory iteration principled derived solely initial algebra semantics data types important helps ensure programs rigorous mathematical foundations used ascertain meaning correctness expressive applicable inductive types every type carrier initial algebra functor rather syntactically defined classes data types polynomial data types correct valid model realisability etc data types interpreted carriers initial algebras induction iteration closely linked induction often used prove properties functions defined iteration correctness induction rules often established reducing iteration may reasonably expect initial algebra semantics used derive principled expressive correct theory induction data types well treatments induction given functor together property proved data type premises induction rule constitute carrier conclusion rule obtained supplying input iteration operator yields function function type obtained however possible characterise carrier without additional assumptions induction rules thus typically derived assumption functors involved certain structure polynomial moreover taking carriers algebras assumes properties represented functions induction rules derived described principled correct expressiveness limited along two dimensions respect data types derived nature properties verify expressive yet still principled correct approach induction given hermida jacobs show lift functor base category types functor category properties types take premises induction rule type hermida jacobs work fibrational setting notion property consider accordingly general indeed accommodate notion property suitably fibred category types overcome one two limitations mentioned hand approach gives sound induction rules polynomial data types limitation class data types treated remains work paper shows remove restriction class data types treated main result derivation sound generic induction rule instantiated every inductive type regardless whether polynomial think important provides counterpart induction existence iteration operator every inductive type take hermida jacobs approach point departure show slightly different assumptions fibration involved lift functor base category fibration functor total category fibration lifting define forms basis generic induction rule generic fibrational induction derivation generic sound induction rule covering inductive types clearly important theoretical result also practical consequences show example generic induction rule instantiated families fibration set fibration often implicitly used type theorists constructing inductive proofs theorem provers derive induction rule rose trees one would intuitively expect data type rose trees lies outside scope hermida jacobs results polynomial hand induction rule rose trees available proof assistant coq although neither one intuitively expect expressive enough prove properties ought amenable inductive proof indeed define rose trees coq node list rose rose coq generates following induction rule forall rose prop forall list rose node forall rose prove property rose tree node must prove property assuming list rose trees without recourse induction hypothesis course presentation rose trees mutual recursion well give expected induction rule coq either intuitively expect induction rule whose premise forall list rose node rule derive rose trees indeed expected one suggests derivation may enable automatic generation useful induction rules coq rather requiring user hand code currently necessary show example generic induction rule instantiated families fibration set derive rule data type finite hereditary sets data type defined terms quotients lies outside current theories induction finally show example generic induction rule instantiated subobject fibration derive rule data type hyperfunctions data type interpreted set fibration families fibration set required case use subobject fibration allows derive induction rule admissible subsets hyperfunctions ability treat data type hyperfunctions thus underscores importance developing results general fibrational framework moreover functor underlying data type hyperfunctions strictly positive ability treat data type also underscores advantage able handle general class functors going beyond simply polynomial functors far know induction rules finite hereditary sets hyperfunctions previously existed general fibrational framework ghani johann fumex although theory induction applicable inductive functors functors initial algebras including giving rise nested types gadts indexed containers dependent types inductive recursive types examples show working general fibrational setting beneficial even restrict attention strictly positive data types however offer preliminary thoughts section potentially delicate issue instantiating general theory fibrations appropriate deriving induction rules specific classes functors interest also worth noting specialisations generic induction rule polynomial functors families fibration set coincide exactly induction rules hermida jacobs structure require fibrations generally slightly different required hermida jacobs theory essence generalisation two strictly speaking incomparable structure require fibrations nevertheless certainly present standard fibrational models type theory see section like hermida jacobs prove generic induction rule correct reducing induction iteration detailed discussion induction rules coincide hermida jacobs given section take purely categorical approach induction paper derive generic induction rule initial algebra semantics data types result work inherently extensional although translating constructions intensional settings may therefore require additional effort expect guidance offered categorical viewpoint support derivation induction rules functors treatable present since use form impredicativity constructions instead use weaker assumption initial algebras exist guidance widely applicable remainder paper structured follows make results accessible possible illustrate section categorical derivation familiar induction rule natural numbers section derive induction rule special case families fibration set also show rule instantiated derive one section ones rose trees finite hereditary sets mentioned section present generic fibrational induction rule establish number results illustrate aforementioned application hyperfunctions approach taken section completely different corresponding one conference version paper allows improve upon extend previous results section concludes discusses possible instantiations generic induction rule functors offers additional directions future research convenient identify isomorphic objects category write rather write canonical singleton set denote single element sections assume types interpreted objects set also denotes unit type sections write identity morphisms category identity functor category familiar induction rule consider inductive data type nat defines natural numbers specified programming language syntax data nat zero succ nat generic fibrational induction observation nat least fixed point functor set category sets functions defined used define following iteration operator foldnat foldnat zero foldnat succ nat foldnat iteration operator foldnat provides uniform means expressing common naturally occurring patterns recursion natural numbers categorically iteration operators foldnat arise initial algebra semantics data types every data type regarded carrier initial algebra functor category functor morphism object call carrier functor collection forms category alg call category alg morphism map following diagram commutes exists initial unique isomorphism least fixed point carrier initiality ensures unique morphism fold gives rise following iteration operator fold inductive type fold fold fold since fold derived initial algebra semantics principled correct also expressive since defined every inductive type fact fold single iteration operator parameterised inductive types rather family iteration operators one type iteration operator foldnat instantiation nat generic iteration operator fold iteration operator foldnat used derive standard induction rule nat coincides standard induction rule natural numbers familiar principle mathematical induction rule says property holds holds whenever holds natural number holds natural numbers representing property natural numbers predicate nat set mapping nat set proofs holds wish represent rule object level function indnat type nat set zero nat succ nat code fragments involve quantification sets properties functors treated categorically inspired within paper quantification objects interpreted set order give formal interpretation code fragments like one would need work category modest sets ability work functors categories set one motivations working general fibrational ghani johann fumex setting section formalising semantics code fragments would obscure central message paper decision treat fragments categorically inspired justified part fact use category theory suggest computational constructions long regarded fruitful within functional programming community see function indnat type takes input property proved proof holds zero function mapping nat proof holds proof holds succ returns function mapping nat proof holds element write indnat terms foldnat thus reduce induction nat iteration nat follows first note indnat obtained instantiating type type foldnat type form specific indnat returns elements types different values types general distinct one another therefore need type containing elements every type informally thought union formally given dependent type nat comprising pairs nat standard approach defining indnat thus apply foldnat carrier nat algebra components given zero succ choose zero succ note oldn tentatively take indnat oldn order know actually gives proof must show fortunately follows easily uniqueness foldnat indeed foldnat nat commutes initiality oldn identity map thus oldn letting second projection dependent pairs involving predicate induction rule nat thus indnat nat set zero nat succ nat indnat foldnat zero succ expected induction rule states every property construct proof holds every nat suffices provide proof holds zero show nat proof holds also proof holds succ use dependent types fundamental formalization induction rule nat possible properties proved taken functions remainder paper uses fibrations generalise treatment induction arbitrary inductive functors arbitrary properties suitably fibred category whose objects interpret types general fibrational setting properties given axiomatically via fibrational structure rather assumed functions generic fibrational induction induction rules predicates set main result paper derivation sound induction rule generic inductive types used verify notion property fibred category whose objects interpret types section assume types modelled sets functors consider set properties consider functions mapping data sets proofs properties hold make assumptions allows present derivation simplest setting possible also type theorists often model properties exactly way makes present section accessible since general fibrational treatment induction seen direct generalisation treatment presented section also easily digestible derivation understood special case although derivation section indeed seen specialisation section families fibration set knowledge fibrations required understand constructions given concretely rather fibrational forms begin considering might naively expect induction rule inductive data type look like derivation nat section suggests general look something like ind set premises denoted generic induction rule ind since want construct term proof term type proof terms substructures since functionality fold operator precisely compute value values substructures natural try equip structure input fold yield mapping element approach quickly hits snag since codomain every predicate set set rather object set applied needed equip structure moreover induction rule obtained applying fold carrier specific suggests try construct term rather considerations led hermida jacobs define category predicates lifting polynomial functor set functor respects structure constructed carrier serve premises induction rules crucial part construction namely lifting polynomial functors proceeds inductively includes clauses construction hermida jacobs general consider functors bicartesian categories rather set represent properties bicartesian fibrations categories instead using specific notion predicate definition hand define liftings polynomial functors construction give section sense orthogonal hermida jacobs focus exclusively functors set particular category predicates show define liftings inductive functors set including ones ghani johann fumex setting induction rule derive properly extends hermida jacobs thus catering variety data types treat next section derive analogous results general fibrational setting allows derive sound induction rules initial algebras functors defined categories set used prove arbitrary properties suitably fibred category interpreting types begin definition predicate definition let set predicate function set mapping set call domain may speak simply predicate domain understood predicate thought mapping element set proofs holds define category predicates definition category predicates predicates objects morphism predicate set predicate set pair functions composition predicate morphisms given diagrammatically set diagram indicates notion morphism require sets proofs equal instead requires existence function maps proof proof denote set forgetful functor mapping predicate set domain predicate morphism alternative definition would take category predicates arrow category set natural lifting setting indicate generalise liftings fibrations indeed properties modelled functions every functor applied property hence every functor lifting general fibrational setting however properties necessarily modelled functions functor general lifting decision use arrow categories model properties thus dictated desire lift functors way indicates liftings constructed general fibrational setting give precise definition lifting definition let functor set lifting set functor following diagram commutes set set decode definition follows object part must map predicate set predicate set thus thought generic fibrational induction function set set set course must also act morphisms functorial manner use definition lifting derive standard induction rule section nat follows example data type natural numbers functor set defined lifting defined sending predicate set set given predicate inl inr carrier nat set given nat nat nat since inl inr see consists element function nat thus second component carrier nat set first component gives premises familiar induction rule example notion predicate comprehension key ingredient lifting begins explain abstractly use theory induction key construct allowing define liftings well polynomial functors definition let predicate comprehension denoted type comprising pairs map taking predicate taking predicate morphism morphism defined defines comprehension functor set position define liftings uniformly functors definition functor set lifting functor given follows every predicate set defined natural transformation given every predicate morphism given definition note inverse image indeed predicate set thus predicate thus predicate lifting set lifting captures modality generalises haskell function lists arbitrary data types similar modality given indexed containers lifting example instantiation construction definition functor set indeed predicate since inverse image coproduct functions coproduct inverse images since since inl inr see similar situation nat holds general functor set second component whose carrier predicate data type whose first component gives premises induction rule used show holds data type ghani johann fumex rest section shows carrier interderivable carrier uses result derive induction rule definition functor set maps set predicate predicate morphism predicate called truth predicate every set proofs holds singleton thus intuitively think predicate set true every therefore consider true exists predicate morphism whose first component functor lifting maps truth predicate set lemma functor set set proof definition isomorphism one proof thus isomorphism well result maps every singleton set therefore fact critical constructions proved include proof completeness establish notation description comprehension right adjoint traced back lawvere lemma left adjoint proof must show predicate set set morphisms bijective correspondence set set morphisms set define maps set set give natural isomorphism set naturality ensures similarly moreover counit adjunction observations used proof lemma lemmas key results relating algebras relating iteration induction special cases theorem include proofs ensure continuity presentation ensure section first show construct lemma functor alg alg proof define two morphism define algebra morphism lemma morphism easy see preserves identities composition generic fibrational induction also construct lemma functor right adjoint proof construct adjoint functor alg alg follows given use fact lemma define specify action morphism define clearly preserves identity composition next show every predicate natural isomorphism morphisms morphisms first observe morphism map morphism map natural isomorphism maps given adjunction lemma must check morphism iff morphism end assume morphism assume must prove definition lemma amounts showing since isomorphism algebra morphism iff naturality ensures previous equality holds iff naturality functoriality definition fact inverses lemma equation thus indeed morphism lemma ensures carrier interderivable carrier example carrier section derived carrier given example since define lifting functor lemma thus shows construct carrier functor predicate corollary functor set predicate carrier initial proof since left adjoint preserves initial objects applying initial gives initial lemma type carrier initial ghani johann fumex derive generic induction rule every predicate every algebra lemma ensures constructs carrier applying iteration operator algebra gives map fold map decomposes two parts fold initiality definition naturality ensure fold recalling second projection dependent pairs involving predicate gives following sound generic induction rule type reduces induction iteration genind set set set fold genind fold notice induction rule actually capable dealing predicates arbitrary sets predicates however initiality ensures fold thus genind specialises expected induction rule inductive data type ind set set set fold ind rule instantiated familiar rules polynomial data types well ones would expect data types rose trees finite hereditary sets lie outside scope hermida jacobs method example data type rose trees given syntax data rose node list rose functor underlying rose list induction rule indrose indrose calculating list rose set rose fold rose set writing component cps list list cps cps list length cps cps whose underlying rose rose thus pair functions type list rose cps list length cps cps node list rose length node generic fibrational induction last equality due surjective pairing dependent products fact length cps length type gives hypotheses induction rule rose trees although finite hereditary sets defined terms quotients thus lie outside scope previously known methods treated example hereditary sets sets whose elements sets core data structures within set theory data type finitary hereditary sets finite powerset functor derive induction rule finite hereditary sets follows set maps set set maps set set carrier set first component therefore second component function type induction rule finite hereditary sets thus indhs generic fibrational induction rules treat general notions predicates using fibrations motivate use fibrations observing semantics data types languages involving recursion effects usually involves categories set circumstances notion predicate longer taken function codomain set iii even working set reasonable notions predicate section example predicate set could subobject moreover future work consider induction rules sophisticated classes data types indexed containers inductive families inductive recursive families see section want develop individual hoc theory induction class instead want appropriately instantiate single generic theory induction want uniform axiomatic approach induction widely applicable abstracts specific choices category functor predicate giving rise different forms induction specific classes data types fibrations support precisely axiomatic approach section therefore generalises constructions previous one general fibrational setting standard model type theory based locally cartesian closed categories arise specific fibration namely codomain fibration set fibration equivalent families fibration set general fibrational setting far flexible moreover locally cartesian closed models type theory predicates types coexist category functor taken lifting general fibrational setting predicates simply functions morphisms properties types coexist category functor taken lifting choice construct lifting scratch treatment induction based solely locally cartesian closed categories would therefore indicate treat induction general fibrations ghani johann fumex another reason working general fibrational setting facilitates direct comparison work hermida jacobs important since approach closely related main difference approach use fibred products coproducts define provably sound induction rules polynomial functors whereas use left adjoints reindexing functors define provably sound induction rules inductive functors section consider situations approaches possible give mild conditions results coincide restricted polynomial functors remainder section organised follows section recall definition fibration expand motivate definition fix basic terminology surrounding fibrations give examples fibrations including families fibration set codomain fibration subobject fibration section recall useful theorem indicates lifting functor category predicates initial algebra key theorem used prove soundness generic fibrational induction rule section construct truthpreserving liftings inductive functors first codomain fibration using intuitions presentation families fibration set studied section general fibrational setting finally section establish number properties liftings hence induction rules derived particular characterise lifting generates induction rules fibrations nutshell section recall notion fibration details fibrations found begin auxiliary definition definition let functor morphism cartesian morphism every exists unique morphism opcartesian morphism every exists unique hard see cartesian morphism morphism codomain unique isomorphism similarly opcartesian morphism generic fibrational induction object write domain codomain capture cartesian opcartesian morphisms diagrammatically follows cartesian morphisms opcartesian morphisms essence fibrations opfibrations introduce fibrations duals since latter prove useful later development speak primarily fibrations understanding dual observations hold opfibrations definition let functor fibration every object every morphism cartesian morphism similarly opfibration every object every morphism opcartesian morphism functor bifibration simultaneously fibration opfibration fibration call base category total category objects total category thought properties objects base category thought types thought mapping property type property one fibration capture many different properties type injective objects say object image similarly morphisms object write fibre subcategory consisting objects morphisms morphism function mapping object extends functor indeed morphism morphism satisfying universal property ensures existence uniqueness call functor reindexing functor induced similar situation ensures opfibrations call functor extends function mapping object opreindexing functor example functor set defined section called families fibration set given function predicate set define cartesian map whose domain comprises pair fibre set predicates set objects morphism set set function type example let category arrow category denoted morphisms arrows objects morphism pair ghani johann fumex morphisms following diagram commutes easy see definition indeed gives category codomain functor cod maps object object morphism pullbacks cod fibration called codomain fibration indeed given object fibre morphism pullback along gives cartesian morphism required fibre object morphisms map objects morphism morphism example category category subobjects denoted sub monomorphisms objects monomorphism called subobject morphism sub pair morphisms map sub sending subobject extends functor pullbacks fibration called subobject fibration indeed pullbacks give cartesian morphisms since pullback monomorphism monomorphism fibre object objects subobjects morphism sub map morphism exists course unique lifting truth comprehension generalise notions lifting truth comprehension general fibrational setting prove setting inductive functor lifting lifting also inductive see inductiveness lifted functor sufficient guarantee soundness generic fibrational induction rule subsection essentially presentation results include forms natural part narrative simply citing material would hinder continuity presentation recall section first step deriving induction rule datatype interpreted set lift functor whose fixed point data type category predicates specifically definition defined lifting functor set set functor use observations generalise notion lifting fibrational setting follows definition let fibration functor lifting respect functor following diagram commutes generic fibrational induction section saw set predicate predicate analogous result general fibrational setting observes lifting object restricts functor analogy results section expect premises fibrational induction rule datatype interpreted constitute order construct conclusion rule need understand axiomatically state predicate true section predicate set considered true morphism truth predicate since mapping set action objects truth functor set definition actually endeavour model truth functor families fibration set axiomatically general fibrational setting modeling truth functor axiomatically amounts understanding universal property since truth functor definition maps set predicate set therefore exactly one morphism fibre predicate gives clear categorical description terminal object fibre leads analogy following definition definition let fibration assume every object fibre terminal object assignment sending object morphism morphism defines fibred truth functor fibred truth functor sometimes called fibred terminal object functor definition following standard result lemma fibred right adjoint interested reader may wish consult literature fibrations definition fibred adjunction formal definition needed instead simply stress fibred adjunction first foremost adjunction observe counit adjunction identity moreover full faithful one simple way guarantee fibration truth functor assume terminal objects maps terminal object terminal object case fact reindexing preserves fibred terminal objects ensures every fibre indeed terminal object second fundamental property liftings used section truthpreserving property easily generalised general fibrational setting definition definition let fibration truth functor let functor let lifting say lifting object final algebraic structure required section comprehension functor set generalise comprehension functor general fibrational setting simply note universal property right adjoint truth functor definition single special attention fibrations whose truth functors right adjoints ghani johann fumex definition let fibration truth functor comprehension category unit right adjoint comprehension category unit call right adjoint comprehension functor denote machinery place hermida jacobs show comprehension category unit lifting inductive case carrier initial proved corollary following abstract theorem theorem let functors natural transformation natural transformation induces functor algf algg given moreover isomorphism right adjoint induces right adjoint algf algg given image adjunction isomorphism hom hom counit adjunction instantiate theorem generalise lemmas theorem let comprehension category unit functor lifting adjunction alg alg moreover proof instantiate theorem letting isomorphism since also theorem thus ensures maps every maps every corollary let comprehension category unit functor lifting inductive moreover proof hypotheses corollary place setting theorem theorem guarantees maps initial carrier since left adjoints preserve initial objects must therefore initial carrier thus exists isomorphic generic fibrational induction theorem let comprehension category unit inductive functor lifting following generic fibrational induction rule sound genfibind genfibind fold alternative presentation genfibind genfibind genfibind fold call genfibind generic fibrational induction rule summary generalised generic induction rule predicates set presented section give sound generic induction rule comprehension categories unit assumption start inductive functor base comprehension category must lifting functor total category comprehension category case specialise genfibind get fibrational induction rule datatype interpreted fibration base category generic fibrational induction rule genfibind however look slightly different generic induction rule predicates section used knowledge specific structure comprehensions predicates extract proofs particular data elements fibrational setting predicates hence comprehensions left abstract therefore take return type general induction scheme genfibind comprehension expectation general theory section instantiated particular fibration interest may possible use knowledge fibration extract comprehension constructed genfibind proof information relevant application hand previously mentioned hermida jacobs provide liftings polynomial functors section define generic lifting inductive functor base category fibration addition comprehension category unit left adjoints reindexing functors gives sound generic fibrational induction rule datatype functor base category fibration constructing liftings light previous subsection natural ask whether liftings exist unique many liftings specific lifting choose others perhaps even universal lifting also ask algebraic structure liftings example liftings preserve sums products functors answers questions given hermida jacobs provided liftings polynomial functors define liftings assume total category base category fibration question products coproducts fibration preserves conditions liftings polynomial functors defined inductively section beyond results hermida jacobs construct liftings inductive functors employ process first building liftings assumption ghani johann fumex fibration interest codomain fibration using intuitions section extend lifting general class fibrations section consider questions previous paragraph algebraic structure liftings liftings codomain fibrations recall example pullbacks codomain fibration functor cod given functor trivial define lifting fibration define functor map object map morphism morphism lifting easily verified verify codomain fibrations comprehension categories unit lifting theorem applied former first observe functor mapping object morphism truth functor fibration fact take isomorphism use observation let denote set morphisms object object fact right adjoint cod established via natural isomorphism cod next show functor dom mapping object morphism comprehension functor codomain fibration dom right adjoint established via natural isomorphism dom finally lifting implicitly given functors category display maps category subfibration codomain fibration category lifting given essentially lifting codomain fibration restricted subfibration question liftings families fibration set section defined every functor set set lifting maps predicate looking closely realise lifting decomposes three parts given predicate first consider projection function next apply functor obtain finally take inverse image get predicate required note functor maps predicate projection function maps predicate morphism predicate set set morphism definition functor sending function inverse predicate morphism predicate morphism three steps defining functorial relationships indicated generic fibrational induction following diagram hold set cod note adjunction equivalence observation however necessary subsequent development particular needed theorem presentation lifting functor families fibration set uses lifting codomain fibration set indeed writing lifting codomain fibration set moreover since see proof lemma since already seen liftings codomain fibrations three constituent functors finally since showed section families fibration set comprehension category unit theorem applied excitingly shall see next subsection presentation lifting functor families fibration set generalises many fibrations liftings fibrations turn attention task constructing liftings fibrations codomain fibrations families fibration set contrast approach outlined conference paper paper based one take uses factorisation like previous subsection codomain fibration specifically let comprehension category unit first define functors construct adjunction relationships indicated following diagram hold cod use adjunction indicated diagram construct lifting codomain fibration define functor generalise definition sections requires work axiomatic characterisation definition comprehension functor right adjoint truth functor counit adjunction natural transformation applying gives natural transformation since fact therefore define indeed functor action object action morphism next turn definition left adjoint see generalise inverse image construction general fibrations first recall example function set extend mapping ghani johann fumex reindexing functor defining define action objects denotes disjoint union operator sets action morphisms taking left adjoint moreover compute recall set singleton clearly equivalent inverse image discussion suggests order generalise inverse image construction general fibration require reindexing functor opreindexing functor left adjoint condition required adjoints following result appears proposition thus allows isolate exact class fibrations sound generic induction rules theorem fibration bifibration iff every morphism reindexing functor left adjoint definition lawvere category bifibration also comprehension category unit construct left adjoint lawvere category follows object morphism define object define action morphisms let morphism pair morphisms following diagram commutes must construct morphism notice also since consider morphism use universal property opcartesian morphism deduce existence morphism difficult using uniqueness morphism prove setting image morphism makes functor fact since cod result page guarantees lawvere category functor exists left adjoint construct lifting lawvere category functor generic fibrational induction theorem let lawvere category functor define functor lifting proof trivial check indeed lifting prove need prove functor object showing preserves fibred terminal objects preserves terminal objects fibre total category domain since terminal object fibre terminal object desired first show preserves fibred terminal objects must show object terminal object fibre isomorphism codomain prove observing unit adjunction isomorphism inverse indeed counit adjunction facts full faithful ensure isomorphism inverse thus isomorphism inverse isomorphism inverse inverse since terminal object terminal object fibre preserves fibred terminal objects hard see preserves fibred terminal objects applying functor isomorphism codomain terminal object fibre gives isomorphism codomain terminal object fibre finally isomorphism left adjoint also right adjoint since right adjoints preserve terminal objects since terminal object terminal object thus preserves fibred terminal objects stress define lifting codomain functor base lawvere category need fibration particular need pullbacks indeed needed construct generic lifting functor existence functors always exists nevertheless present lifting composition presentation shows factored helps motivate definition thereby revealing parallels would otherwise apparent time trades direct presentation elegant modularly structured one makes good use different setting general results comprehension categories due jacobs promised sound generic fibrational induction rule every inductive functor base lawvere category demonstrate flexibility rule derive induction rule data type properties modelled set able derive induction rules fixed points functors categories set key motivation working general fibrational setting example fixed point hyp functor int int data type hyperfunctions since fixed point set interpret category ghani johann fumex strict continuous monotone functions setting property object admissible admissibility means bottom element closed least upper bounds structure forms lawvere category particular routine verify existence opreindexing functor particular constructed continuous map admisible predicate intersection admissible truth functor maps comprehension maps lifting maps least admissible predicate containing image finally derived induction rule states admissible hyp hyp algebra lifting proved lawvere category functor lifting thus following associated sound generic fibrational induction rule genfibind genfibind fold final subsection paper ask kinds algebraic properties lifting operation first result concerns lifting constant functors lemma let lawvere category let object constantly functor isomorphic constantly functor proof object last isomorphism holds next result concerns lifting identity functor requires little additional structure lawvere category interest definition full lawvere category lawvere category full faithful lemma full lawvere category proof discussion following definition since full faithful counit adjunction isomorphism therefore isomorphisms clearly natural therefore generic fibrational induction show lifting coproduct functors coproduct liftings lemma let lawvere category let functors proof third isomorphism holds left adjoint preserves coproducts note statement lemma assert existence either two coproducts mentioned rather whenever exist must equal note also lemma generalises colimit functors unfortunately result analogous lemma yet stated products final result considers whether anything fundamentally special lifting constructed clearly right lifting sense gives expected induction rules liftings might also exist case might hope lifting satisfies universal property fact condition also satisfied liftings hermida jacobs therefore regard reasonable show lifting lifting proof uses line reasoning appears remark lemma let full lawvere category let lifting functor preserves proof finally return question relationship liftings polynomial functors given hermida jacobs liftings derived methods seen constant functors identity functor coproducts functors constructions agree moreover since hermida jacobs liftings preserve lemma guarantees full lawvere category lifting products also coincides conclusion future work given sound induction rule used prove properties data structures inductive types like hermida jacobs give fibrational account induction derive slightly different assumptions fibrations generic induction rule instantiated inductive type rather polynomial ones rule based initial algebra semantics data types parameterised data types properties involved also principled expressive correct ghani johann fumex derivation yields induction rules hermida jacobs specialised polynomial functors families fibration set fibrations also gives induction rules data types rose trees data types finite hereditary sets hyperfunctions fibrational induction rules previously known exist several directions future work immediate instantiate theory give induction rules sophisticated data types nested types exemplified data type perfect trees given syntax follows data ptree set pleaf ptree pnode ptree ptree nested types arise least fixed points functors example type perfect trees functor given appropriate fibration induction rules nested types thus takes category functors set category functors set postcomposition forgetful functor section lifting given inl inr taking premise gives following induction rule perfect trees indp ree set ptree set pleaf set ptree pnode set ptree rule used show example ree functor extending instantiation codomain fibration truly nested types fibrations current work expect able instantiate theory truly nested types gadts indexed containers dependent types inductive recursive types initial investigations show care needed must ascertain fibrations model predicates types since codomain fibration may give useful induction rules well translate rules fibrations give rise intensional setting matthes gives induction rules nested types including truly nested ones intensional type theory handles functors underlie nested types handle functor rank initial algebra insights may help guide choices fibrations truly nested types may turn inform choices gadts indexed containers dependent types induction rules automatically generated many type theories within calculus constructions induction rule data type generated solely inductive structure type generation also key idea coq proof assistant far know generation currently done syntactic classes functors rather inductive functors initial algebras type theories induction schemes added axioms rather generated example attempts generate induction schemes based church encodings calculus constructions proved unsuccessful initiality added system thus giving calculus inductive constructions whereas matthes work based concepts impredicativity induction recursion rather initial algebras reduces induction generic fibrational induction initiality may therefore help lay groundwork extending implementations induction sophisticated data types acknowledgement thank robert atkey dagand peter hancock conor mcbride many fruitful discussions references altenkirch morris indexed containers proceedings logic computer science bird moor algebra programming prentice hall bird meertens nested datatypes proceedings mathematics program construction coquand huet calculus constructions information computation coq proof assistant available dybjer general formulation simultaneous definitions type theory journal symbolic logic ghani abbott altenkirch containers constructing strictly positive types theoretical computer science ghani johann foundations structured programming gadts proceedings principles programming languages ghani johann fumex fibrational induction rules initial algebras proceedings computer science logic hermida jacobs structural induction coinduction fibrational setting information computation jacobs comprehension categories semantics type dependency theoretical computer science jacobs categorical logic type theory north holland johann ghani initial algebra semantics enough proceedings typed lambda calculus applications lawvere equality hyperdoctrines comprehension scheme adjoint functor applications categorical algebra matthes induction principle nested datatypes intensional type theory journal functional programming mendler predicative type universes primitive recursion proceedings logic computer science morris constructing universes generic programming dissertation university nottingham moggi notations computation monads information computation petersson smith programming type theory oxford university press predicates fibrations dissertation university utrecht sheard languages future sigplan notices work licensed creative commons license view copy license visit http send letter creative commons second suite san francisco usa eisenacher strasse berlin germany
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logical methods computer science vol submitted published mar semantics imperative objects jan schwinghammer department computer science saarland university germany address hritcu programming systems lab saarland university germany address jan abstract semantic interpretations types proposed alternative purely syntactic proofs type safety using subject reduction types interpreted sets values indexed number computation steps values guaranteed behave like proper elements type building work ahmed appel others introduce semantics imperative object calculus abadi cardelli providing semantic account calculus using traditional approaches proved challenging due combination dynamically allocated objects store expressive type system show using one interpret rich type discipline object types subtyping recursive bounded quantified types presence state introduction imperative object calculus abadi cardelli small yet expressive language despite extreme simplicity syntax calculus models many important concepts programming well often subtle interaction particular raises interesting questions respect typing contrast common languages imperative object calculus every object comes equipped set methods updated consequence methods need reside store store moreover objects allocated dynamically aliasing possible store present different forms many practical programming languages pointers functions general references sml considerably complicates construction adequate semantic models one reason behaviour programs pointed instance reus purely syntactic arguments subject reduction suffice proving soundness traditional type systems however type systems turned powerful acm subject classification key words phrases formal calculi objects type systems programming language semantics preliminary version paper presented international workshop foundations languages fool january san francisco california logical methods computer science schwinghammer creative commons schwinghammer specification languages like logic objects abadi leino hybrid type system flanagan purely syntactic arguments seem longer appropriate meaning assertions longer obvious since describe code heap believe specifications program behaviour meaning independent particular proof system syntactic preservation proofs rely also argued benton reus schwinghammer case specifications one would ideally prove soundness respect semantic model makes clear distinction semantic validity derivability using syntactic rules however building semantic models challenging currently fully satisfactory semantic account imperative object calculus denotational semantics models employed proving soundness logic abadi leino however existing techniques fall short providing convincing models typed objects reus streicher consider untyped semantics model presented reus schwinghammer handles neither types subtyping depth due store present imperative object calculus models rely techniques recursively defined domains functor categories makes complex establishing properties even specific programs often requires substantial effort equational reasoning gordon develop reasoning principles establishing contextual equivalence untyped objects apply prove correctness compiler optimization jeffrey rathke consider concurrent variant calculus characterize equivalence terms trace sets generated labeled transition system cases semantics limited equational reasoning establishing contextual equivalences programs theory used verify program showing equivalent one trivially correct acts specification however cumbersome practice using program logics established formalism specifying proving correctness programs translations abadi give adequate encoding imperative object calculus lambda calculus records references recursive existential types subtyping together interpretation target language adequate model imperative object calculus could principle obtained however aware adequate models general references impredicative types even model given would still preferable semantics object calculus without added complexity translation solution problem finding adequate models objects could semantic models types introduced appel mcallester alternative subject reduction proofs models based directly operational semantics easy construct existing models types simply interpreted sets syntactic values indexed number computation steps intuitively term belongs certain type behaves like element type number steps every type built sequence increasingly accurate semantic approximations allows one easily deal recursion type safety immediate consequence interpretation types semantic counterparts usual typing rules proved independent lemmas either directly induction index ahmed semantics imperative objects successfully applied generic technique lambda calculus general references impredicative polymorphism recursive types paper extend semantics ahmed object types subtyping use resulting interpretation prove soundness expressive type system imperative object calculus main contribution work novel semantics object types extend semantics two orthogonal ways first adapt self types recursive object types validate usual subtyping rules well strong typing rules structural assumptions second study natural generalization object types results simpler expressive typing rules even though paper concerned safety type system technique restricted types already used equational reasoning proving soundness program logics languages expect therefore eventually become possible use stepindexed model prove soundness expressive program logics imperative object calculus outline next section introduces syntax operational semantics type system consider imperative object calculus section present semantics calculus particular define interpretations types establish semantic properties section properties used prove soundness type system section studies self types section discusses natural generalization object types section gives comparison related work section concludes appendix presents proofs interesting typing subtyping lemmas object types earlier technical report contains additional proofs imperative object calculus recall syntax imperative object calculus recursive types introduce operational semantics calculus equivalent semantics given abadi cardelli syntax let var tvar meth pairwise disjoint countably infinite sets variables type variables method names respectively let range var range tvar let range meth figure defines syntax types terms imperative object calculus objects unordered collections named methods written method binder binds self argument method body self argument used inside method body invoking methods containing object methods arguments self obtained procedure method body methods object invoked updated new methods added existing methods deleted type objects methods named return results type set written variance annotation indicates method considered may used without restriction procedural abstractions sometimes defined imperative object calculus using additional let construct include primitives write procedures schwinghammer top bot type expressions variance annotations variable object creation method invocation method update clone shallow copy procedure application folda recursive folding unfolda recursive unfolding type abstraction type application pack existential package open package opening figure syntax types terms type applications respectively use folda unfolda denote isomorphism recursive type unfolding finally consider bounded universal existential types along introduction elimination forms set free variables term denoted similarly free type variables type identify types terms consistent renaming bound variables use denote singleton map maps finite map variables terms denotes result substitution dom notation used substitution type variables generally function notation denotes function maps otherwise agrees operational semantics let loc countably infinite set heap locations ranged extend set terms representations objects associating heap locations set method names values given grammar val folda pack apart objects values consist procedures values recursive type type abstractions existential packages lambda calculus often consider terms values without free variables denote set closed terms semantics imperative objects clone folda unfolda pack open figure evaluation contexts dom clone dom unfolda foldb open pack figure reduction relation closed values cterm cval respectively program closed term contain locations denote set programs prog heap finite map loc write heap set heaps figure defines set evaluation contexts formalizing strategy write term obtained plugging hole reduction relation defined least relation configurations heap cterm generated rules figure closed following context rule methods actually stored heap procedures object construction allocates new heap storage procedures returns record references upon method invocation corresponding stored procedure retrieved heap applied enclosing object self parameter thus passed like procedure argument identifying methods procedures makes semantics method invocation explicit technically allows use model ahmed modifications variables immutable identifiers methods updated destructively updates modify heap leave object unchanged object fact purpose modelling imperative object calculus would suffice regard procedures kind storable value schwinghammer subtyping subrefl subtop top subtrans subbot subproc bot subvar subobj subobjvar subrec subuniv subexist figure subtyping cloning generates shallow copy object heap last four rules figure lambda calculus denotes reduction relation write configuration irreducible exists configuration note reduction deterministic due arbitrarily chosen fresh locations however still always one uniquely determined redex important consequence reduction order fixed example reduction sequence beginning method invocation ending irreducible configuration sequence split similar decompositions subsequences hold reductions starting term forms easy see operational semantics independent type annotations inside terms also semantic types define section depend syntactic type expressions terms order reduce notational overhead prevent confusion syntax semantics types omit type annotations semantics imperative objects subsumption axioms sub var procedure types lam object types app obj inv upd clone clone recursive types unfold fold bounded quantified types tabs pack open tapp pack open figure typing terms presenting semantics example instead type application merely write type system type system consider features procedure object impredicative bounded quantified types well subtyping corresponds fob fairly standard consists four inductively defined typing judgments describing typing contexts defining types schwinghammer subtyping types typing terms typing context list containing type bindings term variables upper bounds type variables typing context contain duplicate bindings term type variables types appearing type respect context type variables appear figure defines subtyping relation object types allows subtyping width object type methods subtype object type fewer methods long types common methods agree methods object types covariant respectively contravariant subtyping depth allowed subobj furthermore unrestricted methods regarded subtyping either subobjvar since annotations conveniently chosen creation time obj brings much flexibility explained abadi cardelli allows distinguish type system invocations updates done self argument ones done outside main idea type object creation object type methods considered invariant invocations updates self argument internal allowed type preserving rules sub subobjvar applied methods become others enables subsequent weakening types methods using subobj effect allows safe flexible subtyping methods price restricting update invocation methods outside nevertheless internal updates invocations remain unrestricted figure defines typing relation applicability rules method invocation inv method update upd depends variance annotation also notice updates allowed upd finally important note give types heap locations since type system used check programs programs contain locations contrast proof type safety using preservation progress properties would require syntactic judgement also depend heap typing since partially evaluated terms would also need typed semantics objects modelling store necessarily involved treatment firstorder storage since semantic domains become mutually recursive recall heaps store values may procedures turn modeled functions take value initial heap input return value possibly modified heap upon termination suggests following semantic domains values heaps respectively dval dheaps dval dheaps dval dheaps loc dval simple cardinality argument shows solutions denotes set partial functions satisfying equations possible solution use approach done imperative object calculus reus streicher building earlier work kamin reddy semantics imperative objects model typed calculus one also wants interpret types naively taking collection type subsets dval interpretations syntactic types work since values generally depend heap typed model guarantee heap access operations led following approach first order ensure updates also consider heap typings heap typings partial maps heaptyping loc type track set values may stored heap location second collection types refined take heap typings account type consist values paired heap typings describe necessary requirements heaps ideas suggest take type heaptyping dval heaptyping loc type cardinality argument shows impossibility defining sets final obstacle modelling object calculus albeit independent nature heaps due dynamic allocation heap results heap typings may vary course computation reflecting changing shape heap however case many languages object calculus respect inside language possibility deallocating heap locations weak updates allowed consequence extensions changes need considered heap typings intuitively values rely heaps typing also extended heaps extended heap typing reason semantic models dynamic allocation typically lend presentation semantic entities indexed possible worlds drawn set heap typings partially ordered heap typing extension rather trying extend already complex models heap typings dynamic allocation use technique since technique based directly operational semantics provides alternative less mathematical overhead particular need find semantic domains satisfying simply dval set closed values use syntactic procedures place functions moreover relatively easy also model impredicative types model ahmed crucial interpretation object types develop although recently progress finding models languages combine references polymorphic types constructions involved circularity resolved considering stratification based notion execution safety central idea term type approximation assumption proved wrong sense reaching stuck state context executing fewer steps key insight constructing sets satisfying operations heap consume one step thus order determine whether pair heap typing value belongs type approximation sufficient know types stored values relies recorded level true meaning types heap typings obtained taking limit approximations instance heap typing asserts procedure stored location bool bool certainly safe assume schwinghammer pair belongs type procedures however possible contradict assumption taking two reduction steps first step consumed beta reduction second one method selection procedure body involves heap access case steps left observe result computation boolean rather integer consequently value type procedures two computation steps even though actually return integer one course distinguish false positives taking reduction steps preceding considerations formalized building model originally developed ahmed language general references impredicative types apart notational differences definitions section section adds subtyping section deals procedure types section revisits reference types semantics object types presented section constitutes main contribution paper deviate adding bounds types section using instead types section semantic model make circular definition types heap typings work semantics considers triples additional natural number component representing step index rather pairs first inductively define two families pretypek heappretypingk heap heappretypingj cval heappretypingj heappretypingk loc pretypek pretypek set triples set heappretypingj heap drawn depends index clearly pretypek thus heappretypingk possible set pretype heappretypingj cval heappretypingj call elements set rather types since condition proper types must satisfy done definition writing always implicitly assume heappretypingk heappretyping denote set loc pretype finite maps union sets pretypek index appearing elements bounded made explicit following notion semantic approximation stratification invariant definition semantic approximation call approximation define subset containing elements index strictly less definition lifted pointwise partial functions heappretyping dom proposition stratification pretype pretypek moreover semantics imperative objects particular dom pretypej captured following stratification invariant satisfied constructions types ensures whole construction stratification invariant depend beyond approximation indicated order take dynamic allocation account consider possible worlds model intuitively think pair describing state heap lists locations guaranteed allocated contains types stored values approximation course computation three different situations heap state changes new objects allocated heap reflected heap additional locations compared operation affect previously stored objects extension program executes steps without accessing heap reflected heap state forgets precise approximation guarantees heap safe execution steps heap updated way typing guarantees preserved thus updates reflected information forgetting extension previous case however step taken update case necessarily following definition state extension captures possible evolutions state definition state extension state extension relation heappretyping defined dom dom dom technique relies approximation true set values constitute type values behave accordingly unless certain number computation steps taken limiting number available steps able make fewer distinctions moreover instance procedure relies locations heap described state safely apply procedure allocations fact interested safely executing procedure steps heap described state suffice conditions captured precisely state extension require semantic types closed state extension definition semantic types heap typings set type semantic types subset pretype defined type cval also define set heaptyping loc type heap typings ranged following subset heap map semantic types explained ahmed structure may viewed instance kripke models intuitionistic logic states possible worlds state extension schwinghammer reachability relation worlds closure state extension corresponds kripke monotonicity next define particular heap conforms requirements expressed heap typing done respect approximation index definition heap heap respect approximation written dom dom dom semantic types contain values also need associate types terms values two steps first closed terms arbitrary ones closed term certain type approximation respect heap typing heaps respect term behaves like element type computation steps general reducing value term execute steps possibly allocate new heap locations state describing final heap therefore extension state describing initial heap needs safe remaining steps similarly final value needs original type another steps next definition makes precise definition closed term semantic type say closed term type respect state denoted even though terms evaluate closed subterms also reason open terms typing open terms done respect semantic type environment maps variables semantic types reduce typing open terms typing closed instances obtained substituting free variables appropriately typed closed values done value environment finite map variables closed values agrees type environment definition value environment agrees type environment say value environment agrees semantic type environment respect state dom denote definition semantic typing judgement say term possibly free variables containing locations type respect semantic type environment written substituting values free variables obtain closed term type number computation steps precisely dom construction semantic typing judgment enforces terms typable respect produce type errors evaluated definition safe steps call configuration safe steps term get stuck less steps evaluated heap define set configurations safek val semantics imperative objects definition safety call configuration safe get stuck number steps let safe safek theorem safety programs heaps safe proof one first easily shows safek theorem follows observing respect empty heap typing approximation much direct subject reduction proof however unlike subject reduction validity typing rules still needs proved respect semantics two steps remainder section introduce specific semantic interpretations types prove satisfy certain semantic typing lemmas proofs similar spirit proving fundamental theorem kripke logical relations section prove soundness rules initial type system respect typing lemmas even though semantic typing lemmas constructed directly correspond rules original type system big difference two semantic typing lemmas allow logically derive valid semantic judgments using valid judgments premises typing rules syntax used inductive definitions typing subtyping relations subtyping since types interpretation sets satisfying additional constraints natural subtyping relation set inclusion subtyping relation forms complete lattice semantic types infima suprema given intersections unions respectively least element greatest heaptypingj cval obviously satisfy stratification invariant closure state extension condition indeed semantic types easily show standard subsumption property lemma subsumption easy define subtyping way interaction subtyping features type system particular object types far trivial point discussed section procedure types intuitively procedure type computation steps applied argument type produces result type another steps procedure application takes one computation step way use procedure applying argument additionally take account procedure also applied computation steps extend heap every every heap typing applying procedure value type steps respect result must type steps respect computational intuition nicely fits possible worlds reading procedure types intuitionistic implication schwinghammer semlam semapp semsubproc figure typing lemmas procedure types definition procedure types semantic types consists triples heap typings closed values proposition semantic types also semantic type figure contains semantic typing lemmas associated procedure types procedure type constructor course contravariant argument type covariant result type lemma procedure types three semantic typing lemmas shown figure valid implications proof sketch validity semapp semlam proved verifying semsubproc simply matter unfolding definitions revisiting reference types calculus references syntactically use model references construction underlying object types order interpret variance annotations object types additionally introduce readable reference types writable reference types covariant contravariant subtyping respectively heap typing associates allocated location precise type used reading writing heap locations support reading writing certain type locations intuitively readable reference types writable ones precise type locations partially known without additional information one two operations safe meaningful type first recall definition reference types definition reference types semantic type according definition location type type associated heap typing approximately semantic approximation used satisfy stratification invariant operationally justified fact reading location writing takes one computation step type steps values read written type steps readable reference type similar poses less constraints heap typing requires subtype approximation semantics imperative objects semsubcovref semsubconref semsubvarref figure subtyping reference types definition readable reference types semantic type value stored location also type subsumption therefore read safely used value type however true type location general unknown writing value could unsafe true type might empty type nevertheless knowing location type mean write simply means know type values written absence information writing guaranteed type dually type writable references contains locations whose type associated supertype definition writable reference types semantic type safely write value type location type since value also real type location subsumption however real type locations arbitrarily general particular type values thus location know type read safely type definitions place usual reference type definition recovered intersection readable writable reference type hence supertypes also easily shown readable reference type constructor covariant writable reference type constructor contravariant figure usual reference types obviously invariant variance annotation use stand reference type constructor variance note strictly speaking set semantic type since calculus locations values although locations appear object values see section fact definition object types definition next section depend semantic type however order object type constructor yield semantic types crucial closed state extension proposition semantic type closed state extension conceptually different immutable reference types modeled using singleton types schwinghammer object types giving semantics object types much challenging types typing rules section indicate case first adequate interpretation object types must permit subtyping width depth taking variance annotations account second contrast types consider single elimination rule constructed objects support three different operations invocation update cloning definition object types must ensure consistent use object possible future operations requirements invocation update cloning rely must already established object creation time defining object types instructive consider simpler variants fulfill requirements object types decision store methods heap procedures together selfapplication semantics method invocation figure suggest object types somewhat similar recursive types records references holding procedures take enclosing record argument however invariance reference type constructor blocks form subtyping even width look typing rules subtyping recursive types cardelli amber rule appears rule subrec figure suggests position recursion variable covariant instance attempting establish subtyping amber rule one needs show clearly hold even simpler setting without reference types functional object calculus contravariance procedure type constructor first argument would cause subtyping fail combination type recursion existential quantifier uses recursion variable bound would allow enforce covariance positions recursion variable thus subtyping width intuitively viewed true precise type object general type given subtyping essentially idea encodings object types explored abadi subtyping depth respect variance annotations simply use readable writable reference types defined previous section still keeping abstract neither typing rule method invocation inv figure one object cloning clone validated explicitly enforcing definition object types object value fact belongs existentially quantified assumptions become sufficiently strong repair invocation case consistent seeing true type object semantically express using intersection types semantics imperative objects forcing current object value also sufficiently similar values maybe even created yet covers case cloning following definition formalizes construction definition object types let defined set triples type condition stating ensures values object type provide least required methods listed type also provide clearly necessary subtyping width condition postulates existence specific type true type object approximation subsequent conditions stated terms rather condition states requirements methods terms reference type constructors introduced section since existentially quantified might equal one must take care introduce circularity however due use approximation definition reference type constructors condition depends rather ensure construction explained order invoke methods must know belongs specific type steps suffices since application consumes step particular case condition states exactly need general formulation order ensure clones considered object also belong type therefore enforce matter object value constructed belongs type provided satisfies typing assumptions respect possibly extended heap typing allowing state extension necessary since cloning allocates new locations present original also cloning performed intermediate computation steps result allocations show definition object types actually makes sense defines semantic type immediately obvious recursion proposition type also type proof sketch must show recursive definition closed state extension prove one use general results recursive types stepindexed semantics since object type constructor contractive alternatively observation types argue suffices directly definition defines terms closure state extension follows corresponding property types proposition sets proposition transitivity state extension schwinghammer let semobj seminv clone semupd semclone semsubobj semsubobjvar figure typing lemmas object types figure presents semantic typing subtyping lemmas object types lemma object types semantic typing lemmas shown figure valid implications proof sketch semantic typing lemmas proved independently sketch semobj detailed proof well proofs typing lemmas given appendix assuming must show let definition semantic typing judgement must prove equivalently suitable holds let assume irreducible operational semantics clear locations dom choosing easily seen furthermore hypothesis semlam assumption follows definition remains establish achieved proving following general claim induction claim key step choosing equal verifying three conditions definition object types inductive hypothesis used showing semantics imperative objects remark proof establishing directly seem possible generalization claim arises naturally failed proof attempt order prove three conditions definition sensible choice follow easily requires show extension claim states fact inductive argument hidden proof come surprise induction step index resolves recursion inherent objects due self application semantics method invocation bounded quantified types impredicative quantified types previously studied setting ahmed general references follow presentation however unlike work ahmed quantifiers bounds also studying subtyping important note impredicative types reason semantic stratification types needed presence general references opposed syntactic one based nesting reference types setting consider paper need semantic stratification explicitly accommodate quantified types also interpretation object types uses existential types implicitly appel mcallester work type constructor function semantic types semantic types order determine whether term type approximation suffices know type approximation later show lemma type constructors define paper definition type constructor type type nonexpansive types definitions types require applied nonexpansive type constructors condition ensures order determine level universal existential type quantification types pretypek suffices helps avoid circularity otherwise introduced impredicative quantification definition bounded universal types type type type define type definition bounded existential types type type type set defined pack type proposition type type type also types proof sketch proofs minor modifications given additionally take bounds account schwinghammer type type type semtabs type semtapp type pack sempack semopen open type semsubuniv type semsubexist figure typing lemmas bounded quantified types lemma bounded quantified types semantic typing lemmas shown figure valid implications proof sketch first four implications proved additional precondition semtapp sempack serves establish requirements bounds two subtyping lemmas semsubuniv semsubexist easily proved unfolding definitions recursive types contrast previous work models consider rather types folds unfolds explicit syntax consume computation steps types previously considered ahmed relational model lambda calculus simpler sufficient purpose consequence require type constructors opposed stronger contractiveness requirement definition recursive types let type type function define set fold proposition type type type proof sketch follows observation defined terms means relies closure state extension established induction proving type figure presents semantic typing lemmas recursive types consequence expected fixed point property fold lemma recursive types semantic typing lemmas shown figure valid implications proof sketch validity semfold semunfold easy consequences definition semsubrec one shows induction using precondition rule semantics imperative objects type type unfold semunfold fold semfold semsubrec figure typing lemmas recursive types jad figure interpretation types lemma considered type constructors proof sketch easily seen definition uses therefore similar statement holds approximation quantified types since definition depends case object recursive types properties established induction using latter case semantic soundness order prove terms safe evaluate relate syntactic types semantic counterparts use fact semantic typing judgement enforces safety construction theorem approach standard denotational semantics fact none main statements proofs section mentions stepindices explicitly definition interpretation types typing contexts let total function type variables semantic types interpretation type given structurally recursive meaning function defined figure interpretation typing context respect given function maps every note figure type constructors used sides equations simply syntax right refer corresponding semantic constructions defined previous section recall necessary precondition semantic typing lemmas particular depends schwinghammer due use figure begin showing interpretation types map lemma holds induction structure proof sketch show relying lemma semantic type constructions definition let typing context say satisfies written holds appearing show soundness subtyping relation lemma soundness subtyping proof sketch induction derivation case analysis last applied rule case immediately reduced one subtyping lemmas section finally prove semantic soundness syntactic type system respect model theorem semantic soundness whenever follows proof sketch induction derivation case analysis last rule applied case easily reduced one semantic typing lemmas section using standard type substitution lemma derivations ending application fold unfold tapp pack theorems semantic soundness safety proof safety type system section corollary type safety terms safe evaluate self types self types proposed abadi cardelli means reconcile recursive object types proper subtyping self types interesting allow type methods return possibly modified host object clone instance type list nodes filter method produces sublist elements satisfying given predicate lista obj filter bool note similar recursive type filter bool satisfy usual subtyping object types invariance fields semantics imperative objects let clone figure typing lemmas self types semantics self types abadi cardelli show self types understood terms recursive existentially quantified object types via encoding precisely type obj may occur positively stands recursive type bounded existential quantifier introduced encoding gives rise desired subtyping width depth despite type recursion since type system features recursive bounded existential types self types could accommodated via encoding however treatment self types achieved even directly without relying encoding fact almost everything place already recall semantics object types definition employs recursion existential quantification refer true type object condition definition changed take advantage type definition self types assume type type monotonic type constructors let defined set triples type section one shows definition uniquely determines type proof type order ensure defined terms moreover proofs typing lemmas object types see section appendix carry minor modifications show semantic typing lemmas figure hold schwinghammer cases obtained replacing result type throughout proof existentially quantified type condition definition proof uses monotonicity conclude result invocation type fact type given condition proofs universally quantified respective assumptions instantiated since gives necessarily three proofs also use induction shows given way words viewed type constructor lemma still holds interpretation syntactic type expressions given figure extends straightforwardly self types using new type constructor obj jad interpretation semantic typing lemmas figure soundness theorem section extend syntactic type system objects self types similar one derived abadi cardelli encoding also including variance annotations typing rule cloning limitations note exception semantic typing lemmas self types stronger counterparts figure already enough typecheck many examples involving self types however encoding abadi cardelli updating methods one usually full information precise self type host object may proper subtype therefore statement method update figure includes quantification subtypes known type object ensure updated method also works correctly precise type consequence new method body must sufficiently parametric type self parameter overly restrictive abadi cardelli demonstrate limitation example objects provide backup retrieve method obj retrieve backup sensible definition backup method updates retrieve method subsequent invocation retrieve yields clone current object backup let clone let stands usual syntactic sugar let denote interpretation syntactic type backup method correct operational behaviour typecheck method update body using semupdself would need statement statement hold arbitrary subtype therefore semantic typing lemmas stated strong enough prove thus backup holds method body prevents typing object contains method backup backup type using semantic typing lemmas semantics imperative objects abadi cardelli address lack expressiveness modifying calculus two respects first introduce new syntax method update operationally new construct behaves like let let typing rule powerful one induced encoding typing method body assumed precise type object obj second order propagate information typing rules structural assumptions introduced instance inference rule object cloning takes form clone thus applying also case type variable modified system body backup method rewritten backupmod clone judgement backupmod derivable even purely syntactic setting hoc character syntax extension entirely satisfactory semantics types modifications fact problematic first although seems reasonable expect semantic typing lemmas section continue hold change calculus operational semantics would require recheck proofs object types detail fortunately syntax extension seem necessary semantic typing point view already prove semantic soundness rule respect encoding new method update construct let let however rule help typing body backup method introduction rules structural assumptions presents severe difficulty soundness rules relies fact every subtype object type another object type words rule type assumed range object types structural assumptions usually valid semantic models certainly justified respect semantically defined subtype relation set inclusion self types structural assumptions sum previous subsection problem semantic typing lemmas figure weak type certain examples body backup method usual way strengthen rules syntactic setting semantically sound model still backup valid typing judgement method body seen taking closer schwinghammer let clone let figure typing lemmas structural assumptions self types look semantic definition self type essentially suitable substitution substitution instance backup let clone becomes irreducible less steps must object value property asserts existence type property entails becomes bound value type thus eventual update retrieve field valid since new method expected type sufficient approximation similar manual reasoning seems possible cases principled approach let use typing lemmas strong enough avoid explicit reasoning operational semantics step indices facilitate develop semantic counterpart structural assumptions appear syntactic type system abadi cardelli precisely introduce relation semantic types strengthens subtype relation intuitively precise recursive type collection object values object type type acts interface lists permitted operations object values definition self type exposure type relation holds notice essentially states type take place existentially quantified self type object type see definition immediate definition implies note however reflexive general object type could empty intuitively object type obtained union semantics imperative objects figure lists new typing lemmas self types exploit relation compared figure typing lemmas seminvstr use additional assumptions establish precise typing result similarly universally quantifies premise limits type holds finally lets use object within precise type similarly lets type method bodies informative assumption latter two key lemmas introduce assumption proofs using semantic typing lemmas illustration consider example backup method order construct objects backup method establish backup holds backup abbreviates let clone follow backup backup retrieve desugar let construct backup apply lemmas semapp semlam notice order show suffices prove clone using validity former judgement immediate similarly latter follows fact since retrieve method listed variance annotation lemma self types lemmas structural assumptions semantic typing lemmas shown figure valid implications proof sketch proofs straightforward adaptations seminv semclone semupd example give proof lemma appendix interestingly relies following property object types type proof used instantiate universally quantified premise full proof given lemma appendix proof similar one semobj lemma pendix except use heap typing extension recursive record type satisfying conditions validating subtyping property verifying extended heap respect one uses order instantiate assumptions method bodies obtain finally show generated object value type claim strengthened show object value fact subtype since see proposition lemma appendix full proof schwinghammer remark implication type may appear reasonable would entail believe holds problem premise guarantees existence type satisfies requirement general possible construct type limit sequence reason implication pack valid typing lemma avoids problem since needed fixed approximation choice sufficiently large suffices proof lemma appendix hand lemma avoids problem instantiating particular type already known section showed semantics object types naturally extends self types avoiding change syntax operational semantics calculus proved first set typing lemmas natural apply many examples figure lemmas however sufficient typecheck methods achieve developed second set typing lemmas involve object precise type relation figure note latter lemmas fully subsume former ones since relation reflexive leave open problem relating lemmas figure syntactic type system generalizing reference object types semantics described paper generalizes reference types readable writable reference types generalized even reference type constructor takes two types arguments one represents general type used writing reference another specific type read easily expressed using readable writable reference types together intersection types ref unfolding definitions yields ref one would expect generalized reference type constructor contravariant first argument covariant second one ref ref note one takes generalized reference types primitive three reference types section obtained special cases ref ref ref subtyping properties figure still valid figures give graphical representation different reference type constructors figures horizontal axis represents type reference read vertical one gives type written notice different variance read axis goes write axis figure represents usual well readable writable reference types points three edges triangle notice usual references read written type without additional information readable references semantics imperative objects figure reference types figure generalized reference types let clone figure typing lemmas generalized object types written safely type writable ones read type subtyping represented arrows covariant edge readable reference types contravariant writable reference types edge invariant reference type subtyped either readable writable reference type figure illustrates generalization reference types indeed natural generalizing take points edges triangle also points inside reference types furthermore instead three different kinds reference types one subtyping also natural set supertypes reference type cover area rectangle goes point corresponding reference type top reference type ref instance dark gray rectangle figure contains supertypes ref applying idea context imperative object calculus leads expressive subtyping also simplifications since variance annotations longer needed extended object type two types method general type used update given method specific type expected result invoking method defining semantics generalized object types difference respect definition object types condition changed use extended reference type ref schwinghammer let clone bew ber figure typing rules generalized object types figure presents semantic typing lemmas validated definition object types figure gives corresponding syntactic typing rules note complex seemingly rules subtyping object types given figure replaced one rule lemma generalized object types semantic typing lemmas shown figure valid implications proof sketch proof subtyping lemma follows easily lemma subtyping generalized reference types therefore significantly simpler variance annotations involved see lemma appendix semantic typing lemmas proofs basically unchanged see section appendix note generalization object types presented section orthogonal extension self types previous section generalized object types lead type system simpler expressive usual type systems objects generalized object types directly correspond split types bugliesi shown types strictly expressive object types variance annotations example comparison related work models abadi cardelli give semantic model functional object calculus type system comparable one consider types interpreted certain partial equivalence relations untyped domaintheoretic model calculus indication given adapt imperative execution model based earlier work kamin reddy reus construct models imperative object calculus goal proving soundness logic abadi leino store exhibited object calculus requires defining semantic domains recursive equations dynamic allocation addressed interpreting specifications logic kripke relations indexed store specifications similar heap typings used semantics imperative objects building work levy intrinsically typed model imperative object calculus presented second author phd thesis solving domain equations suitable category functors however types considered compared models model present soundly interprets richer type language also easier work way based operational semantics eliminates need explicit continuity conditions admissibility conditions replaced closure state extension usually easy check needed definition types stratification invariant missing model semantic notion equality approximates program equivalence reasoning program equivalence language ahmed recently developed relational model could interesting adapt work setting recently proposed models polymorphism general references suggest adequate semantics imperative objects expressive typing could principle developed also setting detailed comparison stepindexed semantics models would useful make similarities differences two approaches precise interesting see object construction rule obj proved correct models described models directly corresponds recursive predicate whose existence uniqueness must established proof exploits properties underlying recursively defined domain imposes restrictions semantic types besides admissibility types appearing defining equation recursive predicate need satisfy analogue contractiveness property typed functor category model object construction interpreted using recursively defined function correspondingly obj proved fixed point induction case essence proof elementary induction step index suitably generalized induction hypothesis see claim proof sketch lemma page full proof appendix interpretations object types main contribution paper novel interpretation object types model stratification permits construction recursive well impredicative secondorder types key ingredients interpretation object types use recursive existentially quantified types line work object encodings however mainly focused object calculi functional execution model closest work encoding imperative objects imperative variant system updatable records proposed abadi objects interpreted records containing references procedures represent methods case records recursive existentially quantified record type difference two additional record fields included order achieve invocation cloning uninitialized fields used construct recursive record subtyping depth considered encoding functional object calculus however one added target language readable writable reference types use schwinghammer paper encoding imperative object calculus would extend subtyping depth well typing rules self types structural assumptions subtype relation play important role section developed semantic counterpart typing rules structural assumptions order deal polymorphic update selfreturning methods however tailored specifically object types hofmann pierce investigate metatheory subtyping structural assumptions general give elementary presentations two encodings functional objects variant system type destructors may interesting see model variant system found models semantic models introduced appel context foundational code goal construct elementary modular proofs type soundness easily checked automatically primarily interested languages however also applied technique pure recursive types later ahmed successfully extended general references impredicative polymorphism semantic model present extends one ahmed object types subtyping order achieve refine reference types readable writable reference types subtyping semantic model previously considered swadi studied typed machine language setup however much different particular subtle issues concerning subtyping object types original work previous work focuses semantic type systems semantic typing lemmas directly used programs however one considers complex type systems subtyping recursive types polymorphism semantic typing lemmas longer directly correspond usual syntactic rules discrepancies fixed usually cost complex models like one developed swadi track type variables swadi model additional semantic kind system used track contractiveness types free type variables avoid complex model one tracks type variables considering rather types type argument contractive type constructors contractive general identity well type constructor hand type weaker assumption argument type constructors indeed see lemma relatively straightforward use semantic typing lemmas order prove soundness standard syntactic type system consider see theorem type safety proofs abadi cardelli use subject reduction prove safety several type systems similar one considered paper purely syntactic proofs different semantic type safety proof present detailed discussions differences see since type safety built model safety proof neither relies preservation property preservation concluded semantics imperative objects constructing semantics challenging proving progress preservation however particular semantics could reuse model ahmed extend suit needs even though calculus considering quite different one would expect enough general models constructed become easier build new models mixing matching assuming existence adequate model effort needed prove semantic typing lemmas using somewhat comparable one required subject reduction proof since semantic typing lemmas proved isolation resulting type soundness proof modular extensions consider sections illustrate aspect rather well according appel original motivation advantages become even apparent formalizing proofs proof assistant generalized reference object types readable writable reference types define section use modeling object types section similar reference types forsythe programming language channel types generalization reference type constructor taking two arguments described section quite natural also appeared pottier thesis facilitated type inference allowing meets joins distribute reference types idea recently applied craciun inferring variant parametric types java generalized object types introduce section directly correspond split types bugliesi split types also motivated type inference since guarantee existence precise upper lower bounds particular bugliesi show split types strictly expressive object types variance annotations example establish soundness type system split types subject reduction respect functional semantics object calculus functional object calculus initial experiments current topic done context functional object calculus even though functional setting semantic model much simpler models satisfy semantic typing lemmas even syntactic type system considered functional calculus exactly one paper results section directly apply functional object calculus terms get stuck matter whether evaluated functional imperative way would possible directly prove result using subject reduction since subject reduction syntactic typing judgment imperative calculus would also depend heap typing thus different judgment functional calculus however since using subject reduction need partially evaluated terms contain heap locations conclusion presented semantics abadi cardelli imperative object calculus used prove safety type system object types recursive types well subtyping showed semantics extended self types typing lemmas structural assumptions generalized way schwinghammer eliminates need variance annotations time simplifies subtyping rules objects technique however limited type safety proofs already employed general reasoning programs based previous work appel mcallester ahmed built partial equivalence relation model lambda calculus recursive impredicative quantified types showed relational interpretation types sound proving contextual equivalences recently extended significantly reason program equivalence presence general references benton also used technical device together notion orthogonality relating expressions contexts show soundness compositional program logic simple abstract machine also employed framework based relational parametricity specification verification machine code programs hope work paves way similarly compelling semantic investigations program logics imperative object calculus using model possible prove soundness expressive program logics calculus acknowledgements express gratitude anonymous reviewers detailed constructive comments preliminary versions article also thank andreas rossberg pointing work john reynolds forsythe supported fellowship microsoft research international max planck research school computer science references abadi luca cardelli semantics object types proceedings ninth annual ieee symposium logic computer science lics pages ieee computer society press abadi luca cardelli theory objects springer abadi luca cardelli ramesh viswanathan interpretation objects object types proceedings symposium principles programming languages popl pages acm press abadi rustan leino logic programs 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propositions proposition preorder state extension relation reflexive transitive proposition extension proposition relation let closed value exists semantics imperative objects typing lemmas object types lemma semobj object construction object types proof let assume must show thus let value environment heap typing definition semantic typing judgement definition need show equivalently suitable show suppose following three conditions fulfilled operational semantics rule applies means necessarily distinct dom choose show first conjunct holds immediate construction order show second conjunct definition heap first need show dom dom first conjunct definition clear dom dom thus shape definition obtain required inclusion next let dom establish need show distinguish two cases case respectively get thus need show semlam figure lemma assumption already know definition semantic typing judgement obtain since proposition shows implies proposition get together first conjunct transitivity yields since closed state extension latter property imply required schwinghammer case dom respectively get actually need show definition get since closed state extension obtain finally need show third conjunct end prove following general claim claim follows taking observing reflexive proposition claim proved complete induction assume moreover let dom show checking conditions obtained unfolding definition hold choosing yields type next construction together nonexpansiveness procedure types follows definition reference types definition implies lemma subtyping variance annotations semsubvarref figure obtain property finally must prove note last condition holds trivially base case induction assume assumption yield semantics imperative objects moreover transitivity since induction hypothesis claim gives established definition applied object type properties establish indeed finishes inductive proof claim proof lemma lemma seminv method invocation object types proof let assume show end let definition get need show thus let consider heaps term following three conditions fulfilled second third conjunct operational semantics first conjunct together first conjunct definition follows exists heap typing definition object types latter shows exists hold type well using deduce expanding definition definition procedure type means particular must abstraction thus since cval schwinghammer operational semantics obtain reduction sequence form since proposition use property object type instantiate obtain using definition procedure types follows hand second conjunct implies definition proposition closure types state extension moreover irreducible combined definition means exists first conjunct first conjunct using transitivity state extension obtain second third conjuncts definition conclude holds needed show lemma semupd method update object types proof sketch proof similar lemma method invocation existence follows existence corresponding reduction sequence since reduction results configuration proof lemma essentially matter showing semantics imperative objects first note dom dom dom holds next let let dom remains show note definition definition exists type follows prove case distinction location case since assumption yields since subtyping lemma semsubproc procedure types yield monotonicity semantic approximation therefore entails additionally assumption gives since proposition yields closure state extension implies follows case case easier since value heap change location result follows closure state extension lemma semclone object cloning object types clone proof sketch proof similar lemma method update assuming clone halts fewer steps appealing operational semantics assumption one obtains existence clone clone since reduction clone clear distinct dom set follows establish lemma suffices prove observing dom dom satisfied first conjunct proved showing holds dom done case distinction whether dom cases relation follows closure state extension types second conjunct note constructed holds therefore unfolding definition object types condition allows schwinghammer conclude required follows fact holds definition subtyping lemmas object types lemma semsubobj subtyping object types imply proof denote assume prove heap typings values complete induction induction hypothesis equivalently assume definition definition exists type moreover condition holds respect transitivity induction hypothesis get holds moreover entails condition also holds respect object type order conclude therefore remains proven condition choose case analysis variance annotation case deduce thus covariance procedure type constructor second argument semsubproc figure get since know since type constructor covariant semsubcovref figure implies case similarly previous case gives covariance semsubproc infer contravariance semsubconref figure get entails since immediately obtain also semantics imperative objects lemma semsubobjvar subtyping object variances proof proof proceeds similarly proof lemma assume let denote let arbitrary prove complete induction induction hypothesis equivalently assume definition exists type moreover condition holds induction hypothesis transitivity get choice also shows holds respect order show therefore remains proven show case analysis disjunction cases trivial case semsubvarref figure immediate case required statement typing lemmas structural assumptions self types lemma method update structural assumptions object types type proof proof adaptation proof given lemma method update assume let type moreover assume hold show let value environment heap typing must prove let operational semantics sequence induced assumption exists particular form operational semantics particular choosing first last conjuncts yield schwinghammer proposition transitivity closure state extension establish lemma remains show dom follows second conjunct closure state extension interesting case must prove since condition definition self type exposure yields assumption holds definition hence suffices prove follows assumption proposition self type exposure let self type suppose exists type proof suppose definition self types means exists type conditions satisfied choosing clear since conditions rely approximation moreover instantiating obtain proves proposition lemma introducing structural assumptions let suppose type let proof let suppose type must show let thus let definition semantic typing judgement definition must show let equivalently suitable removing syntactic sugar suppose second third conjunct operational semantics first conjunct together assumption first conjunct definition follows exists heap typing particular val operational semantics gives third conjunct proposition exists type first conjunct propositions closure state extension yields semantics imperative objects since instantiating universally quantified type hypothesis obtain therefore gives clearly second conjunct second conjunct shows definition establishes let holds required next define recursive type records type arising recursive record interpretation imperative objects type give rise subtyping show satisfies definition assume type type monotonic type constructors let defined set triples note recursive specification moreover type closed state extension proposition self types proof let arbitrary monotonic type constructors clear conditions definition satisfied definition definition remains prove establish showing complete induction let need show choose type fulfills induction hypothesis conditions definition self types exactly conditions definition definition concludes proof lemma object construction structural assumptions let suppose type proof let assume must show thus let value environment heap typing definition need show equivalently suitable show suppose following three conditions fulfilled schwinghammer operational semantics rule applies means necessarily distinct dom let definition choose show first conjunct holds construction order show second conjunct let dom need show case dom proof proceeds exactly lemma consider case get thus need show proposition obtain instantiate universally quantified obtain semlam figure lemma gives definition obtain since proposition shows implies proposition get together first conjunct transitivity yields since closed state extension latter property imply required finally need show third conjunct end prove following general claim claim last conjunct follows taking observing reflexive proposition since claim proved complete induction assume semantics imperative objects moreover let dom show checking two conditions definition definition satisfied construction together follows definition reference types definition implies lemma subtyping reference types semsubvarref figure obtain property second must prove note last condition holds vacuously base case induction assume assumption yield moreover transitivity since induction hypothesis claim gives established definition applied type properties establish indeed finishes inductive proof claim proof lemma subtyping lemma generalized object types lemma subtyping generalized object types proof denote assume prove heap typings values complete induction induction hypothesis assume definition definition condition instead exists type ref schwinghammer moreover condition holds respect transitivity induction hypothesis get holds moreover entails condition also holds respect object type order conclude remains proven condition ref let since procedure type constructor covariant result type semsubproc figure assumption implies get ref ref together directly implies concludes proof work licensed creative commons license view copy license visit http reative send letter creative commons second suite san francisco usa eisenacher strasse berlin germany
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submitted ieee transactions signal processing efficient analysis via bit flipping panos sandipan shubham dimitris oct department electrical microelectronic engineering rochester institute technology rochester usa panos qualcomm technologies san jose usa sandipan department electrical engineering state university new york buffalo buffalo usa shubhamc pados edics submitted september abstract shown recently principal components data matrix data samples dimensions exactly calculated cost advantageous rank applications large big data large heavy data large costs prohibitive work present novel suboptimal algorithm calculation cost dmin comparable standard analysis theoretical experimental studies show proposed algorithm calculates exact optimal high frequency achieves higher value optimization metric known alternative algorithm comparable computational cost superiority calculated standard singular vectors characterizing potentially faulty demonstrated experiments data dimensionality reduction disease diagnosis genomic data index terms dimensionality reduction data analytics norm norm machine learning outlier resistance principal component analysis subspace signal processing corresponding author part paper presented ieee intern conf speech signal processing icassp florence italy may work supported part national science foundation grant office vice president research rochester institute technology ntroduction analysis pca mainstay signal processing machine learning pattern recognition classification century numerous important applications pca found fields wireless communications computer networks computer vision image processing neuroscience name broadly speaking pca seeks calculate orthogonal directions define subspace wherein data presence maximized traditionally pca quantifies data presence frobenius norm projected data onto subspace equivalently euclidean distance original data subspace representations residual error herein referring standard pca key strengths implementation quadratic number data points means decomposition svd data matrix reliable approximation offers nominal principal data subspace calculated sufficiently many corrupted data points advent era datasets often include grossly corrupted highly deviating irregular data points outliers due variety causes transient sensor malfunctions errors data errors training data labeling data corruption name regretfully standard fragile presence faulty data even appear vanishingly small fraction training set reason objective standard pca minimization error variance maximization squared projection magnitude gives squared importance magnitude every datum thus overemphasizing peripheral data points remedy impact outliers researchers fields data analysis signal processing long focused calculating subspaces minimum absolute error deviations instead minimum error variances important early theoretical studies date back past decade several principal component calculators general label works researchers seek either minimize aggregate absolute data representation error maximize aggregate absolute magnitude projected data points approach shown error surface problem resisting attempts reach exact solution even exponential computational cost therefore suboptimal algorithms exist literature approach problem known works proved fixed data dimension offered first two optimal algorithms literature exact calculation specifically data record size first optimal algorithm solves exactly complexity second optimal algorithm solves exactly complexity rank rank desired number results several suboptimal algorithms existed literature however lack optimal solution absolute performance evaluation algorithms respect metric could conducted made optimal value available today experimental studies indicate existing approximate algorithms yield performance degradation particular one principal component calculated present work introduce based algorithm calculation data matrix complexity dmin proposed algorithm accompanied formal proof convergence theoretical analysis converging points detailed asymptotic complexity derivation iii theoretically proven performance guarantees performance comparisons counterparts comparable computational cost experiments dimensionality reduction foreground motion detection surveillance videos disease diagnosis genomic data studies show outperforms suboptimal counterparts comparable cost respect metric retains high similar optimal thus proposed algorithm may bridge gap computationally efficient principal component analysis rest paper organized follows section presents problem statement reviews briefly pertinent technical background section iii devoted development analysis proposed algorithm extensive experimental studies presented section concluding remarks drawn section roblem tatement background given data matrix rank min interested calculating data subspace dimensionality form orthonormal basis solves argmax denotes argument returns sum absolute values individual entries presented different proofs form formally problem jointly asymptotic suboptimal algorithms performance degradation approximating solution proposed showed first time fixed data dimension presented two optimal algorithms solving review briefly optimal solution suboptimal algorithms exist literature optimal solution matrix admits svd define let denote nuclear norm shown bopt argmax xbopt solution addition kxbopt bopt sgn takes binary quadratic bopt argmax component given xbopt xbopt kxbopt addition kxbopt kxbopt bopt sgn view first optimal algorithm performs exhaustive search candidate set obtain solution bopt returned svd xbopt second optimal algorithm polynomial complexity constructs searches inside subset wherein solution proven exist importantly constant respect cost construct search exhaustively within set seen contrast scalability principle hold argmax general column argmax every nuclear euclidean norm trivially coincide practice optimal algorithm takes advantage invariability negations permutations columns matrix argument searches exhaustively subset wherein solution guaranteed exist therefore problem translated series problems simply projecting onto previous solutions approximate algorithms iterations successive nullspace projections kwak made important early contribution field proposing fixed point iteration approximate solution following formulation notation section algorithm iterative form sgn arbitrary initialization point initialization guaranteed converge fixed point sgn sgn approximated qfp xbfp kxbopt bfp converging point iteratively generated sequence approximated sequentially greedy fashion kth qfp calculated procedure replaced projection onto nullspace previously calculated components qfp complexity algorithm maximum number iterations per component considering bounded linear function practically terminating iterations complexity algorithm expressed iterative alternating optimization nie offered significant advancement case following formulation notation proposed converging iterative algorithm initialized arbitrary orthonormal matrix calculates sgn solution approximated convergence point generated sequence say qao notice algorithm coincides one computational complexity maximum number iterations per component considering bounded linear function practically terminating iterations complexity algorithm becomes sdp successive nullspace projections mccoy tropp suggested novel programming sdp view problem problem rewritten maximize rank set positive matrices specifically opt solution column zopt solution algorithm relaxes rank constraint finds instead solution zsdp convex program maximize present paper notation obtain approximation bopt say bsdp algorithm factorizes zsdp calculates instances sgn vectors drawn instance gaussian randomization bsdp chosen instance maximizes solution approximated qsdp xbsdp kxbsdp similar follows method sequential qsdp nullspace projections calculating kth qsdp sdp complexity solve within accuracy sdp log thus overall computational cost algorithm log iii roposed lgorithm begin new algorithmic developments following proposition svd proposition assume data matrix admits compact decomposition rank min define proof notice positive symmetric matrix thus proposition proposed algorithm attempt find solution solving instead problem bopt argmax bopt argmax takes form reduction loss optimality far contributes low cost proposed algorithm sequel present separately cases calculation component first attempt find efficiently quality approximation bopt say bbf search procedure per qbf xbbf kxbbf suggested component data proposition presented herein first time bounds error metric interest differences proposition corresponding kybopt equality sgn proof proposition kxbopt kybopt kybopt kybopt kybopt equality holds sgn particular proposition performance degradation metric used instead optimal upper bounded performance degradation metric used instead optimal bopt view proposition proposed algorithm operates follows algorithm initializes binary vector employs iterations generate sequence binary vectors iteration step algorithm browses bits flipped kept algorithm negates flips single bit flipped offers highest increase metric bit exists negation increases quadratic metric reset bits become eligible flipping consideration iterations terminate metric increased flipping mathematically tth iteration step optimization objective kykf frobenius euclidean norm factorizing observe contribution nth bit written constant respect flip nth bit setting nth column identity matrix change quadratic metric thus contribution nth bit quadratic negative flipping increase quadratic whereas positive flipping decrease quadratic view analysis initialized arbitrary binary antipodal vector used restrain greediness unconstrained flipping iterations initial bit contributions calculated iteration step find argmin flipped setting lated updated otherwise new solution obtained resetting new solution flipped recalculated updated otherwise iterations terminate algorithm returns bbf qbf ybbf kybbf notice contribution factors end tth iteration step flipped directly updated every given data correlation matrix updating contributions costs ease reference complete code provided fig convergence characterization termination condition iterations convergence summarized met algorithm returns bbf terminates initialization point met iterations converge finite number steps since binary quadratic form maximization finite upper bound every step presented iterations quadratic value increases thus proposed iterations converge argument value characterization point converngence relationship fixed points last point generated sequence bbf satisfies termination condition bbf view equivalently rewritten bbf ybbf therefore every initialization point say sequence converges globally convergences initialization iterations converge set thus importantly following proposition holds true proposition every optimal solution maximization belongs proof let solution outside exists least one entry say nth entry contribution objective value negative consider binary vector derived negation nth entry hence maximizer proposition element necessary condition binary vector solution since point reachable iterations upon appropriate initialization may indeed calculate bopt return optimal solution following proposition establishes important relationship set convergence points proposed iterations set convergence points iterations say proposition every element lies also set proof consider arbitrary assume kyn sgn otherwise kyn sgn hence every vector sgn thus denote set optimal solutions propositions summarized terms set sizes illustration fig demonstrate experimentally relationship cardinalities generate independent instances data matrix element drawn independently gaussian distribution plot average observed cardinality versus number data points see increases remains tight cardinality number possible converging points iterations increases far higher rate fig offers insight proposed iterations expected return optimal solution much often iterations performance bounds performance attained qbf formally lower bounded proposition whose proof given appendix proposition performance qbf metric calculated upon initialization lower bounded qbf kxkf moreover optimal kxbopt max max maximum singular value data matrix loss respect metric experienced qbf bounded qbf kxkf fig execute proposed iterations arbitrary data matrix elements drawn independently plot binary quadratic metric kxb metric per iteration reference also plot optimal line kxbopt lower bound kxkf fig offers vivid numerical illustration proposition initialization next consider initializations may expedite convergence offer superior approximations trivial case bopt arg arg motivated special case initialize iterations sign right singular vector corresponds highest singular value call initialization choice initialization evidently definition sgn experimental studies compare efficiency proposed initialization equiprobable random initializations takes value probability show initialization leads superior approximations respect also reduces significantly actual execution time proposed algorithm reducing number needed convergence illustration generate realizations data matrix entries drawn independently realization run iterations equiprobable binary initialization observe time obtained initialization attains value metric greater equal obtained random initialization also fig plot empirical cdf number needed convergence two initializations observe time initialization already convergence set take place moreover initialization needed convergence empirical probability contrary random initialization may need probability convergence complexity analysis prior iterations calculates chooses initial binary vector complexity dmin svd algorithm calculates stores correlation matrix complexity given evaluates initial contribution factors cost therefore initialization total cost dmin iteration finding costs case repetition maximization entire index set costs additional therefore bit flip costs denoting number bit flips termination criterion met find computational cost calculating bbf dmin finally taking account computation qbf bbf proposed algorithm costs min according experiments empirical probability upper bounded bits need flipped reach bbf therefore total complexity proposed algorithm becomes dmin keeping dominant complexity terms complexity dmin quadratic either linear quadratic depending cost initial svd thus proposed algorithm cost comparable standard pca svd summary calculated complexity provided table multiple initializations metric improvement may also run distinct initialization points sgn sgn obtain corresponding con vergence points bbf qbf xbbf kxbbf return qbf qbf certainly multiple initializations increase complexity algorithm practice termination imposed algorithm constant factor calculation components contrast scalable problem therefore approaches fail return optimal bases optimal demands joint computation principal components procedure increases exponentially computational complexity optimal algorithms section generalize algorithm presented section calculate given data matrix rank given data matrix see proposition proposed algorithm first attempts approximate solution bopt begins initial matrix employs iterations reach approximation bopt say bbf finally per algorithm returns qbf xbbf evidently bbf exact solution qbf exact solution problem calculation bbf via iterations algorithm initializes binary matrix employs iterations generate sequence binary matrices matrix attains highest possible value maximization metric given differs exactly one entry similar case indices bits flipped throughout iterations kept memory initialized bit binary matrix argument corresponding index mathematically tth iteration step search single bit current binary argument whose index flipping increases metric kyb notice flip bit setting therefore tth iteration step algorithm obtains index pair argmax algorithm flips setting updates otherwise algorithm obtains new solution resetting new pair algorithm sets updates otherwise iterations terminate algorithm returns bbf qbf svd xbbf note solve exhaustively one calculate worst case demands dependent calculations simplistic computationally inefficient approach would perform svd scratch cost would yield worst case total cost find present alternative method solve cost evaluations optimal beginning tth iteration step perform decomposition svd cost singular values square roots eigenvalues found constant cost due rotation invariance equal eigenvalues eigenvalue decomposition defining evd evd kym diag easy show consider eigenvalue decomposition evd eigenvalue permutation invariance singular values equal square roots eigenvalues notice matrix needed calculation constant evaluations tth iteration given svd found cost thus computational cost absorbed svd examination bits mth row suffices calculate quantities appearing cost employing algorithm proposed evd diagonal positive semidefinite matrix perturbated symmetric matrix evd calculation costs defining finding eigenvalues per costs additional thus including initial svd total cost examination bits worst case scenario termination guarantee similarly case termination condition guaranteed met finite number iterations binary maximization finite upper bound every step iterations value increases initialization generalizing initialization idea case choose sgn allone vector length motivation lies fact bopt sgn vopt particular initialization ysgn pseudocode proposed algorithm offered fig course generated algorithms fig fig coincide complexity analysis iterations algorithm expends dmin calculate svd svd calculate also initial binary matrix proposed initialization calculated beginning first iteration complexity find solution incur worst case cost setting maximum number iterations run algorithm total complexity calculate bbf worst case dmin calculate qbf bbf costs additional dmin therefore total complexity proposed algorithm dmin quadratic complexity quadratic complexity notice asymptotic complexity algorithm becomes dmin equal algorithmic operations fig summary calculated complexity provided table multiple initializations similar case increase performance proposed scheme running iterations distinct initialization matrices choose sgn initialization initialize randomly sgn obtain corresponding convergence points bbf associated bases qbf return qbf qbf qbf xperimental tudies comparison current algorithms first compare performance algorithms proposed experiment generate arbitrary matrices size entries drawn independently calculate primary iterations initialization algorithm initialization sdp approach gaussian randomization measure performance degradation experienced vector metric fig plot empirical cumulative distribution function cdf qbf qfp qsdp observe time returns exact optimal addition performance degradation attained proposed algorithm empirical probability less hand iterations sdp attain optimal performance empirical probabilities respectively performance degradation sdp much respectively repeat experiment increasing number initializations proposed algorithm number gaussian randomizations sdp fig plot corresponding empirical cdfs qbf qfp qsdp outperforms two counterparts returning exact empirical probability next examine performance four algorithms iterations successive nullspace projections alternating optimization sdp successive nullspace projections proposed iterations regarding sdp approach total computational cost employing gaussian randomization instances log similar running proposed iterations distinct initialization points sought generate arbitrary data matrices size entries drawn independently plot fig empirical cdf corresponding performance degradation ratios qfp qao qsdp qbf iterative methods run single initialization point one instance sdp observe time returns exact optimal performance degradation attained proposed pcs empirical probability less hand pcs calculated attain optimal performance empirical probabilities respectively performance degradation algorithms much evidently nullspace projections violate principle problem significant impact performance next repeat experiment initializations instances sdp fig plot new performance degradation ratios four algorithms time proposed method returns probability high inferior performance also attained iterative algorithm hand algorithms follow greedy approach solving sequence problem instances experience performance degradation values similar fig experiment visual evaluation outlier resistance proposed consider points drawn nominal gaussian distribution organized matrix form xnom use nominal data points extract estimate true subspace line data distribution means standard svd method means proposed method fig plot plain training data points xnom lines described xnom qbf xnom reference also plot alongside true line three lines visually coincide xnom qbf xnom excellent estimates true line next assume instead clean matrix nominal training data points xnom given outliercorrupted data matrix xcor xnom estimating qopt addition previous nominal data points xnom xcor also contains outliers seen fig follow nominal distribution similar treatment xnom calculate xcor xcor qbf xcor respectively fig plot two lines true line nominal data time lines xcor qbf xcor differ significantly sharp contrast xcor strongly attracted four outliers drifting away true line superior performance proposed approximate presence faulty data clearly illustrated classification genomic data prostate cancer diagnosis experiment conduct microrna mirna data prostate cancer diagnosis specifically operate expressions mirnas recently considered differentially expressed malignant normal human prostate tissues mirnas obtain one sample expression one malignant one normal prostate tissue patients patient indexes per swedish watchful waiting cohort mirna expressions malignant normal tissues organized respectively experiment conducted follows collect ntrain points ntrain ntrain points ntrain calculate malignant normal tissue training datasets ntrain ntrain ntrain ntrain respectively next find malignant normal datasets respectively conduct used training every malignant normal bias term run classification experiment distinct configurations calculate receiver operating characteristic roc curve developed classifier identifying detection event classifying correctly malignant tissue false alarm event classifying erroneously normal tissue fig curves information studied data found next consider event training data recalculate roc curves exchanged data points fig plot roc curves designed malignant tissue detectors observe mislabeling two classifiers perform almost identically attaining frequency detection close frequency false alarm ffa less however event training data mislabeling strong resistance proposed dataset contamination apparent classifier outperforms significantly every examined value instance achieves ffa achieves ffa onclusions work presented novel based algorithm named calculate data matrix complexity dmin formal proof convergence theoretical analysis converging points detailed asymptotic complexity derivation carried numerical experiments show proposed algorithm outperforms optimization metric calculators cost comparable almost indistinguishable behavior clean nominal data shows remarkable relative resistance faulty data contamination thus proposed algorithm retaining robustness cost close may bridge gap outlier resistant computationally efficient analysis ppendix proof proposition since bbf imply kybbf kyn addition proposition bbf lies well satisfies bbf sgn ybbf therefore qbf xbbf kxbbf qbf xbbf kxbbf sgn xbbf xbbf kxbbf sgn ybbf xbbf kxbbf kxbbf kybbf kxkf eferences markopoulos karystinos pados options signal processing proc intern symp wireless commun syst iswcs ilmenau germany markopoulos karystinos pados optimal algorithms signal processing ieee trans signal vol kundu markopoulos pados fast computation component data proc ieee int conf speech signal process icassp florence italy may pearson lines planes closest fit systems points space philosophical vol jolliffe principal component analysis new york bishop pattern recognition machine learning new york springer duda hart stork pattern classification new york wiley barnett lewis outliers statistical data new york john wiley sons tukey future data analysis ann math wright robust principal component analysis acm vol art may singleton method minimizing sum absolute values deviations ann math vol karst linear curve fitting using least deviations american stat vol mar barrodale approximation analysis data royal stat app vol barrodale roberts improved algorithm discrete linear approximation siam num vol kanade robust subspace computation using norm internal technical report computer science carnegie mellon pittsburgh usa kanade robust norm factorization presence outliers missing data alternative convex programming proc ieee conf comp vision patt recog cvpr san diego jun brooks hyperplane problem appl math vol brooks boone pure principal component analysis comput stat data vol may kwak principal component analysis based maximization ieee trans patt anal mach vol kwak feature extraction classification problems enhancements biased discriminant analysis patt vol nie huang ding luo wang robust principal component analysis maximization proc int joint conf artif intell ijcai barcelona spain jul mccoy tropp two proposals robust pca using semidefinite programming electron vol jun eriksson hengel efficient computation robust matrix approximations presence missing data using norm proc ieee conf comput vision patt recog cvpr san francisco usa jun zheng kong robust principal component analysis based maximum correntropy criterion ieee trans image vol jun zhang ding efficient algorithm principal component analysis proc ieee int conf acoust speech signal process icassp kyoto japan mar pang yuan ieee trans man cybern vol pang yuan robust tensor analysis ieee trans circuits syst video vol ding zhou zha rotational invariant principal component analysis robust subspace factorization proc int conf mach pittsburgh markopoulos kundu pados robust image recovery severely corrupted copies proc ieee int conf image process icip quebec city canada funatsu kuroki fast parallel processing using gpu computing bases proc ieee tencon fukuoka japan meng zhao improve robustness sparse pca maximization patt vol wang tang zheng common spatial patterns ieee trans biomed vol mar wang block principal component analysis image analysis patt recog vol apr johnson savakis fast robust face recognition proc ieee west new york image signal process workshop wnyispw rochester markopoulos filtering subspaces proc ieee sensor array multichan signal process workshop sam rio janeiro brazil jul markopoulos tsagkarakis pados karystinos direction finding subspaces proc comp sens spie security sens dss baltimore may tsagkarakis markopoulos pados direction finding complex component analysis proc ieee int workshop signal process adv wireless commun spawc stockholm sweden jun nesterov nemirovski algorithms norm minimization acta numerica vol may luo zhang semidefinite relaxation quadratic optimization problems ieee signal process vol may golub modified matrix eigenvalue problems siam vol getz miska lamb peck cordero ebert mak ferrando downing jacks horvitz golub microrna expression profiles classify human cancers nature vol jun gaur jewell liang ridzon moore chen ambros israel characterization microrna expression levels biological correlates human cancer cell lines cancer vol mar carlsson davidsson helenius karlsson lubovac olsson mirna expression signature separates normal malignant prostate tissues cancer cell vol may johansson andersson dickman holmberg magnuson adami natural history early localized prostate cancer amer med vol jun jain duin mao statistical pattern recognition review ieee trans pattern anal mach vol liu classifier distance nearest subspace ieee trans neural vol calculation first principal component input csvd bbf sgn qbf xbbf kxbbf output qbf function length true terminate bfs elseif else break return fig proposed algorithm calculation component data matrix samples dimension csvd returns compact svd argument average cardinality number data points fig average cardinality set convergence set set optimal points versus number data points kxt kxbopt kxt kxb kxkf iteration index fig binary quadratic metric metric per iteration plot along upper bound line kxbopt lower bound line kxkf proposition empirical cdf random initialization initialization number convergence fig empirical cdf number needed convergence random initialization experiments table computational cost calculation principal component real data matrix computational task computational cost min one bit flip iterations qbf bbf total min calculation principal components input data matrix rank csvd sgn bbf bfk svd xbbf qbf output qbf function bfk kky true terminate bfs svd mod evd kym diag fevd fevd diag argmaxm elseif else break return fig proposed algorithm calculation components data matrix samples dimension csvd returns compact svd argument fevd returns evd argument diagonal positive semidefinite matrix perturbated symmetric algorithm table computational complexity calculation principal components real data matrix rank computational task computational cost dmin one bit flip iterations qbf bbf dmin total dmin empirical cdf proposed sdp iterations alternating optimization performance degradation ratio empirical cdf proposed sdp iterations alternating optimization performance degradation ratio fig empirical cdf qbf qfp qsdp initializations empirical cdf proposed sdp iterations alternating optimization performance degradation ratio empirical cdf proposed sdp iterations alternating optimization performance degradation ratio fig empirical cdf qao qfp qsdp qbf initializations nominal training data svd proposed true line nomining training data outliers svd proposed true line fig trained clean data matrix nominal data points outlier corrupted data matrix xcor nominal data points plus outliers frequency detection proposed frequency false alarm fig roc curve malignant tissue detector mislabeled points per training dataset ntrain
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sep struclus structural clustering graph databases till petra mutzel dortmund university dept computer science dortmund germany email dortmund university dept computer science dortmund germany email present structural clustering algorithm datasets small labeled graphs utilizing frequent subgraph sampling strategy set representatives provides intuitive description cluster supports clustering process helps interpret clustering results nature clustering approach allows bypass dimensionality feature extraction problems arise context graph datasets reduced pairwise distances feature vectors achieving high quality human interpretable clusterings runtime algorithm grows linearly number graphs furthermore approach easy parallelize therefore suitable large datasets extensive experimental evaluation synthetic real world datasets demonstrates superiority approach existing structural subspace clustering algorithms runtime quality point view ntroduction molecules protein interaction networks xml documents social media interactions image segments common modeled labeled graphs ability represent topological semantic information causes graphs among versatile data structures computer science age big data huge amounts graph data collected demand analyze increases collection focus special case clustering large sets small labeled graphs main motivation stems need cluster molecular databases drug discovery synthetically constructed databases contain billion molecules however presented approach limited use case clustering techniques aim find homogeneous subsets set objects classical approaches interpret objects directly abstract utilizing intermediate representation feature vectors pairwise distances abstraction pairwise distances beneficial terms generality disadvantageous case intrinsic high dimensional datasets case concentration effect may cause pairwise distances loose relative contrast distances converge towards https https http common value concentration effect closely related bad clusterability sets graphs usually clustered transforming graphs feature vectors using graph theoretic similarity measures typical feature extraction methods graphs counting graphlets small subgraphs counting walks using eigenvectors adjacency matrices spectral graph theory enumeration subgraphs considered intractable even graphs moderate size exist exponentially many subgraphs wrt graph size many efficient clustering algorithms proposed vector space therefore transformation feature vectors might look beneficial first place however mentioned feature extraction methods tend produce large amount distinct often unrelated features results datasets high intrinsic dimensionality additionally extracted features approximate graph structure implies feature vectors transformed back graph hence interpretability clustering algorithms perform vector modifications calculating centroids limited besides utilization various feature extraction techniques possible compare graphs directly using graph theoretic distances maximum common subgraph derived distances graph edit distance computation previously mentioned graph theoretic distances shown application results datasets high intrinsic dimensionality well high quality clustering methods arbitrary metric high dimensional datasets furthermore require superlinear number exact distance computations factors render graph theoretic distance measures combination generic clustering algorithms infeasible datasets subspace projected clustering methods tackle high dimensional datasets identifying subspaces well separated clusters exist however generic subspace algorithms come high runtime burden often limited euclidean vector space structural projection clustering algorithm approaches dimensionality problems explicitly selecting cluster representatives form common subgraphs consider feature mapping binary feature vectors contain one feature subgraph found complete dataset graph feature vector binary entries encoding presence associated substructure graph selecting common subgraph cluster representative another way selecting subspace feature vectors consisting features associated subgraphs main contributions paper present novel structural projection clustering algorithm datasets small labeled graphs scales linearly dataset size set representatives provides intuitive description cluster supports clustering process helps interpret clustering results knowledge first approach actively selecting representative sets cluster based new ranking function candidates representatives constructed using frequent subgraph sampling order speed computation suggest new error bounded sampling strategy support counting context frequent subgraph mining experimental evaluation shows new approach outperforms competitors runtime quality paper structured follows section provides overview related clustering algorithms basic definitions given section iii section presents main algorithm runtime analysis experimental evaluation compare new algorithm scap proclus kernel presented section elated ork several clustering algorithms graph molecule data proposed last years tsuda kudo presented algorithm using binomial mixture model high dimensional binary vectors indicating presence frequent substructures two years later tsuda kurihara presented similar graph clustering algorithm using dirichlet process mixture model pruning set frequent substructures achieve smaller feature vectors like graph clustering algorithm presented ferrer maps graph euclidean space utilizing edit distance pivot elements median graph selected based distance euclidean median furthermore parallel greedy overlapping clustering algorithm presented seeland adds graph cluster whenever common substructure minimum size exists however none previously mentioned algorithms suitable large datasets result high computational complexity xproj uses approach selecting enumerated frequent substructures fixed size cluster representatives approach scales well dataset size limited trees generalization graphs would result huge performance degradation furthermore exist hybrid approaches dataset using representation refine results using structural clustering algorithms relevant respect datasets scap algorithm proposed seeland course many subspace algorithms also applicable using feature extraction method giving comprehensive overview subspace techniques scope article however two algorithms special interest work shown frequent pattern mining used feature selection vector space relates xproj approach way selection graph help frequent substructure another way selecting subspace feature space substructures later evaluation compare ourself proclus algorithm fast projected clustering algorithm noise detection selects features minimizing variance algorithm studied intensively performed well various subspace clustering comparisons iii reliminaries undirected labeled graph consists finite set vertices finite set edges labeling function finite set labels used short term path length sequence vertices let two undirected labeled graphs label preserving subgraph isomorphism injection iff exists subgraph isomorphism say supported subgraph supergraph write exists subgraph isomorphism two graphs isomorphic common subgraph graph subgraph isomorphic furthermore support sup graph set graphs fraction graphs support said frequent iff support larger equal minimum support threshold minsup frequent subgraph maximal iff exists frequent supergraph set graphs write set frequent subgraphs set maximal frequent subgraphs clustering graph dataset partition cluster consists set graphs linked set cluster representatives undirected labeled graphs please note consider graph dataset distinct object result possible isomorphic graphs single set struclus lgorithm high level description struclus algorithm given algorithm initially partitions dataset using lightweight algorithm afterwards clustering refined using optimization loop similar kmeans algorithm order fit number clusters dataset structure achieve good cluster separation homogeneity see sections use cluster splitting merging strategy iteration algorithm struclus algorithm apply section convergent section split clusters section merge clusters section update representatives section assign graphs closest cluster section end sup sup important ingredient algorithm set representatives cluster representatives serve intuitive description cluster define substructures intra cluster similarity measured set chosen every graph exists least one representative subgraph isomorphic supported exception single noise cluster following invariant holds iteration otherwise removed extension possible without violating minimum support threshold maximal frequent subgraph found process justified monotonicity property subgraphs graphs representative set cluster constructed using maximal frequent subgraphs see section representative set instead single representative advantage graphs composed multiple common substructures represented order meaningful human interpretable cardinality limited user defined value rmax figure shows two example clusters representatives real world molecular dataset generated struclus stochastic representative mining construct representative set cluster using maximal frequent subgraphs since set may exponential size wrt maximal graph size restrict subset candidate representatives using randomized maximal frequent connected subgraph sampling technique origami combined new stochastic sampling strategy support counting second step final representative set selected using ranking function see section origami constructs maximal frequent connected subgraph set graphs extending random frequent vertex frequent paths length one leading graph first step frequent vertices frequent paths length one enumerated single scan extension random vertex chosen random label preserving path connected forward creating new vertex backward connecting two existing vertices fashion extension support sup evaluated solving subgraph isomorphism test graphs sup minsup extension permanently added origami greatly improves performance comparison enumeration algorithms subgraph isomorphism tests extension remain major performance bottleneck struclus reason added stochastic sampling strategy support counting initially draw random sample sup estimator parameter binomial distribution sup true probability underlying bernoulli distribution interested true value sup smaller minimum support threshold without loss generality let focus case minsup following take advantage binomial test null hypothesis minsup thereby determine probability error assume sup minsup predefined significance level decide sample gives enough confidence justify assumption discard null hypothesis continue doubling sample size repeat process extreme case therefore calculate exact value sup statistical test repeated extension sample size doubling consequence multiple hypothesis testing correction necessary bound real error maximal frequent substructure proposition let set undirected labeled graphs minimal sample size set frequent paths length one minsup minimum support threshold vmax minsup sorted increasing order graph sizes graph maximal number binomial tests construct maximal frequent substructure bounded vmax vmax log times proof sample size doubled log min test never reaches desired significance level size bounded size supporting graph worst case supported largest graphs graph size smallest supporting graph equal minsup sorted graph sizes increasing order number backward extensions bounded number vertex pairs times number applicable extensions additionally need check forward extensions vertex conclude maximal finally value proposition able apply bonferroni correction significance level afford relative high error selection fig two real world clusters generated struclus grey boxes show cluster representatives final representatives filter bad candidates however used significance level influence runtime one hand high error leads many bad candidates need increase number candidates mine hand low error lead larger sample sizes reject null hypothesis maximal error turned good choice experimental evaluation update representatives representative selection role cluster description good representative explains large portion cluster accordingly supported large fraction cover large fraction vertices edges graph supporting define coverage graph representative cov two criteria closely related cluster homogeneity uniform cluster cluster contains isomorphic graphs achieve optimal values criteria vice versa monotonicity property implies values inhomogeneous clusters least one two criteria homogeneous clusters desired use product two criteria ranking function order discriminate clusters cluster representative cluster specific well thus support rest dataset low reason use following ranking function dataset cluster representative sup sup rank finally select rmax highest ranked sampled subgraphs cluster representatives balancing cluster homogeneity besides representative selection choice minimum support threshold representative mining influence cluster homogeneity fraction unsupported graphs cluster updating cluster representatives bounded minsup bound clearly relates criteria representative ranking see section however unsupported graphs removed cluster assigned different cluster end current iteration see section due monotonicity property process sorting graphs choosing minimum support threshold increase size representatives therefore coverage value criteria decrease minimum support threshold lead increase size representatives subsequently process also reduce cluster cardinality therefore increasing homogeneity optimal value result clustering uniform singleton clusters clearly desired behavior get around aim towards similar homogeneity clusters choose minimum support threshold cluster specific however choosing fixed homogeneity level priori easy task appropriate value depends dataset therefore calculate average coverage score clusters use baseline adjustment ease computation choose slightly simplified coverage approximation acov relcov acov acov finally define linear mapping relative coverage relcov cluster specific minimum support threshold help two predefined tuples parameter denotes lowest highest support value relative coverage value mapped lowest highest minimum support relcov relcov minsup relcov otherwise result minimum support threshold clusters stopping process sorting graphs clustering balanced set values parameters close equal cluster assignment graph dataset assigned similar cluster assignment phase measure similarity summing squared sizes representatives cluster subgraph isomorphic choice similarity justified representative ranking criteria square representative sizes prefer high coverage high number representatives subgraph isomorphic assigned graph mentioned section possible graph supported representative cluster updating representatives situation create single noise cluster graphs clusters supported representative collected minimum support threshold bounded fixed value see section number representatives limited rmax guaranteed find appropriate set representatives likely largely inhomogeneous noise cluster therefore excluded invariant problem finding subgraph isomorphic graphs graph database also known subgraph search problem extensively studied past apply fingerprint technique emerged research topic speed assignment phase enumerates trees circles specified size given graph hashes presence subgraphs binary fingerprint fixed length fingerprint graph bit set unset fingerprint graph conclude subgraph isomorphism exists contains subgraph present calculate fingerprint graph representative perform subgraph isomorphism test assignment phase fingerprint comparison rule presence subgraph isomorphism cluster splitting merging without operation cluster splitting mentioned process creating noise clusters would create one extra cluster iteration main loop large difference initial final number clusters therefore would lead slow convergence struclus algorithm towards final result mentioned also possible representative found noise cluster therefore process sorting graphs noise cluster increase homogeneity stopped completely worst case similar situation occur regular clusters example cluster composed uniform sets graphs require minimum support threshold less equal mint sort smallest possible number graphs reason cluster splitting step necessary see algorithm step clusters relative coverage value priori specified threshold relcovmin merged single set graphs algorithm applied resulting clusters added back clustering contrary cluster splitting focuses cluster homogeneity cluster merging ensures minimum separation clusters separation measured different levels many classical measures define separation minimum distance two cluster elements however type definition suitable projected clustering algorithms comparison take cluster specific subspace account mentioned introduction cluster representatives struclus define subspace cluster additionally serve description graphs inside cluster high coverage value leads accurate cluster description thus define separation two clusters solely representatives sets definition also beneficial runtime perspective separation calculation independent cluster size without cluster merging possible clusters similar representatives exist although ensure initial clusters dissimilar representatives see section may happen two clusters converge towards newly formed clusters similar already existing one therefore merge two clusters whenever representatives similar compare two single representatives calculate size maximum common subgraph mcs use relative size similarity sim max maximum representatives sizes chosen denominator support different clusters subgraph isomorphic representatives differ largely size finally merge two clusters following condition holds sim sim min sim num sim num minimum number representative pairs similarity greater equal sim min note calculated mcs two representatives supported graphs support either subgraph isomorphism relation transitive coverage graphs merged cluster furthermore bounded max cov cov sim reuse mcs representative reason recommend set sim num close number representatives per cluster support large fraction graphs merged cluster parameter sim min furthermore intuitive knob adjust granularity clustering finally representatives merged clusters updated use calculated mcss representatives better representatives may exist decision problem mcs larger threshold exist computationally less demanding calculating mcs serves initial partitioning dataset random partitioning graphs would problematic representatives may found partitions found representatives likely cluster specific result high number clusters merged inhomogeneous clusters slow convergence struclus algorithm dataset compute maximal frequent subgraphs described section fixed minimum support frequent subgraphs serve representative candidates initial clusters avoid similar representatives first greedily construct maximal sets dissimilar graphs measure similarity reuse similarity threshold sim min cluster merging words picking graphs random order add dissimilar set sim sim min process repeated several times largest set dmax used create one cluster single representative afterwards run regular assignment phase described section result sim min expect well separated cluster cluster merging necessary first iteration main loop excluding noise cluster convergence struclus optimizes cluster homogeneity maintaining minimal cluster separation described section homogeneity defined two criteria wrt cluster representatives exception noise cluster representative support criteria dismissed represented graphs sorted cluster discussed optimal homogeneity achieved singleton clusters however later introduced cluster separation constraint limit granularity clustering two similar representatives merged nevertheless might exist several clusterings different granularity respect separation constraint reason objective function balances coverage criteria homogeneity granularity clustering parameter adjusts granularity acov consequence cluster splitting merging objective function fluctuate contain local optima therefore smooth objective function let clustering iteration algorithm terminate first iteration following condition holds averaging width minimum relative increase objective function iterations runtime analysis subgraph isomorphism problem maximum common subgraph problem even decision variants struclus solves problems order calculate support decide clusters need merged furthermore size representative linear sizes supporting graphs thus struclus scales exponentially wrt size graphs dataset nevertheless problems solved sufficiently fast small graphs hundred vertices edges molecular structures therefore consider graph size constant following analysis focus scalability wrt dataset size let vmax maximal number vertices graph cmax maximal number clusters clustering process representative mining described proposition number extensions mine single maximal frequent subgraph set graphs bounded vmax vmax variable vmax obviously constant also bounded vmax lowest minimum support threshold fraction dlsv max edges must isomorphic frequent thus vmax exist frequent paths length one note number distinct labels influence theoretical bound mined maximal frequent subgraph need calculate support involves many subgraph isomorphism tests therefore runtime mine single maximal frequent subgraph bounded representative update updating representatives involves two steps representative mining representative selection representative mining includes calculation cluster specific minimum support help relcov relcov values cluster calculated cmax maintain sum graph sizes cluster update value cluster assignment since cluster sizes sum runtime actual frequent subgraph mining rank representative candidate set value sup value sup calculated single scan calculating whether subgraph isomorphic graph number candidates per cluster constant runtime ranking cmax also overall runtime representative update parts runtimes cluster assignment cmax cluster splitting putable cmax cluster merging cmax converge cmax time overall runtime single iteration algorithm runtime cmax hence linear justified cmax let observation cmax number iterations algorithm terminates overall runtime cmax valuation following evaluation compare struclus scap proclus kernel wrt runtime clustering quality furthermore evaluate influence sampling strategy support counting evaluate parallel scaling implementation hardware software tests performed dual socket numa system intel xeon gib ram applications pinned single numa domain cores gib ram eliminate random memory effects turbo boost deactivated minimize external runtime influences ubuntu linux used operating system java implementations struclus proclus kernel running oracle java hotspot scap compiled gcc optimization level struclus scap shared memory parallelized test setup evaluation measures tests repeated times runtime hours times otherwise quality measured ground truth comparisons use normalized variation information nvi measures struclus proclus kernel nvi established quality measures suited overlapping clusterings produces scap thus purity used comparison scap datasets evaluate algorithms synthetic datasets different sizes three real world datasets synthetic datasets vertices edges average vertex edge labels weights drawn exponential distribution contain clusters random noise graphs cluster seed patterns randomly generated poisson distributed number vertices mean edge probability additionally created common seed pool noise seeds cluster specific graphs generated connecting cluster specific seeds common noise seeds first real world dataset anchorquery molecular database molecule result chemical reaction multiple purchasable building blocks used reaction types class labels second real world dataset heterocyclic similar first one contains heterocyclic compounds distinct reaction types third real world dataset chemdb contains million molecules chemdb chemdb collection purchasable molecules chemical vendors dataset ground truth mainly serve proof able cluster large real world datasets table shows additional statistics real world datasets cluster datasets proclus graph data transformed vectors counting distinct subgraphs size resulting features synthetic datasets application anchorquery heterocyclic datasets results much lower feature counts vectors used features space representation kernel additionally evaluated various graph kernels explicit feature mapping proclus kernel shortest path subtree fixed length random walk kernels however observed feature vectors even higher dimensionality clustering results lower quality reason results graph kernels omitted algorithm configuration parameters struclus set follows maximal error support counting number representative candidates minimum support threshold calculated splitting threshold acov minimum separation sim min sim num convergence determined scap two parameters minimum size common substructure must present single cluster parameter influences granularity set mean seed size synthetic dataset otherwise set highest number results reasonable clustering quality clustering process faster fine grained find clusters boundary synthetic dataset proclus two parameters well number clusters average dimensionality cluster subspaces set number clusters ground truth chemdb dataset ground truth large proclus set got best quality results number clusters http table eal world dataset statistics umulated values given triple min max average dataset size anchorquery heterocyclic chemdb classes vertices edges vertex labels edge labels scaling parameter kernel set exact manner proclus please note priory selection optimal value gives proclus kernel advantage competitors evaluation results shown table struclus outperforms competitors terms quality synthetic datasets differences corresponding mean quality values always larger small multiples standard deviations small synthetic datasets scap fastest algorithm changes larger data set sizes struclus outperforms scap factor size furthermore unable cluster largest dataset scap less days proclus kernel slowest algorithms huge gap next competitor sublinear growth struclus runtime smaller datasets sizes caused sampling strategy support counting running struclus exact support counting yields linear growth runtime important mention clustering quality significantly influenced support counting strategy implementation struclus scales well number cores shown fig cores get speedup including cores get speedup optimal scaling scaling struclus cores fig parallel scaling synthetic dataset size evaluation results wrt real world datasets summarized table iii unable cluster anchorquery chemdb datasets proclus kernel kmeans scap even high values parameter less days chemical reaction datasets struclus also needs time compared synthetic dataset similar size runtime increase partially explained larger graph sizes however hidden parameters datasets size maximal common substructures influence runtime well struclus runtime outliers heterocyclic dataset caused small temporary clusters large vertices representatives high runtime scap algorithm anchorquery dataset bit surprising common substructures processed scap limited size maximum vertices consider larger frequent pattern search space reason runtime increase kernel surprisingly slow heterocyclic dataset took hours single run therefore created random subset size graphs struclus always outperforms competitors wrt quality scores anchorquery dataset struclus created clusters average high score purity measure shows struclus splitted real clusters keeps well inter cluster separation chemdb clustered struclus hours clusters consequence high runtime chemdb measurement lack competitors repeated test times acov value final clustering average highlights ability struclus cluster datasets onclusion paper presented new structural clustering algorithm large scale datasets small labeled graphs help explicitly selected cluster representatives able achieve linear runtime worst case wrt dataset size novel support counting sampling strategy multiple hypothesis testing correction able accelerate algorithm significantly without lowering clustering quality furthermore shown cluster homogeneity balanced dynamic minimum support strategy representative mining cluster merging splitting step introduced achieve well separated clustering even high dimensional pattern space experimental evaluation shown new approach outperforms competitors wrt clustering quality attaining significantly lower runtimes large scale datasets although shown struclus greatly improves clustering performance compared competitors datasets several billion molecules still outside scope work reason consider development distributed variant algorithm future work another consideration improve quality algorithm integrate discriminative frequent subgraph miner representative mining integration discriminative property mining process advantage higher table esults synthetic datasets runtimes given hours est results marked bold coefficient variation value given maximum column values averaged uality measures ormalized variation nformation nvi owlkes allows ndex urity size runtime struclus nvi purity struclus exact support runtime nvi runtime scap purity runtime proclus nvi kernel runtime nvi table iii esults real world datasets runtimes given hours est results marked bold coefficient variation value given maximum column values averaged hem measurement repeated times calculated otherwise meaningless esults ernel eans given random subset eterocylcic dataset uality measures ormalized variation nformation nvi owlkes allows ndex urity dataset runtime anchorquery heterocyclic chemdb struclus nvi purity scap runtime purity runtime subset quality representative candidates mined result lower number necessary candidate patterns positive effect runtime furthermore allows mine highly discriminant subgraphs however extend support counting sampling strategy miners additionally discriminative scores usually subgraph lattice imposes another runtime burden acknowledgment work supported german research foundation dfg priority programme algorithms big data spp would like thank nils kriege providing fast subgraph isomorphism madeleine seeland andreas karwath stefan kramer providing scap implementation eferences kalinski umkehrer weber kolb burdack ross industrial applications mcrs molecular diversity drug discovery generic drug synthesis english molecular diversity vol navarro probabilistic spell curse dimensionality algorithm engineering experimentation third international workshop alenex washington usa january revised papers buchsbaum snoeyink ser lecture 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sound compilation reals eva darulova viktor kuncak epfl epfl sep abstract paper advocate natural ambitious alternative source code programs expressed terms ideal mathematical real numbers system programmer writes program using real data type states desired postconditions specifies uncertainties explicitly postconditions well desired target precision return values trustworthy compiler check taking account uncertainties propagation desired precision soundly realized finite precision implemenation compiler chooses emits one implementation following model derive verification conditions encode reasoning roundoff errors reasoning real numbers verification conditions explicitly separate ideal program without external uncertainties roundoffs actual program actually executed finite precision possibly noisy inputs additionally constraints fully parametric precision thus used different hardware configurations well used determining minimum precision needed summarize view source code functions real numbers several advantages writing accurate numerical software hard many sources unavoidable uncertainties including finite numerical precision implementations present programming model user writes program implementation specification language explicitly includes different types uncertainties present compilation algorithm generates conventional implementation guaranteed meet desired precision respect real numbers verification step generates verification conditions treat different uncertainties unified way encode reasoning roundoff errors reasoning real numbers verification conditions used standardized format verifying precision correctness numerical programs due often nature precise reasoning verification conditions remains difficult show current art smt solvers scale well solving verification conditions propose new procedure combines exact smt solving reals approximate sound affine interval arithmetic show approach overcomes scalability limitations smt solvers providing improved precision affine interval arithmetic using initial implementation show usefullness effectiveness approach several examples including containing computation programmers part reason real arithmetic arithmetic thus achieve separation design algorithms may still approximate realization using finite precision computations verify ideal meaning programs using techniques introduction developed reason real numbers scalable techniques directly deal floating point arithmetic writing numerical programs difficult part programmer needs deal correctness algorithm also unavoidable uncertainties program inputs may exactly known come physical experiments measured embedded sensor computation suffers roundoff errors step use arithmetic addition resources like energy may scarce certain number bits available numerical datatype time computed results consequences used control example vehicle nuclear power plant therefore great importance improve confidence numerical code one first challenges automated reasoning tools work real arithmetic whereas code implemented typically floating point arithmetic many current approaches verifying numerical programs start floatingpoint implementation try verify absense runtime errors however absence errors guarantee program behavior matches desired specification expressed using real numbers fundamentally source code semantics expressed terms data types floating points problematic compiler optimizations associativity law unsound respect source code semantics approach allows quantify deviation tation outputs ideal ones instead merely proving range bounds variables used simpler static analyses compiler reals free optimizations long preserve precision requirements allows compiler apply example associativity arithmetic even select different approximation schemes transcendental functions addition roundoff errors approach also allows developer quantify program behavior face external uncertainties input measurement errors using verification condition generation approach correctness precision compilation small programs directly verified using smt solver see section capability likely continue improve solvers advance however complexity generated verification conditions larger programs still reach solvers believe specialized techniques continue necessary task paper presents two specialized techniques improve feasibility verification task first technique performs local approximation effective different selection numerical algorithms support programming larger code fragments system also supports modular verification technique handles functions inlining function bodies using postconditions thus expect technique applicable larger code bases well possibly refactoring code multiple smaller annotated functions even benchmarks release aware available system would provide guarantees level automation even benchmarks containing nonlinear arithmetic second technique specifically handles conditional expressions solving constraints nonlinear arithmetic poses significant challenge verification directly handled using algorithms embedded inside smt solvers although interesting relevant fragments decidable supported solvers complexity much higher unfortunately arithmetic ubiquitous numerical software furthermore verification conditions add roundoff errors arithmetic expressions resulting constraints grow complexity often becoming reach solvers alternative encoding smt solver input use sound overapproximating arithmetic model interval affine arithmetic however used code approaches yield pessimistic results useful show combine range arithmetic computation smt solving overcome limitations individual techniques applied arithmetic obtain sound precise somewhat scalable procedure range computation technique also checks common problems overflow division zero square root negative number emitting corresponding warnings additionally procedure forward computation suitable automatically generating function summaries containing output ranges errors function point view logical encoding problem range arithmetic becomes specialized method perform approximate quantifier elimination bounded variables describe uncertainty present implementation specification guage numerical programs uncertainties define semantics terms verification constraints induce believe verification conditions used standardized format verifying precision correctness numerical programs develop approximation procedure computing precise range error bounds nonlinear expressions combines smt solving interval arithmetic show approach significantly improves computed range error bounds compared standard interval arithmetic scales better smt solving alone procedure also used independently precise alternative interval arithmetic thus perform forward computation without desired postconditions describe approach soundly computing error bounds presence branches uncertainties ensures soundness compilation case function defined program known continuous sound compilation conditional branches presence uncertainties conditional braches become another verification challenge challenge ideal execution may follow one branch input roundoff errors actual execution follows another behaviour may acceptable however show error output remains within required bounds therefore approach would directly benefit automated analysis continuity advocated previously continuous functions analysis done separately program path absence knowledge continuity present new method check different paths taken floating point versions program still preserves desired precision specification check require continuity becomes difficult prove code instead directly checks difference two values different branches meets required precision technique extends method handling nonlinear arithmetic benefits combination range arithmetic smt solving summary contributions overall contribution approach sound compilation real numbers specifically implemented framework report experience set diverse benchmarks including benchmarks physics biology chemistry control systems results show technique effective achieves synergy techniques relies example demonstrate aspects system example figure methods triangle trianglesorted compute area triangle side lengths notation short consider particular application user may two side lengths given may vary third two functions available computation wants determine whether either satisfy precision requirement line tool determines requirement needs least double floating point precision challenge establish precision sufficient ensure bounds given errors floating code accumulate grow without priori bound tool verifies fully automatically method trianglesorted indeed satisfies postcondition generates source code double datatype also includes precise complete postcondition implementation evaluation implemented compilation verification procedure including verification condition generation analysis possibly expressions handling conditionals system implemented extension verifier functional scala programs implementation relies range arithmetic implementation scala well smt solver evaluated system number diverse benchmarks obtaining promising results releasing benchmarks making available supplementary material reviewers hope benchmarks used compare future tools error quantification help development nonlinear solvers also present challenges aggressive compilation schemes different number representations res res res achieve result tool first checks precondition function call satisfied using solver inlines body function trianglesorted computes sound bound result uncertainty approximation procedure uses show postcondition satisfied error user annotates postconditions explicitly talk uncertainties real represents ideal real numbers without uncertainty allow arithmetic expressions reals standard arithmetic operators together conditionals function calls form body methods tool also supports immutable variable declarations val language allows user define computation real numbers note specification language executable precondition allows user provide specification environment complete environment specification consists lower upper bounds method parameters upper bound uncertainty noise range bounds expressed regular comparison operators uncertainty expressed predicate denotes variable known alternatively programmer specify relative error noise except roundoff present roundoff errors automatically added input variables postcondition specify range maximum uncertainty accepted output addition language allowed precondition postcondition may reference errors inputs directly following way res says maximum acceptable error output variable res bounded times initial error whereas precondition may talk ideal values postcondition also reference actual value directly via allows assert runtime values exceed certain range instance def mainfunction real real real real require val val val area trianglesorted area triangleunstable area ensuring res res def triangle real real real real require val sqrt def trianglesorted real real real real require sqrt else sqrt figure computing area triangle arithmetic tool technique support generated target code precision particular single double precision defined ieee standard also straightforward combine techniques generate arithmetic similarly relies knowing ranges variables assume rounding mode basic arithmetic ations rounded correctly means result operation must closest representable number hence provided overflow result binary operation satisfies tation takes account sound way input uncertainty initial roundoff error inputs propagation roundoff errors commited arithmetic operation additionally due initial roundoff error comparison line exact computations take different branch corresponding computation precisely total error computing condition computed tool values satisfy may take else branch even though corresponding real values would follow branch similarly opposite direction tool verifies difference computed result two branches due divergence real arithmetic floating point arithmetic code remains within precision requirement finally tool uses novel range computation procedure also compute precise output range could obtained interval arithmetic includes complete postcondition generated code fact interval arithmetic alone computes ranges using methods triangle trianglesorted respectively seemingly suggesting two methods perform entirely different computations technique tool computes range methods shows difference absolute error computation method triangle verification fails desired computed error exceeds required precision bound result textbook formula triangles known suffer imprecision flat triangles somewhat rectified method trianglesorted ideal operation real numbers machine epsilon determines upper bound relative error model provides basis roundoff error estimates compiling reals finite precision given specification program reals possible target precision tool generates code numbers satisfy given postconditions figure presents view compilation algorithm tool first analyses entire specification generates one verification condition postcondition proven obtain modular algorithm tool also generates verification conditions check function call precondition called function satisfied methods sorted occuring function calls allows already computed postconditions function calls modular analysis user specified target precision remaining part compilation process perfomed respect precision case user specified several possible precisions tool iteratively select next precise precision attempt compile entire program compilation fails due unsatisfied assertions next precision selected thus one task algorithm programs reals program compiled consists one object methods written functional subset scala programming language methods functions real datatype errx errx errx errx errx sqrt sqrt sqrt val else fresh denote functions roundoff errors step input spec specification reals prec candidate precisions fnc generatevcs fnc prec notproven precision fnc notproven approx getnextapproximation precision approx generatespec fnc generatecode spec output code generated postconditions figure compilation algorithm table semantics specification language automatically determine necessary least precision ensure specification met currently precision iteration computed separately big issue due small number alternative targets identify certain shared computations iterations exploit future efficient compilation order compilation process succeed specification met respect given precision thus principal part algorithm spent verification describe section envision future compilation task also include automatic code optimizations paper concentrate challenging groundwork verifying precision code tool currently generates scala code single double arithmetic precision implemented software provide scala interface generated code case verification part compilation fails nonetheless generate code together failure report best postconditions tool able compute user use generated specifications gain insight program satify requirements techniques restricted precisions however work precision follows abstraction given equation furthermore implemented tool accept specifications domain specific language embedded scala generate code scala techniques apply equally programming languages hardware follow abstraction assume equation res denote input output local variables respectively table summarizes verification constraints generated specification language variable specification corresponds two variables ideal one absence uncertainties roundoff errors actual one computed compiled program note ideal actual variables related error bounds postconditions allows ideal actual executions take different paths method body take account roundoff errors arithmetic operations propagation existing errors system currently supports operations principle extended elementary functions instance encoding via taylor expansions note resulting verification conditions parametric machine epsilon order give feedback developers facilitate automatic modular analysis tool also provides automatic specification generation mean programmer still needs provide environment specification form preconditions tool automatically computes precise postcondition formally rewrite constraint res body res res unknown obtain precise postcondition applying quantifier elimination body res eliminate theory arithmetic reals admits theoretically possible use approach currently use full procedure specification generation expensive clear whether returned expressions would suitable format instead use approximation approach computes ranges maximum errors forward fashion allows compute approximation postcondition form res res verifying real programs describe verification part compilation algorithm following call ideal computation computation absence uncertainties implemented real arithmetic actual computation computation finally executed potentially uncertain inputs verification conditions programs method precondition postcondition implies following verification condition res body res res specification generation difficulty simple encoding smt solvers small functions already prove interesting properties using exact encoding problem described discharging verification constraints consider approximation pipeline support inlining postconditions postconditions computed specification generation procedure still precise enough inline entire function body postcondition inlining implemented replacing function call fresh variable constraining postcondition thus verification suceeds inlining postcondition avoid consider path inlined function separately perform modular verification avoiding potential path explosion problem modular verification feasible postconditions imprecise plan explore generation precise postconditions future one step direction allow postconditions parametric initial errors example operator introduced section tool currently supports postcondition inlining postconditions yet generate automatically def getnextapproximation precision getpatherror inlining postcondition inlining full function functions getrange evalwitherror approx error merging approx error range paths arithmetic first approximation figure approximation pipeline ing code programmer may write implement third basic function commonly used signal processing def real real require ensuring res res res res arithmetic arithmetic part verification constraints generated table essentially divided ideal part actual part includes roundoff errors computation step ideal part determines whether ideal range constraints postcondition satisfied actual part determines whether uncertainty part postcondition satisfied use procedure presented section compute sound approximation result range well uncertainty based tool constructs two approximations first ideal part constraint left untouched actual part replaced computed uncertainty bound effectively removes large number variables many times suffiently simplifies constraint succeed fails tool additionally replaces ideal part computed range constraint note second approximation may enough information prove complex postconditions correlation information lost note computation ranges errors approximations thus trying affect efficiency significantly functions correspoding verification conditions complexity already within possibilities nonlinear solver within complex functions however yet provide answer reasonable time returns unknown whether alternative techniques smt solvers help cases remains seen provide approach based approximation addresses difficulty constraint solving verification approximations order nontheless verify interesting programs developed approximation procedure computes sound overapproximation range expression uncertainty output procedure forward computation also use generate specifications automatically describe approximation procedure detail section assume exists given precondition expression expr computes sound bound output range associated uncertainty paths case several paths program option consider path separately merge results join control flow graph introduces tradeoff efficiency precision since one hand considering path separately leads exponential number paths consider hand merging join looses correlation information variables may necessary prove certain properties approximation pipeline chooses merging first resorting verification case failure believe techniques exploring path space could also integrated tool another possible improvement heuristics select different order approximations depending particular characteristics verification condition err evalwitherror expr res res expr res res err identified three possibiities approximation nonlinear arithmetic function calls paths approximated different levels observed experiments one size fit combination different approximations successful proving verification conditions encountered verification condition thus construct approximations able prove one run approximations report verification failed thus view verification stream approximations proven illustrate pipeline computes different approximations figure routines getpatherror getrange evalwitherror described following sections detail first approximation indicated long arrow figure use alone entire constraint constructed rules table indeed approximation function calls treated uninterpreted functions case noted approach works simple cases uncertainties functions present example illustrate verification algorithm example figure functions sinetaylor verified first since contain function calls verification full verification constraint fails next tool computes errors output suceeds prove resulting constraint ideal part untouched approximation tool directly computes new precise postcondition particular narrow resulting error next tool considers comparison function inlining postcondition enough case computing error approximation inlined functions suceeds verifying postcondition able verify portion independently finally tool verifies preconditions function calls satisfied using alone verification function comparisoninvalid fails approximations tool reports unknown also function calls verification constraint contains function calls first approximation failed tool attempt inline postconditions pass resulting constraint sents possible values variables affine forms def comparisonvalid real real require val sinetaylor val ensuring res denotes central value represented interval noise symbol formal variable denoting deviation central value intended range maximum magnitude noise term given corresponding note sign matter isolation however reflect relative dependence values example take def sinetaylor real real require ensuring res res res res subtracted instead resulting interval would width zero range represented affine form computed def real real require ensuring res res res res rad rad general affine operation consists addition subtraction addition constant multiplication constant expanding affine forms get provides counterexample obtains case valid counterexample additional motivation using affine arithmetic different contributions range represents remain least partly separated information used instance help identify major contributor result uncertainty separate contributions external uncertainties roundoff errors soundness procedure sound constraints overapproximate actual errors furthemore even full constraint generated table roundoff errors overapproximated since assume error bound step ensures soundness also introduces incompleteness may fail validate specification overapproximation large implies counterexamples reported general valid disprove ideal part verification constraint general easy distinguish case verification fails due large error bounds prefers simpler counterexamples complex ones thus chooses error variables zero ideal part violated range computation goal procedure perform determine range nonlinear arithmetic expression given ranges inputs two common possibilities interval affine arithmetic tend overapproximate resulting range especially input intervals sufficiently small order affine arithmetic improves interval arithmetic somewhat tracking linear correlations case nonlinear expressions results become actually worse interval arithmetic solving nonlinear constraints observation nonlinear theorem prover one comes decide relatively good precision whether given bound sound check prover whether expression range sound interval enclosure observation basis range computation given overview approximation pipeline describe computation approximation nonlinear arithmetic corresponds last box figure completeness presentation first review interval affine arithmetic common choices performing sound arithmetic computations also use part technique present novel procedure computing output range nonlinear expression given ranges inputs precise substitute interval affine arithmetic finally continue procedure computes sound overapproximation uncertainty result nonlinear expression one possibility perform guaranteed computations use standard interval arithmetic interval arithmetic computes bounding interval basic operation min max rad figure different polynomial approximations sine def comparisoninvalid real real require val sinetaylor val ensuring res counterexample input algorithm nonlinear expression expr precondition inputs specifies among possibly constraints ranges input variables output interval satisfies following getrange expr res expr res res algorithm computing lower bound range given figure computation upper bound symmetric range computed tool first computes initial sound estimate range interval arithmetic performs initial quick check test whether computed first approximation bounds already tight uses first approximation starting point narrows lower upper bounds analogously square root affine arithmetic originally introduced addresses difficulty interval arithmetic handling correlations variables affine arithmetic additional constraints ideal variables uncertainty specification necessary related ideal actual variables idea procedure execute computation keeping track output range current expression associated errors arithmetic operation propagate existing errors compute upper bound roundoff error add overall errors since roundoff error depends proportionally range values also need keep track ranges precisely possible procedure build abstraction computation ideal computation plus minus uncertainty abstraction roundoff errors chose also follows separation def getrange expr precondition precision maxiterations precondition ainit binit evalinterval expr bound expr precision unsat ainit binit numiterations precision numiterations maxiterations mid expr mid match case sat mid case unsat mid case unknown break anew else anew ainit allows treat uncertainties unified manner procedure builds idea smartfloat datatype uses affine arithmetic track range errors nonlinear operations however computed ranges become pessimistic quickly error computation may also suffer imprecision observed since errors tend relatively small imprecision affect error propagation extent initial errors small less one multiplied nonlinear terms tend even smaller whereas affine terms larger one nonlinear terms grow thus concentrate improving ideal range values use novel range computation procedure part leave error propagation affine arithmetic adaptation represent every variable intermediate computation result datatype following components bnew bound symmetrically return anew bnew figure algorithm computing range expression using binary search step binary search tool uses confirm reject newly proposed bound search stops either fails returns unknown query answer within given timeout difference subsequent bounds smaller precision threshold maximum number iterations reached stopping criterion set dynamically range interval err inef orm range range variable computed described section err affine form representing errors overapproximation actual range including uncertainties given totalrange range err err denotes interval represented affine form additional constraints addition input ranges precondition may also contain constraints variables example consider method triangle figure precondition bounds inputs formula useful valid triangles every two sides together longer third get error latest try take square root negative number approaches consider input intervals satisfy constraint values thus check several possibly many cases approach since using check soundness bounds assert additional constraints subsequent checks performed respect additional initial contraints allows avoid interval subdivisions due imprecisions problem specific constraints triangle example becomes especially valuable presence multiple variables may need exponential number subdivisions roundoff error computation computation step roundoff errors computed maxabs totalrange machine epsilon added err fresh noise term note roundoff error computation makes error computation parametric precision error propagation affine operations addition subtraction multiplication constant factor propagated errors computed thus standard affine arithmetic refer reader details describe propagation nonlinear arithmetic multiplication division square root magnitude errors also depends ranges variables since ranges affine terms propagation adjusted following denote range variable associated error affine form err write err mean interval converted affine form multiplication performed affine arithmetic multiplication computed error approximation describe approximation procedure given expression expr precondition inputs computes range error output formally procedure satisfies following err evalwitherror expr res res expr err err res res err err err err err new roundoff error thus first term contributes ideal range remaining three error affine form larger factors larger finally computed represents expression evaluated arithmetic ideal actual variables precondition specifies ranges uncertainties initial variables errors order keep overapproximation small possible evaluate new range computation division computed err err err err err square root first compute affine approximation square root perform affine multiplication term wise overflows nan procedure allows detect potential overflows division zero square root negative value tool computes ranges intermediate values currently report issues warnings user def computepatherror pre errc evalwitherror pre loat errf loat evalwitherror pre errc real getrange pre return max real loat errf loat figure computing error due diverging paths limitations conditional form would yield different results ideal actual computation nearly input allow specification language actual computation commits certain error computing condition branch error causes executions follow different branch corresponding ideal one would consider case ideal computation evaluates actual one evaluates algorithm gives computation path error case idea compute ranges inputs could diverging final error maximum difference value algorithm extends naturally several variables case several paths program error principle computed combination paths use rule infeasible paths front path error computation performed paths actually feasible currently implemented approach tool case use merging handle paths order avoid consider exponential number path combinations also use higher default precision number iterations threshold binary search range computation computation requires general tight intervals path identify two challenges performing computation limitation approach clearly ability check constraints found capabilities satisfactory although expect performance still significantly improve emphasize difference constraints defined table constraints use add errors step thus number variables reduced significantly also found several transformations helpful rewriting powers multiplying products avoiding comparisons precondition although benefits entirely consistent note step error computation tool computes current range thus even fails tighten bound expressions still compute precise bounds interval arithmetic overall cases ranges remaining subexpressions already computed precisely def getpatherror input pre program cond else val computepatherror pre cond val computepatherror pre cond return max conditional statements section consider difference ideal actual computation due uncertainties computing branch conditions resulting different paths taken note full constraint constructed according section automatically includes error recall ideal actual computations independent except initial conditions possible follow different paths program case approximation however compute error individual paths consider error due diverging paths separately method encodes continous function usual mathematical sense note need quantify errors path separately thus method determine whether function conditional statements continuous approach described far sufficient provide sound error bounds case procedure exist fails provide answer example due nonlinearity function simply continuous propose following algorithm explicitly compute difference ideal actual computation across paths note assume continuity algorithm allows compute error bounds even case functions simplicity present algorithm case one conditional statement soon program multiple variables inputs different branches intervals anymore makes accurate evaluation individual paths difficult standard interval arithmetic inputs two branches thus simply evaluating two branches inputs correct ranges correlated yields pessimistic results computing final difference line overcome first challenge range computation takes account additional constraints second challenge use range computation well however unfortunately fails tighten final range satisfactory precision due timeouts still obtain much better error estimates interval arithmetic alone ranges values individual paths already computed much precisely report section type programs whose verification already reach today else experiments examples figure section provide idea type programs tool currently able verify fully automatically example section largest generalizes readily complex expressions assume condition form indeed benchmark jetengine jetengine verhulst verhulst predatorprey predatorprey carbongas carbongas sine single sine sqrt single sqrt sine order single sine order error variables preconditions path conditions table compares results range computation procedure described section ranges obtained standard interval arithmetic interval arithmetic one methods used range estimation alternative affine arithmetic also experimented affine arithmetic implementation however found affine arithmetic gives pessimistic results computing ranges benchmarks believe due imprecision computing nonlinear operations note however still use affine arithmetic estimate errors given computed ranges set default precision threshold maximum number iterations binary search obtain idea ranges functions also computed lower bound range using simulations random inputs exact rational arithmetic evaluation expressions observe range computation significantly improve standard interval bounds jetengine benchmark notable example interval arithmetic yields bound procedure still provide bounds quite close true range running times seconds complex benchmarks except jetengine runs minute due timeouts intermediate ranges simulated error error computation table compares uncertainties computed tool maximum uncertainties obtained extensive simulation random inputs ran simulation parallel rational corresponding value obtained error taking difference result benchmarks marked added initial uncertainties unless otherwise indicated used double precision knowledge first quantitative comparison error computation precision approximation true errors benchmarks except benchmarks computed uncertainties within order many times even closer underapproximation true errors provided simulation case benchmark believe imprecision mainly due complexity subsequent failures values parentheses second column indicate errors computed ranges arithmetic operation computed using interval arithmetic alone attempted improve affine error computation see cases precise range computation gain improvements full effect imprecision standard range computation appears due imprecision obtain possible errors square root negative number errors first case happens case jetengine benchmark interval arithmetic alone would therefore obtain meaningful result similarly triangle example section without able constrain inputs form valid triangles compute error bound radicand becomes possibly negative table presents another relevant experiment evaluating ability use additional constraints range computation use triangle example section additional constraints allowing increasingly flat triangles setting threshold line different values given first column triangles become flatter observe expected increase uncertainty input since formula becomes prone roundoff errors threshold range computation fails provide necessary precision radicand becomes possibly negative used double precision example well table comparison errors computed procedure simulated errors simulations performed random inputs indicates inputs external uncertainties associated meaningful example able find alone could verify presence uncertainties cases necessary use approximation methods evaluating effectiveness nonlinear expressions evaluate range error computation technique chosen several nonlinear expressions commonly used physics biology chemistry benchmark functions well benchmarks used control systems suitable benchmarks performed desktop computer running ubuntu processor ram running times highly depend timeout used default setting second find much improvement success rate threshold range computation stepwise estimation errors crucially depends estimate ranges variables strength using constraint solver perform estimation taking account precise dependencies benchmark jetengine verhulst predatorprey carbongas sine sqrt sine order approx range interval arithmetic simulated range table comparison ranges computed procedure interval arithmetic simulation simulations performed random inputs ranges rounded outwards benchmark range max abs error def real real require else ensuring res res res res def real real require else sqrt ensuring res res table ranges errors increasingly flat triangles values rounded outwards interval arithmetic alone fails provide result def smartroot real real real real require evaluating errors across program paths figure presents several examples evaluate error computation procedure across different paths section first method used benchmark function computing output range tool verify given postcondition immediately note error result actually large result since method aspect ignored previous work tool detects automatically method also noncontinous function computes square root using approximation small values regular library method otherwise note additional uncertainty input could occur instance method used embedded controller tool verify given spefication change condition line however verification fails fashion use tool determine appropriate branch condition meet precision requirement examples verify seconds finally smartroot method computes one root quadratic equation using precise val discr sqrt discr else sqrt discr else sqrt discr else sqrt discr ensuring res res figure path error computation examples method currently aware automatic tool prove programs continuous presence nonlinear arithmetic need compute error across different paths well tool succeeds verifying postcondition rather long running time due complexity conditions computing error across paths thus longer response time number timeouts timeout means merely ranges tightend best precision future envision optimizations arithmetic make many verification problems feasible techniques alternative approach using linear approximations solve polynomial constraints believe advances largely orthogonal use range arithmetic complement select alternative approximations performed automatically trustworthy compiler related work current approaches verifying code include abstract interpretation interactive theorem proving decision procedures survey section aware work would automatically integrate reasoning uncertainties testing symbolic execution technique generating test inputs use combination search interval constraint solving solve constraints arise whereas combine random search evolutionary techniques test numerical code precision perturbing bits values rewriting expressions idea exagerate initial errors thus make imprecisions visible probabilistic arithmetic similar approach perturbation using different rounding modes also propose testing produce detect accuracy problems instrumentring code perform computation side side regular computations approaches sound respect arithmetic generate check individual inputs thus able verify compute output ranges roundoff errors abstract interpretation abstract domains sound respect computations prove bounds ranges variables work area also quantify roundoff errors tool fluctuat techniques use interval affine arithmetic together required join meet operations may yield pessimistic results improves precision fluctuat refining input domains constraint solver approach viewed approaching problem different end starting exact constraint using approximation solver succeeds unlike tools general system currently handles functional code particular handle loops user provide inductive postconditions still prove code correct general discover focus lies proving precise bounds ranges presence nonlinear computations quantification roundoff errors uncertainties robustness analysis combines abstract interpretation model checking check programs stability tracking evolution width interval representing single input use concolic execution find inputs given maximum deviations inputs maximize deviation outputs two works however use testing approach provide sound guarantees presents framework continuity analysis programs along mathematical definition continuity builds work presents sound robustness analysis framework provides syntactic proof robustness programs reals thus consider approach describes quantitative measure robustness nonlinear programs numbers uncertainties believe complement cited framework theorem proving gappa tool generates proof checkable interactive theorem prover coq source code specifications reason properties reduced reasoning ranges errors targets precise properties specialized functions software implementations elementary functions specification requires expertize proofs human intervention similar approach taken generate verification conditions discharged various theorem provers harisson also done significant work proving programs hol light theorem prover approach makes different compromise precision automation tradeoff less precise automatic gappa approach used complementary detect precision needed gappa employed expert user selected methods results used tool instead automatically computed specifications conclusion presented programming model numerical programs decouples mathematical problem description realization finite precision model uses real data type corresponds mathematical real numbers developer specifies program using reals indicates target precision compiler chooses floating point representation checking desired precision targets met described soundness criteria translating programs precision requirements verification conditions mathematical reals resulting verification conditions natural description problem solved difficult solve using art smt solver therefore developed algorithm combines smt solving range computation notion soundness incorporates full behavior functions taking account due conditionals small differences values lead different paths taken program cases approach estimates sound upper bound total error computation evaluated techniques number benchmarks literature including benchmarks physics biology chemistry control systems found invocation smt solver alone sufficient handle benchmarks due scalability issues whereas use range arithmetic precise enough combining two techniques able show floating point version code conforms version reasonable precision requirements believe results indicate reasonable introduce reals data type following list previously introduced mathematical abstractions programming languages range computation gappa tool constraint solvers internally use interval arithmetic sound range computations whose limitations describes arithmetic based function enclosures use arithmetic based taylor series alternative approach useful checking constraint suitable forward computation ranges errors decision procedures alternative approach verification via range computation decision procedures bitprecise constraints however become large quickly addresses problem using combination underapproximations present alternative approach combining interval constraint solving cdcl algorithm decision procedure nonlinear real arithmetic combining interval constraint solving smt solver linear arithmetic formalizes format approach check ranges numeric variables handle roundoff errors uncertainties compute specifications automatically techniques rely performance hope integration recent new improved solver nonlinear unbounded integers rationals algebraic data types feasibility verified compilation benchmarks suggests realistic decouple verification executable mathematical models reals sound compilation therefore expect methodology help advance rigorous formal verification numerical software enable focus highlevel correctness properties opposed errors alone goubault putot modular static analysis zonotopes sas haller griggio brain kroening deciding floatingpoint logic systematic abstraction fmcad harrison verification using theorem proving formal methods hardware verification international school formal methods design computer communication software systems sfm ivancic ganai sankaranarayanan gupta numerical stability analysis computations using software model checking memocode jeannet apron library numerical abstract domains static analysis cav jiang luk rueckert computation freeform deformations logic applications references anta tabuada sample sample control nonlinear systems ieee transactions automatic control ayad verification programs ijcar bailey hida thompson package http benz hildebrandt hack dynamic program analysis find accuracy problems pldi blanchet cousot cousot feret mauborgne monniaux rival static analyzer large software pldi pages borges amorim anand bushnell pasareanu symbolic execution interval solving search icst borralleras lucas oliveras rubio sat modulo linear arithmetic solving polynomial constraints automated reasoning brillout kroening wahl mixed abstractions floatingpoint arithmetic fmcad pages chaudhuri gulwani lublinerman continuity analysis programs popl chaudhuri gulwani lublinerman navidpour proving programs robust chen wang cousot interval polyhedra abstract domain infer interval linear relationships sas darulova kuncak trustworthy numerical computation scala oopsla darulova kuncak majumdar saha synthesis fixedpoint arithmetic code emsoft dinechin lauter melquiond certifying floatingpoint implementation elementary function using gappa ieee trans figueiredo stolfi numerical methods applications brazil moura efficient smt solver tacas delmas goubault putot souyris tekkal vedrine towards industrial use fluctuat avionics software fmics silva haller kroening tautschnig numeric bounds analysis learning tacas duracz konecny polynomial function enclosures floating point software verification constraints formal feret static analysis digital filters european symposium programming esop gao ganai ivancic gupta sankaranarayanan clarke integrating icp lra solvers deciding nonlinear real arithmetic problems fmcad ghorbal goubault putot logical product approach zonotope intersection cav goldberg every computer scientist know arithmetic acm comput moura solving arithmetic ijcar kahan miscalculating area angles triangle technical report university california berkeley kuznetsov kinder bucur candea efficient state merging symbolic execution pldi lakhotia tillmann harman halleux flopsy floating point constraint solving symbolic execution testing software systems springer berlin heidelberg leroy verified squared critical software deserve verified tools popl linderman dill meng nolan towards program optimization automated analysis numerical precision cgo majumdar saha wang systematic testing control applications memocode makino berz taylor models validated functional inclusion methods international journal pure applied mathematics relational abstract domains detection floatingpoint errors esop moore interval analysis murray mathematical biology introduction springer odersky spoon venners programming scala comprehensive guide artima incorporation ponsini michel rueher refining abstract interpretation based value analysis constraint programming techniques quarteroni saleri gervasio scientific computing matlab octave springer edition wahl theory binary floatingpoint arithmetic informal proceedings international workshop satisfiability modulo theories smt floc scott denis chesneaux numerical health check scientific codes cadna approach computer physics communications society ieee standard arithmetic ieee std tang barr perturbing numerical calculations statistical analysis program stability issta westbrook chaudhuri semantics approximate program transformations corr woodford phillips numerical methods worked examples volume springer
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fundamental concepts cyclus nuclear fuel cycle simulation framework kathryn huffa matthew giddenb robert carlsenb robert flanaganc meghan mcgarryb arrielle opotowskyb erich schneiderc anthony scopatzd paul wilsonb university california berkeley department nuclear engineering berkeley wisconsin madison department nuclear engineering engineering physics madison university texas austin department mechanical engineering nuclear radiation engineering program austin university south carolina nuclear engineering program columbia mar university abstract nuclear power expands technical economic political environmental analyses nuclear fuel cycles simulators increase importance date however current tools often rather discrete restrictively licensed rather open source choices presents challenge modeling fidelity generality efficiency robustness scientific transparency cyclus nuclear fuel cycle simulator framework modeling ecosystem incorporate modern insights simulation science software architecture solve problems challenges nuclear fuel cycle analysis better addressed summary cyclus fuel cycle simulator framework modeling ecosystem presented additionally implementation discussed context motivating challenges nuclear fuel cycle simulation finally current capabilities cyclus demonstrated open closed fuel cycles keywords nuclear fuel cycle simulation agent based modeling nuclear engineering object orientation systems analysis introduction nuclear power expands technical economic political environmental analyses nuclear fuel cycles simulators increase importance merits advanced nuclear technologies fuel cycles shaped myriad physical nuclear chemical industrial political factors nuclear fuel cycle simulators must therefore couple complex models nuclear process physics facility deployment material routing indeed cardinal purpose dynamic nuclear fuel cycle simulator calculate mass flow part fuel cycle dynamic nuclear fuel cycle analysis realistically supports range simulation goals static analysis historically dynamic nuclear fuel cycle simulators calculated fuel cycle mass balances performance metrics derived corresponding author email address huff kathryn huff preprint submitted advances engineering software using software ranging spreadsheetdriven flow calculators highly specialized system dynamics modeling platforms date current tools typically distributed restrictive rather open source licenses developed industrial contexts using commercial software platforms additionally often developed customized applications many possess inflexible architectures never designed enable new features extensions finally many model dynamics facilities materials rather discrete resolution individual agents objects fuel cycles technologies systems analysis campaign developed requirements necessary next generation fuel cycle simulator three main failure modes associated software choices first discourage targeted contribution collaboration among experts next hobble efforts directly compare modeling methodologies finally rendering tools applicable subset desired simulamarch background nuclear fuel cycle simulators drive research development design calculating metrics quantitative measures performance compared among fuel cycle options feasibility technology development deployment strategies comprise fuel cycle option operational features nuclear energy systems dynamics transitions fuel cycles many measures performance expressed terms metrics example economic feasibility often measured levelized cost electricity lcoe combination fuel operating costs normalized electricity generation environmental performance might measured spent fuel volume radiotoxicity mined uranium requirements fuel cycle systems studies identified two dozen unique quantitative metrics spanning economics cost environmental sustainability waste management impacts safety security nonproliferation resource adequacy utilization among others exceptions metrics derived mass balances facility operation histories calculated fuel cycle simulator example nuclear waste repository burden derived ejected fuel masses water pollution land use derived facility operational histories however methods calculating metrics vary among simulators model system facilities economics materials static equilibrium simulators capture dynamics system similarly simulators discretely model batches material individual facilities others aggregate facilities fleets materials streams simulators designed model single aspect fuel cycle great detail neglecting others example simulator created policy modeling might excellent capability economics capabilities tracking transformations material isotopics effects isotopics technology performance neglected code advanced fuel cycles assessment cafca simulator way elected neglect isotopic resolution favor integral effects historically domestic national laboratories driven development regulated use tools verifiable fuel cycle simulation model vision dynamic model nuclear development dymond nuclear fuel cycle simulator nfcsim internationally tion fidelities scales applications three constraints identified presenting significant challenges modeling fidelity generality efficiency robustness scientific transparency field fuel cycle analysis cyclus nuclear fuel cycle simulator framework modeling ecosystem suite agents physics libraries compatible incorporate modern insights simulation science software architecture solve problems modern methods simultaneously enable efficient accurate robust validated analysis fuel cycle simulator result design choices made support access tool fuel cycle analysts users encourage developer extensions enable comparison modeling methodologies address range analysis types levels detail analyst sophistication cyclus dynamic model employs modular architecture open development process discrete agents discrete time arbitrarily detailed isotopic resolution materials experience broader field systems analysis indicates modeling enables flexible simulation control system dynamics without loss generality furthermore openness allows collaboration increases software robustness improves strength quality results peer review cultivates ecosystem modeling options ecosystem modular comprised dynamically loadable interchangeable libraries fuel cycle component process physics vary scope depth fidelity modularity allows users developers customise cyclus analyze cases interest rather custom application simulator originally developed address additionally customizability allows users developers address cases level fidelity necessary application fundamental concepts cyclus nuclear fuel cycle simulator capture modern insights novel challenges nuclear fuel cycle analysis better addressed laboratories created well cosi orion finally simulators initiated national lab setting continued propriety simulators dynamic analysis nuclear energy system strategies daness outside national laboratories researchers created new nuclear fuel cycle simulation tools existing tools available sufficiently general calculate metrics interest limited access national laboratory tools need customize research purposes universities private industry researchers reinvented wheel developing tools scratch tailored needs examples include cafca dynamic analysis nuclear energy systems strategies desae cyclus emerged line tools seeking break practice precursor global evaluation nuclear infrastructure utilization scenarios genius version originated within idaho national laboratory inl sought provide generic regional capability based lessons learned genius version genius version simulator sought provide generality extensible interface facilitate collaboration cyclus project improved upon genius effort implementing increased modularity encapsulation result dynamic simulator treats materials facilities discretely architecture permits multiple variable levels fidelity using framework simulator tracks transformation trade resources autonomous regional institutional entities customizable behavior objectives concepts agentbased resource tracking regional well institutional entities described sections sections respectively together provide capability extension reuse beyond pursued existing fuel cycle simulator essential capability absent previous simulators user customization extensibility design objectives modular open architecture cyclus necessary meet goals sufficient agent interchangeability also required facilitate direct comparison alternative modeling methodologies facility concepts concept core cyclus provides platform users quickly develop capabilities level detail validation necessary unique applications finally cyclus applicable broader range fidelities scales applications simulators due flexibility generality modeling abm paradigm discrete approach structure recognizes specialists utilize time resources modeling specific process associated area expertise reprocessing advanced fuel fabrication without create model entire fuel cycle serve host cyclus supports separating problem modeling physicsdependent supply chain two distinct components simulation kernel archetypes interact kernel responsible supporting deployment interaction logic entities simulation physics calculations customized behaviors entities implemented within archetype classes ultimately modeling evolution physicsdependent international nuclear fuel supply chain problem existing tools support either focused macro effects stocks flows commodities micro effects composition fast reactor fuel focus driven development specialized tools rendering task answering questions macro micro levels challenging within single tool contrast open extensible architecture discrete object tracking cyclus allow creation interchangeability custom archetypes level fidelity fuel cycle analyst motivation cyclus paradigm enables targeted contribution collaboration within nuclear fuel cycle analysis community achieve two important goals lower barrier users include custom nuclear technologies facility types fuel cycle analyses improving ability compare simulations without custom concepts open access development practices proprietary concerns research institutions security constraints data within fuel cycle simulators often restrict access use simulator therefore often limited institution origin necessitating effort duplication institutions thereby squandering broader human resources license agreements institutional proval required current simulators daness desae evolcode nfcsim including orion vision even source code unrestricted platform relies vensim often restricted costly mit cafca software example relies commercially licensed vensim product platform cyclus hand written freely available development tools open standard available cyclus relies open source freely available libraries provides fully free open access users developers foreign domestic moreover technical institutional aspects software development practices employed cyclus community facilitate collaboration technically cyclus employs set tools commonly used collaborative software development reduce effort required comment test ultimately merge individual contributions main development path many simulation platforms adopted previous simulators technical obstacles impeded kind collaboration institutionally cyclus invites participants propose discuss provide input final decision making important changes enter leave simulation time seek trade materials whose specific composition may known priori discrete facilities materials many fuel cycle phenomena aggregate effects captured discrete material tracking cyclus tracks materials discrete objects current fuel cycle simulation tools cosi genius version genius version nfcsim orion also possess ability model discrete materials however even among ability model reactor facilities individually equivalent ability model distinct activities cosi example support modeling reactors individually according recent benchmark models many reactors operating sync refuelling discharging occur simultaneously reactors cyclus allows type aggregation reactor behavior cyclus also enables operations facility vary independently others simulation similarly ability model disruptions facility shutdowns due insufficient feed material insufficient processing storage capacity readily captured software capable tracking operations status discrete facilities fleetbased models vision unable capture gracefully since supply disruptions modeled reduction capacity whole fleet software capable discrete materials notion discrete facilities however handle disruption manner desae example allow shutdown due insufficient feedstock event insufficient fissile material reprocessing desae borrows material storage leaving negative value cyclus framework dictate heuristics rather provides flexible framework either method possible final benefit discreteness facilities materials power combined ability track material history moves one facility another unique cyclus current simulators track materials discrete quanta necessarily preserve identity quantum materials move around fuel cycle coupled cyclus individual facility modeling capacity becomes distinct fuel cycle simulators able modularity extensibility modularity key enabler extending scope fuel cycle analysis within cyclus framework changes required improve fidelity modeling particular agent introduce entirely new agents narrowly confined place new requirements cyclus kernel furthermore assumptions heuristics would otherwise restrict algorithmic complexity used model behavior agents example current simulators describe finite set acceptable cycle constructions limits capability create novel material flows economic scenarios cyclus simulation logic relies market paradigm parameterized user flexibly simulates dynamic responses pricing availability institutional preferences minimal set mutual dependencies kernel agents expressed dynamic resource exchange dre provides level flexibility exist fuel cycle simulators creates potential novel agent archetypes interact existing archetypes cosi identify whether batch discharged reactor originated mixed oxide mox fabrication rather fresh uranium oxide uox fabrication cyclus tracking fresh batches contained material particular discharged batch extension cyclus also report individual facilities batch passed originated ability track single material simulation though might split transmuted merged materials along way allows cyclus answer questions previous simulators unable ask example allows precise tracking specific material diversions queries nonproliferation robustness facility levied either context single event series nefarious acts encapsulated design choice provides two major benefits first analysts take advantage simulation logic api archetype ecosystem apply cyclus specific problem modeler focus creating customizing nuclear facility institution resource toolkit models within specific area technical expertise second cyclus uses modular archetype approach comparing two archetypes straightforward example analyst would like compare effect using different models determine input fuel composition fast reactors fuel fabrication archetypes developed interchanged keeping rest models used simulation fixed software many fuel cycle simulators rely commercial cots software limits performance compatibility resource computing infrastructures cluster cloud computing constrains possible scope simulations increases time necessary conduct parameterized sensitivity analyses studies example large scale sensitivity analyses quantify dependence fuel cycle outcomes feasible massively parallelized environment enable research cyclus primarily written relies libraries supported linux unix including ubuntu osx platforms flexible support parallelization cyclopts cyclusenabled application large compute system uses htcondor computing htc infrastructure study dre performance outcomes example investigation effect solution degeneracy commonly observed phenomenon scenarios individual facilities basic preference definitions performed three different fuel cycles mox fast reactorthermal reactor cycles oxide thox recycle fast reactor cycles objective coefficients generated based two factors pairings facility location population possible values pairings always small size population possible values location effect investigated values zero ten infinity real number study additionally included investigation methodology implementation modular modeling abm approach ideal modeling coupled physicsdependent supply chain problems involving material routing facility deployment regional institutional hierarchies arise fuel cycle analysis additionally choice build cyclus open source libraries modern programming languages enables remote multiprocess execution number platforms section begins describing general design features make cyclus flexible powerful software dynamically loadable libraries abm framework described focusing implementation benefits fuel cycle context discussion treatment discrete resources follows focusing dre support users developers via cycamore library archetypes experimental toolkit also presented lastly methods quality assurance outlined modular software architecture architecture cyclus allows developers define nuclear fuel cycle processes independent simulation logic achieve agents developed represent facilities institutions regions comprising nuclear fuel cycle agents created using cyclus framework application programming interface api set functions protocols assist agent development specify agents defined scaling behavior order quantify magnitude dre solution times function population fuel cycle total cyclopts run jobs comprising compute hours htc infrastructure separately utilized run collect information full cyclus simulations running parallel machines reliably order simulations dynamically loadable libraries cyclus architecture encourages efficient targeted contribution ecosystem archetype libraries cyclus researcher focus generating archetype model within sphere expertise relying contributions others fill technologies simulation similarly individual developers may explore different levels complexity within archetypes including wrapping simulation tools loadable libraries within cyclus cyclus achieves behavior implementing generic apis modular architecture via suite dynamically loadable libraries pictured figure anticipating possible classes information required simulation kernel cyclus apis facilitate information passing agents core framework though common modern software architecture paradigm previously implemented nuclear fuel cycle simulator allows core cyclus framework operate independently libraries dynamically loadable primary mechanism extending cyclus capabilities independent core additional benefit ability contributors choose different distribution licensing strategies contributions users modelers control accessibility archetypes data sets see figure particular since clean architecture loads libraries without modifications cyclus kernel archetypes used simulator alongside open source archetypes without transfer sensitive information architecture allows libraries representing sensitive nuclear processes subject export control developed licensed privately finally dynamically loadable libraries enable cyclus easily handle varying levels simulation complexity hence single simulation engine figure cyclus core provides apis abstract away details kernel allow archetypes loaded simulation modular fashion figure cyclus framework enables fully open partially open fully closed collaborations used users interested policy questions well users focused detailed technical analyses merely choose preferred level modeling depth among available libraries ecosystem critically novel functionality enables comparison agent implementations example archetype inherits region interface figure interchangeable region agent cyclus paradigm cyclus implements modeling paradigm abm enables model development take place agent level rather system level nuclear fuel cycle context example analyst design reactor agent entirely independent fuel fabrication agent defining behavior agents according api contract sufficient interact one another bona fide agents simulation two archetype libraries used simulation without shared knowledge allowing modelers construct simulation building blocks many types origins furthermore abm paradigm flexible intuitive developer user perspective system dynamic approach used current simulators system dynamics popular approach modeling nuclear fuel cycles formally however system dynamics models simply strict subset models system dynamics model translated model abm techniques therefore enable broader range simulations generic fashion cyclus cyclus cyclus cyclus cyclus cyclus figure inheriting cyclus class interfaces agent facility institution region classes abstracts away unnecessary details exposing powerful functionality example dummy archetype simply inherits region order become bona fide regiontype agent way researcher directly compare two different reactor modeling implementations perhaps imaginary classes detailedreactor simplereactor simply exchanging two corresponding archetypes two reactor archetypes inheriting facility class indistinguishable simulation perspective done agent type agents regions institutions regions institutions facilities cyclus provides novel representation entities nuclear fuel cycle reflects reality international nuclear power facilities implementing individual fuel cycle technologies institutions managing facilities regions providing geographical political context institutions facilities simulators provided notion static regional effects cyclus allows regions institutions firstclass agents simulated fuel cycles fundamental interactions entity implemented corresponding archetype class cyclus region class institution class facility class archetype developers build provided functionality inheriting appropriate class cyclus implements rif relationship hierarchy regions parents institutions turn parents agent interchangeability abm inherently agents represent discrete independently acting objects figure illustrates modular nature cyclus archetypes core cyclus simulator creates set classes agent plugins based agent utilize generic core apis define interaction well interaction example use resource exchange paradigm api trading resources one another archetype developer interfaces provide enormous power simply api abstracts away details unnecessary specifying archetype behavior providing necessary functionality interacting cyclus simulation kernel user multiple archetypes inherit apis interchangeable simulation resources materials another use case seeks capture system vulnerability material diversion provenance distinct materials fundamental information unit studies type analysis requires discrete simulation target facility individual materials modified within material risk analysis therefore demands facilities materials discretely modeled objects like cyclus cyclus agents transfer discrete resource objects among one another cyclus supports two types resources facilities words rif hierarchies form directed acyclic graph dag regions root nodes facilities leaf nodes two primary consequences arise structure first institutions nominally responsible deploying decommissioning facilities accordingly advanced logic regarding building decommissioning implemented top behaviors inherited institution interface second facility class implements trader interface participate resource exchange institutions regions respectively adjust resource flow preferences managed facilities importantly capability allows modeling preferential regional trading resources tariffs well preferential institutional trading contracts materials represent typical nuclear materials nuclide compositions products represent measure carbon credits build permits employees etc discrete objects cyclus models facilities institutions regions resources discrete objects discrete resource model allows range modeling granularity macroscopic extreme equivalent continuous flow microscopic extreme model capable representing arbitrarily small material objects isotopic resolution way cyclus applicable across full range modeling fidelity models distinguish discrete facility entities materials however questions require resolution level individual facilities materials result many detailed performance metrics captured previously existing models example meaningful models spent nuclear fuel storage transport disposal strategies require representation discrete casks varying isotopic compositions reasons abm paradigm enables novel simulations multiple use cases require agents regions institutions facilities cyclus must represented discrete objects instance tracking individual shipments viable materials resources tracked discrete objects operations performed every resource object splitting combining decay etc tracked detail performed information includes agent created resource introduced simulation parentage resource also tracked makes possible follow history resources transferred agents cyclus kernel experimental support decay calculations materials store time since last decay agents free invoke decay function desired decay current simulation time cyclus operate decay modes additional mode likely added future release manual currently implemented default mode agents decay materials requested archetype never currently implemented globally turns decay material decay function nothing lazy currently implemented decays material automatically whenever composition observed agent queries information material content dags common graph theoretic structures whose important feature lack cycles single path root node node graph periodic future automatically decays materials simulation fixed frequency decay invoked material checks see contains nuclides decay constants significant respect time change since last decay operation none decay constants significant decay calculation performed material remains unchanged error accumulate next time material decay function invoked time change larger cyclus notion tracked versus untracked nuclides cyclus composition material represented arbitrarily large list potentially thousands nuclides agents free treat nuclides present materials way please including ignoring responsibility archetype developers choose handle potentially compositions large simulations many material objects may change frequently material decay also contribute significantly changes order help avoid unnecessary runtime performance database size impacts compositions cyclus special features particular compositions immutable created allows multiple material objects hold references composition safely additionally new compositions resulting decay cached used avoid redundant decay calculations figure illustrates decay history cache works composition immutability concert decay history caching help eliminate many redundant calculations addition reducing total number composition entries recorded database decay decay decay decay decay figure simple decay line cache holding compositions composition comes decaying time step comes decaying time steps etc black represents cache particular composition chain blue indicates decay operations satisfied cache lookups needs decayed time step quick lookup returns previously computed decaying time steps requires decay calculation compute new composition subsequent requests decaying time steps require recalculation question procedure used materials products step begins polling potential consumers agents define quantity commodity need consume well target isotopics quality posting demand market exchange series requests users may optionally parameterize agent associate collection demand constraints collection requests collections requests may grouped together denoting mutual requests represent demand common purpose example reactor may request uox mox fuel mutually indicating either satisfy demand fuel suppliers respond series requests bid bid supplies notion quantity quality resource match request suppliers may add arbitrary number constraints accompany bids example enriched uox supplier may constrained current inventory natural uranium total capacity provide enrichment separative work units swus attaches constraints bids potential resource transfer bid request may denoted exclusive exclusive dynamic resource exchange dre cyclus simulation paradigm allows discrete agents based archetypes kernel knowledge enter simulation arbitrary times trade discrete resources resources defined priori therefore logic engine defining resource interaction mechanisms among agents crucial dre described detail critical logic engine cyclus simulations built supporting general cyclus philosophy facilities treated black boxes communication framework defined dre consists three steps information gathering resource exchange solution trade execution importantly step agnostic respect exchange resources transfer excludes partial fulfilment must either met fully mode supports concepts trading individual reactor assemblies combination notion mutual requests complex instances supply demand enabled finally requesting facilities institutions regions may apply preferences potential pairing based proposed resource transfer facilities apply arbitrary complex logic rank bids received whether based quantity available bid quality bid consequent implications physics behavior facility addition institution apply higher preference partner congenial similarly region may negate transfers material higher uranium enrichment allowable min set supply nodes set request nodes unit cost flow flow node node aki coefficient constraint rhs constraint given supplier requester requested quantity request node practice lps solve much faster milps however capabilities exist cyclus order provide users desired level fidelity trades agents initiated cyclus kernel solution dre found trade supplying agent notified matched request provides associated resource exchange supplied resources sent corresponding requesting agents cyclus dre executed time step therefore facility request resource met given time step may offer request following time step agent behavior may change arbitrarily exchange executed given time step may unique simulation dre novel simulation concept nuclear fuel cycle domain provides flexible infrastructure supporting dynamic flows resources agents even agents enter leave simulation even agents defined archetypes arbitrary complexity trading agents affected proposed quality resource agent relationships use preferences accordingly wide range possible effects modeled fuel supply international trade agreements figure flow graph representing three suppliers left two requesters right potential flows various commodities among second consumer makes two different requests meanwhile second supplier supply commodities requested consumers provided two bids accordingly given full definition supply demand represented figure flow graph dre may solved either optimally using mathematical program approximately heuristic trade denoted exclusive analyst desires model either heuristic must used exchanges must represented linear program milp exclusive trades exist linear program model may used formulation cyclus form given gidden simulation support users developers build working simulations shortest time possible cyclus ecosystem provides fundamental building blocks basic archetypes toolkit commonly needed functions cycamore library provides suite fundamental region institution facility archetypes cyclus toolkit provides assistance developers classes provide basis deployment determination commodityproducer commodityproducermanager builder buildingmanager commodityproducer class provides interface querying prototypes capacity produce given commodity commodityproducermanager provides interface registering commodityproducers querying current capacity supply commodity builder class provides interface querying prototypes built interacts buildingmanager orders prototypes built buildingmanager uses simple minimum cost algorithm determine many prototype build given demand capacities costs indexes available prototypes perform similar function demand capacity cost carry units defined developer cycamore cycamore cyclus additional module repository provides fundamental set dynamically loadable libraries providing agent archetypes basic simulation functionality within cyclus since cyclus relies external archetypes represent agents within simulation cycamore provides basic archetypes new user needs get started running simple simulations archetypes support minimal set fuel cycle simulation goals provide example guide new developers would seek contribute archetypes outside cycamore version cycamore contains one region archetype two institution archetypes four facility archetypes three additional facilities added version short descriptions functions found table section demonstrate current cycamore release provides basic functionality enabling simple fuel cycle analyses future contributions vetted capabilities cycamore may grow min integer nuclear engineering tools cyclus toolkit provides two useful interfaces querying physical parameters material objects first matquery tool provides basic querying api including atom mass fractions nuclides number moles nuclide material etc composition material enrichment tool provides api determining parameters material including swu natural uranium required enrich material provided knowledge feed product tails assays toolkit addition core functionality cyclus kernel focused set capabilities needed implement simulation dre toolkit provided assist developers users related simulation nuclear engineering tasks toolkit actively developed part cyclus primarily focus supporting interesting situ metric analysis tools toolkit extensions addition already exist new tools emerge archetypes developed community tools gain adoption projects demonstrate utility developer community considered screening adoption kernel toolkit extensions likely extensions include simulation tools series utility classes provided support agent facility deployment example symbolic function representations linear exponential piecewise functions supported via symbfunctionfactory class functions used toolkit classes determine commodity demand power demand user input four fuel cycle metrics calculators supportive data tables policy models entity facility facility facility archetype enrichmentfacility facility facility sink facility source institution managerinst institution deployinst region growthregion functionality facility enriches uranium specified capacity facility fabricates fuel material based separated streams reactor model handles batch refueling based recipes compositions requests number input fuel types transmutes static compositions upon discharge facility separates incoming material specified streams facility capable accepting finite infinite quantity commodity produced simulation facility generates material composition commodity type specified input manager institution manages production commodities among facilities building new ones needed institution deploys specific facilities defined explicitly input file region determines whether need meet certain capacity defined via input time step table archetypes cycamore seek cover large range simple simulation use cases facilities added version marked economic models verficiation cyclus implementation relies software development best practices version control testing continuous integration documentation style guidelines ensure reliability reproducibility sections discuss greater detail software development components comprise cyclus verification strategy allows simulation kernel physics models tested explicitly separately approaches valuable addition entire answer requirements imposed taken together strictly adhered present fortress protect poorly designed otherwise undesirable code quality assurance garner trust broad user archetype developer community cyclus project must implement strategy assure ongoing quality software provides multiple strategies collectively known quality assurance developed scientific software development community mitigate structural algorithmic errors software include verification validation testing others nuclear engineering software quality often governed nuclear quality assurance american society mechanical engineers asme specification whose latest revision appeared cyclus adopted agile development process interpreting manner similar process adopted department energy doe within nuclear engineering advanced modeling simulation neams pyne toolkit validation simulators like cyclus intended give forecasts system behavior uncertain futures well defined experimental benchmarks upon rely instead comparisons fuel cycle simulators use modeling paradigms underway best approach establish confidence cyclus produces correct answers version control automated version control provided git distributed version control tool heart strategy records full history change along metadata author timestamp accompanying message makes possible identify changes introduced related changes made changes necessary also facilitates reversing individual changes long intricate set changes version control also enables code review process descried section testing automated software testing first line defense errors implementation testing directly compares actual results running software versus expected behavior software cyclus three categories tests defined unit tests integration tests regression tests proposed code change allowed cyclus change must covered test either new existing tests must pass version cyclus source code includes requested changes runs complete test suite reporting back whether steps successful successful code author must first identify resolve problems steps performed incoming code prior inclusion broken code never enters main software development branch code review unit tests unit tests verify behavior smallest code unit typically single function class cyclus uses google test framework harness running unit tests sufficient unit tests required proposed change cyclus code base currently cyclus implements unit tests cycamore implements cover approximately respective code bases numbers expected grow time automated testing including necessary sufficient component cyclus system keeps bad code cyclus however cyclus always require human eyes hands let good code main cyclus repository hosted remotely publicly github website part provides tools facilitate culture frequent thorough code review number policies exist ensure proposed set changes known pull request adhere projects standards every pull request must reviewed accepted member cyclus core team involved authoring changes addition reviewing algoirthmic design source code reviewer relies tests ensure correct behavior requires authors adhere style guide provide sufficient documentation standards met proposed changes merged software step repeatedly shown improve code quality successful code reviewer inspects changes proposed software also changes proposed tests long term benefit maintainability project documentation coding style also reviewed part process api must documented required cyclus policy cyclus information aggregated together static websites doxygen sphinx tools accessed http cyclus also strictly enforces use google style guide software contributions means developers cyclus write cyclus code way homogenization may hurdle new developers ultimately improves code legibility therefore robustness integration tests integration tests combine multiple elements cyclus interface test work correctly cyclus cycamore integration tests performed running sample simulations scenarios verifying results match predictions set standard input files run output inspected compared via nose python test framework regression tests regression tests ensure significant unintended changes occur course cyclus development regression tests implemented similarly integration tests category however comparison done output input file run previous version cyclus typically last released version sense regression tests dumb care contents simulation correct whether changed continuous integration continuous integration idea software automatically tested validated developed rather final stage longer development cycle results testing shown code reviewers prior reviewing software changes cyclus project uses circleci service continuous integration code change submitted review circleci builds demonstration already ecosystem growing early crossinstitutional contributions ecosystem demonstrate significant achievement cyclus framework provide basis communitydriven development model success cyclus simulator measured many ways compelling early successes development model demonstrations fundamental simulation capabilities promising growth cyclus ecosystem multiple institutions indicates fuel cycle simulator advance development paradigm additionally simulation results complex recycling scenarios demonstrate cyclus possesses fundamental fuel cycle simulation capabilities contribute field supplementary projects number projects tools outside core simulation kernel developed improve scope diversity capabilities cyclus ecosystem table lists tools projects developed close integration cyclus kernel tools used ease development simulation design cycstub cycic ciclus data visualization analysis cyclist cyan remote execution cloudlus ecosystem cyclus software development model seeks break proliferation specialized simulators instead leverages collective expertise fuel cycle researchers toward single extensible tool targeted contributions researchers ecosystem capabilities emerge cyclus ecosystem collection tools calculation libraries archetypes data input files intended use cyclus simulator members ecosystem include archetype contributions expected common type contribution cyclus contributions new archetypes researchers driven create new archetype need arises improve fidelity simulation represent novel reactor type innovative reprocessing strategy particular governmental institutional behavior utility cyclus part measured breadth diversity archetypes developed way early progress promising many archetypes external cycamore library table developed contribution cyclus ecosystem archetypes provide first examples capabilities add fundamental cycamore archetypes providing reactors separations logic fuel fabrication processing storage facilities expanded institutional paradigms existence diversity contributed archetypes illustrate power potential development approach cyclus taken archetypes provided cycamore repository archetypes created researchers isotopic composition data historical facility deployment data cyclus graphical user interface gui tool cyclist fundamental analysis tools cyclus toolkit tools cyclus optimization parallelization development simulations illustrate flexibility cyclus section discuss creation results range representative fuel cycle simulations previous benchmarks cyclus system dynamics simulators complex problems including transition analyses reported elsewhere cyclus demonstrated satisfactory performance examples presented taken together form ecosystem capabilities time ecosystem grow archetype developers kernel developers even users contribute capabilities developed needs indeed vision cyclus framework predicts ecosystem general specialized capability extensions name cycic cyclist ciclus cycstub cyan cloudlus description input control embedded cyclist interactive data exploration environment continuous integration scripts cyclus skeleton clean slate module development cyclus analysis tool tools running cyclus cloud environment ref table many tools developed outside scope cyclus kernel improved user developer analyst experiences cyclus name nuclear fuel inventory model commodconverter mktdriveninst separationsmatrix streamblender description reactor archetype fuel fabrication archetype flexible reactor analysis module simple commodity converting storage facility archetype institution controls deployment based commodity availability facility elemental separations used fuel facility fuel fabrication reprocessed streams ref table diverse set archetypes development reflect diverse needs researchers various institutions archetypes contributed outside cyclus core cycamore libraries first demonstration development fuel cycle simulator limit cyclus capabilities extended new addition ecosystem discussed rather simulations designed illustrate cyclus matches capabilities simulator variations single cyclus simulations run small changes input specification simplicity current demonstration many simplifying assumptions made respect material compositions fuel transmutation among others three fuel cycles examined recycle mox recycle mox recycle fuel cycles month singlereactor cyclus simulation run addition simulation staggered refueling times number reactors increases system converges toward continuous material flow results cyclus flexibility allows transition examined facility lwr cycamore spent fuel fabrication cycamore separations cycamore repository cycamore uox fabrication cycamore source cycamore parameter type batches batch size cycle length refuelling time requests requests requests requests offers requests offers efficiency separation capacity requests capacity offers capacity offers capacity value lwr mthm months months recycled mox preference enriched uox preference depleted uranium separated fissile material recycled mox spent fuel types separated fissile material waste uox depleted table facility configurations example simulations described table facilities used simulations oped french commissariat atomique aux alternatives cea separations cycamore facility takes kinds spent fuel separates plutonium waste streams efficiency used simulations separated per month repository cycamore facility take types material including separations waste streams spent reactor fuel reactor cycamore reactor model requests number input fuel types transmutes static compositions burnt discharged core reactor configured model light water reactor batch core operating month cycle month refuel time batches core contain metric tons heavy metal mthm heavy metals actinide elements like thorium uranium plutonium reactor also configured take either enriched uox fuel recycled mox fuel uox fabrication cycamore facility provides enriched uox fuel requested facility infinite production capacity source cycamore facility provides depleted uranium requested facility infinite production capacity model three fuel cycles simple adjustments input file specification necessary specifically commodity types trade preferences facilities needed altered one simulation another cases reactor configured request recycled mox fuel higher preference new uox fuel recycle case reactor set offer spent fuel waste recycle case reactor offered spent uox spent fuel market spent mox still offered waste recycle case reactor offers burned fuel spent fuel market separations facility requests spent fuel higher preference repository resulting preferential recycle preferences easy adjust cyclus dynamically handles supply constraints preference resolution notable separations recycle fuel fabrication capacity deployed identically simulations case recycling loop never acquires material reactors always receive fresh uox fuel dre ensures everything operates smoothly cases cyclus discrete materials make recycle straightforward implement reactors keep track fuel discrete batches reactor remembers received batch batch received recycled fuel fabrication facility offer separations spent fuel fabrication cycamore facility requests depleted uranium separated fissile material mixes together recycled mox fuel offers requesters two streams mixed ratio order match simple neutronics properties target fuel closely possible method used based variation equivalence method originally developed baker ross technique also used cosi fuel cycle simulator source source depleted depleted spent fuel fab spent fuel fab enrichment enrichment mox fuel mox fuel uox fuel stream stream lwr lwr spent fuel spent fuel waste separations waste separations uox fuel waste repository waste repository figure full mox recycle fuel cycle material flows figure mox recycle scenario material flows cases enough separated fissile material accumulates fuel fabrication facility generate full recycled batch batch transmuted plutonium burned created results drop total fuel cycle system plutonium inventory pattern repeats roughly every cycles months recycle case every cycles months recycle case recycle scenario material takes fabrication facility cycles longer accumulate full batch fissile material facilities represented individually transact discrete materials discrete events realistic patterns facility behavior affect total system behavior observed using cyclus figure shows plutonium buildup simulations recycle recycle recycle scenarios number reactors staggered refueling increases behavior system approaches steady average reminiscent continuous material flow models fundamental capabilities demonstrated simulations qualify cyclus ecosystem model breadth scenario types expected instead offers waste commodity requested repository material flows recycle case shown figure changing scenario fuel cycle fuel cycle requires oneword change input file regarding output commodity spent mox fuel reactor fuel incommodity mox incommodity outcommodity waste outcommodity outcommodity outcommodity inrecipe inrecipe outrecipe outrecipe fuel results material flows figure similarly trivial change used switch recycle fuel cycle note reactors always transmute fuel fixed compositions error isotopic compositions larger recycle case figure shows plutonium buildup recycle recycle recycle variations onereactor scenario described figure generated directly cyclus output data several batch cycles near month recycle recycle metric tonnes transparency community oversight furthermore abm simulation paradigm ensures generic simulation capability allows cyclus address common analyses flexible fashion enables analyses impossible system dynamics simulators similarly modular cyclus architecture facilitates simulations every level detail simulations relying arbitrarily complex isotopic compositions possible cyclus simulations employing physics physics introduced optional radioactive decay materials use facility libraries support calculation physical processes nuclear facilities cycamore library provides models employing basic physics core fuel cycle facilities extension libraries community support detailed simulations indeed agents varying fidelity even exist simulation researchers longer need reinvent underlying simulator framework order model simulation focused aspects fuel cycle relevant research furthermore capabilities within cyclus cycamore rest ecosystem insufficient adding custom functionality simplified modular architecture clean modern api simplifies customization independent archetype development researchers create models within domain expertise without modifying core simulation kernel throughout cyclus infrastructure architecture choices sought enable collaboration sustainable development ecosystem capabilities already growing may someday reflect full diversity use cases nuclear fuel cycle simulation domain plutonium buildup reactor time month figure system plutonium buildup one reactor plutonium buildup reactors recycle recycle metric tonnes time month figure system plutonium buildup staggered refueling many reactors nuclear fuel cycle simulator tools examples show flexibility provided dre logic within cyclus framework provide example resolution made possible discretefacility modeling fuel cycle simulation acknowledgements work supported number people authors would like thank software contributors zach welch olzhas rakhimov additionally authors grateful many enthusiastic members cyclus community nuclear energy university programs neup collaboration partners university texas university utah university idaho direct support cyclus project received nuclear regulatory commission nrc faculty development program neup conclusions cyclus nuclear fuel cycle framework presents generic flexible alternative existing fuel cycle simulators previous nuclear fuel cycle simulators limited distribution constrained simulation capabilities restricted customizability cyclus emphasizes open strategy access development open strategy improves accessibility also enables research development program national nuclear security administration nnsa consortium verification technology university wisconsin addition students contributing cyclus received fellowship support argonne national laboratory anl lab grad program neup fellowship program national science foundation nsf graduate fellowship program department homeland security dhs nuclear forensics graduate fellowship program petre wilson code review scientists arxiv url http flicker schneider campbell evaluation criteria analyses nuclear fuel cycles transactions american nuclear society vol fuel cycle options analysis iii american nuclear society grange park united states anaheim united states url http poinssot bourg ouvrier combernoux rostaing bruno assessment environmental footprint nuclear energy systems comparison closed open fuel cycles energy url http guerin kazimi impact alternative nuclear fuel cycle options infrastructure fuel requirements actinide waste inventories economics technical report massachusetts institute technology cambridge united states jacobson yacout matthern piet shropshire jeffers schweitzer verifiable fuel cycle simulation model vision tool analyzing nuclear fuel cycle futures nuclear technology yacout jacobson matthern piet moisseytsev modeling nuclear fuel cycle international conference system dynamics society boston url http schneider bathke james nfcsim dynamic fuel burnup fuel cycle simulation tool nuclear technology allan international atomic energy agency international project innovative nuclear reactors fuel cycles guidance application assessment methodology innovative nuclear energy systems inpro manual overview edition international atomic energy agency vienna url http boucher grouiller cosi simulation software pool reactors fuel cycle plants fuel cycle high level waste management beijing china boucher grouiller cosi complete renewal simulation software fuel cycle analysis fuel cycle high level waste management vol asme new york usa miami united states meyer boucher new developments cosi simulation software fuel cycle analysis proceedings global paris france meyer girieud eschbach chabert garzenne barbrault van den durpel duquesnoy carlier lefevre comparison different scenarios deployment fast reactors france results obtained cosi proceedings global references piet dixon jacobson matthern shropshire dynamic simulations advanced fuel cycles nuclear technology url http huff dixon next generation fuel cycle simulator functions requirements document tech idaho national laboratory jul macal simulation system dynamics simulation conference wsc proceedings winter ieee url http jsp cohen modern code review making software really works believe url http fagan design code inspections reduce errors program development software pioneers springer url http donoho maleki rahman shahram stodden reproducible research computational harmonic analysis computing science engineering url http ince hatton case open computer programs nature url http stodden scientific method practice reproducibility computational sciences ssrn electronic url http wicherts bakker molenaar willingness share research data related strength evidence quality reporting statistical results plos one url http grasso monti sumini benchmark fuel cycle scenarios study tech report url http boucher benchmark study nuclear fuel cycle transition scenarios analysis codes tech oecd nuclear energy agency jun gidden cyclopts figshare http doi http url http gidden modeling framework application generic nuclear fuel cycle thesis university wisconsin madison united states mar carlsen gidden huff opotowsky rakhimov scopatz welch wilson cyclus figsharehttp doi http url http jacobson jeffers matthern piet baker grimm vision user verifiable fuel cycle simulation model tech idaho national laboratory inl url http carlsen gidden huff opotowsky rakhimov scopatz wilson cycamore figsharehttp doi http url http boehm software risk management springer asme addenda asme quality assurance requirements nuclear facility applications nuclear quality assurance american society mechanical engineers new york larman agile iterative development manager guide professional neams nuclear energy advanced modeling simulation neams software verification validation plan requirements tech deptartment energy office nuclear energy jul url http biondo scopatz gidden slaybaugh bates cameron wilson quality assurance within pyne open source toolkit american nuclear society winter meeting ans huff fratoni greenberg extensions cyclus ecosystem support transition capability transactions american nuclear society american nuclear society anaheim makuhari japan worrall gregg scenario analyses future fuel cycle options journal nuclear science technology url http van den durpel yacout wade taiwo lauferts daness overview capabilities developments proceedings global paris france guerin feng hejzlar forget kazimi van den durpel yacout taiwo dixon matthern boucher delpech girieud meyer benchmark study computer codes system analysis nuclear fuel cycle technical report massachusetts institute technology center advanced nuclear energy systems nuclear fuel cycle program electric power research institute apr url http andrianova davidenko tsibul skii desae program systems studies growth nuclear power atomic energy url http mccarthy dixon choi boucher ono gonzalez hyland benchmark study nuclear fuel cycle transition codes url http juchau pasamehmetoglu wilson oliver turinsky hays stover global evaluation nuclear infrastructure utilization scenarios genius global advanced nuclear fuel cycles systems september september global advanced nuclear fuel cycles systems american nuclear society boise united states jain wilson transitioning global optimization fuel cycle system study tools winter meeting american nuclear society nov vol transactions american nuclear society american nuclear society albuquerque united states oliver wilson reveillere ahn dunn huff elmore studying international fuel cycle robustness discrete fuel cycle systems analysis tool proceedings global global advanced nuclear fuel cycles systems paris france huff oliver wilson ahn dunn elmore discrete modeling international fuel cycle robustness transactions american nuclear society vol nuclear fuel cycle codes applications american nuclear society washington united states juchau jacobson modeling nuclear fuel cycle nuclear technology url http united states software freedom conservancy git software freedom conservancy url http inc googletest google testing framework technical report google url https pellerin nose nicer testing python technical report nose url https biggar circleci https url https dabbish stuart tsay herbsleb social coding github transparency collaboration open software repository proceedings acm conference computer supported cooperative work acm van heesch doxygen source code documentation generator tool url http brandl sphinx documentation url http weinberger silverstein eitzmann mentovai landray google style guide ttp googlecode xml flanagan schneider input visualization cyclus nuclear fuel cycle simulator cyclus input control proceedings global salt lake city united states livnat gur potter flanagan cyclist url scopatz welch gidden ciclus https url https carlsen gidden huff opotowsky rakhimov scopatz welch wilson cycstub https jun url https carlsen cyan figshare http doi http url http carlsen cloudlus https url https huff streamblender figsharehttp doi http url http huff commodconverter figsharehttp doi http url http flanagan https url https skutnik development reactor analysis module cyclus figsharehttp doi http url http huff mktdriveninst figsharehttp doi http url http djokic scopatz greenberg huff nibbelink fratoni application cyclus fuel cycle transition analysis proceedings global paris france scopatz dynamic fuel cycle benchmarking physics arxiv url http gidden wilson huff carlsen oncethrough benchmarks yclus modular opensource fuel cycle simulator proceedings ans winter conference san diego baker ross comparison value plutonium uranium isotopes fast reactors proc conf 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autonomous racing scale cars alexander liniger alexander domahidi manfred morari nov abstract paper describes autonomous racing race cars based mathematical optimization using dynamical model vehicle control inputs computed receding horizon based controllers objective maximize progress track subject requirement staying track avoiding opponents two different control formulations presented first controller employs structure consisting path planner nonlinear model predictive controller nmpc tracking second controller combines tasks one nonlinear optimization problem nlp following ideas contouring control linear time varying models obtained linearization used build local approximations control nlps form convex quadratic programs qps sampling time resulting qps typical mpc structure solved range milliseconds recent structure exploiting solvers key feasibility overall control scheme obstacle avoidance incorporated means corridor planner based dynamic programming generates convex constraints controllers according current position opponents track layout control performance investigated experimentally using scale race cars driven speeds operating regions saturated rear tire forces drifting algorithms run sampling rate embedded computing platforms demonstrating feasibility high performance approaches autonomous racing ntroduction autonomous car racing challenging task automatic control systems due need handling vehicle close stability limits highly nonlinear operating regimes addition dynamically changing racing situations require advanced path planning mechanisms obstacle avoidance executed fast dynamics constrain sampling time range tens milliseconds severely limits admissible computational complexity algorithms situation even challenging autonomous algorithms shall executed simple embedded computing platforms paper investigate control strategies task racing autonomous vehicle around given track focus methods implemented run embedded control platforms present experimental results using scale kyosho dnano race cars achieve top speeds corresponds upscaled speed high performance proposed controllers operate car friction limits far beyond linear region typically used autonomous driving systems challenging task generally mastered expert drivers lots training contrast approach requires merely map track dynamical model car particular use bicycle model nonlinear tire forces neglecting load transfer coupling lateral longitudinal slip two control schemes presented paper first describe hierarchical control scheme consisting path planner generating feasible trajectories underlying nonlinear model predictive control nmpc trajectory tracking controller second approach combines path planning path following one formulation resulting one nmpc controller based particular formulation known contouring control objective approaches maximize progress track measured projection vehicle position onto center line track linear time varying ltv models obtained linearization employed construct authors automatic control laboratory eth zurich switzerland emails tractable convex optimization problem solved sampling time efficient interior point solvers embedded systems generated forces employed solve resulting optimization problems makes approaches presented paper amenable use embedded systems furthermore demonstrate schemes easily extended incorporate obstacle avoidance adjusting constraints resulting optimization problems according racing situation mechanism place avoidance trajectory optimized yield maximal overall progress translates highly effective overtaking maneuvers automatically planned executed related work safe autonomous driving moderate speeds demonstrated example context autonomous highway systems ahs within californian path programme contests darpa grand challenge urban challenge numerous research groups projects main goal develop autonomous driving systems public traffic situations autonomous racing received less attention impressive advances made recently also discipline instance fully autonomous audi tts reported race high speeds using trajectory tracking controller follows trajectory however best knowledge authors fully automatic racing including obstacle avoidance dynamic racing situations demonstrated yet practice control perspective fundamental building blocks automatic racing grouped three categories drift control reference tracking path planning research associated first group focuses gaining better understanding behavior car near friction limit designing safety control systems try prevent accidents analyzing nonlinear car model shown steady state motions corresponding drifting generated see approaches designed trajectory tracking controlled car drifts circle represents steady state drift equilibrium reference tracking controllers usually designed operate vehicle within linear tire force region maximum safety robustness approaches based nmpc allows one incorporate latter constraint systematic manner deal nonlinearities model either directly nonlinear programming nlp solver use ltv affine pwa approximations resulting convex quadratic program respectively approaches without optimization include nonlinear robust control control however lack ability incorporate constraints one approach explicitly designed operating saturated tire forces center percussion cop used design state feedback steering controller reference tracking cop rear tire forces influence dynamics definition allows simple linear steering controller order avoid obstacles reference tracking schemes rely path planner simple point mass model path planner used limit computational complexity problematic generated trajectories infeasible respect car physical dynamics lead preventable collisions latter reference suggests use library steadystate movements trims parametrizable length generated complex dynamical model connect maneuvers allow transition one trim another similar idea path planning advantage generates feasible reference trajectories low level tracking controller drifting trims included increase agility car however resulting optimization problem program complex solved within typical sampling times embedded platforms consequently approach well suited autonomous racing order avoid relying feasibility path planner approaches investigated optimal trajectories associated control inputs rally racing maneuvers calculated numerically simulation obstacle avoidance incorporated tracking controller using additional cost term penalizes coming close obstacle however controller reported perform worse counterpart especially car diverge reference trajectory approach similar studied simulation obstacle avoidance incorporated problem formulation using spatial reformulation imposing constraints transformed variables solution nlp obtained standard nonlinear solver prohibitively long solve times uses iterations achieve low computation times needed implementation however assumed one side obstacle overtaken may case practical racing situations interesting alternative methods methods example rapidly exploring random trees rrts investigated generate time optimal trajectories curve advantage algorithms tend quickly find feasible trajectories differential flatness system exploited speed convergence algorithm generating trajectories even complex situations however reported computation times yet low enough allow implementation contribution paper describe two novel autonomous racing controllers incorporate obstacle avoidance track constraints actuator limitation ability controlled drift systematic straightforward way feasibility sampling rate demonstrated experimental results fast cars best knowledge authors first implementation autonomous racing controllers obstacle avoidance experimental testbed fundamental idea approaches use receding horizon controller maximizes progress track within horizon performance measure closely related time optimality criterion allows systematic incorporation obstacles constraints particularly effective overtaking opponents controllers seek solution around obstacles order deal track constraints obstacle avoidance situations represent feasible set position car time instance slab defined two parallel linear inequalities ensures tractability subproblems convex programming one hand incorporation track obstacle constraints systematic manner hand deal situations overtaking obstacle left right corridor planning algorithm supplies set appropriate convex constraints controllers constraint set result shortest path problem solved dynamic programming first hierarchical control approach presented paper principle similar complexity low enough fully implemented path planner also uses trims one selected whole prediction horizon set trajectories represent cornering conditions nonlinear model gridded velocity steering angle path planner selects trajectory largest progress leave track hits obstacle moving obstacles inherently dealt new trims generated every sampling time low level nonlinear reference tracking mpc controller tracks selected trim additionally use obstacle avoidance constraints described turns effective practice second controller based model predictive contouring control mpcc formulation combining path generation path tracking one problem mpcc essentially plans progressoptimal path taking account nonlinear projection vehicle position onto center line resulting controller able plan follow path similar path horizon chosen long enough nmpc problems approximated locally linearizing continuous nonlinear system dynamics around state trajectory obtain ltv model hierarchical approach linearize around trajectory obtained path planner approach linearization points given shifted trajectory previous sampling time discretize matrix exponential solve resulting quadratic program tailored interior point solver generated forces computation time solvers scale linearly horizon length hence make use long horizons time steps mpcc example solved first control input applied system process repeated next sampling time approach solving nmpc problems corresponds basic version iterations outline paper organized follows section dynamical model car basic properties described section present hierarchical control approach separate path planner path tracking combined approach presented section section briefly discuss method selecting convex constraints two controllers enabling obstacle avoidance dynamic racing situations section presents testbed setup experimental results controllers otation following notation used throughout paper set reals denoted set real column vectors length denoted set nonnegative strictly positive reals denoted respectively set positive definite matrices size denoted set positive matrices size define vector positive definite matrix column vectors scalars denote concatenation odel bicycle model cars modeled using bicycle model done car modeled one rigid body mass inertia symmetry car used reduce bicycle motions considered pitch roll dynamics well load changes neglected used cars rear wheel driven active brakes longitudinal force front wheel neglected resulting model shown figure together resulting differential equations defining model cos sin sin cos sin mvy cos mvx cos equation motion derived around center gravity cog states position cog inertial frame angle car relative inertial frame kinematic part model kinetic part model derived around frame centered cog states longitudinal lateral velocity car finally yaw rate control inputs pwm duty cycle electric drive train motor ffy fry frx fig schematic drawing car model steering angle subscripts indicate longitudinal lateral forces velocities refer front rear tires respectively finally distance cog front rear wheel tire forces model interaction car road important part dynamics goal cars race model tire forces realistic enough represent car high speeds handling limits lateral forces modeled using simplified pacejka tire model see good approximation measured friction curves practice sin arctan arctan sin arctan arctan parameters define exact shape curve longitudinal force rear wheel modeled using motor model electric motor well friction model rolling resistance drag due size cars currently feasible measure wheel speed makes impossible use combined slip models model identified using combination stationary dynamic identification experiments reported allows identifying rear wheel combined slip effects lateral tire friction model thus identified model suitable represent car also handling limits tire forces saturated close saturation stationary velocity analysis better understanding model stationary velocities model investigated similar elaborate analysis presented objective find points model accelerations zero done different constant forward velocities constant steering angles thus problem becomes nonlinear algebraic system equations two equations two unknowns finding solution algebraic equations different steering angles different initial conditions stationary velocities one forward velocity calculated normal driving understeering oversteering normal driving understeering oversteering rad rad rad rad fig stationary velocities plots left plots right parameterized steering angle resulting stationary velocities shown figure resulting stationary velocities categorized three different regions blue line normal driving region characterized linear dependency yaw rate steering angle well small lateral velocities two regions red green correspond understeering points model understeering region green curves saturated front tire force law prevents car achieving larger yaw rates thus car take sharper turn oversteering region red curves rear tire force law fully saturated car drives high lateral velocity usually called drift dynamics model different regions different example oversteering region car drifting steering angle opposite sign curvature car driving known counter steering occur case normal understeering mode stability analysis lateral velocity model around different stationary velocity points also shows difference dynamics normal driving region understeering region stable drifting region unstable figure visible normal steering region getting smaller car driving faster thus maximal minimal steering angle tire forces saturated reduced increasing velocities time maximal yaw rate decreases velocities constant movement car uniform circular movement relationship yaw rate radius curvature known consequently stationary velocity points correspond car driving circle constant radius model predict drifting along circle iii autonomous acing ontrol main part paper present two formulations task autonomous racing cars based model described section first hierarchical approach presented section followed scheme section briefly discuss method obstacle avoidance controllers section summary algorithms given section table part path planning library description rad left straight slight right hard right hard left drift hierarchical receding horizon controller hierarchical control approach path planner finds feasible trajectory within finite number possibilities horizon sampling times trajectory tracked mpc controller employing ensure feasibility optimization times process repeated fashion hence call controller hierarchical receding horizon controller hrhc follows details individual components given following path planning purpose path planner generate trajectory maximal progress feasible car dynamics based gridding stationary velocities nonlinear system described section solving stationary lateral velocity points given set different longitudinal velocities steering angles library zero acceleration points generated includes stationary velocities normal well drifting region table shows excerpt library library used experiments consists stationary points correspond drifting equilibria stationary points selected uniformly gridding longitudinal velocity steps five nine points selected normal driving region four drifting points selected forward velocities note library change hence generated offline generate trajectories whole horizon stationary points integrated computationally cheap assumption stationary velocity reached within one time step since case accelerations zero procedure allows generate reference trajectories also unstable drifting points integration reasonable stationary velocity points leads countable set possible trajectories constant turning radius horizon visualization set candidate trajectories shown figure proceed finding best trajectory among candidates obtained gridding integration equivalent solving following optimization problem max xnj ynj xjk fkm xjk xtrack position orientation time step generated integrating kinematic part bicycle model using stationary velocity library stationary points integration denoted discrete time version model fkm objective find trajectory largest progress track measured projection operator calculates scalar projection position xkj ykj onto piecewise linear center line parameterized arc length length center fig left generation reference trajectories two different longitudinal velocities black slow green fast right solution path planner red trajectory largest progress leaves track orange one optimal trajectory maximizing line positions trajectory xjk need lie set xtrack set admissible positions inside track boundaries finally demand stationary velocity lies set depends current velocity measurement enforces planned velocity reasonably close measured velocity helps deal assumption stationary velocity reached within one time step otherwise two cases correspond selecting drift trajectories respectively based lateral steady state velocity tuning parameters used determine behavior path planner terms selecting candidate trajectories choosing parameters lead loose constraints result typically physically impossible trajectories result crashes conversely constraints tight limit performance controller planned trajectories conservative problem integer program decision variable solved enumeration practice checking discrete positions local convex inner approximations track set reasonable due relatively small number trajectories one hundred case result optimization problem visualized figure path planning allows efficiently handle drift drifting trajectory selected yields largest progress path planner limitations firstly one hand horizon length quite short otherwise planned trajectories become circles always yield infeasibilities respect track constraints even selected velocity would well suited current state horizon short hand leads poor performance path planner recognizes forthcoming curve late second problem whole horizon velocity lead problems complicated curve combinations chicanes model predictive reference tracking reference trajectory path planner generated simplifying assumptions thus possible directly apply controls correspond stationary velocity point order efficiently track reference following mpc formulation used penalizing deviation reference trajectory using quadratic function min kxn xref pksn ref kxk xref qksk kuk tuning matrices penalties soft constraints discussed capture much nonlinear model possible maintaining computational tractability model linearized around reference trajectory necessary linearize around trajectory rather single operating point since position orientation vary significantly horizon reference trajectory construction feasible respect track order also guarantee trajectory optimal mpc solution satisfies track constraints states limited stay inside track possible achieved limiting point horizon lie within two parallel half spaces leads two affine inequality constraints per time step one right one left border resulting convex feasible set figure depicts border constraints formulate soft constraints order avoid infeasibility problems practice using exact penalty function infinity norm case ensure resulting optimization problem always feasible original solution hard constrained problem recovered case would admit solution finally control inputs limited within physical constraints states limited within reasonable bounds avoid convergence problems solver due free variables problem reformulated convex thus efficiently solved work forces employed generate tailored solves instances parameters initial state matrices defining dynamics along reference trajectory well changing halfspace constraints model predictive contouring control contouring control used various industrial applications machine tools milling turning laser profiling applications challenge compute inputs control movement tool along reference path latter given spatial coordinates associated velocities angles etc calculated imposed control algorithm different tracking controllers controller freedom determine state trajectories follow given path example schedule velocity tracking defined reference trajectory contour following problem formulated predictive control framework incorporate constraints see work adapt particular formulation model predictive contouring control mpcc framework obtain controller autonomous racing maximizes travelled distance reference path within prediction horizon experiments use center line reference path employ merely measure progress selecting low weights tracking contouring error result driven trajectory similar driven expert drivers advantage approach path planning path tracking combined one nonlinear optimization problem solved approximating nlp using local convex approximations sampling time resulting qps fig left halfspace constraints red lines one point horizon large red dot right resulting halfspaces red lines whole horizon approximating track green points center line correspond closest point discrete center line point horizon points borders projection center line points borders formulated multistage structure exploited forces obtain solution times range milliseconds posing mpcc problem preliminaries introduced parameterization reference path definition useful error measures parameterization reference trajectory reference path parameterized arc length using third order spline polynomials total length splines obtained offline fitting center line using parameterization obtain point ref ref center line evaluating third order polynomial argument angle tangent path reference point respect ref arctan ref also readily available parameterization leads accurate interpolation within known points reference path accurate linear parameterization employed hrhc section uses linear interpolation points error measures order formulate mpcc problem error measures needed define deviation car current position desired reference point ref ref use definitions give another derivation let projection operator reference trajectory defined arg min ref ref brevity define orthogonal distance car reference path given contouring error sin ref cos ref defined contouring error depicted left picture figure projection operator well suited use within online optimization algorithms resembles optimization problem thus approximation introduced independent variable determined controller approximation useful necessary link via lag error xref yref xref yref ref ref fig contouring error left lag error right linear approximations measures quality approximation lag error depicted right picture figure red segment reference path order independent projection operator contouring lag error approximated function position approximate projection variables controlled mpcc controller sin ref cos ref cos ref sin ref approximate contouring error approximate lag error defined orthogonal tangential component error ref ref position see right picture figure mpcc problem error measures place formulate model predictive contouring control problem autonomous racing objective quality low contouring error amount progress achieved finite horizon sampling times subject model dynamics input constraints min keck position car time step determined nonlinear model version constant control inputs contouring error eck objective defined associated path parameter ref ref orthogonal projection onto reference path two objectives maximum progress tight path following traded weights respectively constraints parallel half space constraints containing position allowed corridor described section corresponds limiting states inputs physically admissible values due implicit dependency objective projection operator optimization problem nlp complex solve thus idea use approximate projection instead introduced section control approximation quality adding cost lag error objective order allow forming lag error time step prediction horizon necessary introduce integrator state dynamics interpreted projected velocity state progress time respectively approximation reduces optimal control problem form nlp amenable implementation min note approximate contouring error used thep objective furthermore replaced maximization final progress measure equivalent approximation accurate sensible lower upper bounds imposed avoid spurious solutions nlp denoting largest possible progress per sampling time cost lag error links state progress dynamics car ensure accurate progress approximation thus strong coupling cost function car model weight lag error chosen high suggested furthermore cost term rate change inputs added objective order penalize fast changing controls helps obtain smooth control inputs order prevent amplifying unmodeled dynamics solving mpcc problem order solve nonlinear optimal control problem realtime local convex approximations form following qps built sampling time linearization nonlinear terms min qksk formed quadratic part linearized contouring lag error cost function stems linear part respectively order keep linearization error small use ltv approximation dynamics well contouring lag errors nonlinear function linearized around output last iteration shifted one stage measurement used first linearization point last input previous iteration kept constant generate new last input linearization point terminal state calculated simulating nonlinear model one time step track corridor constraints formulated two half space constraints tangential track per time step described section fig left combinatorial nature overtaking problem possible overtake opponent left right side center grid shortest path problem solved dynamic programming allowed visit blue grid points right optimal path black resulting corridor issued hrhc section mpcc section constraints formulated soft constraints slack variables corresponding infinitynorm penalty objective weighted chosen quite high recover behavior hard constrained problem whenever possible note described scheme essentially basic iteration nonlinear optimal control problem solved using multiple shooting technique exponential map integrator every time step one step sequential sqp scheme performed hence convergence properties known iteration hold also mpcc problem local solved using forces due rate costs problem lifted obtain multistage structure forces designed introducing copy previous control input stage obstacle avoidance two controllers presented section plan path within track corridor taking account constraints position car thus possible include obstacle avoidance adapting corridor online depending current constellation opponents generating optimal corridor based position obstacles general hard fundamental problem decision side overtake combinatorial nature one opponent involved situation depicted figure left picture work employ high level path planner based dynamic programming decide side overtake obstacles based result adapted corridor given controllers section section high level path planner solves shortest path problem grid figure incorporate planned path lower level controller grid built based last output path planner case hrhc last solution case mpcc minimizes travelled distance additionally deviation last plan lower level controller path close state prediction lower level controller ensures linearizations stay approximately valid time based optimal trajectory corridor constraints easily identified see right picture figure due space restrictions detail many references exist solve corridor problem different approaches example similar problem solved darpa urban challenge goal autonomous driving city traffic several approaches problem presented example based model predictive trajectory generation algorithms modified rrts summary main part paper presented two approaches automatic racing approaches maximize progress within given time horizon directly incorporate track obstacle constraints using two parallel affine inequalities slabs representing feasible set positions time major difference approaches first section algorithm approach separate path planning mechanism combined reference tracking mpc controller second approach section algorithm formulates tasks path planning path tracking one nonlinear optimal control problem based model predictive contouring control algorithm hrhc mpcc algorithm hrhc get current position velocities compute delay compensation function order djustment shift old planned path one stage get position opposing cars run get new borders sec end function run path planner linearize discretize model build problem solve using forces send first control end time slot step mpcc get current position velocities compute delay compensation augment old output section function order djustment get position opposing cars run get new borders sec end function linearize discretize model cost function build problem solve using forces send first control end time slot step controllers send new control input end sampling period order compensate fixed delay nonlinear model simulated forward time interval beginning algorithm see line algorithm nonlinear model integrated using method current state estimate kalman filter initial condition esults section performance controllers evaluated experimental testbed using scale race cars details experimental setup particular implementation performance given following experimental setup experiments use kyosho dnano cars quite sophisticated front rear suspension rear axle differential cars able reach speeds straights corresponds upscaled speed wide speed range high maximal forward velocity small scale introduce complexity present full size testbeds testbed consists infrared camera tracking system control board custom built race track length measured center line see figure pointgrey camera captures frames per second accuracy wide angle lens infrared filter used capture meter track reflecting markers arranged cars different unique patterns used identify car measure position angle cars extended kalman filter ekf used state estimation filtering vision data estimating longitudinal lateral velocity well yaw rate vision data cars passed ethernet connection control runs one two control algorithms described section iii together corridor planner solving problem calculated inputs sent via bluetooth car original electronics dnano cars feature open digital communication link replaced custom made pcb featuring bluetooth chip current voltage sensors infrared vision system reflecting markers car detection ethernet estimation car controller bluetooth fig visualization control loop picture used track motor actuation arm cortex microcontroller low level control loops steering servo traction motor control implementation hrhc implemented ansi embedded platform using arm chip exynos ghz chip identical one used samsung galaxy smartphones runtime environment ubuntu linux version code compiled gcc compiler option sampling time controller discretization time path planner tracking mpc controller horizon length time steps gives ahead prediction tracking mpc controller results multistage variables per stage variables total use library candidate trajectories drifting stationary velocity points mpcc limitation hrhc path planner words acceleration zero whole horizon thus possible use longer horizons increase performance hence controller runs horizon length sampling time corresponds ahead prediction method presented section discretization sampling time based last solution resulting variables per stage total mpcc implemented newer gereration embedded platform using exynos chip based arm cortex architecture running ghz chip identical one used korean version samsung galaxy smartphone embedded board runs ubuntu linux version ansi code mpcc controller compiled gcc option improve convergence problem data scaled lie objective hessian linear term scaled factor reduce large entries occurring due lag error term note scalings change minimizer optimality criteria forces solver adjusted terminate sufficiently good solution found residuals equality inequality constraints set duality gap set single car racing driven trajectories single car depicted figure velocity profile encoded colors depicted three laps hrhc achieves lap time mpcc yields lap times better understand driven trajectory hrhc path planner hrhc analyzed path planner comparably short prediction horizon velocities whole horizon constant leads turns constant radius path planner works fine even curves however car tends drive outer border curve allows driving wider radius faster longitudinal velocity thus maximizes progress however limitation car drive combination curves position car end first curve essential low overall time problem seen figure example chicane lower left corner furthermore due short lookahead hrhc path planner able prevent ill positioning relative curve ahead due suboptimal position relative curve controller brake compared car ideal line even possible car touches borders see narrow curve center track figure limitations present mpcc approach comparably long horizon employed unlike hrhc velocity fixed horizon computed time step result nlp consequence features mpcc able plan path even complicated combination curves results trajectory closer ideal line least curvature sense especially complicated curves like chicane lower left corner figure curve combination mpcc achieves velocities smaller hrhc reduce velocity nearly drive chicane additionally causes car drive overall longer distance however mpcc disadvantages compared hrhc current implementation mpcc still tracks center line seen first straight top left corner movement center line clearly visible figure clearly due objective function mpcc completely avoided even contouring cost small hrhc hand cost related center line uses measure progress thus car drives straight aforementioned segment track second disadvantage seen lower end track mpcc small shape driven trajectory hrhc hand shows section driven completely straight gives speed benefit probably also attributed contouring error penalty would beneficial drive order follow center line summary results indicate hrhc limited path planner fact whole horizon velocity allowing multiple velocities within one prediction horizon would probably increase performance controller however complexity path planner grows exponentially number different velocity triples horizon mpcc limitation able plan better trajectories also improve closed loop performance however mpcc comes price first terms computational cost factor five expensive hrhc comparing computation times platform see section details computation times second terms robustness experiments relinearization scheme turned sensitive measurement errors model drift particular sudden unexpected skidding car cause problems scheme drifts also responsible high variations different trajectories certain areas track furthermore since mpcc relies results previous iteration compute linearizations sudden jumps corridor path planner might make previous trajectory infeasible situation several time steps might needed recover feasibility planned trajectory although using soft constrained formulation sensible inputs still provided car hrhc hand need information last iteration current position velocity needed helps deal efficiently unexpected behavior fast changing measurements corridors obstacle avoidance mechanism racing multiple cars since controllers plan path around opponents maximizing progress fast safe overtaking maneuvers enabled current work focus static obstacles seems sufficient outperform human drivers robustly overtake dynamically moving obstacles automatic controllers expert drivers would necessary prediction behavior currently available predictions could however systematically incorporated hrhc controller section mpcc controller section fig driven trajectory velocity profile different laps video hrhc controller available https hrhc controller section mpcc controller section fig driven trajectories static obstacles different laps video mpcc obstacle avoidance available https corridor planner outlined section would merely change corridor issued controllers presented section section obstacle avoidance situation shown figure detailed zoom taken three different time points figure zooms show prediction horizon corresponding planned trajectories due limited path planner hrhc able directly plan path around obstacles beginning thus avoids first three cars without predicting overtake next two cars car made enough progress overtaken first group obstacles controller find way two cars mpcc hand plans path around obstacles even reaches first car due long horizon advantage model mismatch significant lead problems planned path optimistic computation times computation times components two controllers given tables iii laps depicted figure single car racing figure avoidance opponents note computation times individual blocks two controllers compared directly hrhc controller section mpcc controller section fig obstacle avoidance different time steps grey indicates corridor chosen obstacle avoidance algorithm directly compare computation times two controllers horizon length identical however due constant velocity limitation hrhc path planner horizon length identical mpcc impossible without using multiple segments constant velocity would make path planning combinatorial complexity thus currently prohibitive implementation vice versa mpcc work robustly short horizon length used hrhc multiple car case significant impact execution times lower level controller path planning hrhc gets slightly complicated however computation time corridor planner approach section varies significantly depending complexity racing situation mpcc limiting factor respect sampling time computation time solving main bottleneck hrhc path planner high maximal computation time caused back rules case feasible trajectory found hand expensive step average large variation computation time makes less critical sampling time missed hrhc controller one sampling instant three laps mpcc controller sampling instants three laps time overall experiments demonstrate schemes implemented soft sampling rates acknowledgment gratefully acknowledge work students course race car project kenneth kuchera samuel zhao florian perrodin fruitful help implementing mpcc approach michael janser contribution obstacle avoidance algorithm sandro merkli nils wenzler marcin dymczyk development embedded control software testbed setup celestine sandro merkli building track filtering software michael table hrhc computation times milliseconds without obstacles mean stdev max obstacles mean stdev max border adjustment border adjustment path planning generation path planning generation forces forces mean stdev max table iii mpcc computation times milliseconds without obstacles mean stdev max obstacles border adjustment border adjustment generation generation forces forces dahinden christian stocker initial design embedded pcb board benjamin keiser current version infrared vision system furthermore thank sean summers nikolaos kariotoglou christian conte john lygeros automatic control laboratory supervising various projects providing important comments methods presented paper special thanks colin jones originated idea race car testbed research leading results received funding via embocon eferences faulwasser kern findeisen model predictive constrained nonlinear systems proceedings ieee conference decision control held jointly chinese control conference lam manzie good model predictive contouring control conference decision control cdc domahidi zgraggen zeilinger morari jones efficient interior point methods multistage problems arising receding horizon control conference decision control cdc maui usa domahidi forces fast optimization control embedded http horowitz varaiya control design automated highway system proceedings ieee vol buehler iagnemma singh darpa grand challenge great robot race vol springer buehler iagnemma singh darpa urban challenge autonomous vehicles city traffic vol springer kritayakirana gerdes using centre percussion design steering controller autonomous race car vehicle system dynamics vol velenis frazzoli tsiotras cornering equilibria 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multiplexing based nonlinear fourier transform illem oossens ansoor yousefi ves artmut afermann mathematical algorithmic sciences lab paris research center huawei technologies france communications electronics department telecom paristech paris france jul abstract multiplexed pdm transmission based nonlinear fourier transform nft proposed optical fiber communication nft algorithms generalized scalar nonlinear equation one polarization manakov system two polarizations transmission performance pdm nonlinear multiplexing nfdm pdm orthogonal multiplexing ofdm determined shown transmission performance terms approximately single polarization nfdm twice data rate dispersion seriously degrade system performance compared achieves gain theory generalized fibers strong coupling regime paving way application nft address nonlinear effects multiplexing optical society america ocis codes fiber optics communications multiplexing nonlinear optics fibers references links hasegawa tappert transmission 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first experimental demonstration nonlinear inverse synthesis transmission transoceanic distances optical fiber communications conference ieee philips prilepsky kamalian ellis harper turitsyn achievable information rate nonlinear inverse synthesis based ofdm transmission european conference optical communication ecoc vde buelow aref demonstration nonlinear frequency division multiplexed transmission optical fiber communication conference optical society america aref schuh idler experimental demonstration nonlinear frequency division multiplexed transmission european conference optical communication ecoc ieee aref buelow demonstration fully nonlinear spectrum modulated system highly nonlinear optical transmission regime european conference optical communication ecoc vde maruta matsuda polarization division multiplexed optical eigenvalue modulation international conference photonics switching ieee yangzhang yousefi alvarado lavery bayvel nonlinear multiplexing focusing regime optical 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solitons instability rogue waves rogue shock waves nonlinear dispersive media springer agrawal nonlinear fiber optics academic press civelli forestieri secondini noise dispersion may seriously hamper nonlinear frequencydivision multiplexing https mollenauer evangelides haus soliton propagation using lumped amplifiers dispersion shifted fiber lightwave technol blow doran average soliton dynamics operation soliton systems lumped amplifiers ieee photonics technol lett wai menyuk chen stability solitons randomly varying birefringent fibers opt lett evangelides mollenauer gordon bergano polarization multiplexing solitons lightwave technol bertolini rossi serena bononi correct nonlinear pmd emulation systems opt fiber technol eberhard braimiotis numerical implementation method varying differentialgroup delay springer boston poole winters nagel dynamical equation polarization dispersion opt lett mumtaz essiambre agrawal nonlinear propagation multimode multicore fibers generalization manakov equations lightwave technol introduction nonlinear multiplexing nfdm elegant method address nonlinear effects optical fiber communication scheme viewed generalization communication using fiber solitons goes back original idea eigenvalue communication approach information encoded nonlinear spectrum signal defined means nonlinear fourier transform nft evolution nonlinear spectral components fiber governed simple independent equations result combined effects dispersion nonlinearity compensated digital domain application filter furthermore communication achieved network environments nonlinear spectrum consists continuous part case anomalous dispersion fiber also discrete part solitonic component principle modulated prior work mostly focused either continuous spectrum modulation discrete spectrum modulation transmission based nft experimentally demonstrated continuous spectrum discrete spectrum well full spectrum modern coherent optical fiber systems based multiplexed pdm transmission improve achievable rates however research data transmission using nft limited scalar nonlinear equation take account polarization effects exception paper overcome limitation generalizing nfdm manakov system proposing develop stable accurate algorithm compute forward inverse nft signal shown based continuous spectrum modulation feasible data rate approximately doubled compared single polarization nfdm compared exhibits peak gain system spans standard fiber qam paper set discrete spectrum zero modulate continuous spectrum even though available degrees freedom modulated achievable rates reach remarkably close upper bound differential group delay randomly varying birefringence give rise temporal pulse broadening due dispersion pmd pmd might compensated conventional systems may even beneficial reducing nonlinear distortions hand pmd changes nonlinear interaction signals two polarizations impact pmd nfdm fully investigated yet paper show linear polarization effects equalized receiver using standard techniques pmd seriously degrade performance channel model two polarizations light propagation two polarizations optical fiber modeled coupled nonlinear equation cnlse fiber birefringence usually varies rapidly randomly along fiber practical systems scale meters assumption averaging nonlinearity cnlse leads equation jones vector containing complex envelopes two polarization components denotes distance along fiber represents time pauli matrix depending state polarization constant numbers term responsible pmd second line represents loss chromatic dispersion kerr nonlinearity factor front nonlinear term stems polarization averaging nft applies integrable equations however system apparently integrable presence loss pmd loss may approximately compensated using ideal distributed raman amplification leading lossless model distributed additive white gaussian noise awgn discrete amplification short spans also modeled similarly modified nonlinearity coefficient depending loss see ignoring pmd moment noise reduced convenient normalize let introducing normalized variables simplified manakov equation normal dispersion defocusing regime anomalous dispersion focusing solitonic regime shown integrable follows develop corresponding nft nft signals section basics nft theory signals well numerical algorithms compute forward inverse nft briefly presented brief review theory equation represented lax pair means operators found correspondence lax equation lax pair manakov equation found manakov operator simplify presentation consider focusing regime eigenvalue problem solved jost function eigenvector assuming signals vanish gives rise six boundary conditions denoted exp exp unit vectors boundary conditions bounded eigenvalue problem boundary conditions solved obtaining six jost functions shown form orthonormal basis solution space thus expand basis called nonlinear fourier coefficients shown focusing regime inner product two jost functions corresponding eigenvalue depend time implies depend time notation suggests nft defined nft solutions important property nft evolves distance according filter remark note set zero manakov equation associated operator equation operator reduced respectively scalar nlse corresponding operator also theory section straightforwardly generalized include number signals instance modes propagating fiber strong coupling regime remark since jost vectors orthonormal times implies unimodularity condition used algorithm forward nft algorithm develop nft algorithms continuous spectrum considered paper forward inverse nft algorithms respectively based discrete layer peeling dlp methods algorithms generalize corresponding ones begin rewriting eigenvalue problem form remainder section suppress dependence coordinate discretize time interval according set similarly vector method discretization follows discretization becomesp periodic function period introduced normalization factor improve numerical stability iterative equation initialized iterations nonlinear fourier coefficients obtained projections onto continuous component nft computed based algorithm efficiently implemented frequency domain let write discretize interval let tilde denote action discrete fourier transform dft respect dft discrete frequency note dft shift shift denotes circular right shift array one element equation frequency domain shift shift shift shift initial condition given dft found recovering inverse dft using inverse nft algorithm inverse nft algorithm consists two steps first compute continuous spectra invert forward iterations second iteration compute substituting unimodularity condition compute absence discrete spectrum log log log since therefore log analytic functions recover phase log denotes hilbert transform inverse iterations obtained inverting matrix dropping terms order yield accuracy forward iterations practice forward nft better invert frequency domain iterations signal obtained follows recall shift using initial conditions forward iterations straightforward show particular first element shift shift obtain nft inft nft inft nft inft nft inft fig comparing nft inft displaced gaussians fixed sampling rate algorithm less accurate higher equations solved testing nft algorithms forward inverse nft numerical algorithms tested follows figure compares signal containing two polarization components nonlinear fourier domain reconstruction successive inft nft operations take two displaced gaussians standard deviation input signals two polarization components set time windows take samples time nonlinear fourier domain discrete layer peeling method signal periodic nonlinear fourier domain period deviations occur amplitude signal increased accuracy nft inft enhanced increasing sampling rate time nonlinear frequency reduces errors due approximation signal find roughly need twice number samples achieve accuracy case transmitter receiver section describe transmitter receiver digital signal processing dsp schematic diagram multiplexed nfdm ofdm transmission systems shown fig dsp produces digital signals quadrature components polarization components fed modulated signals two polarization components combined polarization beam combiner enter transmission line consisting multiple fiber spans consider practically relevant case lumped amplification using fiber amplifiers edfas propagation fiber two polarization components separated polarization beam splitter fed intradyne coherent receiver provides input dsp briefly describe dsp ofdm nfdm first map incoming bits signals taken qam constellation oversample signal time domain introducing zeros frequency domain outside support signal add guard intervals time domain increasing guard interval increases accuracy inft unless stated otherwise increase guard intervals computation inft amplitude increased penalty due inaccuracies algorithm nfdm steps performed called see ssmf xin dsp mod xin mod yin dsp nfdm dsp ofdm ofdm data encoder ofdm data encoder oversample time include guard interval edfa xout yout spans xout qxout yout yout coh dsp nfdm dsp ofdm separate bursts separate bursts oversample time normalize normalize include guard interval manakov nft fft one tap channel cdc dbp dsp manakov inft ifft unnormalize unnormalize remove guard interval remove guard interval combine bursts combine bursts training sequence equalization training sequence equalization downsample time downsample time minimum distance decoder minimum distance decoder fig system diagram nfdm ofdm transmission systems processing steps transmitter receiver dsp steps highlighted purple require joint processing polarization components explanation see text signal subsequently mapped nonlinear fourier domain transformation log step analogue ofdm note contrary case energy signal time domain approximately proportional energy point perform inverse nft case nfdm inverse fft case ofdm obtain unnormalized signal introducing units using finally combine ofdm nfdm bursts obtain signal transmitted output dsp quadrature components signals two polarization components steps dsp performed independently two polarizations except inft requires joint processing invert steps dsp signal processing begins burst separation signal normalization using followed forward nft fft case nfdm ofdm nfdm subsequently equalize channel using phase compensation similarly ofdm either compensate dispersion phase exp nspan perform digital backpropagation dbp fixed number steps per span remove guard intervals time signal time domain obtain output symbols bit error rate ber calculated using minimum distance decoder symbols fig compare time domain signals nfdm ofdm one polarization components use subcarriers burst duration cases parameters ref correspond burst data rate per polarization baudrate nfdm shape signal time domain changes amplitude increased nonlinear fourier domain signal power dbm one see significant amount broadening nfdm signal time domain effect makes difficult control time duration signals nfdm nfdm ofdm jqj jqj nfdm ofdm fig nfdm ofdm signals left panel right panel using ideal model dbm without noise one two polarizations shown nfdm signal broader due nonlinear effects nft time duration signals parameter value length span number spans number carriers burst duration guard interval burst data rate effective data rate bandwidth ghz oversampling factor constellation qam noise figure attenuation dispersion nonlinear index core area table system parameters used simulations pulse broadening time domain due chromatic dispersion estimated signal bandwidth guard time intervals minimizing interaction among bursts approximated using parameters table obtain add margin obtain total symbol duration corresponds effective data rate right panel fig shows time domain signals nfdm ofdm one polarization propagation governed integrable model consider transmission realistic fiber model sec observe amount temporal broadening nfdm ofdm signals simulations employ may added increase data rate simulation results section consider system shown fig compare via simulations taking account loss pmd first consider model loss periodic amplification setting pmd zero namely neglecting terms first line next pmd effects studied sec system parameters used summarized table lossless lossy transf lossless ber osnr fig ber function osnr nfdm lossless lossy models effect loss lumped amplification lumped amplification scheme signal periodically amplified span channel described model including attenuation term manifestly integrable effect lumped amplification nft transmission studied previously illustrate effect attenuation comparing ber function osnr fig four models configuration lossless model propagation without loss lossy model lumped amplification transformedlossless model introduced producing fig artificially introduced awgn receiver exclude effects interaction comparison fix power keep accuracy nft algorithms vary osnr changing noise power see ber lossless models approximately result verifies implementation nft correct also shows effect burst interaction due finite guard interval negligible ber lossy model significantly higher ber lossless models high osnrs ber lossy model seems flatten osnr increased fiber loss taken account nft follows attenuation term eliminated change variable transforms equation integrable equation thus zero loss modified nonlinearity parameter refer resulting model model note power signal model higher power original lossy model yields higher amplitudes according nlse manakov dbm fig comparison nft transmission based nlse single polarization manakov equation denotes total power signal low power offset curves signal power doubles two polarization components propagate higher power interactions decrease gap figure shows proposed scheme loss cancellation indeed effective using inverse forward nfts ber lossy models almost identical transmission performance order assess transmission performance transmission performed system simulations described sec focus deterministic impairments neglect pmd simulate transmission subcarrier nfdm ofdm pulses based constellation nspan spans standard fiber assume ideal amplifiers noise figure system parameters summarized table comparable ref facilitate comparison case first compare nft transmission based nlse polarizationmultiplexed transmission based manakov equation figure shows linear regime two curves offset polarization multiplexing doubles transmission power without increasing signal noise ratio note calculation increased guard band thereby sampling rate frequency domain factor two computation guard band transmission compared case avoid penalties due inaccuracies nft amount approximately peak power guard bands used without penalty peak power roughly cases implies data rates approximately doubled using polarization multiplexing show sec conclusion still holds presence polarization effects fig compare function launch power transmission optimum launch power ofdm significantly smaller nfdm ofdm transmission limited nonlinearity nfdm mainly limited nonlinear interaction optimum launch power significantly higher nfdm ofdm observe overall gain figure shows nfdm ofdm ofdm dbp ofdm dbp ofdm dbp dbm fig comparison ofdm nfdm transmission idealized setting use manakov equation without spans nfdm performs well ofdm dbp full potential nfdm leveraged scenario dbp less efficient fig received constellations ofdm left panel without dbp nfdm right panel respective optimal launch power noise nfdm clearly corresponding received constellations respective optimal launch power fig report results obtained digital backpropagation ofdm transmission fair comparison nfdm use sampling rate dbp apply without prior downsampling ofdm dbp achieves similar performance nfdm around dbp steps per span exceeds steps per span note considered scenario full potential nfdm leveraged network scenario dbp less efficient results good qualitative agreement reported ref case achieving twice data rate effect pmd simulation pmd fiber model describes light propagation linearly birefringent fibers difference propagation constants two polarization states induced fiber imperfections stress real fibers modal birefringence varies randomly resulting pmd simulate pmd method continuous variations ntaps fig effect number taps different values dpmd power signal simulation dbm birefringence approximated large number short fiber sections birefringence kept constant pmd thus emulated distributed fashion use sections length larger typical fiber correlation lengths beginning section polarization randomly rotated new point sphere apply uniform random phase new reference frame latter accounts fact reality birefringence vary sections assumed constant lead random phase relationship two polarization components differential group delay dgd section selected randomly gaussian distribution way artifacts wavelength domain caused fixed delay sections avoided within scattering section equation without pmd terms solved using standard fourier integration speed simulations employ based implementation mex interface matlab resulting dgd fiber maxwell distributed exp distribution mean related rms delay maxwell average dgd varies square root fiber length dpmd dpmd pmd parameter values fibers used telecommunications range modern fibers pmd impact nfdm section consider polarization effects nft transmission since nonlinear term manakov equation invariant polarization rotations equalization done nonlinear fourier domain fig use simple training sequence based equalization algorithm compensate linear polarization effects samples first nfdm symbol used training sequence filter taps equalizer determined using birefringence dbm fig effect pmd used taps equalizer dpmd respectively estimation obtain average ber random realizations pmd data point due statistical nature effects pmd often quantified terms outage probabilities practice system designed tolerate certain amount pmd case fixing number taps equalizer dgd exceeds margin system said outage first determine number taps achieve given outage probability fig shows function number taps different values pmd parameter dpmd random polarization rotations dgd hence interaction nonlinearity pmd use case reference upturn small number taps due fact couple taps required fully compensate polarization rotations presence noise finite pmd curves different pmd values converge result within error bars estimate required number taps maxwell distribution order system fully compensate pmd cases outage probability equalization must cover time interval equal mean plus standard deviations dgd distribution example dpmd interval corresponds using sampling frequency find approximately need taps converge consistent figure note construction equalizer also corrects potential phase rotations therefore least part nonlinear effects order separate effects due interaction nonlinearity pmd performed calculation reversed polarization rotations phases exactly instead using equalizer keeping track randomly generated angles dgd values fiber simulation find small phase rotation consistent smaller one reported case remains negligible peak power fig compare function launch power without pmd effects use number taps determined described observe penalty roughly peak power relative case without pmd dpmd pmd penalty therefore serious typical fibers used telecommunication compared case without pmd effects random birefringence labeled birefringence find penalty roughly peak power case zero pmd due equalization conclusions paper proposed multiplexing based nonlinear fourier transform nft algorithms developed based manakov equations simulations demonstrate feasibility polarization multiplexed nfdm transmission standard fiber results show data rates approximately doubled transmission compared case important step achieve data rates exceed conventional linear technology numerical simulations polarization multiplexed transmission realistic fiber model including randomly varying birefringence polarization mode dispersion shown penalties due pmd real fibers seriously impact system performance fiber simulations based manakov equations next step beyond scope paper results verified experimentally equation governing light propagation modes fibers strong coupling regime written form denotes average group velocity dispersion nft well algorithms presented generalize equation straightforward manner approach therefore paves way combining nonlinear fourier transform multiplexing acknowledgements would like thank wasyhun gemechu helpful discussions acknowledges useful discussions djalal bendimerad majid safari sergei turitsyn huijian zhang
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arxiv feb numerical method solution ordinary differential equations fractional order jacek leszczynski mariusz ciesielski technical university czestochowa institute mathematics computer science dabrowskiego czestochowa poland jale cmariusz abstract paper propose algorithm numerical solution arbitrary differential equations fractional order algorithm obtained using following decomposition differential equation system differential equation integer order connected inverse forms equations algorithm used solution linear equations introduction opposite differential equations integer order derivatives depend local behaviour function fractional differential equations accumulate whole information function weighted form called memory effect many applications physics chemistry engineering etc reason need method solving equations effective applied equations general form however known methods used solution equations disadvantages analytical methods described detail work case arbitrary real order another analytical method uses multivariate function generalizes previous results used linear type equations hand specific differential equations oscillating periodic solution specific numerical methods numerical methods allow solution equations arbitrary real order work properly relatively simple form fractional equations let consider initial value problem fractional differential equation connected initial conditions given function defined interval unknown function solution eqn fractional derivative operator defined sense also definitions operator like caputo regarding riemannliouville operator define operator dtm study also use integral operators defined sense integral operator defined informations concerning operators properties one find literature applying definitions following property represents typical derivative operator integer order according fractional integrals composition rule occurs general case fractional derivatives commute extending considerations integer operator commutes fractional operator dtm opposite property impossible mixed operators derivatives integrals commute following way commute opposite way turn attention composition rule two following ways first found literature fact authors neglect general property fractional derivatives given eqn regarding solution fractional differential equations apply properties next sections mathematical background section concentrate describing method decomposition fractional differential equation system one ordinary differential equation inverse forms equations focus decomposition ways arbitrary fractional differential equation initial conditions considerations classify fractional differential equations dependence fractional derivative occurrence equations occurs one equations occurs much times corollary arbitrary equation initial conditions taken account decomposed following system proof use property introduce new variable call temporary function introduction dtm new variable represents abel integral equation solution found literature left inverse operator dtm dependence integer order introduce initial conditions multiplied term term issues kernel operator system eqns first equation differential equation integer order next equations represent inverse forms equations also found literature different similar approaches solution eqn mathematical point view interesting second class called equations let consider eqn initial conditions previous class use rule following previous corollary apply system new variables itmi decompose eqn system following equations dtmn itmn system final form requires additional transformations multiterm class distinguish two depending follows decomposition independent subclass different derivatives orders integer type dependent subclass one find dependence derivatives order integer type etc proposition taking consideration independent subclass possible formulate unique solution eqn following form proof proof try show dependence temporary functions temporary function assume initial conditions way easily construct proof observe abel integral operators order consider inverse form operators two operators respectively using property obtain neglecting operator obtain following formula arbitrary constant additionally multiply operator eqn applying property obtain differentiating sides operator applying property see literature obtain final dependence general way state depends together additional term represented constants proposition constants eqn equaled zero proof taking consideration formula calculate constants additionally putting initial condition obtain remark according previous considerations initial conditions temporary functions satisfy following dependence practical point view given formula solve ordinary differential equation integer type variable coefficients zeros initial conditions temporary functions let consider next subclass fractional differential equation assume integer orders may happen case belong integer values introduce parameter denotes number temporary functions derivative operator integer type different temporary functions express functions function may observe relationship eqn functions respectively assume equalities integer orders previous relationship becomes following property ordinary differential equation presented system change depends moreover introduce new temporary function following form need find inverse form eqn denotes left inverse operator operator apply known laplace transform eqn find left inverse operator expression formula suitable practical purposes computations proposition connecting expressions solution second subclass fractional differential equation presented follows need proof proposition done previous subclass class numerical treatment examples calculations section propose explicit numerical scheme numerical solution differential equation integer order apply euler method limits extend approach method according results presented oldham spanier use numerical scheme integral operator valid arbitrary also use algorithm given oldham spanier numerical differentiation algorithm depends range literature one may find algorithm highest values limited practical solution eqn found literature babenko method general form eqn consider left inverse operator found simplified formula limited two fractional orders binomial expansion show comparison select popular fractional differential equation literature called equation fig presents qualitative comparison numerical solution generated method analytical solution assume function numerical results podlubny work function fig solution equation comparison results analytical solution podlubny work numerical solution form equation initial conditions respectively analytical solution one may find also found simple way solution created podlubny work fig certifies numerical results behave good comparison analytical solution great advantage would apply approach form fractional differential equation let consider form fractional differential equation initial conditions equation term introduced fig shows behaviour numerical solution eqn see step strong influence solution fractional differential equations need computational tests choose right value step conclusions paper propose new method numerical solution arbitrary differential equations fractional order following blank results main advantage method decomposition fractional differential equation system composed one ordinary differential equation integer order left inverse equations operator distinguish two classes system fractional derivative fractional derivatives comparison certifies method gives quite good results summarizing results say decomposition method general form gives reasonable calculations effective method easy use applied fractional differential equations general form references babenko heat mass transfer khimiya leningrad russian blank numerical treatment differential equations fractional order numerical analysis report manchester centre computational mathematics menchester carpinteri mainardi eds fractals fractional calculus continuum mechanics springer verlag vienna diethelm algorithm numerical solution differential equations fractional order elec transact numer anal goto oldham semiintegral electroanalysis shapes neopolarogeographic plateau anal konguetsof simons construction exponentially fitted methods numerical solution schrodinger equation journal computational methods sciences engineering oldham spanier fractional calculus academic press new york london palczewski ordinary differential equations wnt warsaw polish podlubny fractional differential equations academic press san diego riewe nonconservative lagrangian hamiltonian mechanics phys rev riewe mechanics fractional derivatives phys rev samko kilbas marichev integrals derivatives fractional order applications gordon breach london simons finite difference methods solution schrodinger equation computers chemistry simons atomic structure computations chemical modelling applications theory hinchliffe umist royal society chemistry
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adaptive computation klee measure high dimensions pablo javiel oct departmento ciencias universidad chile chile jeremy jrojas escuela civil universidad chile abstract klee measure boxes volume union computed time within worst case describe three techniques boost computation one based type degeneracy input two ones inherent easiness structure input first technique benefits instances maxima input small size yields solution running time within second technique takes advantage instances hyperplane intersects boxes dimension yields solution running time within log third technique takes advantage instances intersection graph input small treewidth yields algorithm running time within log log general time within log optimal tree decomposition intersection graph given show combine techniques algorithm takes advantage three configurations introduction klee measure set boxes defined volume union boxes set computation first posed victor klee originally considered measure intervals real line bentley generalized problem dimensions described algorithm running time within log several years later overarms yap described solution running time within log remained essentially unbeaten years chan presented algorithm running time within consider additionally domain box given making objective compute klee measure within special cases problem studied hypervolume problem boxes orthants form corresponding author real number solved time within polylog boxes hypercubes solved running time within polylog yildiz suri considered case projection input boxes first dimensions instance hypervolume described algorithm solve time within dimension best lower bound known computational complexity klee measure problem worst case instances size within log far tight dimensions one two best known upper bound dimension chan conjectured purely combinatorial algorithm computing klee measure dimension exists running time within proved klee measure problem solved time one decide whether arbitrary graph contains clique size time within current best combinatorial algorithm finding graph requires time hence conjecture adaptive analysis cost algorithm measured function input size parameters capture inherent simplicity difficulty input instance algorithm said adaptive easy instances solved faster hard ones adaptive algorithms solve classical problems sorting permutation sorting multiset computing convex hull set points plane computing maxima set vectors also adaptive algorithms maximum weight box problem particular interest since dimension maximum weight box problem reduced instance klee measure problem dimensions even though asymptotic complexity best known far klee measure problem many cases solved time within see figures examples easy instances mere particular cases others hints hidden measures difficulty klee measure problem hypothesis difficulty measures gradually separate instances size various classes difficulty easy ones solvable time within log difficult ones best known algorithm solves time within results describe three techniques boost computation klee measure easy instances analyze adaptive model technique identify proper difficulty measure models features technique taking advantage first technique simplest taking advantage degenerated instances second third ones sophisticated fig two easy instances klee measure problem red dashed boxes removed without affecting klee measure within domain yielding much smaller instance solve instance belongs family instances intersection graph tree path particular case solved time within log algorithm first technique described section related classical problem computational geometry computation maxima set vectors vector set called maximal none remaining vectors dominates every component maxima denoted set maximal elements kirkpatrick seidel gave sensitive algorithm problem running time within size maxima extend concept maxima sets boxes describe algorithm computing klee measure time within denotes size maxima input set second technique described section based profile input set amore defined minimum dimensions maximum number boxes intersected axis aligned hyperplane orthogonal algorithm described chan compute klee measure fact sensitive difficulty measure based slightly weaker definition profile improve describing algorithm compute klee measure time within log profile input set third technique described section based treewidth intersection graph input set intersection graph set boxes graph vertices boxes two boxes connected edge intersect treewidth graph measures close graph tree technique yields solution running time within log tree decomposition intersection graph input set given solution running time within log log boxes given section discuss compare combine three techniques section describe potential directions future work maxima filtering first technique considers maxima input set boxes take advantage instances many boxes filtered small time box set called maximal none remaining boxes completely contains maxima set maximal elements one observe definition elements maxima input set klee measure problem removed input set without affecting value klee measure algorithm takes advantage fact compute klee measure time sensitive size maxima input set algorithm maxima adaptive measure input domain box set boxes output klee measure within compute maxima return sdc overmars showed maxima set vectors computed time maxima set boxes computed time within prove overmars expressed box dimensional vector note boxes dominates dominates use result show lemma klee measure computed running time sensitive size maxima input lemma let set boxes box klee measure within computed time within log size maxima proof algorithm achieves bound given lemma consequence overmars result maxima computed step one time within using sensitive algorithm described kirkpatrick seidel second step klee measure size computed time within using algorithm proposed chan result follows note bound lemma base exponent instead bound running time sdc degenerated instances significantly smaller bound lemma significantly better one way improve result remove dominated elements recursive call chan algorithm discuss difficulties analyzing approach section next section describe another boosting technique still reduces chan algorithm less focused degenerated instances partitioning profile set boxes defined maximum number boxes intersected hyperplane orthogonal dimension profile set boxes defined amore showed compute linear time sorting boxes dimension make observation chan algorithm problem adaptive measure slightly different profile weaker improve result describing technique yields solution sensitive profile intrinsic adaptivity chan algorithm simplify divide conquer algorithm sdc short proposed chan compute klee measure already behaves adaptively sense runs faster large families instances let set boxes defined max denotes profile observation let set boxes within domain box algorithm sdc computes klee measure within time within log proof observation quite technical long since result next section subsumes one analysis considerably simpler omit proof observation example class instances small illustrated figure next subsection describe slightly modified version algorithm sdc runs time sensitive profile rather improvement since instances partitioning let set boxes profile domain box given profile algorithm splits slabs measure within equal summation measures within respectively algorithm performs plane sweep one dimensions smallest profile cuts domain hyperplane every endpoints algorithm input domain set boxes profile output partition slabs intersecting one boxes let dimension profile equals let endpoint within split hyperplane let return computing klee measure within summing values one compute klee measure within slabs divided algorithm intersect boxes definition profile boxes intersect boundaries slab since slab contains endpoints boxes completely lie interior used bound running time computation klee measure lemma let set boxes domain box denote profile klee measure within computed time within log proof using algorithm one split domain slabs linear time sorting input measure within slab computed time within using algorithm sdc result follows note bound lemma value exponent profile instead size set instances small profile like ones class illustrated figure bound lemma significantly better upper bound running time sdc next section describe technique based treewidth intersection graph input set measure captures close graph tree technique takes advantage inputs intersection graph small treewidth intersection graph treewidth instances one described figure intersection graph tree minor variant sdc performs time within log independently dimension concept treewidth discovered independently several times different names nice introduction see sections kleinberg tardos book many graph problems nphard general graphs solved polynomial time graphs small treewidth example arnborg proskurowski showed problems linear time algorithms trees algorithms solving time linear size graph exponential treewidth illustrated idea classical optimization problems involving independent sets dominating sets graph coloring hamiltonian circuits network reliability section describe generalize behavior instances general intersection graph taking advantage treewidth intersection graph recall definition basic results treewidth section apply computation klee measurein section preliminaries widely used treewidth definition based tree decompositions introduced robertson seymour definition tree decomposition graph pair family subsets tree node coverage edge coverage coherence simple path one refers elements nodes elements vertices width tree decomposition tree decomposition called optimal width minimum width among tree decompositions treewidth graph width optimal decomposition see figure illustration tree decompositions treewidth fig tree decomposition set boxes intersection graph set optimal tree decomposition graph treewidth use denote vertices associated nodes gvi denote induced property let node suppose components gvd vertices common edges tree decomposition nonredundant edge simple procedure make tree decomposition nonredundant without affecting width edge one contract edge folding node repeating process often necessary one ends nonredundant tree decomposition property nonredundant tree decomposition graph nodes determine treewidth given graph furthermore known algorithm compute approximation optimal tree decomposition polynomial time best polynomial time tion algorithms tree decompositions log approximation algorithm running time within described feige log approximation algorithm running time within log described amir treewidth known constant approximation computed time within log approximation computed time within using two algorithms described bodlaender respectively algorithm sensitive intersection graph treewidth describe algorithm benefits instances intersection graph small treewidth say set boxes treewidth intersection graph treewidth intuitively consider set boxes tree intersection graph klee measure computed fashion reduce problem two dividing intersection graph via vertex removal two roughly equal sizes solve problem independently combine solutions since intersection graph tree one always find vertex divides tree two forests size adding back forests obtain size klee measure original instance sum klee measure minus measure intersection volume box vertex used split procedure yields algorithm running time within log procedure extended computation klee measure instance tree width given tree decomposition intersection graph following lemma shows solutions two combined general solution node denote subset boxes corresponding vertices within lemma let tree decomposition intersection graph set boxes node removed split two nonempty let tree decompositions respectively klee measure equals klee measure plus klee measure minus klee measure proof property know share vertices edge hence box corresponding vertex node intersect box corresponding vertex node therefore intersection fact volume box klee measure box lebesgue measure proves klee measure equals klee measure plus klee measure minus klee measure result follows using lemma apply procedure described trees general tree decompositions algorithm lemma provides upper bound running time new solution algorithm measure input domain box set boxes tree decomposition intersection graph output klee measure within let node measure computed measures sdc return measures else find node removed splits two sizes let let return measure measure measures lemma let set boxes let tree decomposition intersection graph nodes sizes respectively klee measure within given domain box computed time within log proof show algorithm runs time within given bound recursion tree corresponding algorithm leaves since step size problem approximately reduced one half height within log total running time internal level tree within hence log term moreover klee measure boxes within leaf computed making total time computing measure leaves within result follows computing klee measure set boxes whose intersection graph given optimal tree decomposition following result follows corollary let set boxes optimal tree decomposition intersection graph klee measure within given domain box computed time within log treewidth intersection graph proof let denote number nodes property know since besides node tree decomposition vertices result follows replacing lemma note bound log lemma better equal long bound achieved optimal tree decomposition intersection graph required decomposition general needs computed known whether performed polynomial time optimal tree decomposition certain classes graphs found efficiently klee measure sets intersection graph classes computed time depending treewidth describe two examples results corollaries corollary let set boxes ddimensional box intersection graph measure union within computed time within number connected components sizes connected components proof algorithm described edelsbrunner computes connected components time within connected components one easily obtain tree decomposition graph follows create node connected component add edges nodes obtaining arbitrary tree lemma bound follows profile input instance bound lemma achieved using algorithm seen corollary corollary let set boxes box profile within klee measure within computed time within log proof transform set klee measure treewidth within follows split domain slabs using algorithm slab add boxes intersect slab restricted box intersects several slabs box split multiple boxes intersect one slab union original one tree decomposition intersection graph nodes size within obtained follows create node slab add edges nodes arbitrary tree obtained since box intersect box slab tree decomposition valid bound follows applying lemma approximation optimal tree decomposition input set could used obtaining weaker bounds described corollary corollary let set boxes domain box klee measure within computed time within log log proof result follows replacing term bound lemma log approximation obtained amir algorithm see end section details note log log bound general case lower long first term log following section compare techniques described far describe combine combining techniques low profile implies intersection graph low treewidth corollary low treewidth imply low profile instance boxes class illustrated figure profile within treewidth one hand running time algorithm taking advantage profile never worth true running time one sensitive treewith even optimal tree decomposition provided treewidth profile measures independent size maxima example instance boxes class illustrated figure maxima size treewidth respect treewidth hard instance respect maxima size easy contrary instance boxes class illustrated figure maxima size whiles treewidth one since two measures independent combine obtain algorithm sensitive time computing maxima set finding klee measure maxima remaining graph described corollary way proceeding yields algorithm running time improving results lemmas fig two classes instances klee measure problem easy hard depending measure considered instance easy size maxima considered difficult profile treewidth considered instance easy treewidth hard maxima size profile theorem let set boxes box intersection graph klee measure within computed time within log log log size maxima treewidth lower bound known problem respect measures fact believe results obtained improved considering finer versions measures order improve analysis describe preliminary results direction next section discussion three boosting techniques analyzed improved describe preliminary results technique directions well lines research maxima based technique described section yields algorithm running time within log size maxima input set bound improved example instead filtering items maxima done part simplification step algorithm simplify divide conquer sdc expression running time running time becomes still within worst case also within never worse asymptotically better many cases formally analyze improvement still open question profile based technique described section yields solution running time within log algorithm sdc already adaptive profile input set limitation though necessarily cycles dimensions order ensure running time within dimensions profile set small technique performs considerably better sdc technique could improved instead considering upper bound profile use exact value profile subproblem however clear analyze improvement treewidth based technique described section yields algorithm running time within log optimal tree decomposition intersection graph given time within log log note running time bounds always better equal achivied algorithm sdc dependence tree decomposition main weakness approach known whether klee measure computed using algorithm depend explicitly tree decomposition intersection graph finally note techniques described focus structure instance opposed order instance given algorithms described beat log bound even though instances still solved time within one considers order input given special inputs one achieve bound acknowledgement authors partially supported millennium nucleus information coordination networks thank timothy chan helpful comments one anonymous referee esa pointing relation techniques analysis treewidth intersection graph references afshani barbay chan geometric algorithms proceedings annual ieee symposium foundations computer science focs agarwal improved algorithm computing volume union cubes proceedings annual symposium computational geometry socg acm new york usa amir approximation algorithms treewidth algorithmica arnborg proskurowski linear time algorithms problems restricted partial journal discrete applied mathematics jdam barbay time space fast algorithms yield small fast data structures data structures streams algorithms ianfest barbay chan navarro planar boxes time better information processing letters ipl barbay adaptive computation klee measure high dimensions corr http last accessed bentley algorithms klee rectangle problems unpublished notes bodlaender drange dregi fomin lokshtanov pilipczuk algorithm treewidth corr http last accessed bringmann improved algorithm klee measure problem fat boxes computational geometry theory applications cgta chan slightly faster algorithm klee measure problem proceedings annual symposium computational geometry socg chan klee measure problem made easy proceedings annual ieee symposium foundations computer science focs amore nguyen roos widmayer optimal cuts hyperrectangles computing edelsbrunner van leeuwen ottmann wood computing connected components simple rectilinear geometrical objects rairo theoretical informatics applications informatique thorique applications feige hajiaghayi lee improved approximation algorithms vertex separators proceedings annual acm symposium theory computing stoc stoc acm new york usa fredman weide complexity computing measure communications acm cacm jul gavril intersection graphs subtrees trees exactly chordal graphs journal combinatorial theory jct series kirkpatrick seidel sensitive algorithms finding maximal vectors proceedings annual symposium computational geometry socg kirkpatrick seidel ultimate planar convex hull algorithm siam journal computing feb klee measure computed less log steps american mathematical monthly amm kleinberg tardos algorithm design longman publishing boston usa moffat petersson overview adaptive sorting australian computer journal ajc overmars yap new upper bounds klee measure problem siam journal computing overmars equivalence rectangle containment rectangle enclosure technical report university utrecht department computer science robertson seymour graph minors algorithmic aspects yildiz suri klee measure problem grounded boxes proceedings annual symposium computational geometry socg acm new york usa
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uplink coverage performance underlay drone cell temporary events invited paper xiaohui jing salman halim mar research school engineering australian national university canberra act australia emails department systems computer engineering carleton university ottawa canada email halim drone aerial base station abs provide coverage users ground envisaged promising solution beyond fifth generation wireless networks literature date examined downlink cellular networks abss consider uplink cellular network abs specifically analyze use underlay abs provide coverage temporary event sporting event concert stadium using stochastic geometry derive analytical expressions uplink coverage probability terrestrial base station tbs abs results expressed terms laplace transforms interference power distribution tbs abs distance distribution abs independently uniformly distributed abssupported user equipment abs user equipment accuracy analytical results verified monte carlo simulations results show varying abs height leads uplink coverage probability tbs abs addition assuming quality service tbs uplink coverage probability abs achieved abs deployed optimal height typically considered setup ntroduction use drones aerial base stations abss form drone cells provide flexible agile coverage gained enormous interest potential beyond fifth generation wireless network solution eight scenarios identified drone cell deployment drones providing service rural areas urban areas users high mobility congested urban areas congested backhaul temporary events temporary blind spots sensor networks scenarios drone cells investigated literature different perspectives drone channel modeling drone deployment optimization trajectories performance analysis drone enabled cellular systems work focus last aspect motivation literature date focused downlink generally scenarios work focus uplink scenario using abs provide additional coverage temporary events concerts work supported part australian research council discovery project funding scheme project number natural sciences engineering research council canada nserc sports events temporary events happening frequently cities world high number users gathering often leads network congestion poor user experience optimal abs height respect downlink performance necessarily optimize uplink performance example distribution interference powers different uplink downlink cases moreover uplink bottleneck events since increasing number users try share content event social media applications therefore assessing usefulness abss provide uplink coverage temporary events important open problem literature great practical importance well related work downlink performance single static drone single mobile drone underlay users studied analyzed downlink coverage finite network formed multiple drones downlink coverage probability network multiple directional beamforming drones investigated downlink coverage performance network multiple drones urban environment studied authors studied downlink cellular network multiple ground base stations drone user equipment contributions paper analyze use abs provide coverage temporary event best knowledge scenario yet considered literature date using stochastic geometry analyze uplink coverage probability terrestrial base station tbs abs novel contributions paper derive analytical expressions uplink coverage probability tbs abs results terms laplace transforms interference power distribution tbs abs distance distribution abs independently uniformly distributed user equipment asue abs user equipment tsue results show increasing height abs generally improve uplink coverage probability abs degrades uplink coverage probability tbs addition optimal abs height maximizes uplink coverage probability abs assess feasibility establishing underlay abs provide uplink coverage temporary events assuming quality service tbs considered system set uplink coverage probability abs achieved abs optimal height center stadium sufficiently distanced tbs ystem model consider uplink communication network comprised tbs abs network region disk radius tbs located center assume temporary event held inside stadium within network region stadium building area modeled disk radius center distance tbs provide additional resources event abs deployed act abs assumed placed height center stadium shown fig tsues served tbs spatially distributed according binomial point process bpp inside network region excluding stadium time asues ground inside stadium tractability assume asues served abs location asues modelled independent bpp channel model two types communication links considered system model aerial links terrestrial links link tsue abs link asue abs aerial links link tsue tbs link asue tbs terrestrial links analytical tractability similar assume aerial links experience fading asue abs link strong line sight los fading parameter maa tsue abs link weak los fading parameter mta terrestrial links rayleigh fading assumed general model signal power decays rate propagation distance exponent due different characteristics aerial link terrestrial link different exponent assumed terrestrial link aerial link tsue abs aerial link asue abs respectively furthermore tsue tbs link asue abs link experience additive white gaussian noise awgn variance uplink network model spectrum efficiency assume tbs abs share spectrum resource underlay fashion orthogonal multiple current drone regulations prohibit drone flying stadiums sports events expected change future validity using model aerial links verified section comparing simulations using channel model aerial links tbs tsue abs asue tbs tsue abs asue view projection ground fig illustration system model access technique employed work assume number tsues asues sufficiently high say always one tsue one asue served per channel time hence interference among tsues asues interference exists tsues asues work focus analysis one uplink channel since channels share interference statistics reliable successful uplink communication tsue controls transmit power average signal received tbs equal threshold power control deployed asue also set maximum transmit power constraint asue avoid transmit power asue going large abs placed high altitude words asue compensates pathloss keep average signal power abs equal threshold transmit power required pathloss inversion less pmax otherwise asue tries establish uplink connection abs transmitting power pmax therefore instantaneous transmit power asue depends propagation distance asue abs shown top next page euclidean distance asue abs sinr considered setup instantaneous ratio sinr tbs given sinrt tbs tbs tsue transmit power fading power gain tsue tbs asue tbs respectively follow exponential distribution euclidean distance tsue tbs asue tbs respectively transmit power asue given instantaneous sinr abs given sinra fading power gain asue abs tsue abs respectively follow gamma distribution euclidean distance asue abs tsue abs respectively pmax pmax pmax pmax pmax pmax pmax fza lit pmax pmax lit fza lit lit pmax fza iii overage robability abs coverage probability section analyze network performance adopting coverage probability performance metric coverage probability formally defined pbcov sinrb superscript tbs abs sinr threshold sinrt sinra found respectively results coverage probability tbs abs presented next two subsections tbs coverage probability first present two lemmas used deriving coverage probability tbs theorem lemma laplace transform interference power distribution tbs given top page lit cos proof see appendix lemma see transmit power asue distance asue tbs related distance asue abs important distance distribution presented following lemma lemma probability density function pdf distance abs height center asue inside fza proof see appendix theorem based system model section coverage probability tbs lit exp cov tbs lit given lemma proof see appendix substituting obtain coverage probability tbs begin presenting three lemmas used compute coverage probability abs theorem lemma laplace transform interference power distribution abs lia fzt lia lia fzt lia mta cos proof proof follows lines lemma skipped sake brevity pdf distance tsue abs fzt conditional pdf angle ground projection given lemma lemma respectively lemma pdf distance abs height center tsue inside fzt arcsec proof see appendix lemma pdf angle ground projection conditioned proof lemma proved using cosine rule simple trigonometry pcov pmax fza pmax pcov fza cov cov pmax fza pmax cov pmax tbs coverage probability coverage probability tbs coverage probability abs coverage probability simulation atg sinr threshold dbm dbm abs height fig coverage probability tbs versus height abs different distance center stadium tbs different maximum transmit power asue theorem based system model section coverage probability abs given top page exp lia fig coverage probability tbs abs versus sinr threshold different height abs simulation atg aerial link model cov maa lia given lemma pdf distance asue abs fza provided lemma proof see appendix combining lemma theorem calculate coverage probability abs esults section first validate analytical results discuss design insights simulation results generated averaging monte carlo simulation runs simulations terrestrial links follow model described section however aerial links follow atg channel model unless stated otherwise set parameters follows pmax dbm dbm dbm maa mta dbm fig plots coverage probability tbs abs sinr threshold abs height simulation results match well analytical results abs assumed placed directly stadium model likelihood aerial links los high validates use tractable model aerial links analysis figure see abs placed higher altitude coverage probability abs higher coverage probability tbs lower discussed detail next impact abs height fig plots coverage probability tbs height abs different distance center stadium tbs different pmax maximum transmit power asue figure see coverage probability tbs first decreases abs height increase transmit power asue increases abs height whereby interference tbs increases certain abs height coverage probability tbs stays constant due fact asue reached maximum transmit power interference generated tbs keeps average note height coverage probability tbs starts level independent projection distance center stadium tbs height increases asue maximum transmit power fig plots coverage probability abs height abs different maximum transmit power asue see simulation results using atg channel model aerial links match well analytical figure shows analytical performance atg channel model subject ongoing work maximum abs coverage probability abs coverage probability simulation atg tbs coverage tbs coverage tbs coverage abs height distance center stadium tbs fig coverage probability abs versus abs height different asue maximum transmit power simulation atg aerial link model triangle markers denote optimal abs height fig maximum coverage probability abs versus distance center stadium tbs different tbs coverage requirements optimal height maximize coverage probability abs optimal height increases maximum transmit power asue pmax figure find curves different pmax overlapped small comes fact certain range transmit power asue pmax asue abs link full channel inversion full channel inversion average desired signal power received abs stays interference power drops height increases therefore coverage probability increases abs height optimal height asue transmits maximum power pmax received power desired signal reduces height increases hence coverage probability starts drop red green curves fig furthermore larger asue maximum transmit power generally results higher coverage probability abs height increases feasibility previous figures see coverage probability tbs abs fig plots maximum coverage probability achieved abs versus distance center stadium tbs coverage probability tbs required maximum achievable abs coverage probability increases becomes almost flat certain optimal height abs achieved small sufficiently large abs placed optimal height optimal coverage probability abs achieved instance fig find coverage probability tbs coverage probability abs achieved placing abs height center stadium apart tbs therefore abs underlay cellular network feasible practical network setup onclusions paper uplink communication network tbs abs considered derived exact expressions uplink coverage probability tbs abs terms laplace transforms interference power distribution tbs abs distance distribution abs asue abs tsue results demonstrated feasibility establishing underlay abs provide uplink coverage temporary events future work consider impact beamforming abs user scheduling multiple ues per channel ppendix proof lemma following definition laplace transform laplace transform interference power distribution tbs expressed lit eit exp epa exp epa tbs spa third step comes fact follows exponential distribution unit mean conditioned value three possible cases lit laplace transform interference power distribution tbs equals max lit fza tbs fza tbs spmax max max max cos spmax cos max max lit lit pmax expressed terms cosine rule obtained taking expectation conditional pdf following similar steps work laplace transform interference power distribution tbs two cases proof lemma relation length asue abs link projection distance distance distribution ground projection distance ground fra thus get pdf fra fza proof theorem coverage probability tbs given cov sinrt tbs exp exp lit third step use fact link tsue tbs experiences rayleigh fading pdf fha exp proof lemma proof lemma know order find distribution distance tsue abs distribution projection distance needed using similar approach distance distribution derived frt arcsec using pdf auxiliary random variable obtain lemma proof theorem coverage probability abs given cov sinra eza eza exp dsk cov fza third step comes fact follows gamma distribution transmit power asue given eferences yanikomeroglu new frontier ran heterogeneity ieee commun vol chandrasekharan gomez kandeepan rasheed goratti reynaud grace bucaille wirth allsopp designing implementing future aerial communication networks ieee commun vol may amorim nguyen mogensen kovcs wigard srensen radio channel modeling uav communication cellular networks ieee wireless commun vol amorim mogensen sorensen kovacs wigard pathloss measurements modeling uavs connected cellular networks proc ieee vtc spring jun gomez modeling suburban environments appear ieee wireless commun kandeepan jamalipour modeling path loss low altitude platforms urban environments proc ieee globecom fotouhi ding hassan dronecells improving spectral efficiency using flying base stations online available https zeng zhang lim throughput maximization mobile relaying systems ieee trans vol alzenad lagum yanikomeroglu placement unmanned aerial vehicle base station energyefficient maximal coverage ieee wireless 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